guilford public schools math curriculum kindergarten through … curriculum 2013 for w… · math...

Guilford Public Schools

Math Curriculum

Kindergarten through Grade 12

Approved by the Guilford Public Schools Board of Education June 10, 2013

INTRODUCTION

Guilford’s Mathematics Curriculum Guide has been revised and aligned to meet the new Common Core State Standards for Mathematics. The

Principles of Learning; Common Core, Inc.’s math curriculum maps; and the North Carolina Department of Instruction Unpacked Content

documents were used as additional guides. Beginning in fall 2012, committee representatives from each level began their work with these

professional reference materials and others to create a document that reflects the current thinking in mathematics education and one that establishes a

guide for mathematical growth for the students of Guilford, grades K-12.

The guide is divided into four major sections.

(1) Philosophy Statement and Introduction to the Common Core State Standards for Mathematics

(2) Year-Long Curriculum Map: Teachers may use this section to see the distribution of topics over the course of a year.

(3) Curricular Expectations: Teachers will use specific grade level or course sections to determine classroom and student goals.

(4) Glossary: An addenda including math definitions, the Standards for Mathematical Practice, tables with information applicable to multiple

grades, and a list of works consulted in the development of the Common Core State Standards for Mathematics.

The committee will develop supplemental resources for formative assessment to help guide instructional decisions for the teacher and to help build

awareness of learning for the student. These tasks encourage the integration of real-life, problem-solving skills with the deep mathematical concepts

of the standards. We will continue to expand components by receiving feedback during the implementation of the curriculum.

The following are the 2012 committee members:

Gail Whitney, Grade 1, Melissa Jones School Jen Brown, Grade 2, Guilford Lakes School Vinny Mascola, Grade 3, A.W. Cox School Jamie Froelick, Grade 4, Calvin Leete School Anne Lombard, Grade 5, Baldwin Middle School Courtney Barbour, Grade 6, Baldwin Middle School Jessica Gellert, Grade 8, Adams Middle School Barbara Tokarska, Math Teacher, Guilford High School Alison Strzepek, Math Teacher, Guilford High School Donna Pudlinski, Math Chair, Guilford High School Maria Curreri, Math Specialist, Guilford Public Schools

PHILOSOPHY STATEMENT

The ability to think and reason mathematically as well as to communicate and apply mathematical understandings has been the over-riding goal of

Guilford Public School’s Mathematics Curriculum. This goal aligns seamlessly with the new Common Core State Standards. Fundamental to the

mathematics program is the development within our students of an inquisitive mind, a positive attitude, and persistent effort required to solve

complex problems. Mathematical skills are presented as tools to be used both in and out of school in addressing conceptual as well as authentic, real-

life situations. Computational fluency and an understanding of mathematical vocabulary are considered important components of the mathematics

program. Of equal importance is the development of meaningful mathematical concepts that promote high level thinking skills. In the course of

learning mathematics, students must have the opportunity to explore and share multiple strategies for problem solving, connect mathematical

concepts to various content areas, apply mathematics in various situations, and analyze information. In addition, students must effectively convey

their findings orally, in writing, pictorially, graphically, and with models. All aspects of mathematical communication and investigation can and

should be enhanced through the use of technology at all levels.

The Guilford Public School’s Mathematics Curriculum K-12 has been designed to build the mathematical concepts and skills needed by our children

to become good thinkers and problem solvers. The research behind the development of the Common Core State Standards revealed the need for

United States standards to become more focused and coherent in order to improve mathematics achievement in this country. It is important to

recognize that “fewer standards” are no substitute for focused standards. Instead, these standards aim for clarity and specificity. We value the

Common Core State Standards and have used them to guide the development of our district curriculum document.

A willingness to learn, mutual respect and a classroom atmosphere that promotes accountable talk make for healthy learning environments. We

encourage the use of approaches and materials that motivate, support, and challenge our students. We welcome assistance and cooperation between

children, parents, teachers and our district in implementing the mathematics curriculum of Guilford Public Schools. Continual effort by all will

ensure that the students of Guilford are successful in receiving a high quality mathematics education that ensures college and career readiness.

GUIDING PRINCIPLES

Build math knowledge and reasoning skills through problem solving

Build math confidence and perseverance

Integrate the content and practice standards

Make connections with other subjects through content connections and a focus on communication

Communicate mathematical understanding through speaking, reading, and writing

Use technology effectively, pervasively, and appropriately

Develop math understanding through accountable talk

Allow students to struggle to enhance learning and view mistakes as an opportunity to learn

Provide students relevant and real world problems with multiple entry points so that all students have access

Use multiple representations to illustrate problems and their solutions

Arrange classroom environment to support individual, group and class work

Focus on big ideas/concepts and build coherence across time

Use assessment to monitor and adjust instruction

Mathematics Review Committee

Overview of the Common Core State Standards The Common Core State Standards provide a consistent, clear understanding of what students are expected to learn, so teachers and parents know what they need

to do to help them. As shown below through the grade level domains, focuses, and required fluencies, the standards are focused, coherent, and relevant to the real

world, describing the knowledge and skills that students need for success in college and careers.

In K-8 (Kindergarten, Elementary, and Middle School) each grade contains work on several domains, as described in the table below. For example: In Grade 1,

the content includes Operations and Algebraic Thinking, Number and Operations in Base Ten, Measurement and Data, and Geometry.

Grade K 1 2 3 4 5 6 7 8 HS Conceptual Categories

Do

ma

ins

Counting &

Cardinality

Ratios & Proportional

Relationships

Functions Functions

Operations and Algebraic Thinking Expression and Equations Algebra

Number and Operations in Base Ten The Number System Number & Quantity

Fractions

Measurement and Data Statistics and Probability Statistics & Probability

Geometry Geometry Geometry

In High School, the standards are arranged in conceptual categories, such as Algebra or Functions. In each conceptual category there are domains, such as

Creating Equations and Interpreting Functions.

Key Areas of Focus in Mathematics

Grade Focus Areas in Support of Rich Instruction and

Expectations of Fluency and Conceptual Understanding

K-2 Addition and Subtraction—concepts, skills, and problem

solving and place value

3-5 Multiplication and division of whole numbers and

fractions—concepts, skills, and problem solving

6 Ratios and proportional reasoning; early expressions and

equations

7 Ratios and proportional reasoning; arithmetic of rational

numbers

8 Linear algebra

Required Fluencies in K-6

Grade Standard Required Fluency

K K.OA.5 Add/Subtract within 5

1 1.OA.6 Add/Subtract within 10

2 2.OA.2

2.NBT.5

Add/Subtract within 20 (know single-digit sums from

memory)

Add/Subtract within 100

3 3.OA.7

3.NBT.2

Multiply/Divide within 100 (know single-digit

products from memory)

Add/Subtract within 1000

4 4.NBT.4 Add/Subtract within 1,000,000

5 5.NBT.5 Multi-digit multiplication

6 6.NS.2, 3 Multi-digit division

Multi-digit decimal operations

Mathematics

Mathematical Practices

The Standards for Mathematical Practice describe characteristics and traits that mathematics educators at all levels should seek to develop in their

students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these

are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the

strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence,

conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures

flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and

worthwhile, coupled with a belief in diligence and one’s own efficacy). These eight practices can be clustered into the following categories as shown

in the chart below: Habits of Mind of a Productive Mathematical Thinker, Reasoning and Explaining, Modeling and Using Tools, and Seeing

Structure and Generalizing.

H

abit

s o

f M

ind o

f a P

rodu

ctiv

e M

ath

emati

cal

Th

inker

MP

.1 M

ake

sense

of

pro

ble

ms

and p

erse

ver

e in

solv

ing t

hem

.

MP

.6 A

tten

d t

o p

reci

sion.

Reasoning and Explaining

MP. 2 Reason abstractly and quantitatively.

MP. 3 Construct viable arguments and critique the reasoning of others.

Modeling and Using Tools

MP. 4 Model with mathematics.

MP. 5 Use appropriate tools strategically.

Seeing Structure and Generalizing

MP. 7 Look for and make use of structure.

MP. 8 Look for and express regularity in repeated reasoning.

Arizona Department of Education: Standards and Assessment Division

2010

Mathematics

Weeks Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5

1-4 Unit 1: ID & Describe

Shapes

Unit 1: Addition,

Subtraction of Numbers

to 10 and Fluency

Unit 1: + & - #s to 100,

Fluency with + & - to 20

Unit 1: Multiplication

and Division with Factors

of 2, 3, 4, 5 and 10

Unit 1: Place Value,

Rounding, Fluency w/ +

and - Algorithms of

Whole #s

Unit 1: Whole # and

Decimal Fraction Place

Value Unit 2: Classify and

Count Numbers to 10

5-8 Unit 2: Addition and

Subtraction of Length

and Time Measurements

Unit 2: Conversions: +/-

of Measurements

Unit 2: Multi-Digit

Whole Number and

Decimal Fraction

Operations

Unit 2: Problem Solving

with Mass, Time and

Capacity Unit 3: Multiplication

and Division of Up to a

4-Digit Number by Up to

a 1-Digit Number Using

Place Value

9-12 Unit 2: Place Value,

Comparison, Addition

and Subtraction of

Numbers to 20


Counting, and

Comparison of Numbers

to 1000 Unit 3: Multiplication

and Division with Factors

of 6, 7, 8, 9, and

Multiples of 10

Unit 3: Comparison with

Length, Weight and

Numbers to 10

Unit 3: Addition and

Subtraction of Fractions

with Fraction Concepts 13-16

Unit 4: Addition and

Subtraction of Numbers

to 1000 with Problem-

Solving Unit 3: Measuring,

Representing, and

Interpreting Data

Unit 4: Measuring and

Classifying Shapes

Unit 4: Order and

Operations with Fractions 17-20



and Subtraction of

Numbers to 40

Unit 4: Multiplication

and Division of Fractions

and Multi-digit Decimals 21-24 Unit 4: Number Pairs,

Addition and Subtraction

of Numbers to 10

Unit 5: Measuring,

Representing, and

Solving with Money and

Length Data

Unit 5: Fractions as

Numbers on the Number

Line

25-28 Unit 5: Decimal

Fractions Unit 5: Identify,

Compose, and Partition

Shapes

Unit 5: Volume, Area,

and Shapes Unit 6: Preparation for

Multiplication and

Division Concepts

29-32 Unit 6: Collecting and

Displaying Data

Unit 6: Two-dimensional

Shape Exploration and

Problem-solving Unit 5: Numbers 10 – 20,

Counting to 100 by 1 and

10



and Subtraction of

Numbers to 100

Unit 7: Challenging

Problems Unit 6: Graph Points on

the Coordinate Plane to

Solve Problems 33-36 Unit 7: Reasoning about

Shapes and Fractions of

Shapes

Unit 7: Exploring

Multiplication and More

Challenging Problems Unit 6: Analyze,

Compare, Create

KEY: Operations and Algebraic

Thinking

Number Fractions Measurement and

Data

Geometry

Mathematics

Weeks Grade 6 Grade 7 Grade 8 Algebra Geometry Algebra II

1-4 Unit 1: Arithmetic

Operations Including

Dividing by a Fraction

Unit 1: Rational

Numbers

Unit 1: The Number

System and Properties

of Exponents

Unit 1: Patterns Unit 1:

Transformations

Unit 1: Functions and

Inverses

5-8 Unit 2: Linear

Equations and

Inequalities Unit 2: Rational

Numbers

Unit 2: Congruence,

Proof, and

Constructions

Unit 2: Polynomial

Functions Unit 2: Expressions and

Equations

Unit 2: Pythagorean

Theorem

9-12

Unit 3: Functions

Unit 3: Ratios and Unit

Rates Unit 3: Three

Dimensional Geometry

Unit 3: Rational

Expressions and

Functions 13-16 Unit 4: Linear

Functions Unit 3: Ratios and

Proportional

Relationships

Unit 3: Congruence

Unit 4: Exponential

and Logarithmic

Functions 17-20 Unit 4: Similarity Unit 4: Similarity,

Proof, and

Trigonometry Unit 4: Expressions and

Equations Unit 5: Scatter Plots

and Trend Lines Unit 4: Percent and

Proportional

Relationships 21-24 Unit 5: Linear

Equations Unit 5: Trigonometric

Functions Unit 6: Systems of

Equations

Unit 5: Circles and

Other Conic Sections

25-28 Unit 5: Statistics and

Probability Unit 7: Introduction to

Exponential Equations Unit 5: Area, Surface

Area, and Volume

Problems

Unit 6:

Statistics Unit 6: Applications

and Probability 29-32

Unit 6: Geometry Unit 6: Linear

Functions (and in

Geometry)

Unit 8: Quadratic

Functions and

Equations Unit 6: Statistics

Performance Task

33-36 Performance Task

Unit 7: Patterns and

Data Performance Task

KEY: Ratios and Proportional Reasoning Number System Expressions and Equations/ Functions Geometry Statistics and Probability

Mathematics

CC= Counting and Cardinality OA= Operations and Algebraic Thinking NBT= Numbers and Operations in Base Ten

MD= Measurement and Data G= Geometry

Kindergarten

In Kindergarten, instructional time should focus on two critical areas: (1) representing and comparing whole numbers, initially with sets of objects;

(2) describing shapes and space. More learning time in Kindergarten should be devoted to number than to other topics.

(1) Students use numbers, including written numerals, to represent quantities and to solve quantitative problems, such as counting objects in a

set; counting out a given number of objects; comparing sets or numerals; and modeling simple joining and separating situations with sets of

objects, or eventually with equations such as 5 + 2 = 7 and 7 – 2 = 5. (Kindergarten students should see addition and subtraction equations,

and student writing of equations in kindergarten is encouraged, but it is not required.) Students choose, combine, and apply effective

strategies for answering quantitative questions, including quickly recognizing the cardinalities of small sets of objects, counting and

producing sets of given sizes, counting the number of objects in combined sets, or counting the number of objects that remain in a set after

some are taken away.

(2) Students describe their physical world using geometric ideas (e.g., shape, orientation, spatial relations) and vocabulary. They identify,

name, and describe basic two-dimensional shapes, such as squares, triangles, circles, rectangles, and hexagons, presented in a variety of ways

(e.g., with different sizes and orientations), as well as three-dimensional shapes such as cubes, cones, cylinders, and spheres. They use basic

shapes and spatial reasoning to model objects in their environment and to construct more complex shapes.


1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

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Kindergarten: Suggested Distribution of Units in Instructional Days Time Approx.

# of weeks

Unit 1: Identify and Describe Shapes 6% ~ 2 weeks

Unit 2: Classify and Count Numbers to 10 25% ~ 9 weeks

Unit 3: Comparison with Length, Weight and Numbers to 10 25% ~ 9 weeks

Unit 4: Number Pairs, Addition and Subtraction of Numbers to 10 25% ~ 9 weeks

Unit 5: Numbers 10-20, Counting to 100 by 1 and 10 13% ~ 5 weeks

Unit 6: Analyze, Compare, Create and Compose Shapes 6% ~ 2 weeks

Unit 1: Identify and Describe Shapes

6%

Unit 2: Classify and Count Numbers to

10 25%

Unit 3: Comparison with Length, Weight, and

Numbers to 10 25%

Unit 4: Number Pairs, + and - of Numbers to 10

25%

Unit 5: Numbers 10-20, Counting to

100 by 1 and 10 13%

Unit 6: Analyze, Compare,

Create, and Compose

Shapes 6%

Instructional Time

Mathematics



Kindergarten Unit 1: Identify and Describe Shapes (~ 2 weeks)

Unit Overview: Students learn to identify and describe shapes which they can use as a context for working on number fluency in later units. This unit contains

opportunities for students to construct viable arguments and critique the reasoning of others (MP 3) as they explore and discuss shapes based on attributes and not

just what they “look like.”

Guiding Question: What words are most helpful when you describe a shape?

The student will be able to: The teacher will use appropriate instructional strategies/approaches based on the needs of the student.

Component Cluster K.G Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

K.G.1 Describe objects in the environment using names of

shapes, and describe the relative positions of these objects

using terms such as above, below, beside, in front of,

behind, and next to.

Students locate and identify shapes in their environment. For example, a student may look at the tile pattern

arrangement on the hall floor and say, “Look! I see squares! They are next to the triangle.” At first students may

use informal names e.g., “balls,” “boxes,” “cans”. Eventually students refine their informal language by learning

mathematical concepts and vocabulary and identify, compare, and sort shapes based on geometric attributes.*

Students also use positional words (such as those italicized in the standard) to describe objects in the environment,

developing their spatial reasoning competencies. Kindergarten students need numerous experiences identifying the

location and position of actual two-and-three-dimensional objects in their classroom/school prior to describing

location and position of two-and-three-dimension representations on paper.

K.G.2 Correctly name shapes regardless of their

orientations or overall size.

Through numerous experiences exploring and discussing shapes, students begin to understand that certain attributes

define what a shape is called (number of sides, number of angles, etc.) and that other attributes do not (color, size,

orientation). As the teacher facilitates discussions about shapes (“Is it still a triangle if I turn it like this?”), children

question what they “see” and begin to focus on the geometric attributes.

Kindergarten students typically do not yet recognize triangles that are turned upside down as triangles, since they

don’t “look like” triangles. Students need ample experiences manipulating shapes and looking at shapes with various

typical and atypical orientations. Through these experiences, students will begin to move beyond what a shape “looks

like” to identifying particular geometric attributes that define a shape.

K.G.3 Identify shapes as two-dimensional (lying in a

plane, “flat”) or three dimensional (“solid”).

*This unit focuses only on two-dimensional shapes. Three-

dimensional shapes are explored in Unit 6.

Students identify objects as flat (2 dimensional) or solid (3 dimensional). As the teacher embeds the vocabulary into

students’ exploration of various shapes, students use the terms two-dimensional and three-dimensional as they

discuss the properties of various shapes.

Mathematics



Kindergarten Unit 2: Classify and Count Numbers to 10 (~ 9 weeks) Unit Overview: Kindergarten numeracy starts out solidifying the meaning of numbers to 10 with a focus on counting, graphing, relationships to 5, and growth and

shrinking patterns to 10 of “1 more” and “1 less” using models such as the number stairs:

. This unit provides opportunities for students to look for and express regularity in repeated reasoning (MP 8) as students observe the

relationship between counting and cardinality as they develop efficient strategies for naming a given amount.

Guiding Question: What are some good ways to know how many objects are in a group?


Component Cluster K.CC Know number names and the count sequence.*

*within 10 only for this unit

K.CC.1 Count to 100 by ones and by tens.

Students rote count by starting at one and counting to 100. When students count by tens they are only expected to

master counting on the decade (0, 10, 20, 30, 40 …). This objective does not require recognition of numerals. It is

focused on the rote number sequence.

K.CC.2 Count forward beginning from a given number

within the known sequence (instead of having to begin at

1).

Students begin a rote forward counting sequence from a number other than 1. Thus, given the number 4, the student

would count, “4, 5, 6, 7 …” This objective does not require recognition of numerals. It is focused on the rote number

sequence 0-100.

K.CC.3 Write numbers from 0 to 20. Represent a number

of objects with a written numeral 0-20 (with 0 representing

a count of no objects).

Students write the numerals 0-20 and use the written numerals 0-20 to represent the amount within a set. Students

can record the quantity of a set by selecting a number card/tile (numeral recognition) or writing the numeral. Students

can also create a set of objects based on the numeral presented.

(Due to varied development of fine motor and visual development, reversal of numerals is anticipated. While

reversals should be pointed out to students and correct formation modeled in instruction, the emphasis of this

standard is on the use of numerals to represent quantities rather than the correct handwriting formation of the actual

numeral itself.)

Component Cluster K.CC Count to tell the number of objects.*

*within 10 only for this unit K.CC.4 Understand the relationship between numbers and

quantities; connect counting to cardinality.

Students count a set of objects and see sets and numerals in relationship to one another. These connections are

higher-level skills that require students to analyze, reason about, and explain relationships between numbers and sets

of objects. The expectation is that students are comfortable with these skills with the numbers 1-20 by the end of K.

a. When counting objects, say the number names in the

standard order, pairing each object with one and only one

number name and each number name with one and only

one object.

Students implement correct counting procedures by pointing to one object at a time (one-to-one correspondence),

using one counting word for every object (synchrony/ one-to-one tagging), while keeping track of objects that have

and have not been counted. This is the foundation of counting.

Mathematics



b. Understand that the last number name said tells the

number of objects counted. The number of objects is the

same regardless of their arrangement or the order in which

they were counted.

Students answer the question “How many are there?” by counting objects in a set and understanding that the last

number stated when counting a set (…8, 9, 10) represents the total amount of objects: “There are 10 bears in this

pile.” (cardinality). Since an important goal for children is to count with meaning, it is important to have children

answer the question, “How many do you have?” after they count. Often times, children who have not developed

cardinality will count the amount again, not realizing that the 10 they stated means 10 objects in all.

Young children believe what they see. Therefore, they may believe that a pile of cubes that they counted may be

more if spread apart in a line. As children move towards the developmental milestone of conservation of number,

they develop the understanding that the number of objects does not change when the objects are moved, rearranged,

or hidden. Children need many different experiences with counting objects, as well as maturation, before they can

reach this developmental milestone.

c. Understand that each successive number name refers to

a quantity that is one larger.

Another important milestone in counting is inclusion (aka hierarchal inclusion). Inclusion is based on the

understanding that numbers build by exactly one each time and that they nest within each other by this amount. For

example, a set of three objects is nested within a set of 4 objects; within this same set of 4 objects is also a set of two

objects and a set of one. Using this understanding, if a student has four objects and wants to have 5 objects, the

student is able to add one more- knowing that four is within, or a sub-part of, 5 (rather than removing all 4 objects

and starting over to make a new set of 5). This concept is critical for the later development of part/whole

relationships.

Students are asked to understand this concept with and without (0-20) objects. For example, after counting a set of 8

objects, students answer the question, “How many would there be if we added one more object?”; and answer a

similar question when not using objects, by asking hypothetically, “What if we have 5 cubes and added one more.

How many cubes would there be then?”

K.CC.5 Count to answer “how many?” questions about as

many as 20 things arranged in a line, a rectangular array,

or a circle, or as many as 10 things in a scattered

configuration; given a number from 1–20, count out that

many objects.

In order to answer “how many?” students need to keep track of objects when counting. Keeping track is a method of

counting that is used to count each item once and only once when determining how many. After numerous

experiences with counting objects, along with the developmental understanding that a group of objects counted

multiple times will remain the same amount, students recognize the need for keeping track in order to accurately

determine “how many”. Depending on the amount of objects to be counted, and the students’ confidence with

counting a set of objects, students may move the objects as they count each, point to each object as counted, look

without touching when counting, or use a combination of these strategies. It is important that children develop a

strategy that makes sense to them based on the realization that keeping track is important in order to get an accurate

count, as opposed to following a rule, such as “Line them all up before you count”, in order to get the right answer.

As children learn to count accurately, they may count a set correctly one time, but not another. Other times they may

be able to keep track up to a certain amount, but then lose track from then on. Some arrangements, such as a line or

rectangular array, are easier for them to get the correct answer but may limit their flexibility with developing

meaningful tracking strategies, so providing multiple arrangements help children learn how to keep track. Since

scattered arrangements are the most challenging for students, this standard specifies that students only count up to 10

objects in a scattered arrangement and count up to 20 objects in a line, rectangular array, or circle.

Component Cluster K.CC Classify objects and count the number of objects in each category.

K.MD.3 Classify objects into given categories; count the

numbers of objects in each category and sort the categories

by count.

(Limit category counts to be less than or equal to 10)

Students identify similarities and differences between objects (e.g., size, color, shape) and use the identified attributes

to sort a collection of objects. Once the objects are sorted, the student counts the amount in each set. Once each set

is counted, then the student is asked to sort (or group) each of the sets by the amount in each set. Thus, like amounts

are grouped together, but not necessarily ordered.

Mathematics



Kindergarten Unit 3: Comparison with Length, Weight, and Numbers to 10 (~ 9 weeks) Unit Overview: Two of the most crucial themes of students’ math experience begin with measurement in kindergarten: unit and comparison (U3). Students use

different units to measure length, weight and capacity, and explore the relationship of those units. Comparison begins with developing the meaning of the word

“than”: “taller than”, “shorter than”, “heavier than”, “longer than”, etc. With the word “than” concretely understood, the (at least!) 8-year curriculum teaching

sequence for terms “more than” and “less than” can begin (a topic that culminates in middle school with “y is 2 less than 3 times as much as x.”). This unit

requires students to attend to precision (MP6) as they compare numbers and objects paying attention to accurate counting strategies, precise language for specific

measurable attributes (not just “bigger”), and exact measuring technique (lining up the objects).

Guiding Question: What are the skills needed to be good at measuring?


Component Cluster K.CC Compare numbers.

K.CC.6 Identify whether the number of objects in one

group is greater than, less than, or equal to the number of

objects in another group, e.g., by using matching and

counting strategies.1

1Include groups with up to ten objects.

Students use their counting ability to compare sets of objects (0-10). They may use matching strategies (Student 1),

counting strategies (Student 2) or equal shares (Student 3) to determine whether one group is greater than, less than,

or equal to the number of objects in another group.

Student 1

I lined up one square and one

triangle. Since there is one extra

triangle, there are more triangles

than squares.

Student 2

I counted the squares and I

got 4. Then I counted the

triangles and got 5. Since 5

is bigger than 4, there are

more triangles than

squares.

Student 3

I put them in a pile. I then took

away objects. Every time I took a

square, I also took a triangle. When

I had taken almost all of the shapes

away, there was still a triangle left.

That means that there are more

triangles than squares.

K.CC.7 Compare two numbers between 1 and 10

presented as written numerals.

Students apply their understanding of numerals 1-10 to compare one numeral from another. Thus, looking at the

numerals 8 and 10, a student is able to recognize that the numeral 10 represents a larger amount than the numeral 8.

Students need ample experiences with actual sets of objects (K.CC.3 and K.CC.6) before completing this standard

with only numerals.

Component Cluster K.MD Describe and compare measureable attributes.

K.MD.1 Describe measurable attributes of objects, such as

length or weight. Describe several measurable attributes of

a single object.

Students describe measurable attributes of objects, such as length, weight, size, and color. Students often initially

hold undifferentiated views of measurable attributes, saying that one object is “bigger” than another whether it is

longer, or greater in area, or greater in volume, and so forth. Conversations about how they are comparing- one

building may be taller (greater in length) and another may have a larger base (greater in area)- help students learn to

discriminate and name these measureable attributes. As they discuss these situations and compare objects using

different attributes, they learn to distinguish, label, and describe several measureable attributes of a single object.

Thus, teachers listen for and extend conversations about things that are “big”, or “small,” as well as “long,” “tall,” or

“high,” and name, discuss, and demonstrate with gestures the attribute being discussed.

K.MD.2 Directly compare two objects with a measurable Direct comparisons are made when objects are put next to each other, such as two children, two books, two pencils.

Mathematics



attribute in common, to see which object has “more

of”/“less of” the attribute, and describe the difference.

For example, directly compare the heights of two children

and describe one child as taller/shorter.

Students are not comparing objects that cannot be moved and lined up next to each other. Similar to the development

of the understanding that keeping track is important to obtain an accurate count, kindergarten students need ample

experiences with comparing objects in order to discover the importance of lining up the ends of objects in order to

have an accurate measurement.

As this concept develops, children move from the idea that “Sometimes this block is longer than this one and

sometimes it’s shorter (depending on how I lay them side by side) and that’s okay.” to the understanding that “This

block is always longer than this block (with each end lined up appropriately).” Since this understanding requires

conservation of length, a developmental milestone for young children, kindergarteners need multiple experiences

measuring a variety of items and discussing findings with one another. As students develop conservation of length,

learning and using language such as “It looks longer, but it really isn’t longer” is helpful.

Mathematics



Kindergarten Unit 4: Number Pairs, Addition and Subtraction of Numbers to 10 (~ 9 weeks)

Unit Overview: The terms “more” and “less” are abstract later in kindergarten because they refer to numbers (“7 is 2 more than 5”) rather than concrete

measurements (“Jim is taller than John.”). “1 more, 2 more, 3 more” lead into the addition fact fluencies (+1, +2, +3). Comparing numbers leads to looking at the

numbers that make up a number (“3 is less than 7. 3 and 4 make 7. ”). This, in turn, leads naturally to discussions of addition and subtraction (U4). This unit

requires students to model with mathematics (MP 4) as they represent real-life problem situations in multiple ways such as with numbers, words (mathematical

language), drawings, objects, acting out, charts, lists, and number sentences.

Guiding Question: What different ways can you represent the math you find in real-life problems?


Component Cluster K.OA Understand addition as putting together and adding to, and understand subtraction as taking apart and taking

from.

K.OA.1 Represent addition and subtraction with objects,

fingers, mental images, drawings*, sounds (e.g., claps),

acting out situations, verbal explanations, expressions, or

equations.

*Drawings need not show details, but should show the

mathematics in the problem. (This applies wherever

drawings are mentioned in the Standards.)

Students demonstrate the understanding of how objects can be joined (addition) and separated (subtraction) by

representing addition and subtraction situations in various ways. This objective is focused on understanding the

concept of addition and subtraction, rather than reading and solving addition and subtraction number sentences

(equations).

Common Core State Standards for Mathematics states, “Kindergarten students should see addition and subtraction

equations, and student writing of equations in kindergarten is encouraged, but it is not required.” Please note that it is

not until First Grade when “Understand the meaning of the equal sign” is an expectation (1.OA.7).

Therefore, before introducing symbols (+, -, =) and equations, kindergarteners require numerous experiences using

joining (addition) and separating (subtraction) vocabulary in order to attach meaning to the various symbols. For

example, when explaining a solution, kindergartens may state, “Three and two is the same amount as 5.” While the

meaning of the equal sign is not introduced as a standard until First Grade, if equations are going to be modeled and

used in Kindergarten, students must connect the symbol (=) with its meaning (is the same amount/quantity as).

K.OA.2 Solve addition and subtraction word problems,

and add and subtract within 10, e.g., by using objects or

drawings to represent the problem.

Kindergarten students solve four types of problems within 10: Result Unknown/Add To; Result Unknown/Take

From; Total Unknown/Put Together-Take Apart; and Addend Unknown/Put Together-Take Apart (See Table 1 at

end of document for examples of all problem types). Kindergarteners use counting to solve the four problem types

by acting out the situation and/or with objects, fingers, and drawings.

Add To Result Unknown

Take From Result Unknown

Put Together/Take Apart Total Unknown

Put Together/Take Apart Addend Unknown

Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now? 2 + 3 = ?

Five apples were on the table. I ate two apples. How many apples are on the table now? 5 – 2 = ?

Three red apples and two green apples are on the table. How many apples are on the table? 3 + 2 = ?

Five apples are on the table. Three are red and the rest are green. How many apples are green? 3 + ? = 5, 5 – 3 = ?

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K.OA.3 Decompose numbers less than or equal to 10 into

pairs in more than one way, e.g., by using objects or

drawings, and record each decomposition by a drawing or

equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

Students develop an understanding of part-whole relationships as they recognize that a set of objects (5) can be

broken into smaller sub-sets (3 and 2) and still remain the total amount (5). In addition, this objective asks students

to realize that a set of objects (5) can be broken in multiple ways (3 and 2; 4 and 1). Thus, when breaking apart a set

(decompose), students use the understanding that a smaller set of objects exists within that larger set (inclusion).

Example: “Bobby Bear is missing 5 buttons on his jacket. How many ways can you use blue and red buttons to

finish his jacket? Draw a picture of all your ideas.

Students could draw pictures of:

4 blue and 1 red button 3 blue and 2 red buttons 2 blue and 3 red buttons 1 blue and 4 red buttons

In Kindergarten, students need ample experiences breaking apart numbers and using the vocabulary “and” & “same

amount as” before symbols (+, =) and equations (5= 3 + 2) are introduced. If equations are used, a mathematical

representation (picture, objects) needs to be present as well.

K.OA.4 For any number from 1 to 9, find the number that

makes 10 when added to the given number, e.g., by using

objects or drawings, and record the answer with a drawing

or equation.

Students build upon the understanding that a number (less than or equal to 10) can be decomposed into parts

(K.OA.3) to find a missing part of 10. Through numerous concrete experiences, kindergarteners model the various

sub-parts of ten and find the missing part of 10. In addition, kindergarteners use various materials to solve tasks that

involve decomposing and composing 10 such as ten-frame, think addition, or fluently add/subtract.

K.OA.5 Fluently add and subtract within 5.

Students are fluent when they display accuracy (correct answer), efficiency (a reasonable amount of steps in about 3-

5 seconds* without resorting to counting), and flexibility (using strategies such as the distributive property).

Students develop fluency by understanding and internalizing the relationships that exist between and among

numbers. Oftentimes, when children think of each “fact” as an individual item that does not relate to any other

“fact”, they are attempting to memorize separate bits of information that can be easily forgotten. Instead, in order to

fluently add and subtract, children must first be able to see sub-parts within a number (inclusion, K.CC.4.c).

Once they have reached this milestone, children need repeated experiences with many different types of concrete

materials (such as cubes, chips, and buttons) over an extended amount of time in order to recognize that there are

only particular sub-parts for each number. Therefore, children will realize that if 3 and 2 is a combination of 5, then

3 and 2 cannot be a combination of 6.

Traditional flash cards or timed tests have not been proven as effective instructional strategies for developing

fluency.** Rather, numerous experiences with breaking apart actual sets of objects and developing relationships

between numbers help children internalize parts of number and develop efficient strategies for fact retrieval.

* Van de Walle & Lovin (2006). Teaching student centered mathematics K-3 (p.94). Boston: Pearson.

**Burns (2000) About Teaching Mathematics; Fosnot & Dolk (2001) Young Mathematicians at Work; Richardson

(2002) Assessing Math Concepts; Van de Walle & Lovin (2006) Teaching Student-Centered Mathematics

Mathematics



Kindergarten Unit 5: Numbers 10-20, Counting to 100 by 1 and 10 (~ 5 weeks) Unit Overview: With numbers 1-10 on firm ground, numbers 10-20 can be parsed as “10 together with a number from 1-10.” “12 is 2 more than 10”

(U5). Unlike the role of 5 in numbers 6-10, which loses significance as those numbers are shown in different configurations other than “5 and a number,” the

number 10 is special; it is the anchor that will eventually become the “ten” unit in the place value system. This unit provides a rich opportunity for looking for and

making use of structure (MP 7) as students investigate the importance of 10 in the number system.

Guiding Question: What is the connection between the number 10 and the “tricky teen” numbers?


Component Cluster K.CC Know number names and the count sequence.

K.CC.1 Count to 100 by ones and by tens.

See unit 1.

K.CC.2 Count forward beginning from a given number

within the known sequence (instead of having to begin at

1).

K.CC.3 Write numbers from 0 to 20. Represent a number

of objects with a written numeral 0-20 (with 0 representing

a count of no objects).

Component Cluster K.CC Count to tell the number of objects.

K.CC.4 Understand the relationship between numbers and

quantities; connect counting to cardinality.

See unit 1.

a. When counting objects, say the number names in the

standard order, pairing each object with one and only one

number name and each number name with one and only

one object.

b. Understand that the last number name said tells the

number of objects counted. The number of objects is the

same regardless of their arrangement or the order in which

they were counted.

c. Understand that each successive number name refers to

a quantity that is one larger.

K.CC.5 Count to answer “how many?” questions about as

many as 20 things arranged in a line, a rectangular array,

or a circle, or as many as 10 things in a scattered

configuration; given a number from 1–20, count out that

many objects.

Component Cluster K.NBT Work with numbers 11–19 to gain foundations for place value.

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K.NBT.1 Compose and decompose numbers from 11 to 19

into ten ones and some further ones, e.g., by using objects

or drawings, and record each composition or

decomposition by a drawing or equation (e.g., 18 = 10 +

8)*; understand that these numbers are composed of ten

ones and one, two, three, four, five, six, seven, eight, or

nine ones.

* Kindergarten students should see addition and

subtraction equations, and student writing of equations in

kindergarten is encouraged, but it is not required.

Students explore numbers 11-19 using representations, such as manipulatives or drawings. Rather than unitizing a

ten (recognizing that a set of 10 objects is a unit called a “ten”), which is a standard for First Grade (1.NBT.1a),

kindergarteners keep each count as a single unit as they explore a set of 10 objects and leftovers.

Example:

Teacher: “I have some chips here. Do you think they will fit on our ten frame? Why? Why Not?”

Students: Share thoughts with one another.

Teacher: “Use your ten frame to investigate.”

Students: “Look. There’s too many to fit on the ten frame. Only ten chips will fit on it.”

Teacher: “So you have some leftovers?”

Students: “Yes. I’ll put them over here next to the ten frame.”

Teacher: “So, how many do you have in all?”

Student A: “One, two, three, four, five… ten, eleven, twelve, thirteen, fourteen. I have fourteen. Ten fit on and four

didn’t.”

Student B: Pointing to the ten frame, “See them- that’s 10… 11, 12, 13, 14. There’s fourteen.”

Teacher: Use your recording sheet (or number sentence cards) to show what you found out.

Student Recording Sheets Example:

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Kindergarten Unit 6: Analyze, Compare, Create and Compose Objects (~ 2 weeks) Unit Overview: The year rounds out by beginning explorations of concepts in area: that shapes can be composed of smaller shapes (U6). Students have the

opportunity to make sense of problems and persevere in solving them (MP1) as they apply their understanding of objects and attributes to analyze, compare, create,

and compose objects in challenging contexts.

Guiding Question: How can a shape be created from different shapes?


Component Cluster K.G Analyze, Compare, Create, and Compose Objects.

K.G.4 Analyze and compare two- and three-dimensional

shapes, in different sizes and orientations, using informal

language to describe their similarities, differences, parts

(e.g., number of sides and vertices/“corners”) and other

attributes (e.g., having sides of equal length).

Students relate one shape to another as they note similarities and differences between and among 2-D and 3-D shapes

using informal language. Kindergarteners also distinguish between the most typical examples of a shape from

obvious non-examples.

K.G.5 Model shapes in the world by building shapes from

components (e.g., sticks and clay balls) and drawing

shapes.

Students apply their understanding of geometric attributes of shapes in order to create given shapes. For example,

students may roll a clump of play-doh into a sphere or use their finger to draw a triangle in the sand table, recalling

various attributes in order to create that particular shape.

K.G.6 Compose simple shapes to form larger shapes. For

example, “Can you join these two triangles with full sides

touching to make a rectangle?”

This standard moves beyond identifying and classifying simple shapes to manipulating two or more shapes to create

a new shape. This concept begins to develop as students move, rotate, flip, and arrange puzzle pieces to complete a

puzzle. Kindergarteners use their experiences with puzzles to use simple shapes to create different shapes.

Component Cluster K.G Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

K.G.3 Identify shapes as two-dimensional (lying in a

plane, “flat”) or three dimensional (“solid”).

*This unit focuses only on two-dimensional shape were

covered in Unit 1. Three-dimensional shapes are now

explored.

Students identify objects as flat (2 dimensional) or solid (3 dimensional). As the teacher embeds the vocabulary into

students’ exploration of various shapes, students use the terms two-dimensional and three-dimensional as they

discuss the properties of various shapes.

Mathematics

OA= Operations and Algebraic Thinking NBT= Numbers and Operations in Base Ten MD= Measurement and Data G= Geometry

Grade 1 In Grade 1, instructional time should focus on four critical areas: (1) developing understanding of addition, subtraction, and strategies for addition

and subtraction within 20; (2) developing understanding of whole number relationships and place value, including grouping in tens and ones; (3)

developing understanding of linear measurement and measuring lengths as iterating length units; and (4) reasoning about attributes of, and

composing and decomposing geometric shapes.

(1) Students develop strategies for adding and subtracting whole numbers based on their prior work with small numbers. They use a variety of

models, including discrete objects and length-based models (e.g., cubes connected to form lengths), to model add-to, take-from, put-together,

take-apart, and compare situations to develop meaning for the operations of addition and subtraction, and to develop strategies to solve

arithmetic problems with these operations. Students understand connections between counting and addition and subtraction (e.g., adding two

is the same as counting on two). They use properties of addition to add whole numbers and to create and use increasingly sophisticated

strategies based on these properties (e.g., “making tens”) to solve addition and subtraction problems within 20. By comparing a variety of

solution strategies, children build their understanding of the relationship between addition and subtraction.

(2) Students develop, discuss, and use efficient, accurate, and generalizable methods to add within 100 and subtract multiples of 10. They

compare whole numbers (at least to 100) to develop understanding of and solve problems involving their relative sizes. They think of whole

numbers between 10 and 100 in terms of tens and ones (especially recognizing the numbers 11 to 19 as composed of a ten and some ones).

Through activities that build number sense, they understand the order of the counting numbers and their relative magnitudes.

(3) Students develop an understanding of the meaning and processes of measurement, including underlying concepts such as iterating (the

mental activity of building up the length of an object with equal-sized units) and the transitivity principle for indirect measurement.1

(4) Students compose and decompose plane or solid figures (e.g., put two triangles together to make a quadrilateral) and build understanding

of part-whole relationships as well as the properties of the original and composite shapes. As they combine shapes, they recognize them from

different perspectives and orientations, describe their geometric attributes, and determine how they are alike and different, to develop the

background for measurement and for initial understandings of properties such as congruence and symmetry.










1Students should apply the principle of transitivity of measurement to make indirect comparisons, but they need not use this technical term.

Mathematics


Grade 1: Suggested Distribution of Units in Instructional Days Time Approx.

# of weeks Unit 1: Addition, Subtraction of Numbers to 10 and Fluency 22% ~ 8 weeks

Unit 2: Place Value, Comparison, Addition and Subtraction of Numbers to 20 20% ~ 7 weeks

Unit 3: Measuring, Representing, and Interpreting Data 7% ~ 3 weeks


Unit 5: Identify, Compose, and Partition Shapes 11% ~ 4 weeks


Unit 1: + and - of Numbers to 10 and

Fluency 22%

Unit 2: Place Value, Comparison, + and - of Numbers to 20

20%

Unit 3: Measuring, Representing, and Interpreting Data

7%

Unit 4: Place Value, Comparison, + and - of Numbers to 40

20%

Unit 5: Identify, Compose, and

Partition Shapes 11%

Unit 6: Place Value, Comparison, + and

- of Numbers to 100 20%

Instructional Time

Mathematics


Grade 1 Unit 1: Addition, Subtraction of Numbers to 10 and Fluency (~ 9 weeks)

Unit Overview: Work with “numbers to 10” continues to be a major stepping-stone in learning the place value system. Unlike pre-kindergarten and

kindergarten, this year starts out with exploring addition and subtraction within 10. Fluency with addition/subtraction facts, a major gateway to later

grades, also begins right away with the intention of energetically practicing the entire year (U1). This unit provides opportunities for students to

model with mathematics (MP 4) as they use models to support their understanding and explanations of their addition and subtraction strategies.

Guiding Question: What are good strategies for solving addition and subtraction problems to 10?


Component Cluster 1.OA Represent and solve problems involving addition and subtraction.*

*Within 10 only is addressed in Unit 1. Unit 2 will complete the standard to 20.

1.OA.1 Use addition and subtraction within 20 to solve

word problems involving situations of adding to, taking

from, putting together, taking apart, and comparing, with

unknowns in all positions, e.g., by using objects, drawings,

and equations with a symbol for the unknown number to

represent the problem.1

1 See Glossary, Table 1

First grade students extend their experiences in Kindergarten by working with numbers to 20 to solve a new type of

problem situation: Compare (See Table 1 at end of document for examples of all problem types). Compare problems

are more complex than those introduced in Kindergarten. In order to solve compare problem types, First Graders

must think about a quantity that is not physically present and must conceptualize that amount. In addition, the

language of “how many more” often becomes lost or not heard with the language of ‘who has more’. With rich

experiences that encourage students to match problems with objects and drawings can help students master these

challenges.

First Graders also extend the sophistication of the methods they used in Kindergarten (counting) to add and subtract

within this larger range. Now, First Grade students use the methods of counting on, making ten, and doubles +/- 1 or

+/- 2 to solve problems.

In order for students to read and use equations to represent their thinking, they need extensive experiences with

addition and subtraction situations in order to connect the experiences with symbols (+, -, =) and equations (5 = 3 +

2). In Kindergarten, students demonstrated the understanding of how objects can be joined (addition) and separated

(subtraction) by representing addition and subtraction situations using objects, pictures and words. In First Grade,

students extend this understanding of addition and subtraction situations to use the addition symbol (+) to represent

joining situations, the subtraction symbol (-) to represent separating situations, and the equal sign (=) to represent a

relationship regarding quantity between one side of the equation and the other.

1.OA.2 Solve word problems that call for addition of three

whole numbers whose sum is less than or equal to 20, e.g.,

by using objects, drawings, and equations with a symbol

for the unknown number to represent the problem.

First Grade students solve multi-step word problems by adding (joining) three numbers whose sum is less than or

equal to 20, using a variety of mathematical representations (ten-frame, number line, make ten).

Component Cluster 1.OA Understand and apply properties of operations and the relationship between addition and subtraction.* *Within 10 only is addressed in Unit 1. Unit 2 will complete the standard to 20.

1.OA.3 Apply properties of operations as strategies to add

and subtract.2 Examples: If 8 + 3 = 11 is known, then 3 +

8 = 11 is also known. (Commutative property of addition.)

To add 2 + 6 + 4, the second two numbers can be added to

Elementary students often believe that there are hundreds of isolated addition and subtraction facts to be mastered.

However, when students understand the commutative and associative properties, they are able to use relationships

between and among numbers to solve problems. First Grade students apply properties of operations as strategies to

add and subtract. Students do not use the formal terms “commutative” and “associative”. Rather, they use the

Mathematics


make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative

property of addition.)

2 Students need not use formal terms for these properties.

understandings of the commutative and associative property to solve problems.

Students use mathematical tools and representations (e.g., cubes, counters, number balance, number line, ten-frames,

100 chart) to model these ideas.

1.OA.4 Understand subtraction as an unknown-addend

problem.

For example, subtract 10 – 8 by finding the number that

makes 10 when added to 8. Add and subtract within 20.

First Graders often find subtraction facts more difficult to learn than addition facts. By understanding the

relationship between addition and subtraction, First Graders are able to use various strategies to solve subtraction

problems including Think Addition, Build Up Through Ten, and Build Back Through Ten.

Component Cluster 1.OA Add and subtract within 20.* *Within 10 only is addressed in Unit 1. Unit 2 will complete the standard to 20.

1.OA.5 Relate counting to addition and subtraction (e.g.,

by counting on 2 to add 2).

When solving addition and subtraction problems to 20, First Graders often use counting strategies, such as counting

all, counting on, and counting back, before fully developing the essential strategy of using 10 as a benchmark

number. Once students have developed counting strategies to solve addition and subtraction problems, it is very

important to move students toward strategies that focus on composing and decomposing number using ten as a

benchmark number, as discussed in 1.OA.6, particularly since counting becomes a hindrance when working with

larger numbers. By the end of First Grade, students are expected to use the strategy of 10 to solve problems.

Counting All: Students count all objects to determine the total amount.

Counting On & Counting Back: Students hold a “start number” in their head and count on/back from that number.

1.OA.6 Add and subtract within 20, demonstrating fluency

for addition and subtraction within 10. Use strategies such

as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4

= 14); decomposing a number leading to a ten (e.g., 13 – 4

= 13 – 3 – 1 = 10 – 1 = 9); using the relationship between

addition and subtraction (e.g., knowing that 8 + 4 = 12,

one knows 12 – 8 = 4); and creating equivalent but easier

or known sums (e.g., adding 6 + 7 by creating the known

equivalent 6 + 6 + 1 = 12 + 1 = 13).

In First Grade, students learn about and use various strategies to solve addition and subtraction problems. When

students repeatedly use strategies that make sense to them, they internalize facts and develop fluency for addition and

subtraction within 10. When students are able to demonstrate fluency within 10, they are accurate, efficient, and

flexible. First Graders then apply similar strategies for solving problems within 20, building the foundation for

fluency to 20 in Second Grade.

Component Cluster 1.OA Work with addition and subtraction equations.

1.OA.7 Understand the meaning of the equal sign, and

determine if equations involving addition and subtraction

are true or false. For example, which of the following

equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 +

2 = 2 + 5, 4 + 1 = 5 + 2.

In order to determine whether an equation is true or false, First Grade students must first understand the meaning of

the equal sign. This is developed as students in Kindergarten and First Grade solve numerous joining and separating

situations with mathematical tools, rather than symbols. Once the concepts of joining, separating, and “the same

amount/quantity as” are developed concretely, First Graders are ready to connect these experiences to the

corresponding symbols (+, -, =). Thus, students learn that the equal sign does not mean “the answer comes next”, but

that the symbol signifies an equivalent relationship that the left side ‘has the same value as’ the right side of the

equation.

When students understand that an equation needs to “balance”, with equal quantities on both sides of the equal sign,

they understand various representations of equations, such as:

Mathematics


an operation on the left side of the equal sign and the answer on the right side (5 + 8 = 13)

an operation on the right side of the equal sign and the answer on the left side (13 = 5 + 8)

numbers on both sides of the equal sign (6 = 6)

operations on both sides of the equal sign (5 + 2 = 4 + 3).

Once students understand the meaning of the equal sign, they are able to determine if an equation is true (9 = 9) or

false (9 = 8).

1.OA.8 Determine the unknown whole number in an

addition or subtraction equation relating three whole

numbers. For example, determine the unknown number

that makes the equation true in each of the equations 8 + ?

= 11, 5 = _ – 3, 6 + 6 = _.

First Graders use their understanding of and strategies related to addition and subtraction as described in 1.OA.4 and

1.OA.6 to solve equations with an unknown. Rather than symbols, the unknown symbols are boxes or pictures.

Mathematics


Grade 1 Unit 2: Place Value, Comparison, Addition and Subtraction of Numbers to 20 (~ 7 weeks)

Unit Overview: The next major stepping-stone is learning to group “10 ones” as a single unit: 1 ten. Work begins slowly by “adding and subtracting

across a 10”.

Solutions like that shown above for 8 + 5 reinforce the need to “make 10.” This strategy of the “completion of a unit” empowers students in later

grades to understand the “renaming” of the addition algorithm, to add 298 and 37 (i.e., 298 + 2 + 35), and add 4 m., 80 cm. and 50 cm. This unit

provides students with the opportunity to construct viable arguments and critique the reasoning of others (MP 3) as they articulate their solution

strategies and respond to those of other students.

Guiding Question: How are addition and subtraction strategies with numbers to 20 different or similar to strategies for numbers to 10?


Component Cluster 1.OA Represent and solve problems involving addition and subtraction.

*Focus on numbers to 20.

1.OA.1 – 1.OA.2 See Unit 1.

Component Cluster 1.OA Understand and apply properties of operations and the relationship between addition and subtraction.

*Focus on numbers to 20. 1.OA.3 – 1.OA.4 See Unit 1.

Component Cluster 1.OA Add and subtract within 20.


1.OA.5 – 1.OA.6 See Unit 1.

Component Cluster 1.OA Work with addition and subtraction equations.


1.OA.7 – 1.OA.8

See Unit 1.

Component Cluster 1.NBT Understand Place Value.*

*Focus on numbers to 20. 1.NBT.2 Understand that the two digits of a two-digit First Grade students are introduced to the idea that a bundle of ten ones is called “a ten”. This is known as unitizing.

Mathematics


number represent amounts of tens and ones. Understand

the following as special cases:

a. 10 can be thought of as a bundle of ten ones — called

a “ten.”

When First Grade students unitize a group of ten ones as a whole unit (“a ten”), they are able to count groups as

though they were individual objects. For example, 4 trains of ten cubes each have a value of 10 and would be

counted as 40 rather than as 4. This is a monumental shift in thinking, and can often be challenging for young

children to consider a group of something as “one” when all previous experiences have been counting single objects.

This is the foundation of the place value system and requires time and rich experiences with concrete manipulatives

to develop.

A student’s ability to conserve number is an important aspect of this standard. It is not obvious to young children

that 42 cubes is the same amount as 4 tens and 2 left-overs. It is also not obvious that 42 could also be composed of 2

groups of 10 and 22 leftovers. Therefore, first graders require ample time grouping proportional objects (e.g., cubes,

beans, beads, ten-frames) to make groups of ten, rather than using pre-grouped materials (e.g., base ten blocks, pre-

made bean sticks) that have to be “traded” or are non-proportional (e.g., money).

As children build this understanding of grouping, they move through several stages:

Counting By Ones; Counting by Groups & Singles; and Counting by Tens and Ones.

b. The numbers from 11 to 19 are composed of a ten and

one, two, three, four, five, six, seven, eight, or nine

ones.

First Grade students extend their work from Kindergarten when they composed and decomposed numbers from 11 to

19 into ten ones and some further ones. In Kindergarten, everything was thought of as individual units: “ones”. In

First Grade, students are asked to unitize those ten individual ones as a whole unit: “one ten”. Students in first grade

explore the idea that the teen numbers (11 to 19) can be expressed as one ten and some leftover ones. Ample

experiences with a variety of groupable materials that are proportional (e.g., cubes, links, beans, beads) and ten

frames help students develop this concept.

In addition, when learning about forming groups of 10, First Grade students learn that a numeral can stand for many

different amounts, depending on its position or place in a number. This is an important realization as young children

begin to work through reversals of digits, particularly in the teen numbers.

1.NBT.3 Compare two two-digit numbers based on

meanings of the tens and ones digits, recording the results

of comparisons with the symbols >, =, and <.

First Grade students use their understanding of groups and order of digits to compare two numbers by examining the

amount of tens and ones in each number. After numerous experiences verbally comparing two sets of objects using

comparison vocabulary (e.g., 42 is more than 31. 23 is less than 52, 61 is the same amount as 61.), first grade

students connect the vocabulary to the symbols: greater than (>), less than (<), equal to (=).

Mathematics


Grade 1 Unit 3: Measuring, Representing and Interpreting Data (~ 3 weeks)

Unit Overview: A unit on expressing length measurement as numbers (U3) provides a few weeks in which to practice and internalize “making a 10”

during daily measurement and data fluency activities. Introducing measurement early also has the added bonus of opening up the variety and types of

word problems that can be asked throughout the year. Students will need to work on attending to precision (MP 6) in this unit focused on collecting

measurements and data.

Guiding Question: After you collect, measure, organize, and display data, how can we use the information to make sense of the world around us?


Component Cluster 1.MD Measure lengths indirectly and by iterating length units.

1.MD.1 Order three objects by length; compare the lengths

of two objects indirectly by using a third object.

First Grade students continue to use direct comparison to compare lengths. Direct comparison means that students

compare the amount of an attribute in two objects without measurement. Sometimes, a third object can be used as an

intermediary, allowing indirect comparison. Another important set of skills and understandings is ordering a set of

objects by length. Such sequencing requires multiple comparisons (no more than 6 objects). Students need to

understand that each object in a seriation is larger than those that come before it, and shorter than those that come

after.

1.MD.2 Express the length of an object as a whole number

of length units, by laying multiple copies of a shorter

object (the length unit) end to end; understand that the

length measurement of an object is the number of same-

size length units that span it with no gaps or overlaps.

Limit to contexts where the object being measured is

spanned by a whole number of length units with no gaps or

overlaps.

First Graders use objects to measure items to help students focus on the attribute being measured. Objects also lend

itself to future discussions regarding the need for a standard unit.

First Grade students use multiple copies of one object to measure the length of a larger object. They learn to lay

physical units such as centimeter or inch manipulatives end-to-end and count them to measure a length. Through

numerous experiences and careful questioning by the teacher, students will recognize the importance of careful

measuring so that there are not any gaps or overlaps in order to get an accurate measurement. This concept is a

foundational building block for the concept of area in 3rd

Grade.

When students use different sized units to measure the same object, they learn that the sizes of the units must be

considered, rather than relying solely on the amount of objects counted. In addition, understanding that the results of

measurement and direct comparison have the same results encourages children to use measurement strategies.

Component Cluster 1.MD Represent and interpret data.

1.MD.4 Organize, represent, and interpret data with up to

three categories; ask and answer questions about the total

number of data points, how many in each category, and

how many more or less are in one category than in another.

First Grade students collect and use categorical data (e.g., eye color, shoe size, age) to answer a question. The data

collected are often organized in a chart or table. Once the data are collected, First Graders interpret the data to

determine the answer to the question posed. They also describe the data noting particular aspects such as the total

number of answers, which category had the most/least responses, and interesting differences/similarities between the

categories. As the teacher provides numerous opportunities for students to create questions, determine up to 3

categories of possible responses, collect data, organize data, and interpret the results, First Graders build a solid

foundation for future data representations (picture and bar graphs) in Second Grade.

Mathematics


Grade 1 Unit 4: Place Value, Comparison, Addition and Subtraction of Numbers to 40 (~ 7 weeks)

Unit Overview: The focus of adding and subtracting within 40 is on establishing “1 ten” as a new unit (U4). Before, students loosely grouped 10

objects to make 10. Now they transition to thinking of that 10 as a single unit (using 10 linker cubes stuck together, for example). Students begin to

see in problems like 23+6 that they can mentally push the “2 tens” in 23 over to the side and concentrate on the familiar addition problem 3+6. This

unit is strengthened from an explicit focus on looking for and expressing regularity in repeated reasoning (MP 8) as students apply and extend

strategies from addition and subtraction facts within 20 to the larger number system.

Guiding Question: How are strategies for solving addition and subtraction problems to 40 similar to or different than solving problems to 20?


Component Cluster 1.NBT Extend the counting sequence.*


1.NBT.1 Count to 120, starting at any number less than

120. In this range, read and write numerals and represent a

number of objects with a written numeral.

First Grade students rote count forward to 120 by counting on from any number less than 120. First graders develop

accurate counting strategies that build on the understanding of how the numbers in the counting sequence are

related—each number is one more (or one less) than the number before (or after). In addition, first grade students

read and write numerals to represent a given amount.

Component Cluster 1.NBT Understand place value.* *Focus on numbers to 40.

1.NBT.2 Understand that the two digits of a two-digit



a. 10 can be thought of as a bundle of ten ones — called a

“ten.”

c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to


tens (and 0 ones).

First Grade students are introduced to the idea that a bundle of ten ones is called “a ten”. This is known as unitizing.

When First Grade students unitize a group of ten ones as a whole unit (“a ten”), they are able to count groups as

though they were individual objects. For example, 4 trains of ten cubes each have a value of 10 and would be

counted as 40 rather than as 4. This is a monumental shift in thinking, and can often be challenging for young

children to consider a group of something as “one” when all previous experiences have been counting single objects.

This is the foundation of the place value system and requires time and rich experiences with concrete manipulatives

to develop.

A student’s ability to conserve number is an important aspect of this standard. It is not obvious to young children

that 42 cubes is the same amount as 4 tens and 2 left-overs. It is also not obvious that 42 could also be composed of 2

groups of 10 and 22 leftovers. Therefore, first graders require ample time grouping proportional objects (e.g., cubes,

beans, beads, ten-frames) to make groups of ten, rather than using pre-grouped materials (e.g., base ten blocks, pre-

made bean sticks) that have to be “traded” or are non-proportional (e.g., money).

As children build this understanding of grouping, they move through several stages:

Counting By Ones; Counting by Groups & Singles; and Counting by Tens and Ones.

First Grade students apply their understanding of groups of ten as stated in 1.NBT.2b to decade numbers (e.g. 10, 20,

30, 40). As they work with groupable objects, first grade students understand that 10, 20, 30…80, 90 are comprised

of a certain amount of groups of tens with none left-over.



See Unit 1.

Mathematics



Component Cluster 1.NBT Use place value understanding and properties of operations to add and subtract.* *Focus on numbers to 40.

1.NBT.4 Add within 100, including adding a two-digit

number and a one-digit number, and adding a two-digit

number and a multiple of 10, using concrete models or

drawings and strategies based on place value, properties of

operations, and/or the relationship between addition and

subtraction; relate the strategy to a written method and

explain the reasoning used. Understand that in adding two-

digit numbers, one adds tens and tens, ones and ones; and

sometimes it is necessary to compose a ten.

First Grade students use concrete materials, models, drawings and place value strategies to add within 100. They do

so by being flexible with numbers as they use the base-ten system to solve problems. The standard algorithm of

carrying or borrowing is neither an expectation nor a focus in First Grade.

Example: 24 red apples and 8 green apples are on the table. How many apples are on the

table?

Student A:

I used ten frames. I put 24 chips on 3 ten frames. Then, I counted out 8 more chips. 6 of them

filled up the third ten frame. That meant I had 2 left over. 3 tens and 2 left over. That’s 32. So,

there are 32 apples on the table.

Student B:

I used an open number line. I started at 24. I knew that I needed 6 more jumps to get to 30. So, I

broke apart 8 into 6 and 2. I took 6 jumps to land on 30 and then 2 more. I landed on 32. So,

there are 32 apples on the table.

Student C:

I turned 8 into 10 by adding 2 because it’s easier to add.

So, 24 and ten more is 34.

But, since I added 2 extra, I had to take them off again.

34 minus 2 is 32. There are 32 apples on the table.

Example: 63 apples are in the basket. Mary put 20 more apples in the basket. How many

apples are in the basket?

Student A:

I used ten frames. I picked out 6 filled ten frames. That’s 60. I got the ten frame with 3 on it.

That’s 63. Then, I picked one more filled ten frame for part of the 20 that Mary put in. That

24 + 6 = 30

30 + 2 = 32

24 + 6 = 30

30 + 2 = 32

8 + 2 = 10

24 + 10 = 34

34 – 2 = 32

Mathematics


made 73. Then, I got one more filled ten frame to make the rest of the 20 apples from Mary.

That’s 83. So, there are 83 apples in the basket.

Student B:

I used a hundreds chart. I started at 63 and jumped down one row to 73. That means I moved 10

spaces. Then, I jumped down one more row (that’s another 10 spaces) and landed on 83. So,

there are 83 apples in the basket.

Student C:

I knew that 10 more than 63 is 73. And 10 more than 73 is 83. So, there are 83 apples in the

basket.

1.NBT.5 Given a two-digit number, mentally find 10 more

or 10 less than the number, without having to count;

explain the reasoning used.

First Graders build on their counting by tens work in Kindergarten by mentally adding ten more and ten less than any

number less than 100. First graders are not expected to compute differences of two-digit numbers other than

multiples of ten. Ample experiences with ten frames and the number line provide students with opportunities to think

about groups of ten, moving them beyond simply rote counting by tens on and off the decade. Such representations

63 + 10 = 73 73 + 10 = 83

63 + 10 = 73 73 + 10 = 83

63 + 10 = 73 73 + 10 = 83

Mathematics


lead to solving such problems mentally.

1.NBT.6 Subtract multiples of 10 in the range 10-90 from

multiples of 10 in the range 10-90 (positive or zero

differences), using concrete models or drawings and

strategies based on place value, properties of operations,

and/or the relationship between addition and subtraction;

relate the strategy to a written method and explain the

reasoning used.

First Grade students use concrete models, drawings and place value strategies to subtract multiples of 10 from decade

numbers (e.g., 30, 40, 50). They often use similar strategies as discussed in 1.OA.4.

Mathematics


Grade 1 Unit 5: Identify, Compose, and Partition Shapes (~ 3 weeks)

Unit Overview: Placed in between the two heavy-duty number units is a unit on geometry (U5). The geometry unit puts necessary internalization time between

the “within 40” unit (U4) and the “within 100” unit (U6). It also gives students who may be more spatially oriented a chance to build confidence before heading

back into arithmetic. As students work with composing and decomposing shapes (including partitioning, or dividing , clocks, circles, and rectangles), they have an

opportunity to attend to precision (MP 6) with the language they use to describe their explorations.

Guiding Question: What are the most precise words we can use to describe the objects around us?


Component Cluster 1.MD Tell and write time.

1.MD.3 Tell and write time in hours and half-hours using

analog and digital clocks.

For young children, reading a clock can be a difficult skill to learn. In particular, they must understand the

differences between the two hands on the clock and the functions of these hands. By carefully watching and talking

about a clock with only the hour hand, First Graders notice when the hour hand is directly pointing at a number, or

when it is slightly ahead/behind a number. In addition, using language, such as “about 5 o’clock” and “a little bit

past 6 o’clock”, and “almost 8 o’clock” helps children begin to read an hour clock with some accuracy. Through rich

experiences, First Grade students read both analog (numbers and hands) and digital clocks, orally tell the time, and

write the time to the hour and half-hour.

Component Cluster 1.G Reason with shapes and their attributes.

1.G.1 Distinguish between defining attributes (e.g.,

triangles are closed and three-sided) versus non-defining

attributes (e.g., color, orientation, overall size) ; build and

draw shapes to possess defining attributes.

First Grade students use their beginning knowledge of defining and non-defining attributes of shapes to identify,

name, build and draw shapes (including triangles, squares, rectangles, and trapezoids). They understand that defining

attributes are always-present features that classify a particular object (e.g., number of sides, angles, etc.). They also

understand that non-defining attributes are features that may be present, but do not identify what the shape is called

(e.g., color, size, orientation, etc.).

1.G.2 Compose two-dimensional shapes (rectangles,

squares, trapezoids, triangles, half-circles, and quarter-

circles) or three-dimensional shapes (cubes, right

rectangular prisms, right circular cones, and right circular

cylinders) to create a composite shape, and compose new

shapes from the composite shape.1

1 Students do not need to learn formal names such as “right

rectangular prism.”

As first graders create composite shapes, a figure made up of two or more geometric shapes, they begin to see how

shapes fit together to create different shapes. They also begin to notice shapes within an already existing shape.

They may use such tools as pattern blocks, tangrams, attribute blocks, straws, twist ties, or virtual shapes to compose

different shapes.

First graders learn to perceive a combination of shapes as a single new shape (e.g., recognizing that two isosceles

triangles can be combined to make a rhombus, and simultaneously seeing the rhombus and the two triangles). Thus,

they develop competencies that include:

Solving shape puzzles

Constructing designs with shapes

Creating and maintaining a shape as a unit

1.G.3 Partition circles and rectangles into two and four

equal shares, describe the shares using the words halves,

fourths, and quarters, and use the phrases half of, fourth of,

and quarter of. Describe the whole as two of, or four of the

shares. Understand for these examples that decomposing

into more equal shares creates smaller shares.

First Graders begin to partition regions into equal shares using a context (e.g., cookies, pies, pizza). This is a

foundational building block of fractions, which will be extended in future grades. Through ample experiences with

multiple representations, students use the words, halves, fourths, and quarters, and the phrases half of, fourth of, and

quarter of to describe their thinking and solutions. Working with the “the whole”, students understand that “the

whole” is composed of two halves, or four fourths or four quarters.

Mathematics


Grade 1 Unit 6: Place Value, Comparison, and Addition and Subtraction to 100 (~ 7 weeks)

Unit Overview: The focus of the “adding and subtracting within 100” unit (U6) is different than the “within 10” and “within 40” units. Here the new

level of complexity is to also introduce the addition and subtraction algorithms using simple examples and the familiar units of 10 made out of linker

cubes. Students will have the opportunity to look for and make use of structure (MP 7) within our number system.

Guiding Question: Why is the number ten so important in our number sytem?


Component Cluster 1.NBT Extend the counting sequence.*


1.NBT.1 Count to 120, starting at any number less than

120. In this range, read and write numerals and represent a

number of objects with a written numeral.

See Unit 4.

Component Cluster 1.NBT Understand place value.*

*Focus on numbers to 100. 1.NBT.2 Understand that the two digits of a two-digit



a. 10 can be thought of as a bundle of ten ones — called a

“ten.”

d. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to


tens (and 0 ones).

See Unit 4.




Component Cluster 1.NBT Use place value understanding and properties of operations to add and subtract.*


1.NBT.4 Add within 100, including adding a two-digit

number and a one-digit number, and adding a two-digit

number and a multiple of 10, using concrete models or

drawings and strategies based on place value, properties of

operations, and/or the relationship between addition and

subtraction; relate the strategy to a written method and

explain the reasoning used. Understand that in adding two-

digit numbers, one adds tens and tens, ones and ones; and

sometimes it is necessary to compose a ten.

See Unit 4.

Mathematics


1.NBT.5 Given a two-digit number, mentally find 10 more

or 10 less than the number, without having to count;

explain the reasoning used.

1.NBT.6 Subtract multiples of 10 in the range 10-90 from

multiples of 10 in the range 10-90 (positive or zero

differences), using concrete models or drawings and




reasoning used.

Mathematics


Grade 2

In Grade 2, instructional time should focus on four critical areas: (1) extending understanding of base-ten notation; (2) building fluency with addition

and subtraction; (3) using standard units of measure; and (4) describing and analyzing shapes.

(1) Students extend their understanding of the base-ten system. This includes ideas of counting in fives, tens, and multiples of hundreds, tens,

and ones, as well as number relationships involving these units, including comparing. Students understand multi-digit numbers (up to 1000)

written in base-ten notation, recognizing that the digits in each place represent amounts of thousands, hundreds, tens, or ones (e.g., 853 is 8

hundreds + 5 tens + 3 ones).

(2) Students use their understanding of addition to develop fluency with addition and subtraction within 100. They solve problems within

1000 by applying their understanding of models for addition and subtraction, and they develop, discuss, and use efficient, accurate, and

generalizable methods to compute sums and differences of whole numbers in base-ten notation, using their understanding of place value and

the properties of operations. They select and accurately apply methods that are appropriate for the context and the numbers involved to

mentally calculate sums and differences for numbers with only tens or only hundreds.

(3) Students recognize the need for standard units of measure (centimeter and inch) and they use rulers and other measurement tools with the

understanding that linear measure involves an iteration of units. They recognize that the smaller the unit, the more iterations they need to

cover a given length.

(4) Students describe and analyze shapes by examining their sides and angles. Students investigate, describe, and reason about decomposing

and combining shapes to make other shapes. Through building, drawing, and analyzing two- and three-dimensional shapes, students develop a

foundation for understanding area, volume, congruence, similarity, and symmetry in later grades.










Mathematics


Grade 2: Suggested Distribution of Units in Instructional Days Time Approx. # of

weeks

Unit 1: Fluency with Sums and Differences to 20, Add/Subtract Numbers to 100 11% ~ 4 weeks

Unit 2: Adding and Subtraction of Length and Time Measurements 11% ~ 4 weeks

Unit 3: Place Value, Counting, and Comparison of Numbers to 1000 14% ~ 5 weeks

Unit 4: Addition and Subtraction of Numbers to 1000 with Problem-Solving 20% ~ 7 weeks

Unit 5: Measuring, Representing, and Solving with Money and Length Data 17% ~ 6 weeks

Unit 6: Preparation for Multiplication and Division Concepts 17% ~ 6 weeks

Unit 7: Reasoning about Shapes and Fractions of Shapes 10% ~ 4 weeks

Unit 1: +/- Numbers to 100,

Fluency with +/- to 20

11%

Unit 2: + and - of Length, Weight,

Capacity, and Time 11%

Unit 3: Place Value, Counting, and Comparison to

1000 14%

Unit 4: + and - of Numbers to 1000

20%

Unit 5: Measuring, Representing, and

Solving with Money and Length Data

17%

Unit 6:Preparation for Mult. and Div.

Concepts 17%

Unit 7: Reasongin about Shapes and

Fractions of Shapes 10%

Instructional Time

Mathematics


Grade 2 Unit 1: Fluency with Sums and Differences to 20, Add/Subtract Numbers to 100 (~4 weeks)

Unit Overview: Students arrive in grade 2 having an extensive background working with numbers to 10. Unit 1 establishes a motivating, differentiated fluency

program in the first few weeks that will provide each student with enough practice to achieve mastery of the required fluencies (i.e., adding and subtracting within

20 and within 100) by the end of the year. Students learn to represent and solve problems using addition and subtraction: a practice that will also continue

throughout the year. Students look for and make use of structure (MP 7) as they develop strategies for fluency with sums and differences to 20.

Guiding Question: What are good strategies to use when adding and subtracting numbers to 100 with fluency within 20?



*Story problems focus primarily on the positions of result and change unknown.


one- and two-step word problems involving situations of

adding to, taking from, putting together, taking apart, and

comparing, with unknowns in all positions, e.g., by using

drawings and equations with a symbol for the unknown

number to represent the problem.1

1 See Glossary, Table 1.

Second Grade students extend their work with addition and subtraction word problems in two major ways. First, they

represent and solve word problems within 100, building upon their previous work to 20. In addition, they represent

and solve one and two-step word problems of all three types (Result Unknown, Change Unknown, Start Unknown).

Please see Table 1 at end of document for examples of all problem types. One-step word problems use one

operation. Two-step word problems use two operations which may include the same operation or opposite operations.

Two-Step Problems: Because Second Graders are still developing proficiency with the most difficult subtypes

(shaded in white in Table 1 at end of the glossary): Add To/Start Unknown; Take From/Start Unknown;

Compare/Bigger Unknown; and Compare/Smaller Unknown, two-step problems do not involve these sub-types

(Common Core Standards Writing Team, May 2011). Furthermore, most two-step problems should focus on single-

digit addends since the primary focus of the standard is the problem-type.

As second grade students solve one- and two-step problems they use manipulatives such as snap cubes, place value

materials (groupable and pre-grouped), ten frames, etc.; create drawings of manipulatives to show their thinking; or

use number lines to solve and describe their strategies. They then relate their drawings and materials to equations.

By solving a variety of addition and subtraction word problems, second grade students determine the unknown in all

positions (Result unknown, Change unknown, and Start unknown). Rather than a letter (“n”), boxes or pictures are

used to represent the unknown number. See Glossary, Table 1 for examples (found at end of document).

Second Graders use a range of methods, often mastering more complex strategies such as making tens and doubles

and near doubles for problems involving addition and subtraction within 20. Moving beyond counting and counting-

on, second grade students apply their understanding of place value to solve problems.

Component Cluster 2.OA Add and subtract within 20. *From this point forward, fluency practice with addition and subtraction to 20 is part of the students’ on-going experience.

2.OA.2 Fluently add and subtract within 20 using mental

strategies.2 By end of Grade 2, know from memory all

sums of two one-digit numbers.

2See standard 1.OA.6 for a list of mental strategies.

Building upon their work in First Grade, Second Graders use various addition and subtraction strategies in order to

fluently add and subtract within 20:

Mathematics


1.OA.6 Mental Strategies

Counting on

Making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14)

Decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9)

Using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows

12 – 8 = 4)

Creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6

+ 6 + 1 = 12, 12 + 1 = 13

Second Graders internalize facts and develop fluency by repeatedly using strategies that make sense to them. When

students are able to demonstrate fluency they are accurate, efficient, and flexible. Students must have efficient

strategies in order to know sums from memory.

Research indicates that teachers can best support students’ memory of the sums of two one-digit numbers through

varied experiences including making 10, breaking numbers apart, and working on mental strategies. These strategies

replace the use of repetitive timed tests in which students try to memorize operations as if there were not any

relationships among the various facts. When teachers teach facts for automaticity, rather than memorization, they

encourage students to think about the relationships among the facts. (Fosnot & Dolk, 2001)

Component Cluster 2.NBT Use place value understanding and properties of operations to add and subtract.

2.NBT.5 Fluently add and subtract within 100 using


and/or the relationship between addition and subtraction.

There are various strategies that Second Grade students understand and use when adding and subtracting within 100

(such as those listed in the standard). The standard algorithm of carrying or borrowing is neither an expectation nor a

focus in Second Grade. Students use multiple strategies for addition and subtraction in Grades K-3. By the end of

Third Grade students use a range of algorithms based on place value, properties of operations, and/or the relationship

between addition and subtraction to fluently add and subtract within 1000. Students are expected to fluently add and

subtract multi-digit whole numbers using the standard algorithm by the end of Grade 4.

Mathematics


Grade 2 Unit 2: Adding and Subtraction of Length and Time Measurements (~4 weeks)

Unit Overview: In Unit 2, students learn to measure and estimate using standard units for length and solve measurement word problems involving addition and

subtraction of length. A major objective is for students to use measurement tools with the understanding that linear measure involves an iteration of units and that

the smaller a unit, the more iterations are necessary to cover a given length. An underlying goal for this unit is for students to learn the meaning of a “unit” in

different contexts (e.g., length and time). This understanding serves as the foundation of arithmetic, measurement, and geometry in elementary school. In

particular, units play a central role in the next unit and in the addition and subtraction work of Unit 4.

Guiding Question: How does using a different unit change our measurement?


Component Cluster 2.MD Measure and estimate lengths in standard units.

2.MD.1 Measure the length of an object by selecting and

using appropriate tools such as rulers, yardsticks, meter

sticks, and measuring tapes.

Second Graders build upon their non-standard measurement experiences in First Grade by measuring in standard

units for the first time. Using both customary (inches and feet) and metric (centimeters and meters) units, Second

Graders select an attribute to be measured (e.g., length of classroom), choose an appropriate unit of measurement

(e.g., yardstick), and determine the number of units (e.g., yards). As teachers provide rich tasks that ask students to

perform real measurements, these foundational understandings of measurement are developed:

Understand that larger units (e.g., yard) can be subdivided into equivalent units (e.g., inches) (partition).

Understand that the same object or many objects of the same size such as paper clips can be repeatedly used to

determine the length of an object (iteration).

Understand the relationship between the size of a unit and the number of units needed (compensatory principal).

Thus, the smaller the unit, the more units it will take to measure the selected attribute.

By the end of Second Grade, students will have also learned specific measurements as it relates to feet, yards and

meters:

There are 12 inches in a foot.

There are 3 feet in a yard.

There are 100 centimeters in a meter.

2.MD.2 Measure the length of an object twice, using

length units of different lengths for the two measurements;

describe how the two measurements relate to the size of

the unit chosen.

Second Grade students measure an object using two units of different lengths. This experience helps students realize

that the unit used is as important as the attribute being measured. This is a difficult concept for young children and

will require numerous experiences for students to predict, measure, and discuss outcomes.

2.MD.3 Estimate lengths using units of inches, feet,

centimeters, and meters.

Second Grade students estimate the lengths of objects using inches, feet, centimeters, and meters prior to measuring.

Estimation helps the students focus on the attribute being measured and the measuring process. As students estimate,

the student has to consider the size of the unit- helping them to become more familiar with the unit size. In addition,

estimation also creates a problem to be solved rather than a task to be completed. Once a student has made an

estimate, the student then measures the object and reflects on the accuracy of the estimate made and considers this

information for the next measurement.

2.MD.4 Measure to determine how much longer one

object is than another, expressing the length difference in

terms of a standard length unit.

Second Grade students determine the difference in length between two objects by using the same tool and unit to

measure both objects. Students choose two objects to measure, identify an appropriate tool and unit, measure both

objects, and then determine the differences in lengths.

Mathematics


Component Cluster 2.MD Relate addition and subtraction to length.

2.MD.5 Use addition and subtraction within 100 to solve

word problems involving lengths that are given in the same

units, e.g., by using drawings (such as drawings of rulers)


represent the problem.

Second Grade students apply the concept of length to solve addition and subtraction word problems with numbers

within 100. Students should use the same unit of measurement in these problems. Equations may vary depending on

students’ interpretation of the task.

2.MD.6 Represent whole numbers as lengths from 0 on a

number line diagram with equally spaced points

corresponding to the numbers 0, 1, 2, ..., and represent

whole-number sums and differences within 100 on a

number line diagram.

Building upon their experiences with open number lines, Second Grade students create number lines with evenly

spaced points corresponding to the numbers to solve addition and subtraction problems to 100. They recognize the

similarities between a number line and a ruler.

Component Cluster 2.MD Work with time and money. *Focus on time. The portion of the cluster relating to money will be taught in Unit 5.

2.MD.7 Tell and write time from analog and digital clocks

to the nearest five minutes, using a.m. and p.m.

Second Grade students extend their work with telling time to the hour and half-hour in First Grade in order to tell

(orally and in writing) the time indicated on both analog and digital clocks to the nearest five minutes. Teachers help

students make connections between skip counting by 5s (2.NBT.2) and telling time to the nearest five minutes on an

analog clock. Students also indicate if the time is in the morning (a.m.) or in the afternoon/evening (p.m) as they

record the time.

Learning to tell time is challenging for children. In order to read an analog clock, they must be able to read a dial-

type instrument. Furthermore, they must realize that the hour hand indicates broad, approximate time while the

minute hand indicates the minutes in between each hour. As students experience clocks with only hour hands, they

begin to realize that when the time is two o’clock, two-fifteen, or two forty-five, the hour hand looks different- but is

still considered “two”. Discussing time as “about 2 o’clock”, “a little past 2 o’clock”, and “almost 3 o’clock” helps

build vocabulary to use when introducing time to the nearest 5 minutes.

Mathematics


Grade 2 Unit 3: Place Value, Counting, and Comparison of Numbers to 1000 (~5 weeks)

Unit Overview: All arithmetic algorithms are manipulations of place value units: ones, tens, hundreds, etc. In Unit 3 students extend their understanding of base-

ten notation and apply their understanding of place value to count and compare numbers to 1000. In Grade 2 the place value units move from a proportional model

to a non-proportional number model like the disk model (see picture). The place value table with number disks can be used through Grade 5 for modeling very

large numbers and decimals, thus providing students greater facility with and understanding of mental math and algorithms. As students solve challenging place

value problems, they will have to construct viable arguments and critique the reasoning of others (MP 3).

Guiding Question: How does changing the place of a digit change its value?


Component Cluster 2.NBT Understand place value.

2.NBT.1 Understand that the three digits of a three-digit

number represent amounts of hundreds, tens, and ones;

e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand


(See 2.NBT.1a & b)

Second Grade students extend their base-ten understanding to hundreds as they view 10 tens as a unit called a

“hundred”. They use manipulative materials and pictorial representations to help make a connection between the

written three-digit numbers and hundreds, tens, and ones.

As in First Grade, Second Graders’ understanding about hundreds also moves through several stages: Counting By

Ones; Counting by Groups & Singles; and Counting by Hundreds, Tens and Ones.

Counting By Ones: At first, even though Second Graders will have grouped objects into hundreds, tens and left-

overs, they rely on counting all of the individual cubes by ones to determine the final amount. It is seen as the only

way to determine how many.

Counting By Groups and Singles: While students are able to group objects into collections of hundreds, tens and

ones and now tell how many groups of hundreds, tens and left-overs there are, they still rely on counting by ones to

determine the final amount. They are unable to use the groups and left-overs to determine how many.

Counting by Hundreds, Tens & Ones: Students are able to group objects into hundreds, tens and ones, tell how

many groups and left-overs there are, and now use that information to tell how many. Occasionally, as this stage

becomes fully developed, second graders rely on counting to “really” know the amount, even though they may have

just counted the total by groups and left-overs.

Understanding the value of the digits is more than telling the number of tens or hundreds. Second Grade students

Mathematics


who truly understand the position and place value of the digits are also able to confidently model the number with

some type of visual representation. Others who seem like they know, because they can state which number is in the

tens place, may not truly know what each digit represents.

a. 100 can be thought of as a bundle of ten tens — called

a “hundred.”

Second Graders extend their work from first grade by applying the understanding that “100” is the same amount as

10 groups of ten as well as 100 ones. This lays the groundwork for the structure of the base-ten system in future

grades

b. The numbers 100, 200, 300, 400, 500, 600, 700, 800,

900 refer to one, two, three, four, five, six, seven,

eight, or nine hundreds (and 0 tens and 0 ones).

Second Grade students build on the work of 2.NBT.2a. They explore the idea that numbers such as 100, 200, 300,

etc., are groups of hundreds with zero tens and ones. Students can represent this with both groupable (cubes, links)

and pre-grouped (place value blocks) materials.

2.NBT.2 Count within 1000; skip-count by 5s, 10s, and

100s.

Second Grade students count within 1,000. Thus, students “count on” from any number and say the next few

numbers that come afterwards.

Second grade students also begin to work towards multiplication concepts as they skip count by 5s, by 10s, and by

100s. Although skip counting is not yet true multiplication because students don’t keep track of the number of

groups they have counted, they can explain that when they count by 2s, 5s, and 10s they are counting groups of items

with that amount in each group. As teachers build on students’ work with skip counting by 10s in Kindergarten, they explore and discuss with

students the patterns of numbers when they skip count. For example, while using a 100s board or number line,

students learn that the ones digit alternates between 5 and 0 when skip counting by 5s. When students skip count by

100s, they learn that the hundreds digit is the only digit that changes and that it increases by one number.

2.NBT.3 Read and write numbers to 1000 using base-ten

numerals, number names, and expanded form.

Second graders read, write and represent a number of objects with a written numeral (number form or standard form).

These representations can include snap cubes, place value (base 10) blocks, pictorial representations or other concrete

materials. Please be cognizant that when reading and writing whole numbers, the word “and” should not be used

(e.g., 235 is stated and written as “two hundred thirty-five).

Expanded form (125 can be written as 100 + 20 + 5) is a valuable skill when students use place value strategies to

add and subtract large numbers in 2.NBT.7.

2.NBT.4 Compare two three-digit numbers based on

meanings of the hundreds, tens, and ones digits, using >, =,

and < symbols to record the results of comparisons.

Second Grade students build on the work of 2.NBT.1 and 2.NBT.3 by examining the amount of hundreds, tens and

ones in each number. When comparing numbers, students draw on the understanding that 1 hundred (the smallest

three-digit number) is actually greater than any amount of tens and ones represented by a two-digit number. When

students truly understand this concept, it makes sense that one would compare three-digit numbers by looking at the

hundreds place first.

Students should have ample experiences communicating their comparisons in words before using symbols. Students

were introduced to the symbols greater than (>), less than (<) and equal to (=) in First Grade and continue to use them

in Second Grade with numbers within 1,000.

While students may have the skills to order more than 2 numbers, this Standard focuses on comparing two numbers

and using reasoning about place value to support the use of the various symbols.

Mathematics


Grade 2 Unit 4: Addition and Subtraction of Numbers to 1000 with Problem-Solving (~7 weeks)

Unit Overview: In Unit 4, students continue to work with place value units to understand the addition and subtraction algorithms of numbers up to 1000. This

work deepens their understanding of base-ten, place value, and properties of operations. It also challenges them to apply their knowledge to one-step and two-step

word problems. During this unit, students also continue to develop one of the required fluencies of the grade: addition and subtraction within 100. Students will

have many opportunities in this unit to make sense of problems and persevere in solving them (MP 1).

Guiding Question: How does place value relate to addition and subtraction strategies?



*Story problems focus primarily on the positions of result and change unknown.


one- and two-step word problems involving situations of

adding to, taking from, putting together, taking apart, and

comparing, with unknowns in all positions, e.g., by using



1 See Glossary, Table 1.

See Unit 1.

Component Cluster 2.NBT Use place value understanding and properties of operations to add and subtract.



and/or the relationship between addition and subtraction.

See Unit 1.

2.NBT.6 Add up to four two-digit numbers using

strategies based on place value and properties of

operations.

Second Grade students add a string of two-digit numbers (up to four numbers) by applying place value strategies and

properties of operations.

2.NBT.7 Add and subtract within 1000, using concrete

models or drawings and strategies based on place value,

properties of operations, and/or the relationship between

addition and subtraction; relate the strategy to a written

method. Understand that in adding or subtracting three-

digit numbers, one adds or subtracts hundreds and

hundreds, tens and tens, ones and ones; and sometimes it is

necessary to compose or decompose tens or hundreds.

Second graders extend the work from 2.NBT. to two 3-digit numbers. Students should have ample experiences using

concrete materials and pictorial representations to support their work.

This standard also references composing and decomposing a ten. This work should include strategies such as making

a 10, making a 100, breaking apart a 10, or creating an easier problem. The standard algorithm of carrying or

borrowing is not an expectation in Second Grade. Students are not expected to add and subtract whole numbers

using a standard algorithm until the end of Fourth Grade.

2.NBT.8 Mentally add 10 or 100 to a given number 100–

900, and mentally subtract 10 or 100 from a given number

100–900.

Second Grade students mentally add or subtract either 10 or 100 to any number between 100 and 900. As teachers

provide ample experiences for students to work with pre-grouped objects and facilitate discussion, second graders

realize that when one adds or subtracts 10 or 100 that only the tens place or the digit in the hundreds place changes

by 1. As the teacher facilitates opportunities for patterns to emerge and be discussed, students notice the patterns and

connect the digit change with the amount changed.

Mathematics


Opportunities to solve problems in which students cross hundreds are also provided once students have become

comfortable adding and subtracting within the same hundred.

This standard focuses only on adding and subtracting 10 or 100. Multiples of 10 or multiples of 100 can be explored;

however, the focus of this standard is to ensure that students are proficient with adding and subtracting 10 and 100

mentally.

2.NBT.9 Explain why addition and subtraction strategies

work, using place value and the properties of operations.*

*Explanations may be supported by drawings or objects.

Second graders explain why addition or subtraction strategies work as they apply their knowledge of place value and

the properties of operations in their explanation. They may use drawings or objects to support their explanation.

Once students have had an opportunity to solve a problem, the teacher provides time for students to discuss their

strategies and why they did or didn’t work.

Mathematics


Grade 2 Unit 5: Measuring, Representing, and Solving with Money and Length Data (~6 weeks)

Unit Overview: The next unit (U5) gives students another chance to practice their non-standard algorithms and problem solving skills with the most famous and

most interesting units of all: dollars, dimes, and pennies. Measuring, comparing, and estimating length is revisited in this module in the context of units from both

the customary system (e.g., inches and feet) and the metric system (e.g., centimeters and meters). As they study money and length, students represent data given by

measurement and money data using picture graphs, bar graphs, and line plots. Students will have the opportunity to model with mathematics (MP 4) as they take

information from the world around them and solve problems.

Guiding Question: How can you prove that you made a good estimate?


Component Cluster 2.MD Measure and estimate lengths in standard units.

2.MD.1 Measure the length of an object by selecting and

using appropriate tools such as rulers, yardsticks, meter

sticks, and measuring tapes.

See Unit 2.

2.MD.2 Measure the length of an object twice, using

length units of different lengths for the two measurements;

describe how the two measurements relate to the size of

the unit chosen.

2.MD.3 Estimate lengths using units of inches, feet,

centimeters, and meters.

2.MD.4 Measure to determine how much longer one

object is than another, expressing the length difference in

terms of a standard length unit.

Component Cluster 2.MD Relate addition and subtraction to length.

2.MD.5 Use addition and subtraction within 100 to solve

word problems involving lengths that are given in the same

units, e.g., by using drawings (such as drawings of rulers)


represent the problem.

See Unit 2.

2.MD.6 Represent whole numbers as lengths from 0 on a

number line diagram with equally spaced points

corresponding to the numbers 0, 1, 2, ..., and represent

whole-number sums and differences within 100 on a


Component Cluster 2.MD Work with time and money.

*Focus on money. 2.MD.8 Solve word problems involving dollar bills,

quarters, dimes, nickels, and pennies, using $ and ¢

In Second Grade, students solve word problems involving either dollars or cents. Since students have not been

introduced to decimals, problems focus on whole dollar amounts or cents.

Mathematics


symbols appropriately.

Example: If you have 2 dimes and 3 pennies, how many

cents do you have?

This is the first time money is introduced formally as a standard. Therefore, students will need numerous experiences

with coin recognition and values of coins before using coins to solve problems. Once students are solid with coin

recognition and values, they can then begin using the values coins to count sets of coins, compare two sets of coins,

make and recognize equivalent collections of coins (same amount but different arrangements), select coins for a

given amount, and make change.

Solving problems with money can be a challenge for young children because it builds on prerequisite number and

place value skills and concepts. Many times money is introduced before students have the necessary number sense to

work with money successfully.

For these values to make sense, students must have an understanding of 5, 10,

and 25. More than that, they need to be able to think of these quantities without

seeing countable objects… A child whose number concepts remain tied to

counts of objects [one object is one count] is not going to be able to understand

the value of coins. Van de Walle & Lovin, p. 150, 2006

Just as students learn that a number (38) can be represented different ways (3 tens and 8 ones; 2 tens and 18 ones)

and still remain the same amount (38), students can apply this understanding to money. For example, 25 cents can

look like a quarter, two dimes and a nickel, and it can look like 25 pennies, and still all remain 25 cents. This concept

of equivalent worth takes time and requires numerous opportunities to create different sets of coins, count sets of

coins, and recognize the “purchase power” of coins (a nickel can buy the same things a 5 pennies).

As teachers provide students with sufficient opportunities to explore coin values (25 cents) and actual coins (2 dimes,

1 nickel), teachers will help guide students over time to learn how to mentally give each coin in a set a value, place

the random set of coins in order, and use mental math, adding on to find differences, and skip counting to determine

the final amount.


2.MD.9 Generate measurement data by measuring lengths

of several objects to the nearest whole unit, or by making

repeated measurements of the same object. Show the

measurements by making a line plot, where the horizontal

scale is marked off in whole-number units.

Second Graders use measurement data as they move through the statistical process of posing a question, collecting

data, analyzing data, creating representations, and interpreting the results. In second grade students represent the

length of several objects by making a line plot. Students should round their lengths to the nearest whole unit.

2.MD.10 Draw a picture graph and a bar graph (with

single-unit scale) to represent a data set with up to four

categories. Solve simple put-together, take-apart, and

compare problems4 using information presented in a bar

graph. 4 See Glossary, Table 1.

In Second Grade, students pose a question, determine up to 4 categories of possible responses, collect data, represent

data on a picture graph or bar graph, and interpret the results. This is an extension from first grade when students

organized, represented, and interpreted data with up to three categories. They are able to use the graph selected to

note particular aspects of the data collected, including the total number of responses, which category had the

most/least responses, and interesting differences/similarities between the four categories. They then solve simple

one-step problems using the information from the graph.

Mathematics


Grade 2 Unit 6: Preparation for Multiplication and Division Concepts (~6 weeks)

Unit Overview: In Unit 6, students extend their understanding of a unit to build the foundation for multiplication and division. Making groups of 4 apples each

establishes the unit “4 apples” (or just four) that can then be counted: 1 four, 2 fours, 3 fours, etc. Relating the new unit to the one used to create it develops the

idea of multiplication: 3 groups of 4 apples equal 12 apples (or 3 fours is 12). As students analyze organizational structures like arrays, they can look for and make

use of structure (MP 7).

Guiding Question: How does the arrangement of a group of objects affect your ability to count them?


Component Cluster 2.OA Work with equal groups of objects to gain foundations for multiplication.

2.OA.3 Determine whether a group of objects (up to 20)

has an odd or even number of members, e.g., by pairing

objects or counting them by 2s; write an equation to

express an even number as a sum of two equal addends.

Second graders apply their work with doubles to the concept of odd and even numbers. Students should have ample

experiences exploring the concept that if a number can be decomposed (broken apart) into two equal addends or

doubles addition facts (e.g., 10 = 5 +5), then that number (10 in this case) is an even number. Students should explore

this concept with concrete objects (e.g., counters, cubes, etc.) before moving towards pictorial representations such as

circles or arrays.

The focus of this standard is placed on the conceptual understanding of even and odd numbers. An even number is an

amount that can be made of two equal parts with no leftovers. An odd number is one that is not even or cannot be

made of two equal parts. The number endings of 0, 2, 4, 6, and 8 are only an interesting and useful pattern or

observation and should not be used as the definition of an even number. (Van de Walle & Lovin, 2006, p. 292)

2.OA.4 Use addition to find the total number of objects

arranged in rectangular arrays with up to 5 rows and up to

5 columns; write an equation to express the total as a sum

of equal addends.

Second graders use rectangular arrays to work with repeated addition, a building block for multiplication in third

grade. A rectangular array is any arrangement of things in rows and columns, such as a rectangle of square tiles.

Students explore this concept with concrete objects (e.g., counters, bears, square tiles, etc.) as well as pictorial

representations on grid paper or other drawings. Due to the commutative property of multiplication, students can add

either the rows or the columns and still arrive at the same solution.

Component Cluster 2.G Reason with shapes and their attributes. *G.2 is taught before G.1 and G.3 because the array model is so important to the foundation for multiplication.

2.G.2 Partition a rectangle into rows and columns of same-

size squares and count to find the total number of them.

Second graders partition a rectangle into squares (or square-like regions) and then determine the total number of

squares. This work connects to the standard 2.OA.4 where students are arranging objects in an array of rows and

columns.

Mathematics


Grade 2 Unit 7: Reasoning about Shapes and Fractions of Shapes (~4 weeks)

Unit Overview: In Unit 7, students investigate, describe, and reason about the composition and decomposition of shapes to form other shapes. Through building,

drawing, and analyzing two- and three-dimensional shapes, students develop a foundation for understanding area, volume, congruence, similarity, and symmetry in

later grades. As students examine similarities between shapes, they will need to look for and express regularity in repeated reasoning (MP 8).

Guiding Question: How does examining and describing the parts of an object help us to describe the whole?



2.G.1 Recognize and draw shapes having specified

attributes, such as a given number of angles or a given

number of equal faces.* Identify triangles, quadrilaterals,

pentagons, hexagons, and cubes.

*Sizes are compared directly or visually, not compared by

measuring.

Second Grade students identify (recognize and name) shapes and draw shapes based on a given set of attributes.

These include triangles, quadrilaterals (squares, rectangles, and trapezoids), pentagons, hexagons and cubes.

2.G.3 Partition circles and rectangles into two, three, or

four equal shares, describe the shares using the words

halves, thirds, half of, a third of, etc., and describe the

whole as two halves, three thirds, four fourths. Recognize

that equal shares of identical wholes need not have the

same shape.

Second Grade students partition circles and rectangles into 2, 3 or 4 equal shares (regions). Students should be given

ample experiences to explore this concept with paper strips and pictorial representations. Students should also work

with the vocabulary terms halves, thirds, half of, third of, and fourth (or quarter) of. While students are working on

this standard, teachers should help them to make the connection that a “whole” is composed of two halves, three

thirds, or four fourths.

This standard also addresses the idea that equal shares of identical wholes may not have the same shape.

It is important for students to understand that fractional parts may not be symmetrical. The only criteria for

equivalent fractions is that the area is equal.

Mathematics

OA = Operations and Algebraic Thinking NBT= Numbers and Operations in Base Ten NF= Numbers and Operations – Fractions


Grade 3

In Grade 3, instructional time should focus on four critical areas: (1) developing understanding of multiplication and division and strategies for

multiplication and division within 100; (2) developing understanding of fractions, especially unit fractions (fractions with numerator 1); (3)

developing understanding of the structure of rectangular arrays and of area; and (4) describing and analyzing two-dimensional shapes.

(1) Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-

sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations. For

equal-sized group situations, division can require finding the unknown number of groups or the unknown group size. Students use properties of operations

to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems

involving single-digit factors. By comparing a variety of solution strategies, students learn the relationship between multiplication and division.

(2) Students develop an understanding of fractions, beginning with unit fractions. Students view fractions in general as being built out of unit fractions, and

they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a fractional part is relative to the size

of the whole. For example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a larger bucket, but 1/3 of a ribbon is longer than

1/5 of the same ribbon because when the ribbon is divided into 3 equal parts, the parts are longer than when the ribbon is divided into 5 equal parts.

Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve comparing fractions by

using visual fraction models and strategies based on noticing equal numerators or denominators.

(3) Students recognize area as an attribute of two-dimensional regions. They measure the area of a shape by finding the total number of same size units of

area required to cover the shape without gaps or overlaps, a square with sides of unit length being the standard unit for measuring area. Students

understand that rectangular arrays can be decomposed into identical rows or into identical columns. By decomposing rectangles into rectangular arrays of

squares, students connect area to multiplication, and justify using multiplication to determine the area of a rectangle.

(4) Students describe, analyze, and compare properties of two-dimensional shapes. They compare and classify shapes by their sides and angles, and

connect these with definitions of shapes. Students also relate their fraction work to geometry by expressing the area of part of a shape as a unit fraction of

the whole.

Mathematical Practices (MP)









Mathematics




# of weeks

Unit 1: Multiplication and Division with Factors of 2, 3, 4, 5 and 10 13% ~ 5 weeks

Unit 2: Problem Solving with Mass, Time and Capacity 13% ~ 5 weeks

Benchmark Assessment

Unit 3: Multiplication and Division with Factors of 6, 7, 8, 9 and Multiples of 10 13% ~ 5 weeks

Unit 4: Measuring and Classifying Shapes 17% ~ 6 weeks


Unit 5: Fractions as Numbers on the Number Line 22% ~ 7 weeks

Unit 6: Collecting and Displaying Data 5% ~ 2 weeks

State Test

Unit 7: Challenging Problems (Post-Test Unit) 17% ~ 6 weeks

Unit 1: Mult. and Div. with Factors 2, 3, 4, 5, and 10

13%

Unit 2: Problem Solving with

Mass, Time, and Capacity

13%

Unit 3: Mult. and Div. with 6, 7, 8, 9, and Multiples

of 10 13%

Unit 4: Measuring and Classifying

Shapes 17%

Unit 5: Fractions as Numbers on a

# line 22%

Unit 6: Collecting and Displaying

Data 5%

Unit 7: Challenging Problems

17%

Instructional Time

Mathematics



Grade 3 Unit 1: Multiplication and Division with Factors of 2, 3, 4, 5 and 10 (~5 weeks)

Unit Overview: The first unit (U1) builds upon the foundation of multiplicative thinking with units started in grade 2. Students concentrate on the meaning of

multiplication and division and begin fluency for learning products involving factors of 2, 3, 4, 5, and 10. The restricted set of facts keeps learning manageable,

and also provides enough examples to do one- and two-step word problems and to start measurement problems involving weight, capacity and time in the second

unit. This unit provides opportunities for students to model with mathematics (MP 4) as they seek to represent and solve multiplication and division word

problems.

Guiding Question: What are fast strategies for learning the multiplication and division facts? The student will be able to: The teacher will use appropriate instructional strategies/approaches based on the needs of the student.

Component Cluster 3.OA Represent and solve problems involving multiplication and division.

*Limited to factors of 2, 3, 4, 5, and 10 and the corresponding dividends.

3.OA.1 Interpret products of whole numbers, e.g., interpret

5 × 7 as the total number of objects in 5 groups of 7

objects each.

For example, describe a context in which a total number of

objects can be expressed as 5 × 7.

This standard interprets products of whole numbers. Students recognize multiplication as a means to determine the

total number of objects when there are a specific number of groups with the same number of objects in each group or

of an equal amount of objects were added or collected numerous times.. Multiplication requires students to think in

terms of groups of things rather than individual things. Students learn that the multiplication symbol ‘x’ means

“groups of” and problems such as 5 x 7 refer to 5 groups of 7.

3.OA.2 Interpret whole-number quotients of whole

numbers, e.g., interpret 56 ÷ 8 as the number of objects in

each share when 56 objects are partitioned equally into 8

shares, or as a number of shares when 56 objects are

partitioned into equal shares of 8 objects each.

For example, describe a context in which a number

of shares or a number of groups can be expressed as 56 ÷

8.

This standard focuses on two distinct models of division: partition models and measurement (repeated subtraction)

models.

Partition models provide students with a total number and the number of groups. These models focus on the question,

“How many objects are in each group so that the groups are equal?” A context for partition models would be: There

are 12 cookies on the counter. If you are sharing the cookies equally among three bags, how many cookies will go in

each bag?

Measurement (repeated subtraction) models provide students with a total number and the number of objects in each

group. These models focus on the question, “How many equal groups can you make?” A context for measurement

models would be: There are 12 cookies on the counter. If you put 3 cookies in each bag, how many bags will you

fill?

3.OA.3 Use multiplication and division within 100 to

solve word problems in situations involving equal groups,

arrays, and measurement quantities, e.g., by using


number to represent the problem.*

*See Table 2 at the end of this document.

This standard references various problem solving context and strategies that students are expected to use while

solving word problems involving multiplication & division. Students should use a variety of representations for

creating and solving one-step word problems, such as: If you divide 4 packs of 5 brownies among 10 people, how

many cookies does each person receive? (4 x 5 = 20, 20÷ 10 = 2). These representations include drawing pictures,

reasoning mentally or verbally, numberlines to show equal jumps, a variety of pictures to represent the unknown

numbers (variables). Letters are also introduced to represent unknowns in third grade.

Table at end of document gives examples of a variety of problem solving contexts, in which students need to find the

product, the group size, or the number of groups. Students should be given ample experiences to explore all of the

different problem structures.

Mathematics



3.OA.4 Determine the unknown whole number in a

multiplication or division equation relating three whole


that makes the equation true in each of the equations 8 × ?

= 48, 5 = _ ÷ 3, 6 × 6 = ?

This standard refers to Table 2 (table included at the end of this document for your convenience) and equations for

the different types of multiplication and division problem structures. The easiest problem structure includes

Unknown Product (3 x 5 = ? or 15 ÷ 3 = 5). The more difficult problem structures include Group Size Unknown (3 x

? = 15 or 15 ÷ 3 = 5) or Number of Groups Unknown (? x 5 = 15, 15 ÷ 5 = 3). The focus of 3.OA.4 extend beyond

the traditional notion of fact families, by having students explore the inverse relationship of multiplication and

division.

Students extend work from earlier grades with their understanding of the meaning of the equal sign as “the same

amount as” to interpret an equation with an unknown.

Component Cluster 3.OA Understand properties of multiplication and the relationship between multiplication and division. *Limited to factors of 2, 3, 4, 5, and 10 and the corresponding dividends.

3.OA.5 Apply properties of operations as strategies to

multiply and divide.2 Examples: If 6 × 4 = 24 is known,

then 4 × 6 = 24 is also known. (Commutative property of

multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then

15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30.

(Associative property of multiplication.) Knowing that 8 ×

5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) =

(8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)


This standard references properties (rules about how numbers work) of multiplication. This extends past previous

expectations, in which students were asked to identify properties. While students DO NOT need to not use the formal

terms of these properties, student must understand that properties are rules about how numbers work, and they need

to be flexibly and fluently applying each of them in various situations. Students represent expressions using various

objects, pictures, words and symbols in order to develop their understanding of properties. They multiply by 1 and 0

and divide by 1. They change the order of numbers to determine that the order of numbers does not make a difference

in multiplication (but does make a difference in division). Given three factors, they investigate changing the order of

how they multiply the numbers to determine that changing the order does not change the product. They also

decompose numbers to build fluency with multiplication.

3.OA.6 Understand division as an unknown-factor

problem.

For example, find 32 ÷ 8 by finding the number that makes

32 when multiplied by 8.

This standard refers the table at the end of the document and the various problem structures. Since multiplication and

division are inverse operations, students are expected to solve problems and explain their processes of solving

division problems that can also be represented as unknown factor multiplication problems.

Component Cluster 3.OA Multiply and divide within 100. *Limited to factors of 2, 3, 4, 5, and 10 and the corresponding dividends.

3.OA.7 Fluently multiply and divide within 100, using

strategies such as the relationship between multiplication

and division (e.g., knowing that 8 × 5 = 40, one knows 40

÷ 5 = 8) or properties of operations. By the end of Grade 3,

know from memory all products of two one-digit numbers.

This standard uses the word fluently, which means accuracy, efficiency (using a reasonable amount of steps and

time), and flexibility (using strategies such as the distributive property). “Know from memory” should not focus only

on timed tests and repetitive practice, but ample experiences working with manipulatives, pictures, arrays, word

problems, and numbers to internalize the basic facts (up to 9 x 9 by the end of unit 3, but with a focus on 2, 3, 4, 5,

and 10 only for this unit).

By studying patterns and relationships in multiplication facts and relating multiplication and division, students build a

foundation for fluency with multiplication and division facts. Students demonstrate fluency with multiplication facts

through 10 and the related division facts. Multiplying and dividing fluently refers to knowledge of procedures,

knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and

efficiently.

Strategies students may use to attain fluency include:

Multiplication by zeros and ones

Mathematics



Doubles (2s facts), Doubling twice (4s), Doubling three times (8s)

Tens facts (relating to place value, 5 x 10 is 5 tens or 50)

Five facts (half of tens)

Skip counting (counting groups of __ and knowing how many groups have been counted)

Square numbers (ex: 3 x 3)

Nines (10 groups less one group, e.g., 9 x 3 is 10 groups of 3 minus one group of 3)

Decomposing into known facts (6 x 7 is 6 x 6 plus one more group of 6)

Turn-around facts (Commutative Property)

Fact families (Ex: 6 x 4 = 24; 24 ÷ 6 = 4; 24 ÷ 4 = 6; 4 x 6 = 24)

Missing factors

Students should have exposure to multiplication and division problems presented in both vertical and horizontal

forms.

Note that mastering this material, and reaching fluency in single-digit multiplications and related divisions with

understanding, may be quite time consuming because there are no general strategies for multiplying or dividing all

single-digit numbers as there are for addition and subtraction. Instead, there are many patterns and strategies

dependent upon specific numbers. So it is imperative that extra time and support be provided if needed.

(Progressions for the CCSSM; Operations and Algebraic Thinking, CCSS Writing Team, May 2011, page 22)

All of the understandings of multiplication and division situations (See Glossary, Table 2. (page 89 in CCSS)

Table included at the end of this document for your convenience), of the levels of representation and solving, and of

patterns need to culminate by the end of Grade 3 in fluent multiplying and dividing of all single-digit numbers and

10. Such fluency may be reached by becoming fluent for each number (e.g., the 2s, the 5s, etc.) and then extending

the fluency to several, then all numbers mixed together. Organizing practice so that it focuses most heavily on

understood but not yet fluent products and unknown factors can speed learning. To achieve this by the end of Grade

3, students must begin working toward fluency for the easy numbers as early as possible. Because an unknown factor

(a division) can be found from the related multiplication, the emphasis at the end of the year is on knowing from

memory all products of two one-digit numbers. As should be clear from the foregoing, this isn’t a matter of instilling

facts divorced from their meanings, but rather the outcome of a carefully designed learning process that heavily

involves the interplay of practice and reasoning. All of the work on how different numbers fit with the base-ten

numbers culminates in these “just know” products and is necessary for learning products. Fluent dividing for all

single-digit numbers, which will combine just knows, knowing from a multiplication, patterns, and best strategy, is

also part of this vital standard. (Progressions for the CCSSM; Operations and Algebraic Thinking, CCSS Writing

Team, May 2011, page 27)

Component Cluster 3.OA Solve problems involving the four operations, and identify and explain patterns in arithmetic. *Limited to factors of 2, 3, 4, 5, and 10 and the corresponding dividends.

3.OA.8 Solve two-step word problems using the four

operations. Represent these problems using equations with

a letter standing for the unknown quantity. Assess the

reasonableness of answers using mental computation and

estimation strategies including rounding.*

Students in third grade begin the step to formal algebraic language by using a letter for the unknown quantity in

expressions or equations for one and two-step problems. But the symbols of arithmetic, x or . or * for multiplication

and ÷ or / for division, continue to be used in Grades 3, 4, and 5.

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*This standard is limited to problems posed with whole

numbers and having whole-number answers; students

should know how to perform operations in the

conventional order when there are no parentheses to

specify a particular order.

3.OA.9 Identify arithmetic patterns (including patterns in

the addition table or multiplication table), and explain

them using properties of operations.

For example, observe that 4 times a number is always

even, and explain why 4 times a number can be

decomposed into two equal addends.

This standard calls for students to examine arithmetic patterns involving both addition and multiplication. Arithmetic

patterns are patterns that change by the same rate, such as adding the same number. For example, the series 2, 4, 6, 8,

10 is an arithmetic pattern that increases by 2 between each term.

This standards also mentions identifying patterns related to the properties of operations.

Examples:

Even numbers are always divisible by 2. Even numbers can always be decomposed into 2 equal addends

(14 = 7 + 7).

Multiples of even numbers (2, 4, 6, and 8) are always even numbers.

On a multiplication chart, the products in each row and column increase by the same amount (skip counting).

On an addition chart, the sums in each row and column increase by the same amount.

Students need ample opportunities to observe and identify important numerical patterns related to operations. They

should build on their previous experiences with properties related to addition and subtraction. Students investigate

addition and multiplication tables in search of patterns and explain why these patterns make sense mathematically.

Example:

Any sum of two even numbers is even.

Any sum of two odd numbers is even.

Any sum of an even number and an odd number is odd.

The multiples of 4, 6, 8, and 10 are all even because they can all be decomposed into two equal groups.

The doubles (2 addends the same) in an addition table fall on a diagonal while the doubles (multiples of 2) in a

multiplication table fall on horizontal and vertical lines.

The multiples of any number fall on a horizontal and a vertical line due to the commutative property.

All the multiples of 5 end in a 0 or 5 while all the multiples of 10 end with 0. Every other multiple of 5 is a

multiple of 10.

Students also investigate a hundreds chart in search of addition and subtraction patterns. They record and organize all

the different possible sums of a number and explain why the pattern makes sense.

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Grade 3 Unit 2: Problem Solving with Mass, Time and Capacity (~5 weeks)

Unit Overview: The measurement unit again plays the role of providing students with “internalization time” before students start into the remaining facts in the

third unit. The second unit also has students work with place value, comparison and rounding as they work toward the 3rd

grade fluency goal of addition and

subtraction within 1000. An additional goal is to develop students’ number sense well enough to build proportional bar diagrams used in solving word problems

throughout third grade and beyond (e.g. “If this bar represents 62 kg, then a bar representing 35 kg needs to be slightly longer than half the 62 kg bar…”).

Drawing the relative sizes of the lengths of two bars also prepares students to locate fractions on a number line in Unit 5 (where they learn to locate the points 1/3

and 1/5 on the number line relative to each other and relative to the whole unit). Students must attend to precision (MP 6) in their measurement solutions.

Guiding Question: How can you tell if your answer is reasonable when working on a measurement problem?


Component Cluster 3.NBT Use place value understanding and properties of operations to perform multi-digit arithmetic.*

*A range of algorithms may be used.

3.NBT.1 Use place value understanding to round whole

numbers to the nearest 10 or 100.

This standard refers to place value understanding, which extends beyond an algorithm or memorized procedure for

rounding. The expectation is that students have a deep understanding of place value and number sense and can

explain and reason about the answers they get when they round. Students should have numerous experiences using a

number line and a hundreds chart as tools to support their work with rounding.


strategies and algorithms based on place value, properties

of operations, and/or the relationship between addition and

subtraction.

1 A range of algorithms may be used.

This standard refers to fluently, which means accuracy, efficiency (using a reasonable amount of steps and time), and

flexibility (using strategies such as the distributive property). The word algorithm refers to a procedure or a series of

steps. There are other algorithms other than the standard algorithm. Third grade students should have experiences

beyond the standard algorithm.

Problems should include both vertical and horizontal forms, including opportunities for students to apply the

commutative and associative properties. Students explain their thinking and show their work by using strategies and

algorithms, and verify that their answer is reasonable.

Computation algorithm. A set of predefined steps applicable to a class of problems that gives the correct result in

every case when the steps are carried out correctly.

Computation strategy. Purposeful manipulations that may be chosen for specific problems, may not have a fixed

order, and may be aimed at converting one problem into another.

(Progressions for the CCSSM; Number and Operation in Base Ten, CCSS Writing Team, April 2011, page 2)

Component Cluster 3.MD Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

3.MD.1 Tell and write time to the nearest minute and

measure time intervals in minutes. Solve word problems

involving addition and subtraction of time intervals in

minutes, e.g., by representing the problem on a number

line diagram.

This standard calls for students to solve elapsed time, including word problems. Students could use clock models or

number lines to solve. On the number line, students should be given the opportunities to determine the intervals and

size of jumps on their number line. Students could use pre-determined number lines (intervals every 5 or 15 minutes)

or open number lines (intervals determined by students).

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3.MD.2 Measure and estimate liquid volumes and masses

of objects using standard units of grams (g), kilograms

(kg), and liters (l).1 Add, subtract, multiply, or divide to

solve one-step word problems involving masses or

volumes that are given in the same units, e.g., by using

drawings (such as a beaker with a measurement scale) to

represent the problem.2

1 Excludes compound units such as and finding the

geometric volume of a container. 2 Excludes multiplicative comparison problems (problems

involving notions of “times as much”; see Glossary, Table

2 at end of document).

This standard asks for students to reason about the units of mass and volume using units g, kg, and L. Students need

multiple opportunities weighing classroom objects and filling containers to help them develop a basic understanding

of the size and weight of a liter, a gram, and a kilogram. Milliliters may also be used to show amounts that are less

than a liter emphasizing the relationship between smaller units to larger units in the same system. Word problems

should only be one-step and include the same units.

Students are not expected to do conversions between units, but reason as they estimate, using benchmarks to measure

weight and capacity.

Foundational understandings to help with measure concepts:

- Understand that larger units can be subdivided into equivalent units (partition).

- Understand that the same unit can be repeated to determine the measure (iteration).

- Understand the relationship between the size of a unit and the number of units needed (compensatory

principal).

Before learning to measure attributes, children need to recognize them, distinguishing them from other attributes.

That is, the attribute to be measured has to “stand out” for the student and be discriminated from the undifferentiated

sense of amount that young children often have, labeling greater lengths, areas, volumes, and so forth, as “big” or

“bigger.”

These standards do not differentiate between weight and mass. Technically, mass is the amount of matter in an

object. Weight is the force exerted on the body by gravity. On the earth’s surface, the distinction is not important (on

the moon, an object would have the same mass, would weigh less due to the lower gravity).

(Progressions for the CCSSM, Geometric Measurement, CCSS Writing Team, June 2012, page 2)

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Grade 3 Unit 3: Multiplication and Division with Factors of 6, 7, 8, 9, and Multiples of 10 (~5 weeks)

Unit Overview: Students learn the remaining multiplication and division facts in Unit 3 as they continue to develop their understanding of multiplication and

division strategies within 100 and use those strategies to solve two-step word problems. The “2, 3, 4, 5 and 10 facts” Unit (U1) and the “6, 7, 8 and 9 facts” unit

(U3) both provide important, sustained time for work in understanding the structure of rectangular arrays to prepare students for area in Unit 4. This work is

necessary because students initially find it difficult to distinguish the different squares in a rectangular array area model (the third array in the picture below), count

them and recognize that the count is related to multiplication. Units 1 and 3 slowly build up to a rectangular array area model using hands-on rectangular arrays

(i.e., a Rekenrek) and/or pictures of rectangular arrays involving objects only (stars, disks, etc.)— all in the context of learning multiplication and division.

Students have the opportunity to make sense of problems and persevere in solving them (MP 1) as they work to find relationships between facts they know and

facts they are still learning and meet the fluency goal of all multiplication and division facts within 100.

Guiding Question: What strategies and models best help you to learn all of your multiplication facts?



*Limited to factors of 2, 3, 4, 5, and 10 and the corresponding dividends.

3.OA.1 Interpret products of whole numbers, e.g., interpret

5 × 7 as the total number of objects in 5 groups of 7

objects each.

For example, describe a context in which a total number of

objects can be expressed as 5 × 7.

See Unit 1.

3.OA.2 Interpret whole-number quotients of whole

numbers, e.g., interpret 56 ÷ 8 as the number of objects in

each share when 56 objects are partitioned equally into 8

shares, or as a number of shares when 56 objects are

partitioned into equal shares of 8 objects each.

For example, describe a context in which a number

of shares or a number of groups can be expressed as 56 ÷

See Unit 1.

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8.





number to represent the problem.*

*See Table at the end of this document.

See Unit 1.

3.OA.4 Determine the unknown whole number in a

multiplication or division equation relating three whole


that makes the equation true in each of the equations 8 × ?

= 48,

5 = _ ÷ 3, 6 × 6 = ?

See Unit 1.

Component Cluster 3.OA Understand properties of multiplication and the relationship between multiplication and division. *Limited to factors of 2, 3, 4, 5, and 10 and the corresponding dividends.

3.OA.5 Apply properties of operations as strategies to

multiply and divide.2 Examples: If 6 × 4 = 24 is known,

then 4 × 6 = 24 is also known. (Commutative property of

multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then

15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30.

(Associative property of multiplication.) Knowing that 8 ×

5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) =

(8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)


See Unit 1.

3.OA.6 Understand division as an unknown-factor

problem.

For example, find 32 ÷ 8 by finding the number that makes

32 when multiplied by 8.

See Unit 1.

Component Cluster 3.OA Multiply and divide within 100.*

* From this point forward, fluency practice with multiplication and division facts is part of the students’ on-going experience.

3.OA.7 Fluently multiply and divide within 100, using

strategies such as the relationship between multiplication

and division (e.g., knowing that 8 × 5 = 40, one knows 40

÷ 5 = 8) or properties of operations. By the end of Grade 3,

know from memory all products of two one-digit numbers.

See Unit 1.

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Component Cluster 3.OA Solve problems involving the four operations, and identify and explain patterns in arithmetic.*

* After being fully taught in Module 3, this standard (as well as OA.3) continues being practiced throughout the remainder of the school year.





estimation strategies including rounding.*

*This standard is limited to problems posed with whole





See Unit 1.







See Unit 1.

Component Cluster 3.NBT Use place value understanding and properties of operations to perform multi-digit arithmetic.*

*A range of algorithms may be used.

3.NBT.3 Multiply one-digit whole numbers by multiples

of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using


operations.

This standard extends students’ work in multiplication by having them apply their understanding of place value. This

standard expects that students go beyond tricks that hinder understanding such as “just adding zeros” and explain and

reason about their products.

For example, for the problem 50 x 4, students should think of this as 4 groups of 5 tens or 20 tens, and that twenty

tens equals 200.

The special role of 10 in the base-ten system is important in understanding multiplication of one-digit numbers with

multiples of 10. For example, the product 3 x 50 can be represented as 3 groups of 5 tens, which is 15 tens, which is

150. This reasoning relies on the associative property of multiplication: 3 x 50 = 3 x (5 x 10) = (3 x 5) x 10 = 15 x 10

= 150. It is an example of how to explain an instance of a calculation pattern for these products: calculate the product

of the non-zero digits, and then shift the product one place to the left to make the result ten times as large •

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Mathematics



Grade 3 Unit 4: Measuring and Classifying Shapes (~6 weeks)

Unit Overview: By Unit 4, students are ready to investigate area and the formula for the area of a rectangle. They measure the area of a shape by finding the total

number of same-size units of area required to cover the shape without gaps or overlaps. When that shape is a rectangle with whole number side lengths, it is easy to

partition the rectangle into squares with equal areas (as in the third stage of the unit 3 illustration above). The area of each square is then a fraction of the area of

the rectangle, which links this unit with the next. This unit provides students with the opportunity to look for and express regularity in repeated reasoning (MP 8)

as they develop the area formula.

Guiding Question: How are multiplication and addition related to finding the area of a shape?



3.G.1 Understand that shapes in different categories (e.g.,

rhombuses, rectangles, and others) may share attributes

(e.g., having four sides), and that the shared attributes can

define a larger category (e.g., quadrilaterals). Recognize

rhombuses, rectangles, and squares as examples of

quadrilaterals, and draw examples of quadrilaterals that do

not belong to any of these subcategories.

In second grade, students identify and draw triangles, quadrilaterals, pentagons, and hexagons. Third graders build on

this experience and further investigate quadrilaterals (technology may be used during this exploration). Students

recognize shapes that are and are not quadrilaterals by examining the properties of the geometric figures. They

conceptualize that a quadrilateral must be a closed figure with four straight sides and begin to notice characteristics of

the angles and the relationship between opposite sides. Students should be encouraged to provide details and use

proper vocabulary when describing the properties of quadrilaterals. They sort geometric figures (see examples below)

and identify squares, rectangles, and rhombuses as quadrilaterals.

Component Cluster 3.MD Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

3.MD.5 Recognize area as an attribute of plane figures and

understand concepts of area measurement.

a. A square with side length 1 unit, called “a unit

square,” is said to have “one square unit” of area, and

can be used to measure area.

b. A plane figure which can be covered without gaps or

overlaps by n unit squares is said to have an area of n

square units.

These standards call for students to explore the concept of covering a region with “unit squares,” which could include

square tiles or shading on grid or graph paper. Based on students’ development, they should have ample experiences

filling a region with square tiles before transitioning to pictorial representations on graph paper.

4

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5 one unit

3.MD.6 Measure areas by counting unit squares (square

cm, square m, square in, square ft, and improvised units).

Students should be counting the square units to find the area could be done in metric, customary, or non-standard

square units. Using different sized graph paper, students can explore the areas measured in square centimeters and

square inches.

3.MD.7 Relate area to the operations of multiplication and

addition.

a. Find the area of a rectangle with whole-number side

lengths by tiling it, and show that the area is the same as

would be found by multiplying the side lengths.

Students can learn how to multiply length measurements to find the area of a rectangular region. But, in order that

they make sense of these quantities, they must first learn to interpret measurement of rectangular regions as a

multiplicative relationship of the number of square units in a row and the number of rows. This relies on the

development of spatial structuring. To build from spatial structuring to understanding the number of area-units as the

product of number of units in a row and number of rows, students might draw rectangular arrays of squares and learn

to determine the number of squares in each row with increasingly sophisticated strategies, such as skip-counting the

number in each row and eventually multiplying the number in each row by the number of rows. They learn to

partition a rectangle into identical squares by anticipating the final structure and forming the array by drawing line

segments to form rows and columns. They use skip counting and multiplication to determine the number of squares

in the array.

Many activities that involve seeing and making arrays of squares to form a rectangle might be needed to build robust

conceptions of a rectangular area structured into squares.

Students should understand and explain why multiplying the side lengths of a rectangle yields the same measurement

of area as counting the number of tiles (with the same unit length) that fill the rectangle’s interior For example,

students might explain that one length tells how many unit squares in a row and the other length tells how many rows

there are.


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b. Multiply side lengths to find areas of rectangles with

whole-number side lengths in the context of solving real

world and mathematical problems, and represent whole-

number products as rectangular areas in mathematical

reasoning.

Students should solve real world and mathematical problems.

Students might solve problems such as finding all the rectangular regions with whole-number side lengths that have

an area of 12 area-units, doing this for larger rectangles (e.g., enclosing 24, 48, 72 area-units), making sketches rather

than drawing each square. Students learn to justify their belief they have found all possible solutions. (Progressions

for the CCSSM, Geometric Measurement, CCSS Writing Team, June 2012, page 18)

c. Use tiling to show in a concrete case that the area of a

rectangle with whole-number side lengths a and b + c

is the sum of a × b and a × c. Use area models to

represent the distributive property in mathematical

reasoning.

This standard extends students’ work with the distributive property. For example, in the picture below the area of a 7

x 6 figure can be determined by finding the area of a 5 x 6 and 2 x 6 and adding the two sums.

Using concrete objects or drawings students build competence with composition and

decomposition of shapes, spatial structuring, and addition of area measurements, students learn to investigate

arithmetic properties using area models. For example, they learn to rotate rectangular arrays physically and mentally,

understanding that their areas are preserved under rotation, and thus, for example, 4 x 7 = 7 x 4, illustrating the

commutative property of multiplication. Students also learn to understand and explain that the area of a rectangular

region of, for example, 12 length-units by 5 length-units can be found either by multiplying 12 x 5, or by adding two

products, e.g., 10 x 5 and 2 x 5, illustrating the distributive property. (Progressions for the CCSSM, Geometric

Measurement, CCSS Writing Team, June 2012, page 18)

d. Recognize area as additive. Find areas of rectilinear

figures by decomposing them into non-overlapping

rectangles and adding the areas of the non-overlapping

parts, applying this technique to solve real world

problems.

This standard uses the word rectilinear. A rectilinear figure is a polygon that has all right angles.

2 x 6 5 x 6

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How could this figure be

decomposed to help find the area?

This portion

of the decomposed figure is

a 4 x 2.

This portion

of the decomposed figure is 2 x

2.

4 x 2 = 8 and 2 x 2 = 4

So 8 + 4 = 12

Therefore the total area of this figure is 12 square units

Component Cluster 3.NBT Use place value understanding and properties of operations to perform multi-digit arithmetic.

3.NBT.3 Multiply one-digit whole numbers by multiples

of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using


operations.

*Focus here within the context of finding area.

See unit 3.

Component Cluster 3.MD Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and

area measures.

Mathematics



3.MD.8 Solve real world and mathematical problems

involving perimeters of polygons, including finding the

perimeter given the side lengths, finding an unknown side

length, and exhibiting rectangles with the same perimeter

and different areas or with the same area and different

perimeters.

*Focus of Unit 4 is on exploring perimeter separately from

area. Exhibiting rectangles with the same perimeter and

different areas or with the same area and different

perimeters is explored in Unit 7.

Students develop an understanding of the concept of perimeter through various experiences, such as walking around

the perimeter of a room, using rubber bands to represent the perimeter of a plane figure on a geoboard, or tracing

around a shape on an interactive whiteboard. They find the perimeter of objects; use addition to find perimeters; and

recognize the patterns that exist when finding the sum of the lengths and widths of rectangles.

Students should also strategically use tools, such as geoboards, tiles, and graph paper to find all the possible

rectangles that have a given perimeter (e.g., find the rectangles with a perimeter of 14 cm.) They record all the

possibilities using dot or graph paper, compile the possibilities into an organized list or a table, and determine

whether they have all the possible rectangles. Following this experience, students can reason about connections

between their representations, side lengths, and the perimeter of the rectangles.

A perimeter is the boundary of a two-dimensional shape. For a polygon, the length of the perimeter is the sum of the

lengths of the sides. Initially, it is useful to have sides marked with unit length marks, allowing students to count the

unit lengths. Later, the lengths of the sides can be labeled with numerals. As with all length tasks, students need to

count the length-units and not the end-points. Next, students learn to mark off unit lengths with a ruler and label the

length of each side of the polygon. For rectangles, parallelograms, and regular polygons, students can discuss and

justify faster ways to find the perimeter length than just adding all of the lengths.

Rectangles and parallelograms have opposite sides of equal length, so students can double the lengths of adjacent

sides and add those numbers or add lengths of two adjacent sides and double that number. A regular polygon has all

sides of equal length, so its perimeter length is the product of one side length and the number of sides.

Perimeter problems for rectangles and parallelograms often give only the lengths of two adjacent sides or only show

numbers for these sides in a drawing of the shape. The common error is to add just those two numbers. Having

students first label the lengths of the other two sides as a reminder is helpful. Students then find unknown side

lengths in more difficult “missing measurements” problems and other types of perimeter problems.


With strong and distinct concepts of both perimeter and area established, students can work on problems to

differentiate their measures. For example, they can find and sketch rectangles with the same perimeter and different

areas or with the same area and different perimeters and justify their claims. Differentiating perimeter from area is

facilitated by having students draw congruent rectangles and measure, mark off, and label the unit lengths all around

the perimeter on one rectangle, then do the same on the other rectangle but also draw the square units. This enables

students to see the units involved in length and area and find patterns in finding the lengths and areas of non-square

and square rectangles. Students can continue to describe and show the units involved in perimeter and area after they

no longer need these. (Progressions for the CCSSM, Geometric Measurement, CCSS Writing Team, June 2012, page

18)

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Grade 3 Unit 5: Fractions as Numbers on the Number Line (~7 weeks)

Unit Overview: The goal of the fifth unit is for students to transition from thinking of fractions as parts of a figure to points on a number line (U5). To make that

jump students once again have to think of fractions as special types of units: Forming fractional units is exactly the same as what was done for multiplication, but

the “group” is now allowed to be the amount when a whole unit is subdivided equally: “1 fourth” is the length of a segment on the number line such that the length

of 4 concatenated (linked together) fourth segments on the line equals 1. Once the unit “1 fourth” has been established, counting them is as easy as counting whole

numbers: 1 fourth, 2 fourths, 3 fourths, 4 fourths, 5 fourths, etc. Students also compare fractions, find equivalent fractions in special cases, and solve problems that

involve comparing fractions. Students should be given many opportunities to construct viable arguments and critique the reasoning of others (MP 3) as they

employ effective fraction models to explain their reasoning.

Guiding Question: What are good strategies and models for comparing and solving problems with fractions?


Component Cluster 3.NF Develop understanding of fractions as numbers.

3.NF.1 Understand a fraction 1/b as the quantity formed by

1 part when a whole is partitioned into b equal parts;

understand a fraction a/b as the quantity formed by a parts

of size 1/b.

This standard refers to the sharing of a whole being partitioned. Fraction models in third grade include only area

(parts of a whole) models (circles, rectangles, squares) and number lines. Set models (parts of a group) are not

addressed in Third Grade.

In 3.NF.1 students start with unit fractions (fractions with numerator 1), which are formed by partitioning a whole

into equal parts and reasoning about one part of the whole, e.g., if a whole is partitioned into 4 equal parts then each

part is ¼ of the whole, and 4 copies of that part make the whole. Next, students build fractions from unit fractions,

seeing the numerator 3 of ¾ as saying that ¾ is the quantity you get by putting 3 of the ¼’s together. There is no need

to introduce “improper fractions" initially.

(Progressions for the CCSSM; Number and Operation – Fractions, CCSS Writing Team, August 2011, page 2)

Some important concepts related to developing understanding of fractions include:

Understand fractional parts must be equal-sized.

Example Non-example

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These are thirds These are NOT thirds

The number of equal parts tells how many make a whole.

As the number of equal pieces in the whole increases, the size of the fractional pieces decreases.

The size of the fractional part is relative to the whole.

o One-half of a small pizza is relatively smaller than one-half of a large pizza.

When a whole is cut into equal parts, the denominator represents the number of equal parts.

The numerator of a fraction is the count of the number of equal parts.

o ¾ means that there are 3 one-fourths.

o Students can count one fourth, two fourths, three fourths.

Students express fractions as fair sharing or, parts of a whole. They use various contexts (candy bars, fruit, and cakes)

and a variety of models (circles, squares, rectangles, fraction bars, and number lines) to develop understanding of

fractions and represent fractions. Students need many opportunities to solve word problems that require them to

create and reason about fair share.

Initially, students can use an intuitive notion of “same size and same shape” (congruence) to explain why the parts

are equal, e.g., when they divide a square into four equal squares or four equal rectangles.

Students come to understand a more precise meaning for “equal parts” as “parts with equal measurements.” For

example, when a ruler is partitioned into halves or quarters of an inch, they see that each subdivision has the same

length. In area models they reason about the area of a shaded region to decide what fraction of the whole it

represents.

(Progressions for the CCSSM, Number and Operation – Fractions, CCSS Writing Team, August 2011, page 3)

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3.NF.2 Understand a fraction as a number on the number

line; represent fractions on a number line diagram.

a. Represent a fraction 1/b on a number line diagram by

defining the interval from 0 to 1 as the whole and

partitioning it into b equal parts. Recognize that each

part has size 1/b and that the endpoint of the part

based at 0 locates the number 1/b on the number line.

b. Represent a fraction a/b on a number line diagram by

marking off a lengths 1/b from 0. Recognize that the

resulting interval has size a/b and that its endpoint

locates the number a/b on the number line.

The number line diagram is the first time students work with a number line for numbers that are between whole

numbers (e.g., that ½ is between 0 and 1). Students need ample experiences folding linear models (e.g., string,

sentence strips) to help them reason about and justify the location of fractions, such that ½ lies exactly halfway

between 0 and 1.

In the number line diagram below, the space between 0 and 1 is divided (partitioned) into 4 equal regions. The

distance from 0 to the first segment is 1 of the 4 segments from 0 to 1 or ¼ (3.NF.2a). Similarly, the distance from 0

to the third segment is 3 segments that are each one-fourth long. Therefore, the distance of 3 segments from 0 is the

fraction ¾ (3.NF.2b).


3.NF.3 Explain equivalence of fractions in special cases,

and compare fractions by reasoning about their size.

a. Understand two fractions as equivalent (equal) if they

are the same size, or the same point on a number line.

b. Recognize and generate simple equivalent fractions,

e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions

are equivalent, e.g., by using a visual fraction model.

An important concept when comparing fractions is to look at the size of the parts and the number of the parts. For

example,

1

8 is smaller than

1

2 because when 1 whole is cut into 8 pieces, the pieces are much smaller than when 1

whole is cut into 2 pieces.

3.NF.3a and 3.NF.3b These standards call for students to use visual fraction models (area models) and number lines

to explore the idea of equivalent fractions. Students should only explore equivalent fractions using models, rather

than using algorithms or procedures.

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c. Express whole numbers as fractions, and recognize

fractions that are equivalent to whole numbers.

Examples: Express 3 in the form 3 = 3/1; recognize

that 6/1 = 6; locate 4/4 and 1 at the same point of a


This standard includes writing whole numbers as fractions. The concept relates to fractions as division problems,

where the fraction 3/1 is 3 wholes divided into one group. This standard is the building block for later work where

students divide a set of objects into a specific number of groups. Students must understand the meaning of a/1.

d. Compare two fractions with the same numerator or the

same denominator by reasoning about their size.

Recognize that comparisons are valid only when the

two fractions refer to the same whole. Record the

results of comparisons with the symbols >, =, or <,

and justify the conclusions, e.g., by using a visual

fraction model.

This standard involves comparing fractions with or without visual fraction models including number lines.

Experiences should encourage students to reason about the size of pieces, the fact that 1/3 of a cake is larger than ¼

of the same cake. Since the same cake (the whole) is split into equal pieces, thirds are larger than fourths.

In this standard, students should also reason that comparisons are only valid if the wholes are identical. For example,

½ of a large pizza is a different amount than ½ of a small pizza. Students should be given opportunities to discuss

and reason about which ½ is larger.

Previously, in second grade, students compared lengths using a standard measurement unit. In third grade they build

on this idea to compare fractions with the same denominator. They see that for fractions that have the same

denominator, the underlying unit fractions are the same size, so the fraction with the greater numerator is greater

because it is made of more unit fractions. For example, segment from 0 to ¾ is shorter than the segment from 0 to 5/4

because it measures 3 units of ¼ as opposed to 5 units of ¼, therefore ¾ < 5/4.

Students also see that for unit fractions, the one with the larger denominator is smaller, by reasoning, for example,

that in order for more (identical) pieces to make the same whole, the pieces must be smaller. From this they reason

that for fractions that have the same numerator, the fraction with the smaller denominator is greater. For example, 2/5

> 2/7, because 1/7 < 1/5, so 2 lengths of 1/7 is less than 2 lengths of 1/5.

As with equivalence of fractions, it is important in comparing fractions to make sure that each fraction refers to the

same whole.


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3.G.2 Partition shapes into parts with equal areas. Express

the area of each part as a unit fraction of the whole.

For example, partition a shape into 4 parts with equal

area, and describe the area of each part as 1/4 of the area

of the shape.

In third grade students start to develop the idea of a fraction more formally, building on the idea of partitioning a

whole into equal parts. The whole can be a shape such as a circle or rectangle. In Grade 4, this is extended to include

wholes that are collections of objects.

This standard also builds on students’ work with fractions and area. Students are responsible for partitioning shapes

into halves, thirds, fourths, sixths and eighths.

Given a shape, students partition it into equal parts, recognizing that these parts all have the same area. They identify

the fractional name of each part and are able to partition a shape into parts with equal areas in several different ways.

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Grade 3 Unit 6: Collecting and Displaying Data (~2 weeks)

Unit Overview: In Unit 6, students leave the world of exact measurements behind. By applying their knowledge of fractions from Unit 5, they estimate lengths to

the nearest halves and fourths of an inch and record that information in bar graphs and line plots. This unit also prepares students for the multiplicative comparison

problems of Grade 4 by asking students “how many more” and “how many less” questions of scaled bar graphs. Students have the opportunity to reason abstractly

and quantitatively (MP 2) as they move back and forth between real-life measurements and the math skills needed to solve problems about them.

Guiding Question: What are good ways to display data?



3.MD.3 Draw a scaled picture graph and a scaled bar

graph to represent a data set with several categories. Solve

one- and two-step “how many more” and “how many less”

problems using information presented in scaled bar graphs.

For example, draw a bar graph in which each square in

the bar graph might represent 5 pets.

Students should have opportunities reading and solving problems using scaled graphs before being asked to draw

one. Work with scaled graphs builds on students’ understanding of multiplication and division.

The following graphs provided below all use five as the scale interval, but students should experience different

intervals to further develop their understanding of scale graphs and number facts.

While exploring data concepts, students should Pose a question, Collect data, Analyze data, and Interpret data

(PCAI). Students should be graphing data that is relevant to their lives

Example:

Pose a question: Student should come up with a question. What is the typical genre read in our class?

Collect and organize data: student survey

Pictographs: Scaled pictographs include symbols that represent multiple units. Below is an example of a pictograph

with symbols that represent multiple units. Graphs should include a title, categories, category label, key, and data.

How many more books did Juan read than Nancy?

Single Bar Graphs: Students use both horizontal and vertical bar graphs. Bar graphs include a title, scale, scale label,

categories, category label, and data.

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Analyze and Interpret data.


using rulers marked with halves and fourths of an inch.

Show the data by making a line plot, where the horizontal

scale is marked off in appropriate units— whole numbers,

halves, or quarters.

Students in second grade measured length in whole units using both metric and U.S. customary systems. It’s

important to review with students how to read and use a standard ruler including details about halves and quarter

marks on the ruler. Students should connect their understanding of fractions to measuring to one-half and one-quarter

inch. Third graders need many opportunities measuring the length of various objects in their environment.

This standard provides a context for students to work with fractions by measuring objects to a quarter of an inch.

Example:

Measure objects in your desk to the nearest ½ or ¼ of an inch, display data collected on a line plot. How many

objects measured ¼? ½? etc…

In Grade 3, students are beginning to learn fraction concepts (3.NF). They understand fraction equivalence in simple

cases, and they use visual fraction models to represent and order fractions. Grade 3 students also measure lengths

using rulers marked with halves and fourths of an inch. They use their developing knowledge of fractions and

number lines to extend their work from the previous grade by working with measurement data involving fractional

measurement values.

For example, every student in the class might measure the height of a bamboo shoot growing in the classroom,

leading to a data set shown in a table.

To make a line plot from the data in a table, the student can determine the greatest and least values in the data, say:

13 ½ inches and 14 ¾ inches. The student can draw a segment of a number line diagram that includes these

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extremes, with tick marks indicating specific values on the measurement scale. This is just like part of the scale on a

ruler. Having drawn the number line diagram, the student can proceed through the data set recording each

observation by drawing a symbol, such as a dot, above the proper tick mark. As with Grade 2 line plots, if a particular

data value appears many times in the data set, dots will “pile up” above that value. There is no need to sort the

observations, or to do any counting of them, before producing the line plot. Students can pose questions about data

presented in line plots, such as how many students obtained measurements larger than 14 ¼ inches.

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Grade 3 Unit 7: Challenging Word Problems* (~6 weeks) *The seemingly eclectic set of standards in Module 7 allows for a new level of challenging word problems, as promised by the module title.

Unit Overview: The year rounds out with plenty of time to solve two-step word problems involving the four operations, and to improve fluency for concepts and

skills initiated earlier in the year. In Unit 7, students also describe, analyze, and compare properties of two-dimensional shapes. By now, students have done

enough work with both linear and area measurement models to study that there is no relationship in general between the perimeter and area of a figure, one of the

concepts of the last unit. Students will have many opportunities to make sense of problems and persevere in solving them (MP 1) as they put their new skills and

understandings to use.

Guiding Question: What strategies do good mathematicians use to see if their answers to challenging problems make sense?








1 See Glossary, Table 2. at end of document

This standard references various problem solving context and strategies that students are expected to use while

solving word problems involving multiplication & division. Students should use a variety of representations for

creating and solving one-step word problems, such as: If you divide 4 packs of 9 brownies among 6 people, how

many cookies does each person receive? (4 x 9 = 36, 36 ÷ 6 = 6).

Glossary page 89, Table 2 (table also included at the end of this document for your convenience) gives examples of a

variety of problem solving contexts, in which students need to find the product, the group size, or the number of

groups. Students should be given ample experiences to explore all of the different problem structures.

Examples of multiplication:

There are 24 desks in the classroom. If the teacher puts 6 desks in each row, how many rows are there?

This task can be solved by drawing an array by putting 6 desks in each row. This is an array model

This task can also be solved by drawing pictures of equal groups.

4 groups of 6 equals 24 objects

A student can also reason through the problem mentally or verbally, “I know 6 and 6 are 12. 12 and 12 are 24.

Therefore, there are 4 groups of 6 giving a total of 24 desks in the classroom.”

A number line could also be used to show equal jumps.

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Students in third grade should use a variety of pictures, such as stars, boxes, flowers to represent unknown numbers

(variables). Letters are also introduced to represent unknowns in third grade.

Examples of Division:

There are some students at recess. The teacher divides the class into 4 lines with 6 students in each line. Write a

division equation for this story and determine how many students are in the class

in the class).

Determining the number of objects in each share (partition model of division, where the size of the groups is

unknown):

Example:

The bag has 92 hair clips, and Laura and her three friends want to share them equally. How many hair clips will each

person receive?

Determining the number of shares (measurement division, where the number of groups is unknown)

Example:

Max the monkey loves bananas. Molly, his trainer, has 24 bananas. If she gives Max 4 bananas each day, how many

days will the bananas last?

Starting Day 1 Day 2 Day 3 Day 4 Day 5 Day 6

24 24 – 4 = 20 20 – 4 = 16 16 – 4 = 12 12 – 4 = 8 8 – 4 = 4 4 – 4 = 0

Solution: The bananas will last for 6 days.

Component Cluster 3.OA Solve problems involving the four operations, and identify and explain patterns in arithmetic.




Students in third grade begin the step to formal algebraic language by using a letter for the unknown quantity in

expressions or equations for one and two-step problems. But the symbols of arithmetic, x or . or * for multiplication

and ÷ or / for division, continue to be used in Grades 3, 4, and 5. (Progressions for the CCSSM; Operations and

Algebraic Thinking, CCSS Writing Team, May 2011, page 27)

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estimation strategies including rounding.3

3 This standard is limited to problems posed with whole





This standard refers to two-step word problems using the four operations. The size of the numbers should be limited

to related 3rd

grade standards (e.g., 3.OA.7 and 3.NBT.2). Adding and subtracting numbers should include numbers

within 1,000, and multiplying and dividing numbers should include single-digit factors and products less than 100.

This standard calls for students to represent problems using equations with a letter to represent unknown quantities.

Example:

Mike runs 2 miles a day. His goal is to run 25 miles. After 5 days, how many miles does Mike have left to run in

order to meet his goal? Write an equation and find the solution (2 x 5 + m = 25).

This standard refers to estimation strategies, including using compatible numbers (numbers that sum to 10, 50, or

100) or rounding. The focus in this standard is to have students use and discuss various strategies. Students should

estimate during problem solving, and then revisit their estimate to check for reasonableness.

Example:

Here are some typical estimation strategies for the problem:

On a vacation, your family travels 267 miles on the first day, 194 miles on the second day and 34 miles on the third

day. How many total miles did they travel?

Student 1

I first thought about

267 and 34. I noticed

that their sum is about

300. Then I knew that

194 is close to 200.

When I put 300 and

200 together, I get 500.

Student 2

I first thought about 194. It is

really close to 200. I also have 2

hundreds in 267. That gives me

a total of 4 hundreds. Then I

have 67 in 267 and the 34. When

I put 67 and 34 together that is

really close to 100. When I add

that hundred to the 4 hundreds

that I already had, I end up with

500.

Student 3

I rounded 267 to 300.

I rounded 194 to 200.

I rounded 34 to 30.

When I added 300,

200 and 30, I know

my answer will be

about 530.

The assessment of estimation strategies should only have one reasonable answer (500 or 530), or a range (between

500 and 550). Problems should be structured so that all acceptable estimation strategies will arrive at a reasonable

answer.

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(Progressions for the CCSSM; Operations and Algebraic Thinking, CCSS Writing Team, May 2011, page 28)

In the diagram above, Carla’s bands are shown using 4 equal-sized bars that represent 4x8 or 32 bands. Agustin’s

bands are directly below showing that the number that August in has plus 15 = 32. The diagram can also be drawn

like this:

8

8

8

8

15

?







This standard calls for students to examine arithmetic patterns involving both addition and multiplication. Arithmetic

patterns are patterns that change by the same rate, such as adding the same number. For example, the series 2, 4, 6, 8,

10 is an arithmetic pattern that increases by 2 between each term.

This standards also mentions identifying patterns related to the properties of operations.

Examples:

Even numbers are always divisible by 2. Even numbers can always be decomposed into 2 equal addends

(14 = 7 + 7).

Multiples of even numbers (2, 4, 6, and 8) are always even numbers.

On a multiplication chart, the products in each row and column increase by the same amount (skip counting).

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On an addition chart, the sums in each row and column increase by the same amount.

Students need ample opportunities to observe and identify important numerical patterns related to operations. They

should build on their previous experiences with properties related to addition and subtraction. Students investigate

addition and multiplication tables in search of patterns and explain why these patterns make sense mathematically.

Example:

Any sum of two even numbers is even.

Any sum of two odd numbers is even.

Any sum of an even number and an odd number is odd.

The multiples of 4, 6, 8, and 10 are all even because they can all be decomposed into two equal groups.

The doubles (2 addends the same) in an addition table fall on a diagonal while the doubles (multiples of 2) in a

multiplication table fall on horizontal and vertical lines.

The multiples of any number fall on a horizontal and a vertical line due to the commutative property.

All the multiples of 5 end in a 0 or 5 while all the multiples of 10 end with 0. Every other multiple of 5 is a

multiple of 10.

Students also investigate a hundreds chart in search of addition and subtraction patterns. They record and organize all

the different possible sums of a number and explain why the pattern makes sense.



using rulers marked with halves and fourths of an inch.

Show the data by making a line plot, where the horizontal

scale is marked off in appropriate units— whole numbers,

halves, or quarters.

See unit 6.

Component Cluster 3.MD Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and

area measures.

3.MD.8 Solve real world and mathematical problems

involving perimeters of polygons, including finding the

perimeter given the side lengths, finding an unknown side

length, and exhibiting rectangles with the same perimeter

and different areas or with the same area and different

perimeters.*

*Complete the standard here with an exploration of the

relationship or non-relationship between area and

perimeter.

See Unit 4 and below:

Given a perimeter and a length or width, students use objects or pictures to find the missing length or width. They

justify and communicate their solutions using words, diagrams, pictures, numbers, and an interactive whiteboard.

Students use geoboards, tiles, graph paper, or technology to find all the possible rectangles with a given area (e.g.

find the rectangles that have an area of 12 square units.) They record all the possibilities using dot or graph paper,

compile the possibilities into an organized list or a table, and determine whether they have all the possible rectangles.

Students then investigate the perimeter of the rectangles with an area of 12.

Area Length Width Perimeter

12 sq. in. 1 in. 12 in. 26 in.

12 sq. in. 2 in. 6 in. 16 in.

12 sq. in 3 in. 4 in. 14 in.

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12 sq. in 4 in. 3 in. 14 in.

12 sq. in 6 in. 2 in. 16 in.

12 sq. in 12 in. 1 in. 26 in.

The patterns in the chart allow the students to identify the factors of 12, connect the results to the commutative

property, and discuss the differences in perimeter within the same area. This chart can also be used to investigate

rectangles with the same perimeter. It is important to include squares in the investigation.


With strong and distinct concepts of both perimeter and area established, students can work on problems to

differentiate their measures. For example, they can find and sketch rectangles with the same perimeter and different

areas or with the same area and different perimeters and justify their claims. Differentiating perimeter from area is

facilitated by having students draw congruent rectangles and measure, mark off, and label the unit lengths all around

the perimeter on one rectangle, then do the same on the other rectangle but also draw the square units. This enables

students to see the units involved in length and area and find patterns in finding the lengths and areas of non-square

and square rectangles. Students can continue to describe and show the units involved in perimeter and area after they

no longer need these. (Progressions for the CCSSM, Geometric Measurement, CCSS Writing Team, June 2012, page

18)


3.G.1 Understand that shapes in different categories (e.g.,

rhombuses, rectangles, and others) may share attributes

(e.g., having four sides), and that the shared attributes can

define a larger category (e.g., quadrilaterals). Recognize

rhombuses, rectangles, and squares as examples of

quadrilaterals, and draw examples of quadrilaterals that do

not belong to any of these subcategories.

See Unit 4.

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Grade 4

In Grade 4, instructional time should focus on three critical areas: (1) developing understanding and fluency with multi-digit multiplication, and

developing understanding of dividing to find quotients involving multi-digit dividends; (2) developing an understanding of fraction equivalence,

addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; (3) understanding that geometric

figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and

symmetry.

(1) Students generalize their understanding of place value to 1,000,000, understanding the relative sizes of numbers in each place. They apply

their understanding of models for multiplication (equal-sized groups, arrays, area models), place value, and properties of operations, in

particular the distributive property, as they develop, discuss, and use efficient, accurate, and generalizable methods to compute products of

multi-digit whole numbers. Depending on the numbers and the context, they select and accurately apply appropriate methods to estimate or

mentally calculate products. They develop fluency with efficient procedures for multiplying whole numbers; understand and explain why the

procedures work based on place value and properties of operations; and use them to solve problems. Students apply their understanding of

models for division, place value, properties of operations, and the relationship of division to multiplication as they develop, discuss, and use

efficient, accurate, and generalizable procedures to find quotients involving multi-digit dividends. They select and accurately apply

appropriate methods to estimate and mentally calculate quotients, and interpret remainders based upon the context.

(2) Students develop understanding of fraction equivalence and operations with fractions. They recognize that two different fractions can be

equal (e.g., 15/9 = 5/3), and they develop methods for generating and recognizing equivalent fractions. Students extend previous

understandings about how fractions are built from unit fractions, composing fractions from unit fractions, decomposing fractions into unit

fractions, and using the meaning of fractions and the meaning of multiplication to multiply a fraction by a whole number.

(3) Students describe, analyze, compare, and classify two-dimensional shapes. Through building, drawing, and analyzing two-dimensional

shapes, students deepen their understanding of properties of two-dimensional objects and the use of them to solve problems involving

symmetry.










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# of Weeks

Unit 1: Place Value, Rounding, Fluency with Addition and Subtraction Algorithms of Whole Numbers 12% ~ 4 weeks

Unit 2: Unit Conversions: Addition and Subtraction of Length, Weight, and Capacity 5% ~ 2 weeks


Unit 3: Multiplication and Division of Up to a 4-Digit Number by Up to a 1-Digit Number Using Place Value 25% ~ 9 weeks

Unit 4: Order and Operations with Fractions 25% ~ 9 weeks


Unit 5: Decimal Fractions 11% ~ 4 weeks

State Test

Unit 6: Two-dimensional Shape Exploration and Problem-solving (Post-Test Unit) 11% ~ 4 Weeks

Unit 7: Exploring Multiplication and More Challenging Problems (Post-Test Unit) 11% ~ 4 Weeks

Unit 1: Place Value, +/-

Algorithms 12%

Unit 2: Unit Conversions

5%

Unit 3: Mult./Div. with Place Value

25%

Unit 4: Order and Operations with Fractions

11%

Unit 5: Decimal Fractions

25%

Unit 6: 2-d shapes and problem-

solving 11%

Unit 7: Exploring Multiplication and

Challenges 11%

Instructional Time

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Grade 4 Unit 1: Place Value, Rounding, Fluency with Addition and Subtraction Algorithms of Whole Numbers (~ 4 weeks)

Unit Overview: Unit 1 begins with a study of large numbers. Students are familiar with big units. For example, in today’s world, “big units” are quite common in

our daily lives. For example, movies take about a gigabyte (1,000,000,000 bytes) to store on a computer while songs take about 1 megabyte (1,000,000 bytes). To

understand these big numbers, the students rely upon previous mastery of rounding and the addition and subtraction algorithms. In a sense the algorithms have

come full circle: In grades 2 and 3 the algorithms were the abstract idea students were trying to learn, but by grade 4 the algorithms have become the concrete

knowledge students are relying upon to understand new ideas (U1). This unit lends itself to opportunities for attending to precision (MP 6) in both the standard

algorithm as well as place value language. This unit also provides opportunities for looking for and expressing regularity in repeated reasoning (MP8) as students

explain calculations and understand how algorithms for addition and subtraction work.

Guiding Question: How do algorithms for addition and subtraction and rounding skills affect our understanding of and ability to solve problems involving real-life numbers?


Component Cluster 4.OA Use the four operations with whole numbers to solve problems.

4.OA.1 Interpret a multiplication equation as a

comparison, e.g., interpret 35 = 5 × 7 as a statement that 35

is 5 times as many as 7 and 7 times as many as 5.

Represent verbal statements of multiplicative comparisons

as multiplication equations.

A multiplicative comparison is a situation in which one quantity is multiplied by a specified number to get another

quantity (e.g., “a is n times as much as b”). Students should be able to identify and verbalize which quantity is being

multiplied and which number tells how many times. Students should be given opportunities to write and identify

equations and statements for multiplicative comparisons.

4.OA.2 Multiply or divide to solve word problems

involving multiplicative comparison, e.g., by using


number to represent the problem, distinguishing

multiplicative comparison from additive comparison.*

* See Glossary, Table 2.

This standard calls for students to translate comparative situations into equations with an unknown and solve.

Students need many opportunities to solve contextual problems. Refer to Glossary, Table 2. In an additive

comparison, the underlying question is what amount would be added to one quantity in order to result in the other. In

a multiplicative comparison, the underlying question is what factor would multiply one quantity in order to result in

the other. The focus in this standard is to have students use and discuss various strategies. It refers to estimation

strategies, including using compatible numbers (numbers that sum to 10 or 100) or rounding. Problems should be

structured so that all acceptable estimation strategies will arrive at a reasonable answer. Students need many

opportunities solving multistep story problems using all four operations.

4.OA.3 Solve multistep word problems posed with whole

numbers and having whole number answers using the four

operations, including problems in which remainders must

be interpreted. Represent these problems using equations

with a letter standing for the unknown quantity. Assess the


estimation strategies including rounding.

The focus in this standard is to have students use and discuss various strategies. It refers to estimation strategies,

including using compatible numbers (numbers that sum to 10 or 100) or rounding. Problems should be structured so

that all acceptable estimation strategies will arrive at a reasonable answer. Students need many opportunities solving

multistep story problems using all four operations.

Component Cluster 4.NBT Generalize place value understanding for multi-digit whole numbers.

4.NBT.1 Recognize that in a multi-digit whole number, a

digit in one place represents ten times what it represents in

the place to its right. For example, recognize that 700 ÷ 70

= 10 by applying concepts of place value and division.

This standard calls for students to extend their understanding of place value related to multiplying and dividing by

multiples of 10. In this standard, students should reason about the magnitude of digits in a number. Students should

be given opportunities to reason and analyze the relationships of numbers that they are working with.

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4.NBT.2 Read and write multi-digit whole numbers using

base-ten numerals, number names, and expanded form.

Compare two multidigit numbers based on meanings of the

digits in each place, using >, =, and < symbols to record

the results of comparisons.

This standard refers to various ways to write numbers. Students should have flexibility with the different number

forms. Traditional expanded form is 285 = 200 + 80 + 5. Written form or number name is two hundred eighty-five.

However, students should have opportunities to explore the idea that 285 could also be 28 tens plus 5 ones or 1

hundred, 18 tens, and 5 ones. To read numerals between 1,000 and 1,000,000, students need to understand the role of

commas. Students should also be able to compare two multi-digit whole numbers using appropriate symbols.

4.NBT.3 Use place value understanding to round multi-

digit whole numbers to any place.

This standard refers to place value understanding, which extends beyond an algorithm or procedure for rounding. The

expectation is that students have a deep understanding of place value and number sense and can explain and reason

about the answers they get when they round. Students should have numerous experiences using a number line and a

hundreds chart as tools to support their work with rounding.


4.NBT.4 Fluently add and subtract multi-digit whole

numbers using the standard algorithm.

Students build on their understanding of addition and subtraction, their use of place value and their flexibility with

multiple strategies to make sense of the standard algorithm. They continue to use place value in describing and

justifying the processes they use to add and subtract.

This standard refers to fluency, which means accuracy, efficiency (using a reasonable amount of steps and time), and

flexibility (using a variety strategies such as the distributive property). This is the first grade level in which students

are expected to be proficient at using the standard algorithm to add and subtract. However, other previously learned

strategies are still appropriate for students to use. Students should know that it is mathematically possible to subtract

a larger number from a smaller number but that their work with whole numbers does not allow this as the difference

would result in a negative number.

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Grade 4 Unit 2: Unit Conversions: Addition and Subtraction of Length, Weight, and Capacity (~2 weeks) Unit Overview: The algorithms from Unit 1 continue to play a part in this next unit (U2) on unit conversions. This unit is intentionally designed to be repetitive to

help students draw similarities between: 10 ones = 1 ten, 100 ones = 1 hundred, 100 cm = 1 m, 1000 ones = 1 thousand, 1000 m = 1 km, 1000 g = 1 kg, 1000 mL =

1 L. Measurement problems again act as the “glue” that binds knowledge of the algorithms, mental math, place value, and real-world applications together into a

coherent whole. Students will need to make sense of problems and persevere in solving them (MP1) in this unit. This unit of application of concepts will also

require students to model with mathematics (MP 4) as they look for the mathematics represented in real-life measurement situations.

Guiding Question: How can changing a unit improve a measurement in a real-world problem?


Component Cluster 4.MD Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.*

*The focus of this unit is on the metric system to reinforce place value, compound units, and word problems with unit conversions. Decimal and fraction word problems wait until Units4 and 6.

4.MD.1 Know relative sizes of measurement units within

one system of units including km, m, cm; kg, g; lb, oz.; l,

ml; hr, min, sec. Within a single system of measurement,

express measurements in a larger unit in terms of a smaller

unit. Record measurement equivalents in a two-column

table.

For example, know that 1 ft is 12 times as long as 1 in.

Express the length of a 4 ft snake as 48 in. Generate a

conversion table for feet and inches listing the number

pairs (1, 12), (2, 24), (3, 36), ...

The units of measure that have not been addressed in prior years are cups, pints, quarts, gallons, pounds, ounces,

kilometers, millimeter, milliliters, and seconds. Students’ prior experiences were limited to measuring length, mass

(metric and customary systems), liquid volume (metric only), and elapsed time. Students did not convert

measurements.

Students develop benchmarks and mental images about a meter (e.g., about the height of a tall chair) and a kilometer

(e.g., the length of 10 football fields including the end zones, or the distance a person might walk in about 12

minutes), and they also understand that “kilo” means a thousand, so 3000 m is equivalent to 3 km. Expressing larger

measurements in smaller units within the metric system is an opportunity to reinforce notions of place value. There

are prefixes for multiples of the basic unit (meter or gram), although only a few (kilo-, centi-, and milli-) are in

common use. Tables such as the one below are an opportunity to develop or reinforce place value concepts and skills

in measurement activities. Relating units within the metric system is another opportunity to think about place value.

For example, students might make a table that shows measurements of the same lengths in centimeters and meters.

Relating units within the traditional system provides an opportunity to engage in mathematical practices, especially

“look for and make use of structure” and “look for and express regularity in repeated reasoning” For example,

students might make a table that shows measurements of the same lengths in feet and inches.

(Progressions for the CCSSM, Geometric Measurement, CCSS Writing Team, June 2012, page20)

4.MD.2 Use the four operations to solve word problems

involving distances, intervals of time, liquid volumes,

masses of objects, and money, including problems

involving simple fractions or decimals, and problems that

require expressing measurements given in a larger unit in

terms of a smaller unit. Represent measurement quantities

using diagrams such as number line diagrams that feature a

measurement scale.

This standard includes multi-step word problems related to expressing measurements from a larger unit in

terms of a smaller unit (e.g., feet to inches, meters to centimeter, and dollars to cents). Students should

have ample opportunities to use number line diagrams to solve word problems.

Mathematics



Grade 4 Unit 3: Multiplication and Division of Up to a 4-Digit Number by Up to a 1-Digit Number Using Place Value (~9

weeks)

Unit Overview: In Unit 3, measurements provide the concrete foundation behind the distributive property in the multiplication algorithm: 4 × (1 m + 2 cm) can be

made physical using ribbon, where it is easy to see the 4 copies of 1 m and the 4 copies of 2 cm. Likewise, 4 × (1 ten + 2 ones) = 4 tens + 8 ones. Students then

turn to the place value table with number disks to develop efficient procedures for multiplying and dividing one-digit whole numbers and use the table with

number disks to understand and explain why the procedures work. Students also solve word problems throughout the unit where they select and accurately apply

appropriate methods to estimate, mentally calculate, or use the procedures they are learning to compute products and quotients. Students will need to use

appropriate tools strategically (MP 5) as they make sense of problems and persevere in solving them (MP 1).

Guiding Question: How can you use a model to solve a real-world problem quickly and to prove your reasoning?










The focus in this standard is to have students use and discuss various strategies. It refers to estimation strategies,

including using compatible numbers (numbers that sum to 10 or 100) or rounding. Problems should be structured so

that all acceptable estimation strategies will arrive at a reasonable answer. Students need many opportunities solving

multistep story problems using all four operations.

Component Cluster 4.OA Gain familiarity with factors and multiplies.

4.OA.4 Find all factor pairs for a whole number in the

range 1–100. Recognize that a whole number is a multiple

of each of its factors. Determine whether a given whole

number in the range 1–100 is a multiple of a given one-

digit number. Determine whether a given whole number in

the range 1–100 is prime or composite.

This standard requires students to demonstrate understanding of factors and multiples of whole numbers. This

standard also refers to prime and composite numbers. Prime numbers have exactly two factors, the number one and

their own number. For example, the number 17 has the factors of 1 and 17. Composite numbers have more than two

factors. For example, 8 has the factors 1, 2, 4, and 8.

A common misconception is that the number 1 is prime, when in fact; it is neither prime nor composite. Another

common misconception is that all prime numbers are odd numbers. This is not true, since the number 2 has only 2

factors, 1 and 2, and is also an even number.

Prime vs. Composite:

A prime number is a number greater than 1 that has only 2 factors, 1 and itself. Composite numbers have more than 2

factors.

Students investigate whether numbers are prime or composite by

building rectangles (arrays) with the given area and finding which numbers have more than two rectangles

(e.g. 7 can be made into only 2 rectangles, 1 x 7 and 7 x 1, therefore it is a prime number)

finding factors of the number

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Students should understand the process of finding factor pairs so they can do this for any number 1 -100,

Example:

Factor pairs for 96: 1 and 96, 2 and 48, 3 and 32, 4 and 24, 6 and 16, 8 and 12.

Multiples can be thought of as the result of skip counting by each of the factors. When skip counting, students should

be able to identify the number of factors counted e.g., 5, 10, 15, 20 (there are 4 fives in 20).

To determine if a number between 1-100 is a multiple of a given one-digit number, some helpful hints include the

following:

all even numbers are multiples of 2

all even numbers that can be halved twice (with a whole number result) are multiples of 4

all numbers ending in 0 or 5 are multiples of 5


4.NBT.5 Multiply a whole number of up to four digits by a

one-digit whole number, and multiply two two-digit

numbers, using strategies based on place value and the

properties of operations. Illustrate and explain the

calculation by using equations, rectangular arrays, and/or

area models.

Students who develop flexibility in breaking numbers apart have a better understanding of the importance of place

value and the distributive property in multi-digit multiplication. Students use base ten blocks, area models,

partitioning, compensation strategies, etc. when multiplying whole numbers and use words and diagrams to explain

their thinking. They use the terms factor and product when communicating their reasoning. Multiple strategies enable

students to develop fluency with multiplication and transfer that understanding to division. Use of the standard

algorithm for multiplication is an expectation in the 5th

grade.

Another part of understanding general base-ten methods for multi-digit multiplication is understanding the role

played by the distributive property. This allows numbers to be decomposed into base-ten units, products of the units

to be computed, and then combined. By decomposing the factors into like base-ten units and applying the distributive

property, multiplication computations are reduced to single-digit multiplications and products of numbers with

multiples of 10, of 100, and of 1000. Students can connect diagrams of areas or arrays to numerical work to develop

understanding of general base-ten multiplication methods. Computing products of two two-digit numbers requires

using the distributive property several times when the factors are decomposed into base-ten units.

4.NBT.6 Find whole-number quotients and remainders

with up to four-digit dividends and one-digit divisors,

using strategies based on place value, the properties of

operations, and/or the relationship between multiplication

and division. Illustrate and explain the calculation by using

equations, rectangular arrays, and/or area models.

In fourth grade, students build on their third grade work with division within 100. Students need opportunities to

develop their understandings by using problems in and out of context.

General methods for computing quotients of multi-digit numbers and one-digit numbers rely on the same

understandings as for multiplication, but cast in terms of division. One component is quotients of multiples of 10,

100, or 1000 and one-digit numbers. For example, 42 ÷ 6 is related to 420 ÷ 6 and 4200 ÷ 6. Students can draw on

their work with multiplication and they can also reason that 4200 ÷ 6 means partitioning 42 hundreds into 6 equal

groups, so there are 7 hundreds in each group. Another component of understanding general methods for multi-digit

division computation is the idea of decomposing the dividend into like base-ten units and finding the quotient unit by

unit, starting with the largest unit and continuing on to smaller units. As with multiplication, this relies on the

distributive property. This can be viewed as finding the side length of a rectangle (the divisor is the length of the

other side) or as allocating objects (the divisor is the number of groups).

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Multi-digit division requires working with remainders. In preparation for working with remainders, students can

compute sums of a product and a number, such as 4 x 8 + 3. In multi-digit division, students will need to find the

greatest multiple less than a given number. For example, when dividing by 6, the greatest multiple of

6 less than 50 is 6 x 8 = 48. Students can think of these “greatest multiples” in terms of putting objects into groups.

For example, when 50 objects are shared among 6 groups, the largest whole number of objects that can be put in each

group is 8, and 2 objects are left over. (Or when 50 objects are allocated into groups of 6, the largest whole number of

groups that can be made is 8, and 2 objects are left over.) The equation 6 x 8 + 2 = 50 (or 8 x 6 + 2 = 50) corresponds

with this situation.

Cases involving 0 in division may require special attention.


This standard calls for students to explore division through various strategies.

Component Cluster 4.MD Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

4.MD.3 Apply the area and perimeter formulas for

rectangles in real world and mathematical problems.

For example, find the width of a rectangular room given

the area of the flooring and the length, by viewing the area

formula as a multiplication equation with an unknown

factor.

Based on work in third grade students learn to consider perimeter and area of rectangles. In fourth grade, from

multiplication, spatially structuring arrays, and area, they abstract the formula for the area of a rectangle A = l x w

Students generate and discuss advantages and disadvantages of various formulas for the perimeter length of a

rectangle that is l units by w units (P = 2l + 2w, P = 2(l + w), P = l + w + l + w) .

Giving verbal summaries of these formulas is also helpful. Specific numerical instances of other formulas or mental

calculations for the perimeter of a rectangle can be seen as examples of the properties of operations, e.g., 2l + 2w =

2(l + w) illustrates the distributive property.

Perimeter problems often give only one length and one width, thus remembering the basic formula can help to

prevent the usual error of only adding one length and one width.

Such abstraction and use of formulas underscores the importance of distinguishing between area and perimeter in

Grade 3 and maintaining the distinction in Grade 4 and later grades, where rectangle perimeter and area problems

may get more complex and problem solving can benefit from knowing or being able to rapidly remind oneself of how

to find an area or perimeter. By repeatedly reasoning about how to calculate areas and perimeters of rectangles,

students can come to see area and perimeter formulas as summaries of all such calculations. (Progressions for the

CCSSM, Geometric Measurement, CCSS Writing Team, June 2012, page 21)

Students learn to apply these understandings and formulas to the solution of real-world and mathematical problems.

Students should be challenged to solve multistep problems.

In fourth grade and beyond, the mental visual images for perimeter and area from third grade can support students in

problem solving with these concepts. “Apply the formula” does not mean write down a memorized formula and put

in known values because in fourth grade students do not evaluate expressions (they begin this type of work in Grade

6). In fourth grade, working with perimeter and area of rectangles is still grounded in specific visualizations and

numbers.

Mathematics



Grade 4 Unit 4: Order and Operations with Fractions* (~9 weeks) * Tenths and hundredths are important fractions in this module, represented in decimal form in Unit 6.

Unit Overview: Unit 4 centers on equivalent fractions and operations with fractions. We use fractions when there is a given unit, the whole unit, but we want to

measure using a smaller unit, called the fractional unit. To prepare students to explore the relationship between a fractional unit and its whole unit, examples of

such relationships in different contexts were already carefully established earlier in the year: 100 cm in 1 meter, 1000 g in 1 kilogram, 1000 mL in 1 liter, etc. The

beauty of fractional units is that, once defined, they behave just like whole number units:

• “3 fourths + 5 fourths = 8 fourths” like “3 apples + 5 apples = 8 apples,” and

• “3 fourths × 4 = 12 fourths” like “3 apples × 4 = 12 apples.”

This unit also includes measuring and plotting fractional numbers and adding/subtracting those measurements. In Grade 2, fractions were mostly used as adjectives

(for example, half cup, third of an hour, etc.). As students do basic fraction arithmetic in Grade 4, they gradually come to understand fractions as numbers.

Students should be given many opportunities to construct viable arguments and critique the reasoning of others (MP 3) as they employ effective fraction models to

explain their reasoning.

Guiding Question: What are good strategies and models for comparing and solving problems with fractions?


Component Cluster 4.NF Extend understanding of fraction equivalence and ordering.

4.NF.1 Explain why a fraction a/b is equivalent to a

fraction (n × a)/(n × b) by using visual fraction models,

with attention to how the number and size of the parts

differ even though the two fractions themselves are the

same size. Use this principle to recognize and generate

equivalent fractions.

This standard refers to visual fraction models. This includes area models, number lines or it could be a collection/set

model. This standard extends the work in third grade by using additional denominators. (5, 10, 12 and 100)

This standard addresses equivalent fractions by examining the idea that equivalent fractions can be created by

multiplying both the numerator and denominator by the same number or by dividing a shaded region into various

parts. Students should begin to notice connections between the models and fractions in the way both the parts and

wholes are counted and begin to generate a rule for writing equivalent fractions.

4.NF.2 Compare two fractions with different numerators

and different denominators, e.g., by creating common

denominators or numerators, or by comparing to a

benchmark fraction such as 1/2. Recognize that

comparisons are valid only when the two fractions refer to

the same whole. Record the results of comparisons with

symbols >, =, or <, and justify the conclusions, e.g., by

using a visual fraction model.

This standard calls students to compare fractions by creating visual fraction models or finding common denominators

or numerators. Students’ experiences should focus on visual fraction models rather than algorithms. When tested,

models may or may not be included. Students should learn to draw fraction models to help them compare. Students

must also recognize that they must consider the size of the whole when comparing fractions. In fifth grade students

who have learned about fraction multiplication can see equivalence as “multiplying by 1":

However, although a useful mnemonic device, this does not constitute a valid argument at fourth grade, since

students have not yet learned fraction multiplication.

Component Cluster 4.NF Build fractions from unit fractions by applying and extending previous understandings of operations on whole

numbers.

4.NF.3 Understand a fraction a/b with a > 1 as a sum of

fractions 1/b.

A fraction with a numerator of one is called a unit fraction. When students investigate fractions other than unit

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a. Understand addition and subtraction of fractions as

joining and separating parts referring to the same

whole.

b. Decompose a fraction into a sum of fractions with the

same denominator in more than one way, recording

each decomposition by an equation. Justify

decompositions, e.g., by using a visual fraction model.

Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2

1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.

c. Add and subtract mixed numbers with like

denominators, e.g., by replacing each mixed number

with an equivalent fraction, and/or by using properties

of operations and the relationship between addition

and subtraction.

d. Solve word problems involving addition and

subtraction of fractions referring to the same whole

and having like denominators, e.g., by using visual

fraction models and equations to represent the

problem.

fractions, such as 2/3, they should be able to join (compose) or separate (decompose) the fractions of the same whole.

Students need multiple opportunities to work with mixed numbers and be able to decompose them in more than one

way. Students may use visual models to help develop this understanding. Students should justify their breaking apart

(decomposing) of fractions using visual fraction models. The concept of turning mixed numbers into improper

fractions needs to be emphasized using visual fraction models. Similarly, converting an improper fraction to a mixed

number is a matter of decomposing the fraction into a sum of a whole number and a number less than 1. Students can

draw on their knowledge from third grade of whole numbers as fractions.

A separate algorithm for mixed numbers in addition and subtraction is not necessary. Students will tend to add or

subtract the whole numbers first and then work with the fractions using the same strategies they have applied to

problems that contained only fractions. Mixed numbers are introduced for the first time in Fourth Grade. Students

should have ample experiences of adding and subtracting mixed numbers where they work with mixed numbers or

convert mixed numbers so that the numerator is equal to or greater than the denominator. Converting a mixed

number to a fraction should not be viewed as a separate technique to be learned by rote, but simply as a case of

fraction addition.

4.NF.4 Apply and extend previous understandings of

multiplication to multiply a fraction by a whole number.

a. Understand a fraction a/b as a multiple of 1/b.

For example, use a visual fraction model to represent

5/4 as the product 5 × (1/4), recording the conclusion

by the equation 5/4 = 5 × (1/4).

b. Understand a multiple of a/b as a multiple of 1/b, and

use this understanding to multiply a fraction by a

whole number.

For example, use a visual fraction model to express 3

× (2/5) as 6 × (1/5), recognizing this product as 6/5.

(In general, n × (a/b) = (n × a)/b.)

c. Solve word problems involving multiplication of a

fraction by a whole number, e.g., by using visual


problem.

For example, if each person at a party will eat 3/8 of a

pound of roast beef, and there will be 5 people at the

This standard builds on students’ work of adding fractions and extending that work into multiplication. Students

should use the number line and area model. Students should see a fraction as the numerator times the unit fraction

with the same denominator.

This standard extended the idea of multiplication as repeated addition. For example, 3 x (2/5) = 2/5 + 2/5 + 2/5 = 6/5

= 6 x (1/5). Students are expected to use and create visual fraction models to multiply a whole number by a fraction.

The same thinking, based on the analogy between fractions and whole numbers, allows students to give meaning to

the product of whole number and a fraction.

When introducing this standard, make sure student use visual fraction models to solve word problems related to

multiplying a whole number by a fraction.

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party, how many pounds of roast beef will be needed?

Between what two whole numbers does your answer

lie?


4.MD.4 Make a line plot to display a data set of

measurements in fractions of a unit (1/2, 1/4, 1/8). Solve

problems involving addition and subtraction of fractions

by using information presented in line plots.

For example, from a line plot find and interpret the

difference in length between the longest and shortest

specimens in an insect collection.

This standard provides a context for students to work with fractions by measuring objects to an eighth of an inch.

Students are making a line plot of this data and then adding and subtracting fractions based on data in the line plot.

Mathematics



Grade 4 Unit 5: Decimal Fractions* (~4 weeks)

*In this unit, we continue to work with fractions, now including decimal form.

Unit Overview: Decimals (U5) start with the realization that decimal place value units are just special fractional units: 1 tenth = 1/10, 1 hundredth = 1/100, etc.

Fluency plays an important role in both of these topics as students learn to relate 3/10 = 0.3 = 3 tenths. Students have the opportunity to look for and make use of

structure (MP 7) as they make connections between fractions and our extended place value system.

Guiding Question: How do decimals and fractions relate to each other in the real world?


Component Cluster 4.NF Understand decimal notation for fractions, and compare decimal fractions.

4.NF.5 Express a fraction with denominator 10 as an

equivalent fraction with denominator 100, and use this

technique to add two fractions with respective

denominators 10 and 100.*

For example, express 3/10 as 30/100, and add 3/10 +

4/100 = 34/100.

* Students who can generate equivalent fractions can

develop strategies for adding fractions with unlike

denominators in general. But addition and subtraction with

unlike denominators in general is not a requirement at this

grade.

This standard continues the work of equivalent fractions by having students change fractions with a 10 in the

denominator into equivalent fractions that have a 100 in the denominator. In order to prepare for work with decimals

(4.NF.6 and 4.NF.7), experiences that allow students to shade decimal grids (10x10 grids) can support this work.

Student experiences should focus on working with grids rather than algorithms.

Students can also use base ten blocks and other place value models to explore the relationship between fractions with

denominators of 10 and denominators of 100.

Students in fourth grade work with fractions having denominators 10 and 100. Because it involves partitioning into

10 equal parts and treating the parts as numbers called one tenth and one hundredth, work with these fractions can be

used as preparation to extend the base-ten system to non-whole numbers. This work in fourth grade lays the

foundation for performing operations with decimal numbers in fifth grade.

4.NF.6 Use decimal notation for fractions with

denominators 10 or 100. For example, rewrite 0.62 as

62/100; describe a length as 0.62 meters; locate 0.62 on a


Decimals are introduced for the first time. Students should have ample opportunities to explore and reason about the

idea that a number can be represented as both a fraction and a decimal.

Students make connections between fractions with denominators of 10 and 100 and the place value chart. By reading

fraction names, students say 32/100 as thirty-two hundredths and rewrite this as 0.32 or represent it on a place value

model as shown below.

Hundreds Tens Ones Tenths Hundredths

3 2

Students use the representations explored in 4.NF.5 to understand 32/100 can be expanded to 3/10 and 2/100.

Students represent values such as 0.32 or 32/100 on a number line. 32/100 is more than 30/100 (or 3/10) and less than

40/100 (or 4/10). It is closer to 30/100 so it would be placed on the number line near that value.

Mathematics



4.NF.7 Compare two decimals to hundredths by reasoning

about their size. Recognize that comparisons are valid only

when the two decimals refer to the same whole. Record the

results of comparisons with the symbols >, =, or <, and

justify the conclusions, e.g., by using a visual model.

Students should reason that comparisons are only valid when they refer to the same whole. Visual models include

area models, decimal grids, decimal circles, number lines, and meter sticks.

Mathematics



Grade 4 Unit 6: Two-dimensional Shape Exploration and Problem-solving (~4 weeks)

Unit Overview: Unit 6 focuses both on building, drawing, and analyzing two-dimensional shapes in geometry as well as on solving unknown angle problems

using letters and equations as it does. To clarify the significance of this second unit focus, reflect on the following. Students have already used letters and equations

to solve word problems in earlier grades. They continue to do so in Grade 4, and now they also learn to solve unknown angle problems: work that challenges

students to build and solve equations to find unknown angle measures. First, students learn the definition of degree and learn how to measure angles in degrees

using a protractor. From the definition of degree and the fact that angle measures are additive, the following rudimentary facts about angles naturally follow:

1. vertical angles are equal,

2. the sum of angle measurements on a line is 180 degrees, and

3. the sum of angle measurements around a point is 360 degrees.

Armed with just these three facts (and the obvious one that angle measures of adjoining angles add), students are able to generate and solve equations that make

sense:

Geometry is the key that unlocks algebra for students because it is visual. The x clearly stands for a specific number: If a student wanted to, he or she could place a

protractor down on that angle and measure it to find x. But doing so destroys the joy of solving the puzzle and deducing the answer for themselves. Students have

a real opportunity here to use appropriate tools strategically (MP 5) as they first explore the uses of a protractor and then determine whether or not they need it to

solve angle measurement problems.

Guiding Question: When you inspect and describe a shape, what are useful attributes to notice?


Component Cluster 4.MD Geometric Measurement: understand concepts of angle and measure angles.

4.MD.5 Recognize angles as geometric shapes that are

formed wherever two rays share a common endpoint, and

understand concepts of angle measurement:

a. An angle is measured with reference to a circle with

its center at the common endpoint of the rays, by

considering the fraction of the circular arc between the

points where the two rays intersect the circle. An

angle that turns through 1/360 of a circle is called a

“one-degree angle,” and can be used to measure

angles.

This standard brings up a connection between angles and circular measurement (360 degrees).

Angle measure is a “turning point” in the study of geometry. Students often find angles and angle measure to be

difficult concepts to learn, but that learning allows them to engage in interesting and important mathematics.

Mathematics



b. An angle that turns through n one-degree angles is

said to have an angle measure of n degrees.

This standard calls for students to explore an angle as a series of “one-degree turns.”

4.MD.6 Measure angles in whole-number degrees using a

protractor. Sketch angles of specified measure.

Before students begin measuring angles with protractors, they need to have some experiences with benchmark

angles. They transfer their understanding that a 360º rotation about a point makes a complete circle to recognize and

sketch angles that measure approximately 90º and 180º. They extend this understanding and recognize and sketch

angles that measure approximately 45º and 30º. They use appropriate terminology (acute, right, and obtuse) to

describe angles and rays (perpendicular). Students should then measure angles and sketch angles. As with other

concepts students need varied examples and explicit discussions to avoid learning limited ideas about measuring

angles (e.g., misconceptions that a right angle is an angle that points to the right, or two right angles represented with

different orientations are not equal in measure).

4.MD.7 Recognize angle measure as additive. When an

angle is decomposed into non-overlapping parts, the angle

measure of the whole is the sum of the angle measures of

the parts. Solve addition and subtraction problems to find

unknown angles on a diagram in real world and

mathematical problems, e.g., by using an equation with a

symbol for the unknown angle measure.

This standard addresses the idea of decomposing (breaking apart) an angle into smaller parts. Students can develop

more accurate and useful angle and angle measure concepts if presented with angles in a variety of situations. They

learn to find the common features of superficially different situations such as turns in navigation, slopes, bends,

corners, and openings. With guidance, they learn to represent an angle in any of these contexts as two rays, even

when both rays are not explicitly represented in the context.

Component Cluster 4.G Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

4.G.1 Draw points, lines, line segments, rays, angles (right,

acute, obtuse), and perpendicular and parallel lines.

Identify these in two-dimensional figures.

This standard asks students to draw two-dimensional geometric objects and to also identify them in two-dimensional

figures. This is the first time that students are exposed to rays, angles, and perpendicular and parallel lines. Examples

of points, line segments, lines, angles, parallelism, and perpendicularity can be seen daily. Students may not easily

identify lines and rays because they are more abstract.

Student should be able to use side length to classify triangles as equilateral, equiangular, isosceles, or scalene; and

can use angle size to classify them as acute, right, or obtuse. They then learn to cross-classify. Students also learn to

apply these concepts in varied contexts. Analyzing the shapes in order to construct them requires students to

explicitly formulate their ideas about the shapes.

4.G.2 Classify two-dimensional figures based on the

presence or absence of parallel or perpendicular lines, or

the presence or absence of angles of a specified size.

Recognize right triangles as a category, and identify right

triangles.

Two-dimensional figures may be classified using different characteristics such as, parallel or perpendicular lines or

by angle measurement. This standard calls for students to sort objects based on parallelism, perpendicularity and

angle types.

Fourth grade students have built a firm foundation of several shape categories, these categories can be the raw

material for thinking about the relationships between classes. Students should classify shapes by attributes and

drawing shapes that fit specific categories.

Example: students can form larger, categories, such as the class of all shapes with four sides, or quadrilaterals, and

recognize that it includes other categories, such as squares, rectangles, rhombuses, parallelograms, and trapezoids.

They also recognize that there are quadrilaterals that are not in any of those subcategories.

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4.G.3 Recognize a line of symmetry for a two-dimensional

figure as a line across the figure such that the figure can be

folded along the line into matching parts. Identify line-

symmetric figures and draw lines of symmetry.

Students need experiences with figures which are symmetrical and non-symmetrical. Figures include both regular

and non-regular polygons. Folding cut-out figures will help students determine whether a figure has one or more lines

of symmetry. This standard only includes line symmetry not rotational symmetry.

Mathematics



Grade 4 Unit 7: Exploring Multiplication and More Challenging Problems (~4 weeks)

Unit Overview: The year ends with an exploratory module on multiplication (M6). Students have been practicing the algorithm for multiplying by a 1-digit

number since the third module. The goal here is to structure opportunities for them to “discover” ways to multiply 2-digit × 2- digit numbers by using their tools

(place value tables, area models, bar models, number disks, the distributive property and equations, etc.). Students also solve challenging problems including

fraction and area problems that involve customary measurements (inches and feet, etc.) and pattern generating problems. Students will be challenged to make

sense of problems and persevere in solving them (MP 1) as they apply previous learning to more challenging contexts.

Guiding Question: What is the connection between 2-digit by 2-digit multiplication and fraction multiplication?










See Unit 1.

Component Cluster 4.OA Generate and analyze patterns.

4.OA.5 Generate a number or shape pattern that follows a

given rule. Identify apparent features of the pattern that

were not explicit in the rule itself. For example, given the

rule “Add 3” and the starting number 1, generate terms in

the resulting sequence and observe that the terms appear

to alternate between odd and even numbers. Explain

informally why the numbers will continue to alternate in

this way.

Patterns involving numbers or symbols either repeat or grow. Students need multiple opportunities creating and

extending number and shape patterns. Numerical patterns allow students to reinforce facts and develop fluency with

operations.

Patterns and rules are related. A pattern is a sequence that repeats the same process over and over. A rule dictates

what that process will look like. Students investigate different patterns to find rules, identify features in the patterns,

and justify the reason for those features.

After students have identified rules and features from patterns, they need to generate a numerical or shape pattern

from a given rule.

This standard calls for students to describe features of an arithmetic number pattern or shape pattern by identifying

the rule, and features that are not explicit in the rule. A t-chart is a tool to help students see number patterns.

This standard begins with a small focus on reasoning about a number or shape pattern, connecting a rule for a given

pattern with its sequence of numbers or shapes. Patterns that consist of repeated sequences of shapes or growing

sequences of designs can be appropriate for the grade. For example, students could examine a sequence of dot

designs in which each design has 4 more dots than the previous one and they could reason about how the dots are

organized in the design to determine the total number of dots in the 100th

design. In examining numerical sequences,

fourth graders can explore rules of repeatedly adding the same whole number or repeatedly multiplying by the same

whole number. Properties of repeating patterns of shapes can be explored with division. For example, to determine

Mathematics



the 100th

shape in a pattern that consists of repetitions of the sequence “square, circle, triangle,” the fact that when we

divide 100 by 3 the whole number quotient is 33 with remainder 1 tells us that after 33 full repeats, the 99th shape

will be a triangle (the last shape in the repeating pattern), so the 100th shape is the first shape in the pattern, which is

a square. Notice that the Standards do not require students to infer or guess the underlying rule for a pattern, but

rather ask them to generate a pattern from a given rule and identify features of the given pattern. (Progressions for

the CCSSM; Operations and Algebraic Thinking , CCSS Writing Team, May 2011, page 31)

Component Cluster 4.NBT Use place value understanding and properties of operations to perform multi-digit arithmetic.* * In Unit 7, the focus is on multiplying two two-digit numbers.

4.NBT.5 Multiply a whole number of up to four digits by a

one-digit whole number, and multiply two two-digit

numbers, using strategies based on place value and the

properties of operations. Illustrate and explain the

calculation by using equations, rectangular arrays, and/or

area models.

Another part of understanding general base-ten methods for multi-digit multiplication is understanding the role

played by the distributive property. This allows numbers to be decomposed into base-ten units, products of the units

to be computed, and then combined. By decomposing the factors into like base-ten units and applying the distributive

property, multiplication computations are reduced to single-digit multiplications and products of numbers with

multiples of 10, of 100, and of 1000. Students can connect diagrams of areas or arrays to numerical work to develop

understanding of general base-ten multiplication methods. Computing products of two two-digit numbers requires

using the distributive property several times when the factors are decomposed into base-ten units. This standard calls

for students to multiply numbers using a variety of strategies. Students explain this strategy and the one below with

base 10 blocks, drawings, or numbers.

Component Cluster 4.NF Build fractions from unit fractions by applying and extending previous understandings of operations on whole

numbers.


multiplication to multiply a fraction by a whole number.

a. Understand a fraction a/b as a multiple of 1/b.

For example, use a visual fraction model to represent

5/4 as the product 5 × (1/4), recording the conclusion

by the equation 5/4 = 5 × (1/4).

b. Understand a multiple of a/b as a multiple of 1/b, and

use this understanding to multiply a fraction by a whole

number.

For example, use a visual fraction model to express 3

× (2/5) as 6 × (1/5), recognizing this product as 6/5.

(In general, n × (a/b) = (n × a)/b.)

c. Solve word problems involving multiplication of a

fraction by a whole number, e.g., by using visual


This standard builds on students’ work of adding fractions and extending that work into multiplication. Students

should use the number line and area model. Students should see a fraction as the numerator times the unit fraction

with the same denominator.

This standard extended the idea of multiplication as repeated addition. For example, 3 x (2/5) = 2/5 + 2/5 + 2/5 = 6/5

= 6 x (1/5). Students are expected to use and create visual fraction models to multiply a whole number by a fraction.

The same thinking, based on the analogy between fractions and whole numbers, allows students to give meaning to

the product of whole number and a fraction.

When introducing this standard, make sure student use visual fraction models to solve word problems related to

multiplying a whole number by a fraction.

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problem.

For example, if each person at a party will eat 3/8 of a

pound of roast beef, and there will be 5 people at the

party, how many pounds of roast beef will be needed?

Between what two whole numbers does your answer

lie?

Component Cluster 4.MD Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.*

* The focus now is on customary units in word problems for application of fraction concepts and multiplication of two two-digit numbers.

4.MD.1 Know relative sizes of measurement units within

one system of units including km, m, cm; kg, g; lb, oz.; l,

ml; hr, min, sec. Within a single system of measurement,

express measurements in a larger unit in terms of a smaller

unit. Record measurement equivalents in a two-column

table.

For example, know that 1 ft is 12 times as long as 1 in.

Express the length of a 4 ft snake as 48 in. Generate a

conversion table for feet and inches listing the number

pairs (1, 12), (2, 24), (3, 36), ...

See Unit 2.

4.MD.2 Use the four operations to solve word problems

involving distances, intervals of time, liquid volumes,

masses of objects, and money, including problems

involving simple fractions or decimals, and problems that

require expressing measurements given in a larger unit in

terms of a smaller unit. Represent measurement quantities

using diagrams such as number line diagrams that feature a

measurement scale.

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Grade 5 In Grade 5, instructional time should focus on three critical areas: (1) developing fluency with addition and subtraction of fractions, and developing

understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole

numbers divided by unit fractions); (2) extending division to 2-digit divisors, integrating decimal fractions into the place value system and

developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations; and (3)

developing understanding of volume.

(1) Students apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with unlike

denominators as equivalent calculations with like denominators. They develop fluency in calculating sums and differences of fractions, and

make reasonable estimates of them. Students also use the meaning of fractions, of multiplication and division, and the relationship between

multiplication and division to understand and explain why the procedures for multiplying and dividing fractions make sense. (Note: this is

limited to the case of dividing unit fractions by whole numbers and whole numbers by unit fractions.)

(2) Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of operations.

They finalize fluency with multi-digit addition, subtraction, multiplication, and division. They apply their understandings of models for

decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. They develop fluency in these

computations, and make reasonable estimates of their results. Students use the relationship between decimals and fractions, as well as the

relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to

understand and explain why the procedures for multiplying and dividing finite decimals make sense. They compute products and quotients of

decimals to hundredths efficiently and accurately.

(3) Students recognize volume as an attribute of three-dimensional space. They understand that volume can be measured by finding the total

number of same-size units of volume required to fill the space without gaps or overlaps. They understand that a 1-unit by 1-unit by 1-unit

cube is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve

estimating and measuring volume. They decompose three-dimensional shapes and find volumes of right rectangular prisms by viewing them

as decomposed into layers of arrays of cubes. They measure necessary attributes of shapes in order to determine volumes to solve real world

and mathematical problems.










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# of weeks

Unit 1: Whole Number and Decimal Fraction Place Value 11% ~ 4 Weeks

Unit 2: Multi-Digit Whole Number and Decimal Fraction Operations 19% ~ 7 Weeks


Unit 3: Addition and Subtraction of Fractions with Fraction Concepts 22% ~8 Weeks

Benchmark Assessment Moment

Unit 4: Multiplication and Division of Fractions and Multi-Digit Decimal Numbers 17% ~6 Weeks

Unit 5: Volume, Area, and Shapes (Complete Post-Test) 17% ~6 Weeks

State Test

Unit 6: Graph Points on the Coordinate Plane to Solve Problems (Post-Test Unit) 14% ~ 5 Weeks

Unit 1: Whole # and Decimal P.V.

11%

Unit 2: Whole # and Decimal

Fraction Operations

19%

Unit 3: Addition and Subtraction of

Fractions 22%

Unit 4: Mult and Div of Fractions and Decimals

17%

Unit 5: Volume, Area, and Shapes

17%

Unit 6: Graphing on Coordinate Plane

14%

Instructional Time

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Grade 5 Unit 1: Whole Number and Decimal Fraction Place Value (~ 4 weeks)

Unit Overview: Students’ experiences with the algorithms as ways to manipulate place value units in grades 2-4 really begin to pay dividends in grade 5. Whole

number patterns on the place value table are easily generalized to decimal numbers (U1). As students work word problems with measurements in the metric

system, where the same patterns occur, they begin to appreciate the value and the meaning of decimals. Decimal fractions of the form 1/10, 1/100, 1/1000 also

play a prominent role in the first unit and are used in investigating patterns on the place value table. This unit provides students with the opportunity to look for

and make use of structure (MP 7) as they explore and explain the expanding number system.

Guiding Question: How does changing the place of a number change its value?


Component Cluster 5.NBT Understand the place value system.

5.NBT.1 Recognize that in a multi-digit number, a digit in

one place represents 10 times as much as it represents in

the place to its right and 1/10 of what it represents in the

place to its left.

Students extend their understanding of the base-ten system to the relationship between adjacent places, how numbers

compare, and how numbers round for decimals to thousandths. This standard calls for students to reason about the

magnitude of numbers. Students should work with the idea that the tens place is ten times as much as the ones place,

and the ones place is 1/10th

the size of the tens place.

In fourth grade, students examined the relationships of the digits in numbers for whole numbers only. This standard

extends this understanding to the relationship of decimal fractions. Students use base ten blocks, pictures of base ten

blocks, and interactive images of base ten blocks to manipulate and investigate the place value relationships. They

use their understanding of unit fractions to compare decimal places and fractional language to describe those

comparisons.

Before considering the relationship of decimal fractions, students express their understanding that in multi-digit

whole numbers, a digit in one place represents 10 times what it represents in the place to its right and 1/10 of what it

represents in the place to its left.

5.NBT.2 Explain patterns in the number of zeros of the

product when multiplying a number by powers of 10, and

explain patterns in the placement of the decimal point

when a decimal is multiplied or divided by a power of 10.

Use whole-number exponents to denote powers of 10.

New at Grade 5 is the use of whole number exponents to denote powers of 10. Students understand why multiplying

by a power of 10 shifts the digits of a whole number or decimal that many places to the left. Patterns in the number

of 0s in products of a whole numbers and a power of 10 and the location of the decimal point in products of decimals

with powers of 10 can be explained in terms of place value. Because students have developed their understandings of

and computations with decimals in terms of multiples rather than powers, connecting the terminology of multiples

with that of powers affords connections between understanding of multiplication and exponentiation. (Progressions

for the CCSSM, Number and Operation in Base Ten, CCSS Writing Team, April 2011, page 16)

This standard includes multiplying by multiples of 10 and powers of 10, including 102 which is 10 x 10=100, and 10

3

which is 10 x 10 x 10=1,000. Students should have experiences working with connecting the pattern of the number of

zeros in the product when you multiply by powers of 10. Students need to be provided with opportunities to explore

this concept and come to this understanding; this should not just be taught procedurally.

5.NBT.3 Read, write, and compare decimals to

thousandths.

a. Read and write decimals to thousandths using base-ten

numerals, number names, and expanded form, e.g.,

This standard references expanded form of decimals with fractions included. Students should build on their work

from Fourth Grade, where they worked with both decimals and fractions interchangeably. Expanded form is included

to build upon work in 5.NBT.2 and deepen students’ understanding of place value.

Students build on the understanding they developed in fourth grade to read, write, and compare decimals to

thousandths. They connect their prior experiences with using decimal notation for fractions and addition of fractions

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347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 x

(1/100) + 2 x (1/1000)

b. Compare two decimals to thousandths based on

meanings of the digits in each place, using >, =, and <

symbols to record the results of comparisons.

with denominators of 10 and 100. They use concrete models and number lines to extend this understanding to

decimals to the thousandths. Models may include base ten blocks, place value charts, grids, pictures, drawings,

manipulatives, technology-based, etc. They read decimals using fractional language and write decimals in fractional

form, as well as in expanded notation. This investigation leads them to understanding equivalence of decimals (0.8 =

0.80 = 0.800).

Comparing decimals builds on work from fourth grade. Students need to understand the size of decimal numbers and

relate them to common benchmarks such as 0, 0.5 (0.50 and 0.500), and 1. Comparing tenths to tenths, hundredths to

hundredths, and thousandths to thousandths is simplified if students use their understanding of fractions to compare

decimals.

5.NBT.4 Use place value understanding to round decimals

to any place.

This standard refers to rounding. Students should go beyond simply applying an algorithm or procedure for rounding.

The expectation is that students have a deep understanding of place value and number sense and can explain and

reason about the answers they get when they round. Students should have numerous experiences using a number line

to support their work with rounding. Students should use benchmark numbers to support this work. Benchmarks are

convenient numbers for comparing and rounding numbers. 0., 0.5, 1, 1.5 are examples of benchmark numbers.

Component Cluster 5.MD Convert like measurement units within a given measurement system.* *The focus of this unit is on the metric system to reinforce place value and its compound units. Customary conversions appear in Unit 3.

5.MD.1 Convert among different-sized standard

measurement units within a given measurement system

(e.g., convert 5 cm to 0.05 m), and use these conversions

in solving multi-step, real world problems.

This standard calls for students to convert measurements within the same system of measurement in the context of

multi-step, real-world problems. Both customary and standard measurement systems are included; students worked

with both metric and customary units of length in second grade. In third grade, students work with metric units of

mass and liquid volume. In fourth grade, students work with both systems and begin conversions within systems in

length, mass and volume.

Students should explore how the base-ten system supports conversions within the metric system.

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Grade 5 Unit 2: Multi-Digit Whole Number and Decimal Fraction Operations (~ 7 weeks)

Unit Overview: The second unit (U2) starts with giving students a chance to sharpen their understanding and skills with adding and subtracting decimals and with

multiplying and dividing whole and decimal numbers by 1-digit whole numbers. They are now ready to generalize the 1-digit algorithms to the multi-digit whole

number versions (multi-digit decimal multiplication such as 4.1 × 3.4 and division such as 4.5 ÷ 1.5 are studied in Unit 4). For multiplication, students must

grapple with and fully understand the distributive property (one of the key reasons for teaching the multi-digit algorithm). While the multi-digit multiplication

algorithm is a straightforward generalization of the 1-digit multiplication algorithm, the division algorithm with 2-digit divisor requires far more care to teach

because students have to also learn estimation strategies, error correction strategies, and the idea of successive approximation (all of which are central concepts in

math, science, and engineering). Learning the multiplication and division algorithms provides students with an opportunity to reason abstractly and quantitatively

(MP 2) as they utilize familiar story problem contexts in conjunction with new faster algorithms in order to make sense of their problem-solving (MP 1).

Guiding Question: What are the connections between the division and multiplication models from 4th grade and the new faster strategies of 5

th grade?


Component Cluster 5.OA Write and interpret numerical expressions.

5.OA.1 Use parentheses, brackets, or braces in

numerical expressions, and evaluate expressions with

these symbols.

The order of operations is introduced in third grade and is continued in fourth. This standard calls for

students to evaluate expressions with parentheses ( ), brackets [ ] and braces { }. In upper levels of

mathematics, evaluate means to substitute for a variable and simplify the expression. However at this level

students are to only simplify the expressions because there are no variables.

This standard builds on the expectations of third grade where students are expected to start learning the

conventional order. Students need experiences with multiple expressions that use grouping symbols

throughout the year to develop understanding of when and how to use parentheses, brackets, and braces.

First, students use these symbols with whole numbers. Then the symbols can be used as students add,

subtract, multiply and divide decimals and fractions.

To further develop students’ understanding of grouping symbols and facility with operations, students

place grouping symbols in equations to make the equations true or they compare expressions that are

grouped differently.

In fifth grade students begin working more formally with expressions. They write expressions to express a

calculation, e.g., writing 2 x (8 + 7) to express the calculation “add 8 and 7, then multiply by 2.” They also

evaluate and interpret expressions, e.g., using their conceptual understanding of multiplication to interpret

3 x (18932 x 921) as being three times as large as 18932 + 921, without having to calculate the indicated

sum or product. Thus, students in Grade 5 begin to think about numerical expressions in ways that

prefigure their later work with variable expressions (e.g., three times an unknown length is 3 . L). In Grade

5, this work should be viewed as exploratory rather than for attaining mastery; for example, expressions

should not contain nested grouping symbols, and they should be no more complex than the expressions

one finds in an application of the associative or distributive property, e.g., (8 + 27) + 2 or (6 x 30) (6 x 7).

Note however that the numbers in expressions need not always be whole numbers.

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5.OA.2 Write simple expressions that record

calculations with numbers, and interpret numerical

expressions without evaluating them.

For example, express the calculation “add 8 and 7,

then multiply by 2” as 2 × (8 + 7). Recognize that 3

× (18932 + 921) is three times as large as 18932 +

921, without having to calculate the indicated sum or

product.

This standard refers to expressions. Expressions are a series of numbers and symbols (+, -, x, ÷) without an

equals sign. Equations result when two expressions are set equal to each other (2 + 3 = 4 + 1).

This standard calls for students to verbally describe the relationship between expressions without actually

calculating them. This standard calls for students to apply their reasoning of the four operations as well as

place value while describing the relationship between numbers. The standard does not include the use of

variables, only numbers and signs for operations.

Component Cluster 5.NBT Understand the place value system.* * These standards, taught in Module 1 in the context of place value, are now taught in the context of the operations with whole numbers.

5.NBT.1 Recognize that in a multi-digit number, a digit in

one place represents 10 times as much as it represents in

the place to its right and 1/10 of what it represents in the

place to its left.

See unit 1.

5.NBT.2 Explain patterns in the number of zeros of the

product when multiplying a number by powers of 10, and

explain patterns in the placement of the decimal point

when a decimal is multiplied or divided by a power of 10.

Use whole-number exponents to denote powers of 10.

Component Cluster 5.NBT Perform operations with multi-digit whole numbers and with decimals to hundredths.*

*Focus on decimal multiplication of a single-digit, whole number factor times a multi digit number with up to 2 decimal places (e.g. 3 x64.98). Restrict decimal division to a single digit whole

number divisor with a multi digit dividend with up to 2 decimal places. (e.g. 64.98 ÷ 3). The balance of the standard is taught in Unit 4.

5.NBT.5 Fluently multiply multi-digit whole numbers

using the standard algorithm.

This standard refers to fluency which means accuracy (correct answer), efficiency (a reasonable amount of steps), and

flexibility (using strategies such as the distributive property or breaking numbers apart also using strategies according to the

numbers in the problem, 26 x 4 may lend itself to (25 x 4 ) + 4 where as another problem might lend itself to making an

equivalent problem 32 x 4 = 64 x 2)). This standard builds upon students’ work with multiplying numbers in third and fourth

grade. In fourth grade, students developed understanding of multiplication through using various strategies. While the

standard algorithm is mentioned, alternative strategies are also appropriate to help students develop conceptual

understanding. The size of the numbers should NOT exceed a three-digit factor by a two-digit factor.

5.NBT.6 Find whole-number quotients of whole numbers

with up to four-digit dividends and two-digit divisors,

using strategies based on place value, the properties of

operations, and/or the relationship between multiplication

and division. Illustrate and explain the calculation by using

equations, rectangular arrays, and/or area models.

This standard references various strategies for division. Division problems can include remainders. Even though this

standard leads more towards computation, the connection to story contexts is critical. Make sure students are exposed

to problems where the divisor is the number of groups and where the divisor is the size of the groups. In fourth grade,

students’ experiences with division were limited to dividing by one-digit divisors. This standard extends students’

prior experiences with strategies, illustrations, and explanations. When the two-digit divisor is a “familiar” number, a

student might decompose the dividend using place value.

5.NBT.7 Add, subtract, multiply, and divide decimals to

hundredths, using concrete models or drawings and


This standard builds on the work from fourth grade where students are introduced to decimals and compare them. In

fifth grade, students begin adding, subtracting, multiplying and dividing decimals. This work should focus on

concrete models and pictorial representations, rather than relying solely on the algorithm. The use of symbolic

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reasoning used.

notations involves having students record the answers to computations (2.25 x 3= 6.75), but this work should not be

done without models or pictures. This standard includes students’ reasoning and explanations of how they use

models, pictures, and strategies.

This standard requires students to extend the models and strategies they developed for whole numbers in grades 1-4

to decimal values. Before students are asked to give exact answers, they should estimate answers based on their

understanding of operations and the value of the numbers.

Students should be able to express that when they add decimals they add tenths to tenths and hundredths to

hundredths. So, when they are adding in a vertical format (numbers beneath each other), it is important that they

write numbers with the same place value beneath each other. This understanding can be reinforced by connecting

addition of decimals to their understanding of addition of fractions. Adding fractions with denominators of 10 and

100 is a standard in fourth grade.

Mathematics



Grade 5 Unit 3: Addition and Subtraction of Fractions with Fraction Concepts (~ 8 weeks)

Unit Overview: Work with place value units in the first two units paves the path to fractions and arithmetic with fractions in Unit 3 as elementary math’s place

value emphasis shifts to a focus on the larger set of fractional units for algebra. Like units are added to and subtracted from like units:

- Place value units: 1.5 + 0.8 = 1

+

= 15 tenths + 8 tenths = 23 tenths = 2 and 3 tenths = 2

= 2.3

- Fractional units: 1

+

= 14 ninths + 8 ninths = 22 ninths = 2 and 4 ninths = 2

The new complexity is that if units are not equivalent, they must be changed for smaller equal units so that they can be added or subtracted. Probably the best

model for showing this is the rectangular fraction model pictured below. The equivalence is then represented symbolically as students engage in active meaning-

making rather than obeying the perhaps mysterious command to “multiply the top and bottom by the same number.”

2 boys + 1 girl = 2 children + 1 child = 3 children

2 thirds + 1 fourth = 8 twelfths + 3 twelfths = 11 twelfths

Relating different fractional units to one another requires extensive work with area and number line diagrams providing students with the opportunity

to model with mathematics (MP 4). Tape diagrams are used often in word problems. Tape diagrams, which students began using in the early grades

and which become increasingly useful as students applied them to a greater and greater variety of word problems, hit their full strength as a model

when applied to fraction word problems. At the heart of a tape diagram is the now-familiar idea of forming units. In fact, forming units to solve word

problems is one of the most powerful examples of the unit theme and is particularly helpful for understanding fraction arithmetic, as in the following

example: Jill had $32. She gave

of her money to charity and

of her money to her brother. How much did she give altogether?

Guiding Question: What visual models and strategies help to make sense of fraction problems and prove your answers to others?


Component Cluster 5.OA Write and interpret numerical expressions.* *These skills are applied to fractions in this module.

5.OA.1 Use parentheses, brackets, or braces in numerical

expressions, and evaluate expressions with these symbols.

See Unit 2.

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5.OA.2 Write simple expressions that record calculations

with numbers, and interpret numerical expressions without

evaluating them.

For example, express the calculation “add 8 and 7, then

multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932

+ 921) is three times as large as 18932 + 921, without

having to calculate the indicated sum or product.

Component Cluster 5.NF Use equivalent fractions as a strategy to add and subtract fractions.* *Examples in this module also include tenths and hundredths in fraction and decimal form.

5.NF.1 Add and subtract fractions with unlike

denominators (including mixed numbers) by replacing

given fractions with equivalent fractions in such a way as

to produce an equivalent sum or difference of fractions

with like denominators.

For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In

general, a/b + c/d = (ad + bc)/bd.)

This standard builds on the work in fourth grade where students add fractions with like denominators. In fifth grade, the

example provided in the standard 2/3 + ¾ has students find a common denominator by finding the product of both

denominators. This process should come after students have used visual fraction models (area models, number lines, etc.) to

build understanding before moving into the standard algorithm described in the standard The use of these visual fraction

models allows students to use reasonableness to find a common denominator prior to using the algorithm.

Students should apply their understanding of equivalent fractions and their ability to rewrite fractions in an equivalent

form to find common denominators. They should know that multiplying the denominators will always give a

common denominator but may not result in the smallest denominator.

Fifth grade students will need to express both fractions in terms of a new denominator with adding unlike

denominators. It is not necessary to find a least common denominator to calculate sums of fractions, and in fact the

effort of finding a least common denominator is a distraction from understanding adding fractions.

5.NF.2 Solve word problems involving addition and

subtraction of fractions referring to the same whole,

including cases of unlike denominators, e.g., by using

visual fraction models or equations to represent the

problem. Use benchmark fractions and number sense of

fractions to estimate mentally and assess the

reasonableness of answers. For example, recognize an

incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 <

1/2.

This standard refers to number sense, which means students’ understanding of fractions as numbers that lie between

whole numbers on a number line. Number sense in fractions also includes moving between decimals and fractions to

find equivalents, also being able to use reasoning such as 7/8 is greater than ¾ because 7/8 is missing only 1/8 and ¾

is missing ¼ so 7/8 is closer to a whole Also, students should use benchmark fractions to estimate and examine the

reasonableness of their answers. Example here such as 5/8 is greater than 6/10 because 5/8 is 1/8 larger than ½(4/8)

and 6/10 is only 1/10 larger than ½ (5/10)

Students make sense of fractional quantities when solving word problems, estimating answers mentally to see if they

make sense. Estimation skills include identifying when estimation is appropriate, determining the level of accuracy

needed, selecting the appropriate method of estimation, and verifying solutions or determining the reasonableness of

situations using various estimation strategies. Estimation strategies for calculations with fractions extend from

students’ work with whole number operations and can be supported through the use of physical models.

Component Cluster 5.NF Apply and extend previous understandings of multiplication and division to multiply and divide fractions. *The balance of this cluster is taught in Module 4.

5.NF.3 Interpret a fraction as division of the numerator by

the denominator (a/b = a ÷ b). Solve word problems

involving division of whole numbers leading to answers in

the form of fractions or mixed numbers, e.g., by using

Fifth grade student should connect fractions with division, understanding that 5 ÷ 3 = 5/3

Students should explain this by working with their understanding of division as equal sharing.

Students should also create story contexts to represent problems involving division of whole numbers.

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visual fraction models or equations to represent the

problem.

For example, interpret 3/4 as the result of dividing 3 by 4,

noting that 3/4 multiplied by 4 equals 3, and that when 3

wholes are shared equally among 4 people each person

has a share of size 3/4. If 9 people want to share a 50-

pound sack of rice equally by weight, how many pounds of

rice should each person get? Between what two whole

numbers does your answer lie?

This standard calls for students to extend their work of partitioning a number line from third and fourth grade.

Students need ample experiences to explore the concept that a fraction is a way to represent the division of two

quantities.

Students are expected to demonstrate their understanding using concrete materials, drawing models, and explaining

their thinking when working with fractions in multiple contexts. They read 3/5 as “three fifths” and after many

experiences with sharing problems, learn that 3/5 can also be interpreted as “3 divided by 5.”

Component Cluster 5.MD Convert like measurement units within a given measurement system.* *The focus of 5.MD.1 in this unit is on the customary system of units as a means of introducing fractions (e.g. 1 inch is 1/12 foot, 1 foot is 1/3 yard, etc).

5.MD.1 Convert among different-sized standard

measurement units within a given measurement system

(e.g., convert 5 cm to 0.05 m), and use these conversions

in solving multi-step, real world problems.

This standard calls for students to convert measurements within the same system of measurement in the context of

multi-step, real-world problems. Both customary and standard measurement systems are included; students worked

with both metric and customary units of length in second grade. In third grade, students work with metric units of

mass and liquid volume. In fourth grade, students work with both systems and begin conversions within systems in

length, mass and volume.

Students should explore how the base-ten system supports conversions within the metric system.

In Grade 5, students extend their abilities from Grade 4 to express measurements in larger or smaller units within a

measurement system. This is an excellent opportunity to reinforce notions of place value for whole numbers and

decimals, and connection between fractions and decimals (e.g., 2 ½ meters can be expressed as 2.5 meters or 250

centimeters). For example, building on the table from Grade 4, Grade 5 students might complete a table of equivalent

measurements in feet and inches. Grade 5 students also learn and use such conversions in solving multi-step, real

world problems.

Component Cluster 5.MD Represent and interpret data

5. MD.2 Make a line plot to display a data set of

measurements in fractions of a unit (1/2, 1/4, 1/8). Use

operations on fractions for this grade to solve problems

involving information presented in line plots.

For example, given different measurements of liquid in

identical beakers, find the amount of liquid each beaker

would contain if the total amount in all the beakers were

redistributed equally.

This standard provides a context for students to work with fractions by measuring objects to one-eighth of a unit.

This includes length, mass, and liquid volume. Students are making a line plot of this data and then adding and

subtracting fractions based on data in the line plot.

Example:

Students measured objects in their desk to the nearest

½, ¼, or 1/8 of an inch then displayed data collected

on a line plot. How many object measured ¼? ½? If

you put all the objects together end to end what would

be the total length of all the objects?

Mathematics



Grade 5 Unit 4: Multiplication and Division of Fractions and Multi-Digit Decimal Numbers (~ 6 weeks)

Unit Overview: In Unit 4, students will relate different fractional units to a common fractional unit: Three-fourths of the class is boys. Two-thirds of the boys are

wearing tennis shoes. What fraction of the class are boys with tennis shoes? 2/3 of the boys x ¾ of the class = ½ of the class are boys with tennis shoes. Relating

different fractional units together back to the whole unit requires extensive work with area and number line models, fluency, and bar diagrams used in word

problems as students learn to multiply and divide with fractions.

In the second half of Unit 4, students use the insight gathered from fractions and whole numbers operations to begin to explore multi-digit decimal multiplication

and division. In multiplying 2.1 × 3.8, for example, students have multiple strategies they can rely on to locate the decimal point in the final answer, including:

• unit awareness: 2.1 × 3.8 = 21 tenths × 38 tenths = 798 hundredths,

• estimation (through rounding): 2.1 × 3.8 ≈ 2 × 4 = 8, so 2.1 × 3.8 = 7.98,

• fraction multiplication: 21/10 × 38/10 = (21 × 38)/(10 × 10).

Similar strategies enrich students understanding of division and help them to see multi-digit decimal division as “whole number division in a different unit.” For

example, we divide to find, “How many groups of 3 apples are there in 45 apples?” and write 45 apples ÷ 3 apples = 15. Similarly, 4.5 ÷ 0.3 can be written as “45

tenths ÷ 3 tenths” with the same answer: there are 15 groups of 0.3 in 4.5. The same idea was used to introduce fraction division earlier in the unit, thus gluing

division with whole numbers, fractions, and decimals together through an understanding of units. Students have an opportunity to employ looking for and

extending regularity in repeated reasoning (MP 8) as they continue to see connections between whole number and decimal operations.

Guiding Question: When do you use the context of a word problem during the solution process?

The student will be

able to:

The teacher will use appropriate instructional strategies/approaches based on the needs of the student.

Component Cluster 5.NBT Perform operations with multi-digit whole numbers and with decimals to hundredths.*

*Teach problems such as 2.7 x 2.1 and 4.5 ÷ 1.5. See “Progressions” pgs. 17 – 18

http://commoncoretools.files.wordpress.com/2011/04/ccss_progression_nbt_2011_04_073.pdf

5.NBT.7 Add, subtract, multiply, and divide decimals to

hundredths, using concrete models or drawings and




reasoning used.

See Unit 2.

Component Cluster 5.NF Apply and extend previous understandings of multiplication and division to multiply and divide fractions. * Include problems involving decimal fractions throughout the cluster.


multiplication to multiply a fraction or whole number by a

fraction.

a. Interpret the product (a/b) × q as a parts of a partition

of q into b equal parts; equivalently, as the result of a

sequence of operations a × q ÷ b.

Students need to develop a fundamental understanding that the multiplication of a fraction by a whole number could

be represented as repeated addition of a unit fraction (e.g., 2 x (1/4) = 1/4 + ¼

This standard extends student’s work of multiplication from earlier grades. In fourth grade, students worked with

recognizing that a fraction such as 3/5 actually could be represented as 3 pieces that are each one-fifth (3 x (1/5)).

This standard references both the multiplication of a fraction by a whole number and the multiplication of two

fractions. Visual fraction models (area models, tape diagrams, number lines) should be used and created by students

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For example, use a visual fraction model to show (2/3)

× 4 = 8/3, and create a story context for this equation.

Do the same with (2/3) × (4/5) = 8/15. (In general,

(a/b) × (c/d) = ac/bd.)

b. Find the area of a rectangle with fractional side

lengths by tiling it with unit squares of the appropriate

unit fraction side lengths, and show that the area is the

same as would be found by multiplying the side

lengths. Multiply fractional side lengths to find areas

of rectangles, and represent fraction products as

rectangular areas.

during their work with this standard.

This standard extends students’ work with area. In third grade students determine the area of rectangles and

composite rectangles. In fourth grade students continue this work. The fifth grade standard calls students to continue

the process of covering (with tiles). Grids can be used to support this work.

5.NF.5 Interpret multiplication as scaling (resizing), by:

a. Comparing the size of a product to the size of one

factor on the basis of the size of the other factor,

without performing the indicated multiplication.

b. Explaining why multiplying a given number by a

fraction greater than 1 results in a product greater than

the given number (recognizing multiplication by

whole numbers greater than 1 as a familiar case);

explaining why multiplying a given number by a

fraction less than 1 results in a product smaller than

the given number; and relating the principle of fraction

equivalence a/b = (n × a)/(n × b) to the effect of

multiplying a/b by 1.

This standard calls for students to examine the magnitude of products in terms of the relationship between two types

of problems. This extends the work with 5.OA.1.

This standard asks students to examine how numbers change when we multiply by fractions. Students should have

ample opportunities to examine both cases in the standard: a) when multiplying by a fraction greater than 1, the

number increases and b) when multiplying by a fraction less the one, the number decreases. This standard should be

explored and discussed while students are working with 5.NF.4, and should not be taught in isolation.

5.NF.6 Solve real world problems involving multiplication

of fractions and mixed numbers, e.g., by using visual

fraction models or equations to represent the problem.

This standard builds on all of the work done in this cluster. Students should be given ample opportunities to use

various strategies to solve word problems involving the multiplication of a fraction by a mixed number. This

standard could include fraction by a fraction, fraction by a mixed number or mixed number by a mixed number.


division to divide unit fractions by whole numbers and

whole numbers by unit fractions.1

a. Interpret division of a unit fraction by a non-zero

whole number, and compute such quotients.

5.NF.7 is the first time that students are dividing with fractions. In fourth grade students divided whole numbers, and

multiplied a whole number by a fraction. The concept unit fraction is a fraction that has a one in the numerator. For

example, the fraction 3/5 is 3 copies of the unit fraction 1/5. 1/5 + 1/5 + 1/5 = 3/5 = 1/5 x 3 or 3 x 1/5

Example:

Knowing the number of groups/shares and finding how many/much in each group/share

Four students sitting at a table were given 1/3 of a pan of brownies to share. How much of a pan will each student get

if they share the pan of brownies equally?

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For example, create a story context for (1/3) ÷ 4, and

use a visual fraction model to show the quotient. Use

the relationship between multiplication and division to

explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 =

1/3.

b. Interpret division of a whole number by a unit

fraction, and compute such quotients. For example,

create a story context for 4 ÷ (1/5), and use a visual

fraction model to show the quotient. Use the

relationship between multiplication and division to

explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.

c. Solve real world problems involving division of unit

fractions by non-zero whole numbers and division of

whole numbers by unit fractions, e.g., by using visual


problem.

For example, how much chocolate will each person

get if 3 people share ½ lb of chocolate equally? How

many 1/3-cup servings are 2 cups of raisins?

1 Students able to multiply fractions in general can develop

strategies to divide fractions in general, by reasoning about

the relationship between multiplication and division. But

division of a fraction by a fraction is not a requirement at

this grade.

The diagram shows the 1/3 pan divided into 4 equal shares with each share equaling 1/12 of the pan.

5.NF.7a This standard asks students to work with story contexts where a unit fraction is divided by a non-zero whole

number. Students should use various fraction models and reasoning about fractions.

Student 1: Expression 1/ 8 ÷ 3

0 3/24 8/24 16/24 24/24

1/8

5.NF.7b This standard calls for students to create story contexts and visual fraction models for division situations

where a whole number is being divided by a unit fraction.

Example:

Create a story context for 5 ÷ 1/6. Find your answer and then draw a picture to prove your answer and use multiplication

to reason about whether your answer makes sense. How many 1/6 are there in 5?

Student :

The bowl holds 5 Liters of water. If we use a scoop that holds 1/6 of a Liter, how many scoops will we need in order to

fill the entire bowl?

I created 5 boxes. Each box represents 1 Liter of water. I then divided each box into sixths to represent the size of the

scoop. My answer is the number of small boxes, which is 30. That makes sense since 6 x 5 = 30.

1 = 1/6 + 1/6 + 1/6 + 1/6 + 1/6 a whole has 6/6 so five wholes would be 6/6 + 6/6 + 6/6 + 6/6 + 6/6 =30/6

Mathematics



Grade 5 Unit 5: Volume, Area, and Shapes (~ 6 weeks)

Unit Overview: The fraction unit prepares students through the daily use of area models for an in depth discussion of area in this next unit (U5). But the unit on

area and volume also reinforces the work done in the fraction unit: questions can now be asked about how the area changes when a rectangle is scaled by a whole

or fractional scale factor. Measuring volume once again highlights the unit theme as a unit cube is chosen to represent a volume unit and used to measure the

volume of simple shapes made out of rectangular prisms. Students must attend to precision (MP 6) as they analyze their answers for accuracy and use specific

vocabulary for measurement and geometry situations.

Guiding Question: How do multiplication and addition relate to volume and area?


Component Cluster 5.NF Apply and extend previous understandings of multiplication and division to multiply and divide fractions.*

* In this module 5.NF.4b is applied to multiplying to find volume and area. 5.NF.4b certainly includes decimal fraction side lengths of sides of a rectangle (in both fraction and decimal form).


multiplication to multiply a fraction or whole number by a

fraction.

a. Interpret the product (a/b) × q as a parts of a

partition of q into b equal parts; equivalently, as

the result of a sequence of operations a × q ÷ b.

For example, use a visual fraction model to show (2/3)

× 4 = 8/3, and create a story context for this equation.

Do the same with (2/3) × (4/5) = 8/15. (In general,

(a/b) × (c/d) = ac/bd.)

b. Find the area of a rectangle with fractional side

lengths by tiling it with unit squares of the

appropriate unit fraction side lengths, and show

that the area is the same as would be found by

multiplying the side lengths. Multiply fractional

side lengths to find areas of rectangles, and

represent fraction products as rectangular areas.

Students need to develop a fundamental understanding that the multiplication of a fraction by a whole number could

be represented as repeated addition of a unit fraction (e.g., 2 x (1/4) = 1/4 + ¼

This standard extends student’s work of multiplication from earlier grades. In fourth grade, students worked with

recognizing that a fraction such as 3/5 actually could be represented as 3 pieces that are each one-fifth (3 x (1/5)).

This standard references both the multiplication of a fraction by a whole number and the multiplication of two

fractions. Visual fraction models (area models, tape diagrams, number lines) should be used and created by students

during their work with this standard.

This standard extends students’ work with area. In third grade students determine the area of rectangles and

composite rectangles. In fourth grade students continue this work. The fifth grade standard calls students to continue

the process of covering (with tiles). Grids can be used to support this work.

Component Cluster 5.MD Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

5. MD.3 Recognize volume as an attribute of solid figures

and understand concepts of volume measurement.

a. A cube with side length 1 unit, called a “unit cube,” is

said to have “one cubic unit” of volume, and can be

used to measure volume.

5. MD.3, 5.MD.4, and 5. MD.5 These standards represent the first time that students begin exploring the concept of

volume. In third grade, students begin working with area and covering spaces. The concept of volume should be

extended from area with the idea that students are covering an area (the bottom of cube) with a layer of unit cubes

and then adding layers of unit cubes on top of bottom layer. Students should have ample experiences with concrete

manipulatives before moving to pictorial representations. Students’ prior experiences with volume were restricted to

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b. A solid figure which can be packed without gaps or

overlaps using n unit cubes is said to have a volume of

n cubic units.

liquid volume. As students develop their understanding of volume, they understand that a 1-unit by 1-unit by 1-unit

cube is the standard unit for measuring volume. This cube has a length of 1 unit, a width of 1 unit and a height of 1

unit and is called a cubic unit. This cubic unit is written with an exponent of 3 (e.g., in3, m

3). Students connect this

notation to their understanding of powers of 10 in our place value system. Models of cubic inches, centimeters, cubic

feet, etc are helpful in developing an image of a cubic unit. Students’ estimate how many cubic yards would be

needed to fill the classroom or how many cubic centimeters would be needed to fill a pencil box.

The major emphasis for measurement in Grade 5 is volume. Volume not only introduces a third dimension and thus a

significant challenge to students’ spatial structuring, but also complexity in the nature of the materials measured. That

is, solid units are “packed,” such as cubes in a three-dimensional array, whereas a liquid “fills” three-dimensional

space, taking the shape of the container. The unit structure for liquid measurement may be psychologically one

dimensional for some students.

“Packing” volume is more difficult than iterating a unit to measure length and measuring area by tiling. Students

learn about a unit of volume, such as a cube with a side length of 1 unit, called a unit cube.5.MD.3 They pack cubes

(without gaps) into right rectangular prisms and count the cubes to determine the volume or build right rectangular

prisms from cubes and see the layers as they build.5.MD.4 They can use the results to compare the volume of right

rectangular prisms that have different dimensions. Such experiences enable students to extend their spatial structuring

from two to three dimensions. That is, they learn to both mentally decompose and recompose a right rectangular

prism built from cubes into layers, each of which is composed of rows and columns. That is, given the prism, they

have to be able to decompose it, understanding that it can be partitioned into layers, and each layer partitioned into

rows, and each row into cubes. They also have to be able to compose such as structure, multiplicatively, back into

higher units. That is, they eventually learn to conceptualize a layer as a unit that itself is composed of units of units—

rows, each row composed of individual cubes—and they iterate that structure. Thus, they might predict the number of

cubes that will be needed to fill a box given the net of the box.

Another complexity of volume is the connection between “packing” and “filling.” Often, for example, students will

respond that a box can be filled with 24 centimeter cubes, or build a structure of 24 cubes, and still think of the 24 as

individual, often discrete, not necessarily units of volume. They may, for example, not respond confidently and

correctly when asked to fill a graduated cylinder marked in cubic centimeters with the amount of liquid that would

fill the box. That is, they have not yet connected their ideas about filling volume with those concerning packing

volume. Students learn to move between these conceptions, e.g., using the same container, both filling (from a

graduated cylinder marked in ml or cc) and packing (with cubes that are each 1 cm3). Comparing and discussing the

volume-units and what they represent can help students learn a general, complete, and interconnected

conceptualization of volume as filling three-dimensional space.

Students then learn to determine the volumes of several right rectangular prisms, using cubic centimeters, cubic

inches, and cubic feet. With guidance, they learn to increasingly apply multiplicative reasoning to determine

volumes, looking for and making use of structure. That is, they understand that multiplying the length times the width

of a right rectangular prism can be viewed as determining how many cubes would be in each layer if the prism were

packed with or built up from unit cubes.5.MD.5a They also learn that the height of the prism tells how many layers

would fit in the prism. That is, they understand that volume is a derived attribute that, once a length unit is specified,

can be computed as the product of three length measurements or as the product of one area and one length

measurement.

5. MD.4 Measure volumes by counting unit cubes, using

cubic cm, cubic in, cubic ft, and improvised units.

5. MD.5 Relate volume to the operations of multiplication and

addition and solve real world and mathematical problems

involving volume.

a. Find the volume of a right rectangular prism with whole-

number side lengths by packing it with unit cubes, and

show that the volume is the same as would be found by

multiplying the edge lengths, equivalently by multiplying

the height by the area of the base. Represent threefold

whole-number products as volumes, e.g., to represent the

associative property of multiplication.

b. Apply the formulas V = l × w × h and V = b × h for

rectangular prisms to find volumes of right rectangular

prisms with whole-number edge lengths in the context of

solving real world and mathematical problems.

c. Recognize volume as additive. Find volumes of solid

figures composed of two non-overlapping right

rectangular prisms by adding the volumes of the non-

overlapping parts, applying this technique to solve real

world problems.

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Then, students can learn the formulas V =l x w x h and V = B x h for right rectangular prisms as efficient methods for

computing volume, maintaining the connection between these methods and their previous work with computing the

number of unit cubes that pack a right rectangular prism.5.MD.5b They use these competencies to find the volumes

of right rectangular prisms with edges whose lengths are whole numbers and solve real-world and mathematical

problems involving such prisms.

Students also recognize that volume is additive and they find the total volume of solid figures composed of two right

rectangular prisms.5.MD.5c For example, students might design a science station for the ocean floor that is composed

of several rooms that are right rectangular prisms and that meet a set criterion specifying the total volume of the

station. They draw their station and justify how their design meets the criterion.

5. MD.5a & b These standards involve finding the volume of right rectangular prisms.Students should have

experiences to describe and reason about why the formula is true. Specifically, that they are covering the bottom of a

right rectangular prism (length x width) with multiple layers (height). Therefore, the formula (length x width x

height) is an extension of the formula for the area of a rectangle.

5.MD.5c This standard calls for students to extend their work with the area of composite figures into the context of

volume. Students should be given concrete experiences of breaking apart (decomposing) 3-dimensional figures into

right rectangular prisms in order to find the volume of the entire 3-dimensional figure.

Students need multiple opportunities to measure volume by filling rectangular prisms with cubes and looking at the

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relationship between the total volume and the area of the base. They derive the volume formula (volume equals the

area of the base times the height) and explore how this idea would apply to other prisms. Students use the associative

property of multiplication and decomposition of numbers using factors to investigate rectangular prisms with a given

number of cubic units.

Component Cluster 5.G Classify two-dimensional figures into categories based on their properties

5.G.3 Understand that attributes belonging to a category of

two-dimensional figures also belong to all subcategories of

that category. For example, all rectangles have four right

angles and squares are rectangles, so all squares have

four right angles.

This standard calls for students to reason about the attributes (properties) of shapes. Student should have experiences

discussing the property of shapes and reasoning.

The notion of congruence (“same size and same shape”) may be part of classroom conversation but the concepts of

congruence and similarity do not appear until middle school.

5.G.4 Classify two-dimensional figures in a hierarchy

based on properties.

This standard builds on what was done in 4th

grade.

Figures from previous grades: polygon, rhombus/rhombi, rectangle, square, triangle, quadrilateral, pentagon,

hexagon, cube, trapezoid, half/quarter circle, circle, kite, parallelograms

A kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are beside (adjacent

to) each other.

Student should be able to reason about the attributes of shapes by examining: What are ways to classify triangles?

Why can’t trapezoids and kites be classified as parallelograms? Which quadrilaterals have opposite angles congruent

and why is this true of certain quadrilaterals?, and How many lines of symmetry does a regular polygon have?

Mathematics



Grade 5 Unit 6: Graph Points on the Coordinate Plane to Solve Problems (~ 5 weeks)

Unit Overview: Scaling is revisited in the last unit on the coordinate plane. Ever since kindergarten, students have been using bar graphs to display data and

patterns. Extensive bar-graph work has set the stage for line plots, which are both the natural extension of bar graphs and the precursor to linear functions. It is in

this final unit of K-5 that a simple line plot of a straight line is presented on a coordinate plane and students are asked about the scaling relationship between the

increase in the units of the vertical axis for 1 unit of increase in the horizontal axis. This is the first hint of slope and marks the beginning of the major theme of

middle school: ratios and proportions. Students have the opportunity to construct viable arguments and critique the reasoning of others (MP 2) as they explore

patterns based on given rules.

Guiding Question: How are numerical expressions, real-world situations, tables, and graphs all connected?


Component Cluster 5.OA Write and interpret numerical expressions.

5.OA.2 Write simple expressions that record calculations

with numbers, and interpret numerical expressions without

evaluating them.

For example, express the calculation “add 8 and 7, then

multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932

+ 921) is three times as large as 18932 + 921, without

having to calculate the indicated sum or product.

This standard refers to expressions. Expressions are a series of numbers and symbols (+, -, x, ÷) without an equals

sign. Equations result when two expressions are set equal to each other (2 + 3 = 4 + 1).

Example:

4(5 + 3) is an expression.

When we compute 4(5 + 3) we are evaluating the expression. The expression equals 32.

4(5 + 3) = 32 is an equation.

This standard calls for students to verbally describe the relationship between expressions without actually calculating

them. This standard calls for students to apply their reasoning of the four operations as well as place value while

describing the relationship between numbers. The standard does not include the use of variables, only numbers and

signs for operations.

Component Cluster 5.OA Analyze patterns and relationships.

5.OA.3 Generate two numerical patterns using two given

rules. Identify apparent relationships between corresponding

terms. Form ordered pairs consisting of corresponding terms

from the two patterns, and graph the ordered pairs on a

coordinate plane.

For example, given the rule “Add 3” and the starting number

0, and given the rule “Add 6” and the starting number 0,

generate terms in the resulting sequences, and observe that

the terms in one sequence are twice the corresponding terms

in the other sequence. Explain informally why this is so.

This standard extends the work from Fourth Grade, where students generate numerical patterns when they are given

one rule. In Fifth Grade, students are given two rules and generate two numerical patterns. The graphs that are

created should be line graphs to represent the pattern. This is a linear function which is why we get the straight lines.

The Days are the independent variable, Fish are the dependent variables, and the constant rate is what the rule

identifies in the table.

Example:

Make a chart (table) to represent the number of fish that Sam and Terri catch.

Days Sam’s Total

Number of Fish

Terri’s Total

Number of Fish

0 0 0

1 2 4

2 4 8

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Describe the pattern:

Since Terri catches 4 fish each day, and Sam catches 2 fish, the amount of Terri’s fish is always greater. Terri’s fish

is also always twice as much as Sam’s fish. Today, both Sam and Terri have no fish. They both go fishing each day.

Sam catches 2 fish each day. Terri catches 4 fish each day. How many fish do they have after each of the five days?

Make a graph of the number of fish.

Plot the points on a coordinate plane and make a line graph, and then interpret the graph.

Student:

My graph shows that Terri always has more fish than Sam. Terri’s fish increases at a higher rate since she catches 4

fish every day. Sam only catches 2 fish every day, so his number of fish increases at a smaller rate than Terri.

Important to note as well that the lines become increasingly further apart. Identify apparent relationships between

corresponding terms. Additional relationships: The two lines will never intersect; there will not be a day in which

boys have the same total of fish, explain the relationship between the number of days that has passed and the number

of fish a boy has (2n or 4n, n being the number of days).

3 6 12

4 8 16

5 10 20

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Example:

Use the rule “add 3” to write a sequence of numbers. Starting with a 0, students write 0, 3, 6, 9, 12, . . .

Use the rule “add 6” to write a sequence of numbers. Starting with 0, students write 0, 6, 12, 18, 24, . . .

After comparing these two sequences, the students notice that each term in the second sequence is twice the

corresponding terms of the first sequence. One way they justify this is by describing the patterns of the terms. Their

justification may include some mathematical notation (See example below). A student may explain that both

sequences start with zero and to generate each term of the second sequence he/she added 6, which is twice as much as

was added to produce the terms in the first sequence. Students may also use the distributive property to describe the

relationship between the two numerical patterns by reasoning that 6 + 6 + 6 = 2 (3 + 3 + 3).

0, +3

3, +3

6, +3

9, +3

12, . . .

0, +6

6, +6

12, +6

18, +6

24, . . .

Once students can describe that the second sequence of numbers is twice the corresponding terms of the first

sequence, the terms can be written in ordered pairs and then graphed on a coordinate grid. They should recognize that

each point on the graph represents two quantities in which the second quantity is twice the first quantity.

Ordered pairs

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Component Cluster 5.G Graph points on the coordinate plane to solve real-world and mathematical problems.

5.G.1 Use a pair of perpendicular number lines, called

axes, to define a coordinate system, with the intersection of

the lines (the origin) arranged to coincide with the 0 on

each line and a given point in the plane located by using an

ordered pair of numbers, called its coordinates. Understand

that the first number indicates how far to travel from the

origin in the direction of one axis, and the second number

indicates how far to travel in the direction of the second

axis, with the convention that the names of the two axes

and the coordinates correspond (e.g., x-axis and x-

coordinate, y-axis and y-coordinate).

5.G.1 and 5.G.2 These standards deal with only the first quadrant (positive numbers) in the coordinate plane.

Although students can often “locate a point,” these understandings are beyond simple skills. For example, initially,

students often fail to distinguish between two different ways of viewing the point (2, 3), say, as instructions: “right 2,

up 3”; and as the point defined by being a distance 2 from the y-axis and a distance 3 from the x-axis. In these two

descriptions the 2 is first associated with the x-axis, then with the y-axis.

5.G.2 Represent real world and mathematical problems by

graphing points in the first quadrant of the coordinate

plane, and interpret coordinate values of points in the

context of the situation.

This standard references real-world and mathematical problems, including the traveling from one point to another and

identifying the coordinates of missing points in geometric figures, such as squares, rectangles, and parallelograms.

(0, 0) (3, 6) (6, 12) (9, 18) (12, 24)

Mathematics

RP = Ratios and Proportional Reasoning NS= The Number System EE= Expressions and Equations G= Geometry SP= Statistics and Probability

Grade 6

In Grade 6, instructional time should focus on four critical areas: (1) connecting ratio and rate to whole number multiplication and division and using

concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system

of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing

understanding of statistical thinking.

(1) Students use reasoning about multiplication and division to solve ratio and rate problems about quantities. By viewing equivalent ratios

and rates as deriving from, and extending, pairs of rows (or columns) in the multiplication table, and by analyzing simple drawings that

indicate the relative size of quantities, students connect their understanding of multiplication and division with ratios and rates. Thus students

expand the scope of problems for which they can use multiplication and division to solve problems, and they connect ratios and fractions.

Students solve a wide variety of problems involving ratios and rates.

(2) Students use the meaning of fractions, the meanings of multiplication and division, and the relationship between multiplication and

division to understand and explain why the procedures for dividing fractions make sense. Students use these operations to solve problems.

Students extend their previous understandings of number and the ordering of numbers to the full system of rational numbers, which includes

negative rational numbers, and in particular negative integers. They reason about the order and absolute value of rational numbers and about

the location of points in all four quadrants of the coordinate plane.

(3) Students understand the use of variables in mathematical expressions. They write expressions and equations that correspond to given

situations, evaluate expressions, and use expressions and formulas to solve problems. Students understand that expressions in different forms

can be equivalent, and they use the properties of operations to rewrite expressions in equivalent forms. Students know that the solutions of an

equation are the values of the variables that make the equation true. Students use properties of operations and the idea of maintaining the

equality of both sides of an equation to solve simple one-step equations. Students construct and analyze tables, such as tables of quantities that

are in equivalent ratios, and they use equations (such as 3x = y) to describe relationships between quantities.

(4) Building on and reinforcing their understanding of number, students begin to develop their ability to think statistically. Students recognize

that a data distribution may not have a definite center and that different ways to measure center yield different values. The median measures

center in the sense that it is roughly the middle value. The mean measures center in the sense that it is the value that each data point would

take on if the total of the data values were redistributed equally, and also in the sense that it is a balance point. Students recognize that a

measure of variability (interquartile range or mean absolute deviation) can also be useful for summarizing data because two very different sets

of data can have the same mean and median yet be distinguished by their variability. Students learn to describe and summarize numerical data

sets, identifying clusters, peaks, gaps, and symmetry, considering the context in which the data were collected.

Students in Grade 6 also build on their work with area in elementary school by reasoning about relationships among shapes to determine area,

surface area, and volume. They find areas of right triangles, other triangles, and special quadrilaterals by decomposing these shapes,

rearranging or removing pieces, and relating the shapes to rectangles. Using these methods, students discuss, develop, and justify formulas for

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areas of triangles and parallelograms. Students find areas of polygons and surface areas of prisms and pyramids by decomposing them into

pieces whose area they can determine. They reason about right rectangular prisms with fractional side lengths to extend formulas for the

volume of a right rectangular prism to fractional side lengths. They prepare for work on scale drawings and constructions in Grade 7 by

drawing polygons in the coordinate plane.










Mathematics


Grade 6: Suggested Distribution of Units in Instructional Weeks Time Approximate

# of Weeks

Unit 1: Arithmetic Operations Including Dividing by a Fraction 14 % ~ 5 weeks

Unit 2: Rational Numbers 14 % ~ 5 weeks


Unit 3: Ratios and Unit Rates 19 % ~ 7 weeks

Unit 4: Expressions and Equations 25 % ~ 9 weeks


Unit 5: Area, Surface Area, and Volume Problems (Half Pre-/Half Post-Test) 14% ~ 5 weeks

State Test

Unit 6: Statistics (Post-Test Unit) 14% ~ 5 Weeks

Unit 1: Arithmetic Operations incl. Div. by Fractions

14%

Unit 2: Rational Numbers

14%

Unit 3: Ratios and Units Rates

19% Unit 4: Expressions

and Equations 25%

Unit 5: Area, Surface Area, and Volume Problems

14%

Unit 6: Statistics 14%

Instructional Time

Mathematics


Grade 6 Unit 1: Arithmetic Operations Including Dividing by Fractions (~5 weeks)

Unit Overview: Students expand their understanding of the number system and build their fluency in arithmetic operations in Unit 1. Students learned in Grade 5

to divide whole numbers by unit fractions and unit fractions by whole numbers. Now, they apply and extend their understanding of multiplication and division to

divide fractions by fractions. The meaning of this operation is connected to real‐world problems as students are asked to create and solve fraction division word

problems. Students continue (from fifth grade) to build fluency with adding, subtracting, multiplying, and dividing multi-digit decimal numbers using the standard

algorithms. Students must attend to precision (MP 6) and make sense of problems and persevere in solving them (MP 1) as they solve a variety of challenging

problems that utilize all of their previously learned and newly acquired arithmetic skills.

Guiding Question: What are good strategies for making sense of multi-digit whole number, fraction, and decimal problems with all four operations?


Component Cluster 6.NS Apply and extend previous understands of multiplication and division to divide fractions by fractions.

6.NS.1 Interpret and compute quotients of fractions, and

solve word problems involving division of fractions by

fractions, e.g., by using visual fraction models and

equations to represent the problem. For example, create a

story context for (2/3) ÷ (3/4) and use a visual fraction

model to show the quotient; use the relationship between

multiplication and division to explain that (2/3) ÷ (3/4) =

8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) =

ad/bc.) How much chocolate will each person get if 3

people share 1/2 lb of chocolate equally? How many 3/4-

cup servings are in 2/3 of a cup of yogurt? How wide is a

rectangular strip of land with length 3/4 mi and area 1/2

square mi

In 5th

grade students divided whole numbers by unit fractions and divided unit fractions by whole numbers. Students

continue to develop this concept by using visual models and equations to divide whole numbers by fractions and

fractions by fractions to solve word problems. Students develop an understanding of the relationship between

multiplication and division.

Component Cluster 6.NS Compute fluently with multi-digit numbers and find common factors and multiples.

6.NS.2 Fluently divide multi-digit numbers using the

standard algorithm.*

*This fluency standard begins in this module and is practiced

throughout the remainder of the year.

In the elementary grades, students were introduced to division through concrete models and various strategies to

develop an understanding of this mathematical operation (limited to 4-digit numbers divided by 2-digit numbers). In

6th

grade, students become fluent in the use of the standard division algorithm, continuing to use their understanding

of place value to describe what they are doing. Place value has been a major emphasis in the elementary standards.

This standard is the end of this progression to address students’ understanding of place value.

6.NS.3 Fluently add, subtract, multiply, and divide multi-

digit decimals using the standard algorithm for each

operation.*

*This fluency standard begins in this module and is practiced

throughout the remainder of the year.

Procedural fluency is defined by the Common Core as “skill in carrying out procedures flexibly, accurately,

efficiently and appropriately”. In 4th

and 5th

grades, students added and subtracted decimals. Multiplication and

division of decimals were introduced in 5th

grade (decimals to the hundredth place). At the elementary level, these

operations were based on concrete models or drawings and strategies based on place value, properties of operations,

and/or the relationship between addition and subtraction. In 6th

grade, students become fluent in the use of the

standard algorithms of each of these operations.

The use of estimation strategies supports student understanding of decimal operations.

Mathematics


6.NS.4 Find the greatest common factor of two whole

numbers less than or equal to 100 and the least common

multiple of two whole numbers less than or equal to 12.

Use the distributive property to express a sum of two

whole numbers 1–100 with a common factor as a multiple

of a sum of two whole numbers with no common factor.

For example, express 36 + 8 as 4 (9 + 2).

*Focus on the first half of this standard. The remainder

will be covered in Unit 4.

In elementary school, students identified primes, composites and factor pairs (4.OA.4). In 6th

grade students will find

the greatest common factor of two whole numbers less than or equal to 100. Students also understand that the

greatest common factor of two prime numbers is 1.

Mathematics


Grade 6 Unit 2: Rational Numbers (~ 5 weeks)

Unit Overview: Major themes of Unit 2 are to understand rational numbers as points on the number line and to extend previous understandings of numbers to the

system of rational numbers, which now include negative numbers. Students extend coordinate axes to represent points in the plane with negative number

coordinates and, as part of doing so, see that negative numbers can represent quantities in real‐world contexts. They use the number line to order numbers and to

understand the absolute value of a number. They begin to solve real‐world and mathematical problems by graphing points in all four quadrants, a concept that

continues throughout to be used into high school and beyond. Students can look for and make use of structure (MP 7) as they extend the number system. ** Work

in this unit must not be limited to integers. Students should be working with positive and negative fractions and decimals as well as whole numbers.

Guiding Question: How are value and absolute value similar and different when you look at rational numbers on a number line?


Component Cluster 6.NS Apply and extend previous understandings of numbers to the system of rational numbers.

6.NS.5 Understand that positive and negative numbers are

used together to describe quantities having opposite

directions or values (e.g., temperature above/below zero,

elevation above/below sea level, credits/debits,

positive/negative electric charge); use positive and

negative numbers to represent quantities in real-world

contexts, explaining the meaning of 0 in each situation.

Students use rational numbers (fractions, decimals, and integers) to represent real-world contexts and understand the

meaning of 0 in each situation.

6.NS.6 Understand a rational number as a point on the

number line. Extend number line diagrams and coordinate

axes familiar from previous grades to represent points on

the line and in the plane with negative number coordinates.

a. Recognize opposite signs of numbers as indicating

locations on opposite sides of 0 on the number line;

recognize that the opposite of the opposite of a

number is the number itself, e.g., – (–3) = 3, and that 0

is its own opposite

b. Understand signs of numbers in ordered pairs as

indicating locations in quadrants of the coordinate

plane; recognize that when two ordered pairs differ

only by signs, the locations of the points are related by

reflections across one or both axes.

c. Find and position integers and other rational numbers

on a horizontal or vertical number line diagram; find

and position pairs of integers and other rational

numbers on a coordinate plane.

In elementary school, students worked with positive fractions, decimals and whole numbers on the number line and

in quadrant 1 of the coordinate plane. In 6th

grade, students extend the number line to represent all rational numbers

and recognize that number lines may be either horizontal or vertical (i.e. thermometer) which facilitates the

movement from number lines to coordinate grids. Students recognize that a number and its opposite are equidistance

from zero (reflections about the zero). The opposite sign (–) shifts the number to the opposite side of 0. For

example, – 4 could be read as “the opposite of 4” which would be negative 4. In the example,

– (–6.4) would be read as “the opposite of the opposite of 6.4” which would be 6.4. Zero is its own opposite.

Students worked with Quadrant I in elementary school. As the x-axis and y-axis are extending to include negatives,

students begin to with the Cartesian Coordinate system. Students recognize the point where the x-axis and y-axis

intersect as the origin. Students identify the four quadrants and are able to identify the quadrant for an ordered pair

based on the signs of the coordinates. For example, students recognize that in Quadrant II, the signs of all ordered

pairs would be (–, +).

Students understand the relationship between two ordered pairs differing only by signs as reflections across one or

both axes. For example, in the ordered pairs (-2, 4) and (-2, -4), the y-coordinates differ only by signs, which

represents a reflection across the x-axis. A change is the x-coordinates from (-2, 4) to (2, 4), represents a reflection

across the y-axis. When the signs of both coordinates change, [(2, -4) changes to (-2, 4)], the ordered pair has been

reflected across both axes. Students place numbers on a number line to justify this order.

6.NS.7 Understand ordering and absolute value of rational Students use inequalities to express the relationship between two rational numbers, understanding that the value of

Mathematics


numbers.

a. Interpret statements of inequality as statements about

the relative position of two numbers on a number line.

For example, interpret –3 > –7 as a statement that –3

is located to the right of –7 on a number line

oriented from left to right.

numbers is smaller moving to the left on a number line.

Common models to represent and compare integers include number line models, temperature models and the profit-

loss model. On a number line model, the number is represented by an arrow drawn from zero to the location of the

number on the number line; the absolute value is the length of this arrow. The number line can also be viewed as a

thermometer where each point of on the number line is a specific temperature. In the profit-loss model, a positive

number corresponds to profit and the negative number corresponds to a loss. Each of these models is useful for

examining values but can also be used in later grades when students begin to perform operations on integers.

Operations with integers are not the expectation at this level.

In working with number line models, students internalize the order of the numbers; larger numbers on the right

(horizontal) or top (vertical) of the number line and smaller numbers to the left (horizontal) or bottom (vertical) of the

number line. They use the order to correctly locate integers and other rational numbers on the number line. By

placing two numbers on the same number line, they are able to write inequalities and make statements about the

relationships between two numbers.

Students recognize the distance from zero as the absolute value or magnitude of a rational number. Students need

multiple experiences to understand the relationships between numbers, absolute value, and statements about order.

b. Write, interpret, and explain statements of order for

rational numbers in real-world contexts. For example,

write –3oC > –7

oC to express the fact that –3

oC is

warmer than –7oC.

Students write statements using < or > to compare rational numbers in context. However, explanations should

reference the context rather than “less than” or “greater than”.

Although 6.NS.7a is limited to two numbers, this part of the standard expands the ordering of rational numbers to

more than two numbers in context.

c. Understand the absolute value of a rational number as

its distance from 0 on the number line; interpret

absolute as magnitude for a positive or negative

quantity in a real-world situation. For example, for an

account balance of –30 dollars, write |–30| = 30 to

describe the size of the debt in dollars.

Students understand absolute value as the distance from zero and recognize the symbols | | as representing absolute

value. In real-world contexts, the absolute value can be used to describe size or magnitude. For example, for an

ocean depth of 900 feet, write | –900| = 900 to describe the distance below sea level.

d. Distinguish comparisons of absolute value from

statements about order. For example, recognize that

an account balance less than –30 dollars represents a

debt greater than 30 dollars.

When working with positive numbers, the absolute value (distance from zero) of the number and the value of the

number is the same; therefore, ordering is not problematic. However, negative numbers have a distinction that

students need to understand. As the negative number increases (moves to the left on a number line), the value of the

number decreases. For example, –24 is less than –14 because –24 is located to the left of –14 on the number line.

However, absolute value is the distance from zero. In terms of absolute value (or distance) the absolute value of –24

is greater than the absolute value of –14. For negative numbers, as the absolute value increases, the value of the

negative number decreases.

6.NS.8 Solve real-world and mathematical problems by

graphing points in all four quadrants of the coordinate

plane. Include use of coordinates and absolute value to

find distances between points with the same first

coordinate or the same second coordinate.

Students find the distance between points when ordered pairs have the same x-coordinate (vertical) or same y-

coordinate (horizontal). Coordinates can be in two quadrants and include rational numbers.

Students graph coordinates for polygons and find missing vertices based on properties of triangles and quadrilaterals.

Mathematics


Grade 6 Unit 3: Ratios and Unit Rates (~ 7 weeks)

Unit Overview: In Unit 3, students build on their prior work in measurement and in multiplication and division as they study the concepts and language of ratios

and unit rates. They use proportional reasoning to solve problems. In particular, students solve ratio and rate using tape diagrams, tables of equivalent ratios,

double number line diagrams, and equations. They plot pairs of values generated from a ratio or rate on the first quadrant of the coordinate plane. Students will

have the opportunity to construct viable arguments and critique the reasoning of others (MP 3) as they use ratio and rate reasoning to solve real-world and

mathematical problems.

Guiding Question: What are the different ways to express relationships between numbers in real-life situations?


Component Cluster 6.RP Understand ratio concepts and use ratio reasoning to solve problems.

6.RP.1 Understand the concept of a ratio and use ratio

language to describe a ratio relationship between two

quantities. For example, “The ratio of wings to beaks in

the bird house at the zoo was 2:1, because for every 2

wings there was 1 beak.” “For every vote candidate A

received, candidate C received nearly three votes.”

A ratio is the comparison of two quantities or measures. The comparison can be part-to-whole (ratio of guppies to all

fish in an aquarium) or part-to-part (ratio of guppies to goldfish). Students should be able to identify and describe

any ratio using “For every _____ ,there are _____.”

NOTE: Ratios are often expressed in fraction notation, although ratios and fractions do not have identical meaning.

For example, ratios are often used to make “part-part” comparisons but fractions are not.

6.RP.2 Understand the concept of a unit rate a/b associated

with a ratio a:b with b = 0, and use rate language in the

context of a ratio relationship. For example, “This recipe

has a ratio of 3 cups of flour to 4 cups of sugar, so there is

¾ cup of flour for each cup of sugar.” “We paid $75 for

15 hamburgers, which is a rate of $5 per hamburger.”1

1 Expectations for unit rates in this grade are limited to

non-complex fractions.

A unit rate expresses a ratio as part-to-one, comparing a quantity in terms of one unit of another quantity. Common

unit rates are cost per item or distance per time.

Students are able to name the amount of either quantity in terms of the other quantity. Students will begin to notice

that related unit rates (i.e. miles / hour and hours / mile) are reciprocals as in the second example below. At this

level, students should use reasoning to find these unit rates instead of an algorithm or rule.

In 6th

grade, students are not expected to work with unit rates expressed as complex fractions. Both the numerator

and denominator of the original ratio will be whole numbers.

6.RP.3 Use ratio and rate reasoning to solve real-world

and mathematical problems, e.g., by reasoning about tables

of equivalent ratios, tape diagrams, double number line

diagrams, or equations.

a. Make tables of equivalent ratios relating quantities

with whole- number measurements, find missing

values in the tables, and plot the pairs of values on the

coordinate plane. Use tables to compare ratios.

Ratios and rates can be used in ratio tables and graphs to solve problems. Previously, students have used additive

reasoning in tables to solve problems. To begin the shift to proportional reasoning, students need to begin using

multiplicative reasoning. Scaling up or down with multiplication maintains the equivalence. To aid in the

development of proportional reasoning the cross-product algorithm is not expected at this level. When working with

ratio tables and graphs, whole number measurements are the expectation for this standard.

Students use tables to compare ratios. Writing equations is foundational for work in 7th

grade. The numbers in the

table can be expressed as ordered pairs (number of books, cost) and plotted on a coordinate plane.

b. Solve unit rate problems including those involving

unit pricing and constant speed. For example, if it took

7 hours to mow 4 lawns, then at that rate, how many

lawns could be mowed in 35 hours? At what rate were

lawns being mowed?

Students recognize the use of ratios, unit rate and multiplication in solving problems, which could allow for the use

of fractions and decimals.

c. Find a percent of a quantity as a rate per 100 (e.g.,

30% of a quantity means 30/100 times the quantity);

solve problems involving finding the whole, given a

This is the students’ first introduction to percents. Percentages are a rate per 100. Models, such as percent bars or 10

x 10 grids should be used to model percents.

Mathematics


part and the percent.

Students use ratios to identify percents. Students use percentages to find the part when given the percent, by

recognizing that the whole is being divided into 100 parts and then taking a part of them (the percent). Students also

determine the whole amount, given a part and the percent.

d. Use ratio reasoning to convert measurement units;

manipulate and transform units appropriately when

multiplying or dividing quantities.

A ratio can be used to compare measures of two different types, such as inches per foot, milliliters per liter and

centimeters per inch. Students recognize that a conversion factor is a fraction equal to 1 since the numerator and

denominator describe the same quantity. For example, 12 inches is a conversion factor since the numerator and

1 foot

denominator equal the same amount. Since the ratio is equivalent to 1, the identity property of multiplication allows

an amount to be multiplied by the ratio. Also, the value of the ratio can also be expressed as

1 foot allowing for the conversion ratios to be expressed in a format so that units will “cancel”.

12 inches

Students use ratios as conversion factors and the identity property for multiplication to convert ratio units. Note:

Conversion factors will be given. Conversions can occur both between and across the metric and English systems.

Estimates are not expected.

Mathematics


Grade 6 Unit 4: Expressions and Equations (~ 9 weeks)

Unit Overview: With their sense of number expanded to include negative numbers, in Unit 4 students begin formal study of algebraic expressions and equations.

Students learn equivalent expressions by continuously relating algebraic expressions back to arithmetic and the properties of arithmetic (commutative, associative,

distributive). They write, interpret, and use expressions and equations as they reason about and solve one‐variable equations and inequalities and analyze

quantitative relationships between two variables. As students do this work of continually relating the algebra back to their previous arithmetic understanding, they

move between reasoning abstractly and quantitatively (MP 2) in order to construct viable arguments and critique the reasoning of others (MP 3).

Guiding Question: How can two different expression be equivalent?


Component Cluster 6.EE Apply and extend previous understanding of arithmetic to algebraic expressions.

6.EE.1 Write and evaluate numerical expressions

involving whole-number exponents.

1 Students demonstrate the meaning of exponents to write and evaluate numerical expressions with whole number

exponents. The base can be a whole number, positive decimal or a positive fraction (i.e. 5 can be written

• •

• • which has the same value as ). Students recognize that an expression with a variable represents the

same mathematics (ie. x5 can be written as x • x • x • x • x) and write algebraic expressions from verbal expressions.

Order of operations is introduced throughout elementary grades, including the use of grouping symbols, ( ), { }, and [

] in 5th

grade. Order of operations with exponents is the focus in 6th

grade.

6.EE.2 Write, read, and evaluate expressions in which

letters stand for numbers.

a. Write expressions that record operations with numbers

and with letters standing for numbers. For example,

express the calculation “Subtract y from 5” as 5 – y.

Students write expressions from verbal descriptions using letters and numbers, understanding order is important in

writing subtraction and division problems. Students understand that the expression “5 times any number, n” could be

represented with 5n and that a number and letter written together means to multiply. All rational numbers may be

used in writing expressions when operations are not expected. Students use appropriate mathematical language to

write verbal expressions from algebraic expressions. It is important for students to read algebraic expressions in a

manner that reinforces that the variable represents a number.

Example Set 1:

Students read algebraic expressions:

r + 21 as “some number plus 21” as well as “r plus 21”

n 6 as “some number times 6” as well as “n times 6”

and s ÷ 6 as “as some number divided by 6” as well as “s divided by 6”

Example Set 2:

Students write algebraic expressions:

7 less than 3 times a number

Solution: 3x – 7

3 times the sum of a number and 5

Solution: 3 (x + 5)

7 less than the product of 2 and a number

Solution: 2x – 7

Twice the difference between a number and 5

b. Identify parts of an expression using mathematical

terms (sum, term, product, factor, quotient,

coefficient); view one or more parts of an expression

as a single entity. For example, describe the

expression 2 (8 + 7) as a product of two factors; view

(8 + 7) as both a single entity and a sum of two terms.

1

2

1

2

1

2

1

2

1

2

1

2

1

32

Mathematics


Solution: 2(z – 5)

The quotient of the sum of x plus 4 and 2

Solution: x + 4

2

Students can describe expressions such as 3 (2 + 6) as the product of two factors: 3 and (2 + 6). The quantity

(2 + 6) is viewed as one factor consisting of two terms.

Terms are the parts of a sum. When the term is an explicit number, it is called a constant. When the term is a product

of a number and a variable, the number is called the coefficient of the variable.

Students should identify the parts of an algebraic expression including variables, coefficients, constants, and the

names of operations (sum, difference, product, and quotient). Variables are letters that represent numbers. There are

various possibilities for the number they can represent.

Consider the following expression:

x2 + 5y + 3x + 6

The variables are x and y.

There are 4 terms, x2, 5y, 3x, and 6.

There are 3 variable terms, x2, 5y, 3x. They have coefficients of 1, 5, and 3 respectively. The coefficient of x

2 is 1,

since x2 = 1x

2. The term 5y represent 5y’s or 5 y.

There is one constant term, 6.

The expression represents a sum of all four terms.

c. Evaluate expressions at specific values of their

variables. Include expressions that arise from formulas

used in real-world problems. Perform arithmetic

operations, including those involving whole- number

exponents, in the conventional order when there are

no parentheses to specify a particular order (Order of

Operations). For example, use the formulas V = s3

and A = 6 s2 to find the volume and surface area of a

cube with sides of length s = ½.*

*6.EE.2c is also taught in Unit5 in the context of geometry.

Students evaluate algebraic expressions, using order of operations as needed. Problems require students to understand

that multiplication is understood when numbers and variables are written together and to use the order of operations

to evaluate.

In 5th

grade students worked with the grouping symbols ( ), [ ], and { }. Students understand that the fraction bar

can also serve as a grouping symbol (treats numerator operations as one group and denominator operations as another

group) as well as a division symbol.

Given a context and the formula arising from the context, students could write an expression and then evaluate for

any number.

6.EE.3 Apply the properties of operations to generate

equivalent expressions. For example, apply the

distributive property to the expression 3 (2 + x) to produce

the equivalent expression 6 + 3x; apply the distributive

property to the expression 24x + 18y to produce the

equivalent expression 6 (4x + 3y); apply properties of

operations to y + y + y to produce the equivalent

expression 3y.

Students use the distributive property to write equivalent expressions. Using their understanding of area models from

elementary students illustrate the distributive property with variables.

Properties are introduced throughout elementary grades (3.OA.5); however, there has not been an emphasis on

recognizing and naming the property. In 6th

grade students are able to use the properties and identify by name as

used when justifying solution methods.

When given an expression representing area, students need to find the factors.

6.EE.4 Identify when two expressions are equivalent (i.e.,

when the two expressions name the same number

regardless of which value is substituted into them). For

example, the expressions y + y + y and 3y are equivalent

Students demonstrate an understanding of like terms as quantities being added or subtracted with the same variables

and exponents. For example, 3x + 4x are like terms and can be combined as 7x; however, 3x + 4x2 are not like terms

since the exponents with the x are not the same.

This concept can be illustrated by substituting in a value for x. For example, 9x – 3x = 6x not 6. Choosing a value for

Mathematics


because they name the same number regardless of which

number y stands for.

x, such as 2, can prove non-equivalence.

Students can also generate equivalent expressions using the associative, commutative, and distributive properties.

They can prove that the expressions are equivalent by simplifying each expression into the same form.

Component Cluster 6.EE Reason about and solve one-variable equations and inequalities.* *Except for 6.EE.8, this cluster is also taught in Unit 5 in the context of geometry.

6.EE.5 Understand solving an equation or inequality as a

process of answering a question: which values from a

specified set, if any, make the equation or inequality true?

Use substitution to determine whether a given number in a

specified set makes an equation or inequality true.

In elementary grades, students explored the concept of equality. In 6th

grade, students explore equations as

expressions being set equal to a specific value. The solution is the value of the variable that will make the equation

or inequality true. Students use various processes to identify the value(s) that when substituted for the variable will

make the equation true: Reasoning, Use knowledge of fact families to write related equations, Use knowledge of

inverse operations, Scale model, and Bar model.

6.EE.6 Use variables to represent numbers and write

expressions when solving a real-world or mathematical

problem; understand that a variable can represent an

unknown number, or, depending on the purpose at hand,

any number in a specified set.

Students write expressions to represent various real-world situations. Given a contextual situation, students define

variables and write an expression to represent the situation.

No solving is expected with this standard; however, 6.EE.2c does address the evaluating of the expressions.

Students understand the inverse relationships that can exist between two variables. For example, if Sally has 3 times

as many bracelets as Jane, then Jane has the amount of Sally. If S represents the number of bracelets Sally has, the

s or represents the amount Jane has.

Connecting writing expressions with story problems and/or drawing pictures will give students a context for this

work. It is important for students to read algebraic expressions in a manner that reinforces that the variable represents

a number.

6.EE.7 Solve real-world and mathematical problems by

writing and solving equations of the form x + p = q and px

= q for cases in which p, q and x are all nonnegative

rational numbers.

Students have used algebraic expressions to generate answers given values for the variable. This understanding is

now expanded to equations where the value of the variable is unknown but the outcome is known. For example, in

the expression, x + 4, any value can be substituted for the x to generate a numerical answer; however, in the equation

x + 4 = 6, there is only one value that can be used to get a 6. Problems should be in context when possible and use

only one variable.

Students write equations from real-world problems and then use inverse operations to solve one-step equations based

on real world situations. Equations may include fractions and decimals with non-negative solutions. Students

recognize that dividing by 6 and multiplying by produces the same result. For example, = 9 and

x = 9 will produce the same result. Beginning experiences in solving equations require students to understand the

meaning of the equation and the solution in the context of the problem.

6.EE.8 Write an inequality of the form x > c or x < c to

represent a constraint or condition in a real-world or

Many real-world situations are represented by inequalities. Students write inequalities to represent real world and

mathematical situations. Students use the number line to represent inequalities from various contextual and

1

3

1

3

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3

1

6

x

6

1

6

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mathematical problem. Recognize that inequalities of the

form x > c or x < c have infinitely many solutions;

represent solutions of such inequalities on number line

diagrams.

mathematical situations.

A number line diagram is drawn with an open circle when an inequality contains a < or > symbol to show solutions

that are less than or greater than the number but not equal to the number. The circle is shaded, as in the example

above, when the number is to be included. Students recognize that possible values can include fractions and

decimals, which are represented on the number line by shading. Shading is extended through the arrow on a number

line to show that an inequality has an infinite number of solutions.

Component Cluster 6.EE Represent and analyze quantitative relationships between dependent and independent variables.

6.EE.9 Use variables to represent two quantities in a real-

world problem that change in relationship to one another;

write an equation to express one quantity, thought of as the

dependent variable, in terms of the other quantity, thought

of as the independent variable. Analyze the relationship

between the dependent and independent variables using

graphs and tables, and relate these to the equation. For

example, in a problem involving motion at constant speed,

list and graph ordered pairs of distances and times, and

write the equation d = 65t to represent the relationship

between distance and time.

The purpose of this standard is for students to understand the relationship between two variables, which begins with

the distinction between dependent and independent variables. The independent variable is the variable that can be

changed; the dependent variable is the variable that is affected by the change in the independent variable. Students

recognize that the independent variable is graphed on the x-axis; the dependent variable is graphed on the y-axis.

Students recognize that not all data should be graphed with a line. Data that is discrete would be graphed with

coordinates only. Discrete data is data that would not be represented with fractional parts such as people, tents,

records, etc. For example, a graph illustrating the cost per person would be graphed with points since part of a person

would not be considered. A line is drawn when both variables could be represented with fractional parts.

Students are expected to recognize and explain the impact on the dependent variable when the independent variable

changes (As the x variable increases, how does the y variable change?) Relationships should be proportional with

the line passing through the origin. Additionally, students should be able to write an equation from a word problem

and understand how the coefficient of the dependent variable is related to the graph and /or a table of values.

Students can use many forms to represent relationships between quantities. Multiple representations include

describing the relationship using language, a table, an equation, or a graph. Translating between multiple

representations helps students understand that each form represents the same relationship and provides a different

perspective.

Component Cluster 6.NS Compute fluently with multi-digit numbers and find common factors and multiples.*

6.NS.4 Find the greatest common factor of two whole

numbers less than or equal to 100 and the least common

multiple of two whole numbers less than or equal to 12.

Use the distributive property to express a sum of two

whole numbers 1–100 with a common factor as a multiple

of a sum of two whole numbers with no common factor.

For example, express 36 + 8 as 4 (9 + 2).

In elementary school, students identified primes, composites and factor pairs (4.OA.4). In 6th

grade students will find

the greatest common factor of two whole numbers less than or equal to 100. Students also understand that the

greatest common factor of two prime numbers is 1.

Given various pairs of addends using whole numbers from 1-100, students should be able to identify if the two

numbers have a common factor. If they do, they identify the common factor and use the distributive property to

rewrite the expression. They prove that they are correct by simplifying both expressions.

Mathematics


Grade 6 Unit 5: Area, Surface Area, and Volume Problems (~ 5 weeks)

Unit Overview: Unit 5 is an opportunity to practice the material learned in Unit 4 in the context of geometry; students apply their newly acquired capabilities with

expressions and equations to solve for unknowns in area, surface area, and volume problems. They find the area of triangles and other two-dimensional figures and

use the formulas to find the volumes of right rectangular prisms with fractional edge lengths. Students use negative numbers in coordinates as they draw lines and

polygons in the coordinate plane. They also find the lengths of sides of figures, joining points with the same first coordinate or the same second coordinate and

apply these techniques to solve real‐world and mathematical problems. Students will need to model with mathematics (MP 4) and attend to precision (MP 6) as

they work to solve the real-life and mathematical situations in this unit.

Guiding Question: How and when do you use the real-life context of a problem when you are solving it?


Component Cluster 6.EE* Apply and extend previous understanding of arithmetic to algebraic expressions.

*The EE topics taught in Unit 4 are practiced in this unit in the context of geometry. 6.EE.2 Write, read, and evaluate expressions in which

letters stand for numbers.

c. Evaluate expressions at specific values of their

variables. Include expressions that arise from formulas

used in real-world problems. Perform arithmetic

operations, including those involving whole- number

exponents, in the conventional order when there are no

parentheses to specify a particular order (Order of

Operations). For example, use the formulas V = s3 and A

= 6 s2 to find the volume and surface area of a cube with

sides of length s = ½.

Students evaluate algebraic expressions, using order of operations as needed. Problems require students to understand

that multiplication is understood when numbers and variables are written together and to use the order of operations

to evaluate.

In 5th

grade students worked with the grouping symbols ( ), [ ], and { }. Students understand that the fraction bar

can also serve as a grouping symbol (treats numerator operations as one group and denominator operations as another

group) as well as a division symbol.

Given a context and the formula arising from the context, students could write an expression and then evaluate for

any number.

Component Cluster 6.EE Reason about and solve one-variable equations and inequalities.

6.EE.5 Understand solving an equation or inequality as a

process of answering a question: which values from a

specified set, if any, make the equation or inequality true?

Use substitution to determine whether a given number in a

specified set makes an equation or inequality true.

In elementary grades, students explored the concept of equality. In 6th

grade, students explore equations as

expressions being set equal to a specific value. The solution is the value of the variable that will make the equation

or inequality true. Students use various processes to identify the value(s) that when substituted for the variable will

make the equation true: Reasoning, Use knowledge of fact families to write related equations, Use knowledge of

inverse operations, Scale model, and Bar model.

6.EE.6 Use variables to represent numbers and write

expressions when solving a real-world or mathematical

problem; understand that a variable can represent an

unknown number, or, depending on the purpose at hand,

any number in a specified set.

Students write expressions to represent various real-world situations. Given a contextual situation, students define

variables and write an expression to represent the situation.

No solving is expected with this standard; however, 6.EE.2c does address the evaluating of the expressions.

Students understand the inverse relationships that can exist between two variables. For example, if Sally has 3 times

as many bracelets as Jane, then Jane has the amount of Sally. If S represents the number of bracelets Sally has, the

s or represents the amount Jane has.

1

3

1

3

s

3

Mathematics


Connecting writing expressions with story problems and/or drawing pictures will give students a context for this

work. It is important for students to read algebraic expressions in a manner that reinforces that the variable represents

a number.

6.EE.7 Solve real-world and mathematical problems by

writing and solving equations of the form x + p = q and px

= q for cases in which p, q and x are all nonnegative

rational numbers.

Students have used algebraic expressions to generate answers given values for the variable. This understanding is

now expanded to equations where the value of the variable is unknown but the outcome is known. For example, in

the expression, x + 4, any value can be substituted for the x to generate a numerical answer; however, in the equation

x + 4 = 6, there is only one value that can be used to get a 6. Problems should be in context when possible and use

only one variable.

Students write equations from real-world problems and then use inverse operations to solve one-step equations based

on real world situations. Equations may include fractions and decimals with non-negative solutions. Students

recognize that dividing by 6 and multiplying by produces the same result. For example, = 9 and

x = 9 will produce the same result.

Beginning experiences in solving equations require students to understand the meaning of the equation and the

solution in the context of the problem.

Component Cluster 6.G Solve real-world and mathematical problems involving area, surface area, and volume.

6.G.1 Find the area of right triangles, other triangles,

special quadrilaterals, and polygons by composing into

rectangles or decomposing into triangles and other shapes;

apply these techniques in the context of solving real-world

and mathematical problems.

Students continue to understand that area is the number of squares needed to cover a plane figure. Students should

know the formulas for rectangles and triangles. “Knowing the formula” does not mean memorization of the formula.

To “know” means to have an understanding of why the formula works and how the formula relates to the measure

(area) and the figure. This understanding should be for all students.

Finding the area of triangles is introduced in relationship to the area of rectangles – a rectangle can be decomposed

into two congruent triangles. Therefore, the area of the triangle is ½ the area of the rectangle. The area of a rectangle

can be found by multiplying base x height; therefore, the area of the triangle is ½ bh or (b x h)/2.

The following site helps students to discover the area formula of triangles.

http://illuminations.nctm.org/LessonDetail.aspx?ID=L577

Students decompose shapes into rectangles and triangles to determine the area. For example, a trapezoid can be

decomposed into triangles and rectangles (see figures below). Using the trapezoid’s dimensions, the area of the

individual triangle(s) and rectangle can be found and then added together. Special quadrilaterals include rectangles,

squares, parallelograms, trapezoids, rhombi, and kites.

1

6

x

6

1

6

http://illuminations.nctm.org/LessonDetail.aspx?ID=L577

Mathematics


Note: Students recognize the marks on the isosceles trapezoid indicating the two sides have equal measure.

6.G.2 Find the volume of a right rectangular prism with

fractional edge lengths by packing it with unit cubes of the

appropriate unit fraction edge lengths, and show that the

volume is the same as would be found by multiplying the

edge lengths of the prism. Apply the formulas V = l w h

and V = b h to find volumes of right rectangular prisms

with fractional edge lengths in the context of solving real-

world and mathematical problems.

Previously students calculated the volume of right rectangular prisms (boxes) using whole number edges. The use of

models was emphasized as students worked to derive the formula V = Bh (5.MD.3, 5.MD.4, 5.MD.5)

The unit cube was 1 x 1 x 1.

In 6th

grade the unit cube will have fractional edge lengths. (ie. ½ • ½ • ½ ) Students find the volume of the right

rectangular prism with these unit cubes.

Students need multiple opportunities to measure volume by filling rectangular prisms with blocks and looking at the

relationship between the total volume and the area of the base. Through these experiences, students derive the

volume formula (volume equals the area of the base times the height). Students can explore the connection between

filling a box with unit cubes and the volume formula using interactive applets such as the Cubes Tool on NCTM’s

Illuminations (http://illuminations.nctm.org/ActivityDetail.aspx?ID=6).

In addition to filling boxes, students can draw diagrams to represent fractional side lengths, connecting with

multiplication of fractions. This process is similar to composing and decomposing two-dimensional shapes.

6.G.3 Draw polygons in the coordinate plane given

coordinates for the vertices; use coordinates to find the

length of a side joining points with the same first

coordinate or the same second coordinate. Apply these

techniques in the context of solving real-world and


Students are given the coordinates of polygons to draw in the coordinate plane. If both x-coordinates are the same (2,

-1) and (2, 4), then students recognize that a vertical line has been created and the distance between these coordinates

is the distance between -1 and 4, or 5. If both the y-coordinates are the same (-5, 4) and (2, 4), then students

recognize that a horizontal line has been created and the distance between these coordinates is the distance between -

5 and 2, or 7. Using this understanding, student solve real-world and mathematical problems, including finding the

area and perimeter of geometric figures drawn on a coordinate plane.

This standard can be taught in conjunction with 6.G.1 to help students develop the formula for the triangle by using

the squares of the coordinate grid. Given a triangle, students can make the corresponding square or rectangle and

realize the triangle is ½.

Students progress from counting the squares to making a rectangle and recognizing the triangle as ½ to the

development of the formula for the area of a triangle.

6.G.4 Represent three-dimensional figures using nets made

up of rectangles and triangles, and use the nets to find the

surface area of these figures. Apply these techniques in the

context of solving real-world and mathematical problems.

A net is a two-dimensional representation of a three-dimensional figure. Students represent three-dimensional

figures whose nets are composed of rectangles and triangles. Students recognize that parallel lines on a net are

congruent. Using the dimensions of the individual faces, students calculate the area of each rectangle and/or triangle

and add these sums together to find the surface area of the figure.

Right trapezoid Isosceles trapezoid

http://illuminations.nctm.org/ActivityDetail.aspx?ID=6

Mathematics


Students construct models and nets of three-dimensional figures, describing them by the number of edges, vertices,

and faces. Solids include rectangular and triangular prisms. Students are expected to use the net to calculate the

surface area.

Students can create nets of 3D figures with specified dimensions using the Dynamic Paper Tool on NCTM’s

Illuminations (http://illuminations.nctm.org/ActivityDetail.aspx?ID=205).

Students also describe the types of faces needed to create a three-dimensional figure. Students make and test

conjectures by determining what is needed to create a specific three-dimensional figure.


Mathematics


Grade 6 Unit 6: Introduction to Statistics (~ 5 weeks)

Unit Overview: In Unit 6, students develop an understanding of statistical variability and apply that understanding as they summarize, describe, and display

distributions. In particular, careful attention is given to measures of center and variability. Students will need to use appropriate tools strategically (MP 5) as they

select appropriate measures of center and variability.

Guiding Question: What tools do best help you to analyze data?


Component Cluster 6.SP Develop understanding of statistical variability.

6.SP.1 Recognize a statistical question as one that

anticipates variability in the data related to the question

and accounts for it in the answers. For example, “How old

am I?” is not a statistical question, but “How old are the

students in my school?” is a statistical question because

one anticipates variability in students’ ages.

Statistics are numerical data relating to a group of individuals; statistics is also the name for the science of collecting,

analyzing and interpreting such data. A statistical question anticipates an answer that varies from one individual to

the next and is written to account for the variability in the data. Data are the numbers produced in response to a

statistical question. Data are frequently collected from surveys or other sources (i.e. documents).

Students differentiate between statistical questions and those that are not. A statistical question is one that collects

information that addresses differences in a population. The question is framed so that the responses will allow for the

differences. For example, the question, “How tall am I?” is not a statistical question because there is only one

response; however, the question, “How tall are the students in my class?” is a statistical question since the responses

anticipates variability by providing a variety of possible anticipated responses that have numerical answers.

Questions can result in a narrow or wide range of numerical values.

6.SP.2 Understand that a set of data collected to answer a

statistical question has a distribution, which can be

described by its center, spread, and overall shape.

The distribution is the arrangement of the values of a data set. Distribution can be described using center (median or

mean), and spread. Data collected can be represented on graphs, which will show the shape of the distribution of the

data. Students examine the distribution of a data set and discuss the center, spread and overall shape with dot plots,

histograms and box plots.

NOTE: Mode as a measure of center and range as a measure of variability are not addressed in the CCSS and as such

are not a focus of instruction. These concepts can be introduced during instruction as needed.

6.SP.3 Recognize that a measure of center for a numerical

data set summarizes all of its values with a single number,

while a measure of variation describes how its values vary

with a single number.

Data sets contain many numerical values that can be summarized by one number such as a measure of center. The

measure of center gives a numerical value to represent the center of the data (ie. midpoint of an ordered list or the

balancing point). Another characteristic of a data set is the variability (or spread) of the values. Measures of

variability are used to describe this characteristic.

Component Cluster 6.SP Summarize and describe distributions.

Mathematics


6.SP.4 Display numerical data in plots on a number line,

including dot plots, histograms, and box plots.

Students display data graphically using number lines. Dot plots, histograms and box plots are three graphs to be

used. Students are expected to determine the appropriate graph as well as read data from graphs generated by others.

Dot plots are simple plots on a number line where each dot represents a piece of data in the data set. Dot plots are

suitable for small to moderate size data sets and are useful for highlighting the distribution of the data including

clusters, gaps, and outliers.

A histogram shows the distribution of continuous data using intervals on the number line. The height of each bar

represents the number of data values in that interval. In most real data sets, there is a large amount of data and many

numbers will be unique. A graph (such as a dot plot) that shows how many ones, how many twos, etc. would not be

meaningful; however, a histogram can be used. Students group the data into convenient ranges and use these intervals

to generate a frequency table and histogram. Note that changing the size of the bin changes the appearance of the

graph and the conclusions may vary from it.

A box plot shows the distribution of values in a data set by dividing the set into quartiles. It can be graphed either

vertically or horizontally. The box plot is constructed from the five-number summary (minimum, lower quartile,

median, upper quartile, and maximum). These values give a summary of the shape of a distribution. Students

understand that the size of the box or whiskers represents the middle 50% of the data.

Students can use applets to create data displays. Examples of applets include the Box Plot Tool and Histogram Tool

on NCTM’s Illuminations.

Box Plot Tool - http://illuminations.nctm.org/ActivityDetail.aspx?ID=77

Histogram Tool -- http://illuminations.nctm.org/ActivityDetail.aspx?ID=78

6.SP.5 Summarize numerical data sets in relation to their

context, such as by:

a. Reporting the number of observations.

b. Describing the nature of the attribute under

investigation, including how it was measured and its

units of measurement.

c. Giving quantitative measures of center (median and/or

mean) and variability (interquartile range and/or mean

absolute deviation), as well as describing any overall

pattern and any striking deviations from the overall

pattern with reference to the context in which the data

were gathered.

d. Relating the choice of measures of center and

variability to the shape of the data distribution and the

context in which the data were gathered.

Students summarize numerical data by providing background information about the attribute being measured,

methods and unit of measurement, the context of data collection activities (addressing random sampling), the number

of observations, and summary statistics. Summary statistics include quantitative measures of center (median and

median) and variability (interquartile range and mean absolute deviation) including extreme values (minimum and

maximum), mean, median, mode, range, and quartiles.

Students record the number of observations. Using histograms, students determine the number of values between

specified intervals. Given a box plot and the total number of data values, students identify the number of data points

that are represented by the box. Reporting of the number of observations must consider the attribute of the data sets,

including units (when applicable).

Measures of Center

Given a set of data values, students summarize the measure of center with the median or mean. The median is the

value in the middle of a ordered list of data. This value means that 50% of the data is greater than or equal to it and

that 50% of the data is less than or equal to it.

The mean is the arithmetic average; the sum of the values in a data set divided by how many values there are in the

data set. The mean measures center in the sense that it is the value that each data point would take on if the total of

the data values were redistributed equally, and also in the sense that it is a balance point.

Students develop these understandings of what the mean represents by redistributing data sets to be level or fair



Mathematics


(equal distribution) and by observing that the total distance of the data values above the mean is equal to the total

distance of the data values below the mean (balancing point).

Students use the concept of mean to solve problems. Given a data set represented in a frequency table, students

calculate the mean. Students find a missing value in a data set to produce a specific average.

Measures of Variability

Measures of variability/variation can be described using the interquartile range or the Mean Absolute Deviation. The

interquartile range (IQR) describes the variability between the middle 50% of a data set. It is found by subtracting

the lower quartile from the upper quartile. It represents the length of the box in a box plot and is not affected by

outliers.

Students find the IQR from a data set by finding the upper and lower quartiles and taking the difference or from

reading a box plot.

Mean Absolute Deviation (MAD) describes the variability of the data set by determining the absolute deviation (the

distance) of each data piece from the mean and then finding the average of these deviations.

Both the interquartile range and the Mean Absolute Deviation are represented by a single numerical value. Higher

values represent a greater variability in the data.

Students understand how the measures of center and measures of variability are represented by graphical displays.

Students describe the context of the data, using the shape of the data and are able to use this information to determine

an appropriate measure of center and measure of variability. The measure of center that a student chooses to describe

a data set will depend upon the shape of the data distribution and context of data collection. The mode is the value in

the data set that occurs most frequently. The mode is the least frequently used as a measure of center because data

sets may not have a mode, may have more than one mode, or the mode may not be descriptive of the data set. The

mean is a very common measure of center computed by adding all the numbers in the set and dividing by the number

of values. The mean can be affected greatly by a few data points that are very low or very high. In this case, the

median or middle value of the data set might be more descriptive. In data sets that are symmetrically distributed, the

mean and median will be very close to the same. In data sets that are skewed, the mean and median will be different,

with the median frequently providing a better overall description of the data set.

Mathematics


Grade 7 In Grade 7, instructional time should focus on four critical areas: (1) developing understanding of and applying proportional relationships; (2) developing

understanding of operations with rational numbers and working with expressions and linear equations; (3) solving problems involving scale drawings and

informal geometric constructions, and working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume; and (4)

drawing inferences about populations based on samples.

(1) Students extend their understanding of ratios and develop understanding of proportionality to solve single- and multi-step problems. Students use their

understanding of ratios and proportionality to solve a wide variety of percent problems, including those involving discounts, interest, taxes, tips, and

percent increase or decrease. Students solve problems about scale drawings by relating corresponding lengths between the objects or by using the fact that

relationships of lengths within an object are preserved in similar objects. Students graph proportional relationships and understand the unit rate informally

as a measure of the steepness of the related line, called the slope. They distinguish proportional relationships from other relationships.

(2) Students develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal representation), and

percents as different representations of rational numbers. Students extend addition, subtraction, multiplication, and division to all rational numbers,

maintaining the properties of operations and the relationships between addition and subtraction, and multiplication and division. By applying these

properties, and by viewing negative numbers in terms of everyday contexts (e.g., amounts owed or temperatures below zero), students explain and

interpret the rules for adding, subtracting, multiplying, and dividing with negative numbers. They use the arithmetic of rational numbers as they formulate

expressions and equations in one variable and use these equations to solve problems.

(3) Students continue their work with area from Grade 6, solving problems involving the area and circumference of a circle and surface area of three-

dimensional objects. In preparation for work on congruence and similarity in Grade 8 they reason about relationships among two-dimensional figures

using scale drawings and informal geometric constructions, and they gain familiarity with the relationships between angles formed by intersecting lines.

Students work with three-dimensional figures, relating them to two-dimensional figures by examining cross-sections. They solve real-world and

mathematical problems involving area, surface area, and volume of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons,

cubes and right prisms.

(4) Students build on their previous work with single data distributions to compare two data distributions and address questions about differences between

populations. They begin informal work with random sampling to generate data sets and learn about the importance of representative samples for drawing

inferences.










Mathematics


Grade 7: Suggested Distribution of Units in Instructional Days Time Approximate

# of Weeks

Unit 1: Rational Numbers 17% ~ 6 weeks

Unit 2: Expressions and Equations 19% ~ 7 weeks


Unit 3: Ratios and Proportional Relationships 17% ~ 6 weeks

Unit 4: Percent and Proportional Relationships 14% ~ 5 weeks


Unit 5: Statistics and Probability (Finish post-test) 14% ~ 5 weeks

State Test

Unit 6: Geometry 19% ~7 Weeks

Unit 1: Ratios and Proportional Relationships

17%


17%

Unit 3: Expressions and Equations

19%

Unit 4: Percent and Proportional Relationships

14%

Unit 5: Statistics and Probability

14%

Unit 6: Geometry 19%

Instructional Time

Mathematics


Grade 7 Unit 1: Rational Numbers (~ 9 weeks)

Unit Overview: Students continue to build an understanding of the number line in Unit 1 from their work in Grade 6. They learn to add, subtract, multiply, and

divide rational numbers. Unit 1 includes rational numbers as they appear in expressions and equations—work that is continued in Unit 2. This unit provides

opportunities for students to look for and make use of structure (MP7) as they apply their previous understandings of addition and subtraction to rational numbers.

Guiding Question: How is addition and subtraction with only positive numbers from elementary school both different as well as similar to addition and subtraction with rational

numbers?


Component Cluster 7.NS Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide

rational numbers.

7.NS.1 Apply and extend previous understandings of

addition and subtraction to add and subtract rational

numbers; represent addition and subtraction on a

horizontal or vertical number line diagram.

a. Describe situations in which opposite quantities

combine to make 0. For example, a hydrogen atom has 0

charge because its two constituents are oppositely charged.

b. Understand p + q as the number located a distance |q|

from p, in the positive or negative direction depending on

whether q is positive or negative. Show that a number and

its opposite have a sum of 0 (are additive inverses).

Interpret sums of rational numbers by describing real-

world contexts.

c. Understand subtraction of rational numbers as adding

the additive inverse, p – q = p + (–q). Show that the

distance between two rational numbers on the

number line is the absolute value of their difference, and

apply this principle in real world contexts.

d. Apply properties of operations as strategies to add and

subtract rational numbers.

Students add and subtract rational numbers. Visual representations may be helpful as students begin this work; they

become less necessary as students become more fluent with these operations. The expectation of the CCSS is to build

on student understanding of number lines developed in 6th grade.

In 6th grade, students found the distance of horizontal and vertical segments on the coordinate plane. In 7th

grade,

students build on this understanding to recognize subtraction is finding the distance between two numbers on a

number line.


multiplication and division and of fractions to multiply and

divide rational numbers.

a. Understand that multiplication is extended from

fractions to rational numbers by requiring that operations

continue to satisfy the properties of operations, particularly

the distributive property, leading to products such as (–

1)(–1) = 1 and the rules for multiplying signed numbers.

Students understand that multiplication and division of integers is an extension of multiplication and division of

whole numbers. Students recognize that when division of rational numbers is represented with a fraction bar, each

number can have a negative sign.

Using long division from elementary school, students understand the difference between terminating and repeating

decimals. This understanding is foundational for the work with rational and irrational numbers in

8th grade. Identify which fractions will terminate (the denominator of the fraction in reduced form only has factors

of 2 and/or 5)

Mathematics


Interpret products of rational numbers by describing real-

world contexts.

b. Understand that integers can be divided, provided that

the divisor is not zero, and every quotient of integers (with

non-zero divisor) is a rational number. If p and q are

integers, then –(p/q) = (–p)/q = p/(–

q). Interpret quotients of rational numbers by describing

real-world contexts.

c. Apply properties of operations as strategies to multiply

and divide rational numbers.

d. Convert a rational number to a decimal using long

division; know that the decimal form of a rational number

terminates in 0s or eventually repeats.

7.NS.3 Solve real-world and mathematical problems

involving the four operations with rational numbers.1

1Computations with rational numbers extend the rules for

manipulating fractions to complex fractions.*

*Computations with rational numbers extend the rules for

manipulating fractions to complex fractions.

Students use order of operations from 6th grade to write and solve problem with all rational numbers.

Component Cluster 7.EE Solve real-life and mathematical problems using numerical and algebraic expressions and equations.*

*The balance of this cluster is taught in Unit 2. 7.EE.4 Use variables to represent quantities in a real-

world or mathematical problem, and construct simple

equations and inequalities to solve problems by reasoning

about the quantities.*

a. Solve word problems leading to equations of the form px

+ q = r and p(x + q) = r, where p, q, and r are specific

rational numbers. Solve equations of these forms fluently.

Compare an algebraic solution to an arithmetic solution,

identifying the sequence of the operations used in each

approach.

For example, the perimeter of a rectangle is 54 cm. Its

length is 6 cm. What is its width?

*In this unit the equations include negative rational

numbers.

7.EE.4a and b Students write an equation or inequality to model the situation. Students explain how they determined

whether to write an equation or inequality and the properties of the real number system that you used to find a

solution. In contextual problems, students define the variable and use appropriate units.

7.EE.4a

Students solve multi-step equations derived from word problems. Students use the arithmetic from the problem to

generalize an algebraic solution.

Mathematics


Grade 7 Unit 2: Expressions and Equations (~ 7 weeks)

Unit Overview: Unit 2 consolidates and expands students’ previous work with generating equivalent expressions and solving equations. Students solve real life

and mathematical problems using numerical and algebraic expressions and equations. Their work with expressions and equations is applied to finding unknown

angles and problems involving area, volume, and surface area, formulas learned in 6th grade. This unit provides the opportunity for students to reason abstractly

and quantitatively (MP2) as they move back and forth between the context of real life math problems and the algebraic representations they are using to solve the

problems.

Guiding Question: How does reflecting on the real-life context/initial math problem help to make sense of a problem?


Component Cluster 7.EE Use properties of operations to generate equivalent expressions.

7.EE.1 Apply properties of operations as strategies to add,

subtract, factor, and expand linear expressions with

rational coefficients.

This is a continuation of work from 6th grade using properties of operations (table 3, pg. 90) and combining like

terms. Students apply properties of operations and work with rational numbers (integers and positive / negative

fractions and decimals) to write equivalent expressions.

7.EE.2 Understand that rewriting an expression in

different forms in a problem context can shed light on the

problem and how the quantities in it are related. For

example, a + 0.05a = 1.05a means that “increase by 5%”

is the same as “multiply by 1.05.”

Students understand the reason for rewriting an expression in terms of a contextual situation. For example, students

understand that a 20% discount is the same as finding 80% of the cost, c (0.80c).

Component Cluster 7.EE Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

7.EE.3 Solve multi-step real-life and mathematical

problems posed with positive and negative rational

numbers in any form (whole numbers, fractions, and

decimals), using tools strategically. Apply properties of

operations to calculate with numbers in any form; convert

between forms as appropriate; and assess the


estimation strategies. For example: If a woman making

$25 an hour gets a 10% raise, she will make an additional

1/10 of her salary an hour, or $2.50, for a new salary of

$27.50. If you want to place a towel bar 9 3/4 inches long

in the center of a door that is 27 ½ inches wide, you will

need to place the bar about 9 inches from each edge; this

estimate can be used as a check on the exact

computation.*

* Problems in this module take on any form but percent,

which is included in Unit 4.

Students solve contextual problems and mathematical problems using rational numbers. Students convert between

fractions, decimals, and percents as needed to solve the problem. Students use estimation to justify the

reasonableness of answers.

Estimation strategies for calculations with fractions and decimals extend from students’ work with whole number

operations. Estimation strategies include, but are not limited to:

• front-end estimation with adjusting (using the highest place value and estimating from the front end making

adjustments to the estimate by taking into account the remaining amounts),

• clustering around an average (when the values are close together an average value is selected and

multiplied by the number of values to determine an estimate),

• rounding and adjusting (students round down or round up and then adjust their estimate depending on how much

the rounding affected the original values),

• using friendly or compatible numbers such as factors (students seek to fit numbers together - i.e., rounding to

factors and grouping numbers together that have round sums like 100 or 1000), and

• using benchmark numbers that are easy to compute (students select close whole numbers for fractions or decimals

to determine an estimate).

7.EE.4 Use variables to represent quantities in a real-



about the quantities.




7.EE.4a

Mathematics







approach.



b. Solve word problems leading to inequalities of the form

px + q > r or px + q < r, where p, q, and r are specific

rational numbers. Graph the solution set of the inequality

and interpret it in the context of the problem. For example:

As a salesperson, you are paid $50 per week plus $3 per

sale. This week you want your pay to be at least $100.

Write an inequality for the number of sales you need to

make, and describe the solutions.



7.EE.4b Students solve and graph inequalities and make sense of the inequality in context. Inequalities may have negative

coefficients. Problems can be used to find a maximum or minimum value when in context.

Component Cluster 7.G Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.*

* Emphasis of 7.G.5 and 7.G.6 in this unit is on solving equations with formulas students have learned in previous grades. The standards are returned to in Unit 6. 7.G.5 Use facts about supplementary, complementary,

vertical, and adjacent angles in a multi-step problem to

write and solve simple equations for an unknown angle in

a figure.

Students use understandings of angles and deductive reasoning to write and solve equations.

7.G.6 Solve real-world and mathematical problems

involving area, volume and surface area of two and three-

dimensional objects composed of triangles, quadrilaterals,

polygons, cubes, and right prisms.

Students continue work from 5th and 6th grade to work with area, volume and surface area of two-dimensional and

three-dimensional objects. (composite shapes) Students will not work with cylinders, as circles are not polygons. At

this level, students determine the dimensions of the figures given the area or volume.

“Know the formula” does not mean memorization of the formula. To “know” means to have an understanding of why

the formula works and how the formula relates to the measure (area and volume) and the figure. This understanding

should be for all students.

Surface area formulas are not the expectation with this standard. Building on work with nets in the 6th

grade, students

should recognize that finding the area of each face of a three-dimensional figure and adding the areas will give the

surface area. No nets will be given at this level; however, students could create nets to aid in surface area

calculations. Students understanding of volume can be supported by focusing on the area of base times the height to

calculate volume. Students solve for missing dimensions, given the area or volume.

Mathematics


Grade 7 Unit 3: Ratios and Proportional Relationships (~ 6 weeks)

Unit Overview: In Unit 3, students build on their Grade 6 experiences with ratios, unit rates, and fraction division to analyze proportional relationships. They

decide whether two quantities are in a proportional relationship, identify constants of proportionality, and represent the relationship by equations. These skills are

then applied to real‐world problems including scale drawings. Students have the opportunity to attend to precisions (MP6) as they solve real-world and


Guiding Question: What are the strengths and weaknesses of the different representations of proportional relationships: tables, graphs, equations, diagrams, and verbal

descriptions?


Component Cluster 7.RP Analyze proportional relationships and use them to solve real-world and mathematical problems.*

*Percent and proportional relationships are covered in Unit 4.

7.RP.1 Compute unit rates associated with ratios of

fractions, including ratios of lengths, areas and other

quantities measured in like or different units. For example,

if a person walks 1/2 mile in each 1/4 hour, compute the

unit rate as the complex fraction 1/2/1/4 miles per hour,

equivalently 2 miles per hour.

Students continue to work with unit rates from 6th grade; however, the comparison now includes fractions compared

to fractions. The comparison can be with like or different units. Fractions may be proper or improper.

7.RP.2 Recognize and represent proportional relationships

between quantities.

a. Decide whether two quantities are in a proportional

relationship, e.g., by testing for equivalent ratios in a table

or graphing on a coordinate plane and observing whether

the graph is a straight line through the origin.

b. Identify the constant of proportionality (unit rate) in

tables, graphs, equations, diagrams, and verbal

descriptions of proportional relationships.

c. Represent proportional relationships by equations. For

example, if total cost t is proportional to the number n of

items purchased at a constant price p, the relationship

between the total cost and the number of items can be

expressed as t = pn.

d. Explain what a point (x, y) on the graph of a

proportional relationship means in terms of the situation,

with special attention to the points (0, 0) and (1, r) where r

is the unit rate.

Students’ understanding of the multiplicative reasoning used with proportions continues from 6th

grade. Students

determine if two quantities are in a proportional relationship from a table. Fractions and decimals could be used with

this standard.

Note: This standard focuses on the representations of proportions. Solving proportions is addressed in 7.SP.3.

Students graph relationships to determine if two quantities are in a proportional relationship and to interpret the

ordered pairs. If the amounts from a table are graphed, the pairs will form a straight line through the origin,

indicating that these pairs are in a proportional relationship. The y-coordinate when x = 1 will be the unit rate. The

constant of proportionality is the unit rate. Students identify this amount from tables, graphs, equations and verbal


Students write equations from context and identify the coefficient as the unit rate which is also the constant of

proportionality.

A common error is to reverse the position of the variables when writing equations. Students may find it useful to use

variables specifically related to the quantities rather than using x and y. Constructing verbal models can also be

helpful. A student might describe the situation as “the number of packs of gum times the cost for each pack is the

total cost in dollars”. They can use this verbal model to construct the equation. Students can check their equation by

substituting values and comparing their results to the table. The checking process helps student revise and recheck

their model as necessary.

7.RP.3 Use proportional relationships to solve multistep

ratio and percent problems.

*Ratio problems only.

In 6th grade, students used ratio tables and unit rates to solve problems. Students expand their understanding of

proportional reasoning to solve problems that are easier to solve with cross-multiplication. Students understand the

mathematical foundation for cross-multiplication.

Mathematics


Finding the percent error is the process of expressing the size of the error (or deviation) between two measurements.

To calculate the percent error, students determine the absolute deviation (positive difference) between an actual

measurement and the accepted value and then divide by the accepted value. Multiplying by 100 will give the percent

error. (Note the similarity between percent error and percent of increase or decrease)

% error = | estimated value - actual value | x 100 %

actual value

The use of proportional relationships is also extended to solve percent problems involving sales tax, markups and

markdowns simple interest (I = prt, where I = interest, p = principal, r = rate, and t = time (in years)), gratuities and

commissions, fees, percent increase and decrease, and percent error.

Students should be able to explain or show their work using a representation (numbers, words, pictures, physical

objects, or equations) and verify that their answer is reasonable. Students use models to identify the parts of the

problem and how the values are related. For percent increase and decrease, students identify the starting value,

determine the difference, and compare the difference in the two values to the starting value.











approach.



* In this unit, the equations are derived from ratio

problems.

7.EE.4a and b Students write an equation or inequality to model the situation. Students explain how they

determined whether to write an equation or inequality and the properties of the real number system that

you used to find a solution. In contextual problems, students define the variable and use appropriate units.

7.EE.4a

Students solve multi-step equations derived from word problems. Students use the arithmetic from the

problem to generalize an algebraic solution.

Component Cluster 7.G Draw, construct, and describe geometrical figures and describe the relationships between them.* *7.G.1 is also covered in Unit 4. The balance of this cluster is taught in Unit 6.

7.G.1 Solve problems involving scale drawings of

geometric figures, including computing actual lengths and

areas from a scale drawing and reproducing a scale

drawing at a different scale.

Students determine the dimensions of figures when given a scale and identify the impact of a scale on actual length

(one-dimension) and area (two-dimensions). Students identify the scale factor given two figures. Using a given scale

drawing, students reproduce the drawing at a different scale. Students understand that the lengths will change by a

factor equal to the product of the magnitude of the two size transformations.

Mathematics


Grade 7 Unit 4: Application of Proportional Relationships (~ 5 weeks)

Unit Overview: Unit 4 parallels Unit 3’s coverage of ratio and proportion, but this time with a concentration on percent. Problems in this unit include simple

interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, and percent error. Additionally, this unit includes percent

problems about populations, which prepare students for probability models about populations covered in the next unit. This unit provides students with the

opportunity to continue practicing modeling with mathematics (MP4) as they look for the mathematics in real-life situations.

Guiding Question: What are good strategies for making sure that your solution strategy and answer make sense when working on a proportional relationship, real-world

problem?


Component Cluster 7.RP Analyze proportional relationships and use them to solve real-world and mathematical problems.










between quantities.
















is the unit rate.


grade. Students


this standard.

Note: This standard focuses on the representations of proportions. Solving proportions is addressed in 7.SP.3.







proportionality.








ratio and percent problems. Examples: simple interest, tax,

markups and markdowns, gratuities and commissions,

fees, percent increase and decrease, percent error






Mathematics





actual value













operations to calculate with numbe rs in any form; convert










computation.*

Students solve contextual problems and mathematical problems using rational numbers. Students convert

between fractions, decimals, and percents as needed to solve the problem. Students use estimation to

justify the reasonableness of answers.

Estimation strategies for calculations with fractions and decimals extend from students’ work with whole

number operations. Estimation strategies include, but are not limited to:

• front-end estimation with adjusting (using the highest place value and estimating from the front end

making adjustments to the estimate by taking into account the remaining amounts),



• rounding and adjusting (students round down or round up and then adjust their estimate depending on

how much the rounding affected the original values),

• using friendly or compatible numbers such as factors (students seek to fit numbers together - i.e.,

rounding to factors and grouping numbers together that have round sums like 100 or 1000), and

• using benchmark numbers that are easy to compute (students select close whole numbers for fractions or

decimals to determine an estimate).

Component Cluster 7.G Draw, construct, and describe geometrical figures and describe the relationships between them.*

*7.G.1 is introduced in Unit 1. The balance of this cluster is taught in Unit 6.









Mathematics


Grade 7 Unit 5: Statistics and Probability (~ 5 weeks)

Unit Overview: In Unit 5, students learn to draw inferences about populations based on random samples. Through the study of chance processes, students learn to

develop, use and evaluate probability models. This unit provides an opportunity for students to construct viable arguments and critique the reasoning of others

(MP3) as they use math to reason about real-life events.

Guiding question: What is the relationship between theoretical and actual probability? Is there a connection between this idea and the idea of random sampling?


Component Cluster 7.SP Use random sampling to draw inferences about a population. 7.SP.1 Understand that statistics can be used to gain

information about a population by examining a sample of

the population; generalizations about a population from a

sample are valid only if the sample is representative of that

population. Understand that random sampling tends to

produce representative samples and support valid

inferences.

Students recognize that it is difficult to gather statistics on an entire population. Instead a random sample can be

representative of the total population and will generate valid predictions. Students use this information to draw

inferences from data. A random sample must be used in conjunction with the population to get accuracy. For

example, a random sample of elementary students cannot be used to give a survey about the prom.

7.SP.2 Use data from a random sample to draw inferences

about a population with an unknown characteristic of

interest. Generate multiple samples (or simulated samples)

of the same size to gauge the variation in estimates or

predictions. For example, estimate the mean word length

in a book by randomly sampling words from the book;

predict the winner of a school election based on randomly

sampled survey data. Gauge how far off the estimate or

prediction might be.

Students collect and use multiple samples of data to make generalizations about a population. Issues of variation in

the samples should be addressed.

Component Cluster 7.SP Draw informal comparative inferences about two populations. 7.SP.3 Informally assess the degree of visual overlap of

two numerical data distributions with similar variabilities,

measuring the difference between the centers by

expressing it as a multiple of a measure of variability. For

example, the mean height of players on the basketball team

is 10 cm greater than the mean height of players on the

soccer team, about twice the variability (mean absolute

deviation) on either team; on a dot plot, the separation

between the two distributions of heights is noticeable.

This is the students’ first experience with comparing two data sets. Students build on their understanding of graphs,

mean, median, Mean Absolute Deviation (MAD) and interquartile range from 6th grade. Students understand that

1. a full understanding of the data requires consideration of the measures of variability as well as mean or median,

2. variability is responsible for the overlap of two data sets and that an increase in variability can increase the overlap,

and

3. median is paired with the interquartile range and mean is paired with the mean absolute deviation .

The mean absolute deviation (MAD) is calculated by taking the mean of the absolute deviations for each data point.

The difference between each data point and the mean is recorded in the second column of the table. The difference

between each data point and the mean is recorded in the second column of the table. The absolute deviation, absolute

value of the deviation, is recorded in the third column. The absolute deviations are summed and divided by the

number of data points in the set.

7.SP.4 Use measures of center and measures of variability

for numerical data from random samples to draw informal

comparative inferences about two populations. For

example, decide whether the words in a chapter of a

seventh-grade science book are generally longer than the

Students compare two sets of data using measures of center (mean and median) and variability MAD and IQR).

Showing the two graphs vertically rather than side by side helps students make comparisons.

Mathematics


words in a chapter of a fourth-grade science book.

Component Cluster 7.SP Investigate chance processes and develop, use, and evaluate probability models.

7.SP.5 Understand that the probability of a chance event is

a number between 0 and 1 that expresses the likelihood of

the event occurring. Larger numbers indicate greater

likelihood. A probability near 0 indicates an unlikely

event, a probability around 1/2 indicates an event that is

neither unlikely nor likely, and a probability near

1indicates a likely event.

This is the students’ first formal introduction to probability. Students recognize that the probability of any single

event can be can be expressed in terms such as impossible, unlikely, likely, or certain or as a number between 0 and

1, inclusive, as illustrated on the number line below.

The closer the fraction is to 1, the greater the probability the event will occur. Larger numbers indicate greater

likelihood. For example, if someone has 10 oranges and 3 apples, you have a greater likelihood of selecting an orange

at random. Students also recognize that the sum of all possible outcomes is 1.

7.SP.6 Approximate the probability of a chance event by

collecting data on the chance process that produces it and

observing its long-run relative frequency, and predict the

approximate relative frequency given the probability. For

example, when rolling a number cube 600 times, predict

that a 3 or 6 would be rolled roughly 200 times, but

probably not exactly 200 times.

Students collect data from a probability experiment, recognizing that as the number of trials increase, the

experimental probability approaches the theoretical probability. The focus of this standard is relative frequency --

The relative frequency is the observed number of successful events for a finite sample of trials. Relative frequency is

the observed proportion of successful event, expressed as the value calculated by dividing the number of times an

event occurs by the total number of times an experiment is carried out.

Students can collect data using physical objects or graphing calculator or web-based simulations. Students can

perform experiments multiple times, pool data with other groups, or increase the number of trials in a simulation to

look at the long-run relative frequencies. Students try the experiment and compare their predictions to the

experimental outcomes to continue to explore and refine conjectures about theoretical probability.

7.SP.7 Develop a probability model and use it to find

probabilities of events. Compare probabilities from a

model to observed frequencies; if the agreement is not

good, explain possible sources of the discrepancy.

a. Develop a uniform probability model by assigning equal

probability to all outcomes, and use the model to

determine probabilities of events. For example, if a student

is selected at random from a class, find the probability that

Jane will be selected and the probability that a girl will be

selected.

b. Develop a probability model (which may not be

uniform) by observing frequencies in data generated from

a chance process. For example, find the approximate

probability that a spinning penny will land heads up or

that a tossed paper cup will land open-end down. Do the

outcomes for the spinning penny appear to be equally

likely based on the observed frequencies?

Probabilities are useful for predicting what will happen over the long run. Using theoretical probability, students

predict frequencies of outcomes. Students recognize an appropriate design to conduct an experiment with simple

probability events, understanding that the experimental data give realistic estimates of the probability of an event but

are affected by sample size.

Students need multiple opportunities to perform probability experiments and compare these results to theoretical

probabilities. Critical components of the experiment process are making predictions about the outcomes by applying

the principles of theoretical probability, comparing the predictions to the outcomes of the experiments, and

replicating the experiment to compare results. Experiments can be replicated by the same group or by compiling class

data. Experiments can be conducted using various random generation devices including, but not limited to, bag pulls,

spinners, number cubes, coin toss, and colored chips. Students can collect data using physical objects or graphing

calculator or web-based simulations. Students can also develop models for geometric probability (i.e. a target).

Mathematics


7.SP.8 Find probabilities of compound events using

organized lists, tables, tree diagrams, and simulation.

a. Understand that, just as with simple events, the

probability of a compound event is the fraction of

outcomes in the sample space for which the compound

event occurs.

b. Represent for compound events using methods such as

organized lists, tables and tree diagrams.

For an event described in everyday language (e.g., “rolling

double sixes”), identify the outcomes in the sample space

which compose the event.

c. Design and use a simulation to generate frequencies for

compound events. For example,

use random digits as a simulation tool to approximate the

answer to the question: If 40% of donors

have type A blood, what is the probability that it will take

at least 4 donors to find one with type A blood?

Students use tree diagrams, frequency tables, and organized lists, and simulations to determine the probability of

compound events.

Mathematics


Grade 7 Unit 6: Geometry (~ 7 weeks)

Unit Overview: The year concludes with students drawing and constructing geometrical figures in Unit 6. They also revisit unknown angle, area, volume, and

surface area problems, which now include problems involving percentages of areas or volumes. This unit provides the opportunity for students to use tools

strategically (MP5) as they choose tools ranging from actual concrete tools, technological support, or tools in the form of helpful formulas.

Guiding Question: What does it mean to “know” a formula in geometry?


Component Cluster 7.G Draw, construct, and describe geometrical figures and describe the relationships between them.

7.G.2 Draw (freehand, with ruler and protractor, and with

technology) geometric shapes with given conditions. Focus

on constructing triangles from three measures of angles or

sides, noticing when the conditions determine a unique

triangle, more than one triangle, or no triangle.

Students draw geometric shapes with given parameters. Parameters could include parallel lines, angles, perpendicular

lines, line segments, etc.

Students understand the characteristics of angles and side lengths that create a unique triangle, more than one triangle

or no triangle. Through exploration, students recognize that the sum of the angles of any triangle will be 180°.

7.G.3 Describe the two-dimensional figures that result

from slicing three-dimensional figures, as in plane sections

of right rectangular prisms and right rectangular pyramids.

Students need to describe the resulting face shape from cuts made parallel and perpendicular to the bases of right

rectangular prisms and pyramids. Cuts made parallel will take the shape of the base; cuts made perpendicular will

take the shape of the lateral (side) face. Cuts made at an angle through the right rectangular prism will produce a

parallelogram;

If the pyramid is cut with a plane (green) parallel to the base, the intersection of the pyramid and the plane is a square

cross section (red).

If the pyramid is cut with a plane (green) passing through the top vertex and perpendicular to the base, the

intersection of the pyramid and the plane is a triangular cross section (red).

Mathematics


If the pyramid is cut with a plane (green) perpendicular to the base, but not through the top vertex, the intersection of

the pyramid and the plane is a trapezoidal cross section (red).

http://intermath.coe.uga.edu/dictnary/descript.asp?termID=95

Component Cluster 7.G Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. 7.G.4 Know the formulas for the area and

circumference of a circle and use them to solve

problems; give an informal derivation of the

relationship between the circumference and area of a

circle.

Students understand the relationship between radius and diameter. Students also understand the ratio of

circumference to diameter can be expressed as pi. Building on these understandings, students generate the

formulas for circumference and area.

The illustration shows the relationship between the circumference and area. If a circle is cut into wedges

and laid out as shown, a parallelogram results. Half of an end wedge can be moved to the other end a

rectangle results. The height of the rectangle is the same as the radius of the circle. The base length is the

circumference (2Πr). The area of the rectangle (and therefore the circle) is found by the following

calculations:

A rect = Base x Height

Area = 1/2 (2Πr) x r

Area = Πr x r

Area = Πr2


Mathematics




Area = Πr x r

Area = Πr2

http://mathworld.wolfram.com/Circle.html

Students solve problems (mathematical and real-world) involving circles or semi-circles.

Note: Because pi is an irrational number that neither repeats nor terminates, the measurements are

approximate when 3.14 is used in place of Π.

Students build on their understanding of area from 6th grade to find the area of left-over materials when

circles are cut from squares and triangles or when squares and triangles are cut from circles. 7.G.5 Use facts about supplementary, complementary,



a figure.













grade, students






Mathematics


Grade 7 Accelerated In Grade 7 Accelerated, instructional time should focus on the following critical areas: (1) developing understanding of and applying proportional relationships; (2)

developing understanding of operations with rational numbers and working with expressions and linear equations; (3) formulating and reasoning about expressions

and equations and solving linear equations; (4) solving problems involving scale drawings and informal geometric constructions, and working with two- and three-

dimensional shapes to solve problems involving area, surface area, and volume; and (5) drawing inferences about populations based on samples.

(1) Students extend their understanding of ratios and develop understanding of proportionality to solve single- and multi-step problems. Students use their

understanding of ratios and proportionality to solve a wide variety of percent problems, including those involving discounts, interest, taxes, tips, and

percent increase or decrease. Students solve problems about scale drawings by relating corresponding lengths between the objects or by using the fact that

relationships of lengths within an object are preserved in similar objects. Students graph proportional relationships and understand the unit rate informally

as a measure of the steepness of the related line, called the slope. They distinguish proportional relationships from other relationships.

(2) Students develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal representation), and

percents as different representations of rational numbers. Students extend addition, subtraction, multiplication, and division to all rational numbers,

maintaining the properties of operations and the relationships between addition and subtraction, and multiplication and division. By applying these

properties, and by viewing negative numbers in terms of everyday contexts (e.g., amounts owed or temperatures below zero), students explain and

interpret the rules for adding, subtracting, multiplying, and dividing with negative numbers.

(3) Students use the arithmetic of rational numbers as they formulate expressions and equations in one variable and use these equations to solve problems.

They deepen their focus on linear equations and systems of linear equations as they learn to represent, analyze, and solve a variety of problems. Students

recognize equations for proportions (y/x = m or y = mx) as special linear equations (y = mx + b), understanding that the constant of proportionality (m) is

the slope, and the graphs are lines through the origin. They understand that the slope (m) of a line is a constant rate of change, so that if the input or x-

coordinate changes by an amount A, the output or y-coordinate changes by the amount m·A.

Students strategically choose and efficiently implement procedures to solve linear equations in one variable, understanding that when they use the

properties of equality and the concept of logical equivalence, they maintain the solutions of the original equation.

(4) Students continue their work with area from Grade 6, solving problems involving the area and circumference of a circle and surface area of three-

dimensional objects. In preparation for work on congruence and similarity in Grade 8 they reason about relationships among two-dimensional figures

using scale drawings and informal geometric constructions, and they gain familiarity with the relationships between angles formed by intersecting lines.

Students work with three-dimensional figures, relating them to two-dimensional figures by examining cross-sections. They solve real-world and

mathematical problems involving area, surface area, and volume of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons,

cubes and right prisms. Students complete their work on volume by solving problems involving cones, cylinders, and spheres.

(5) Students build on their previous work with single data distributions to compare two data distributions and address questions about differences between

populations. They begin informal work with random sampling to generate data sets and learn about the importance of representative samples for drawing

inferences.

Mathematics











Mathematics


Grade 7 Accelerated: Suggested Distribution of Units in Instructional Days Time Approximate

# of Weeks

Unit 1: Rational Numbers 17% ~ 6 weeks

Unit 2: The Number System and Properties of Exponents 17% ~ 6 weeks


Unit 3: Expressions and Equations 17% ~ 6 weeks

Unit 4: Ratios, Percents, and Proportional Relationships 13% ~ 5 weeks


Unit 5: Linear Equations 8% ~ 3 weeks

Unit 6: Statistics and Probability 14% ~ 5 weeks

State Test

Unit 7: Geometry 14% ~5 Weeks


17%

Unit 2: The Number System and

Properties of Exponents

17%

Unit 3: Expressions and Equations

17%

Unit 4: Ratios and Proportional Relationships

13%

Unit 5: Percent and Proportional Relationships

8%

Unit 6: Statistics and Probability

14%


Instructional Time

Mathematics


Grade 7 Accelerated Unit 1: Rational Numbers (~ 5 weeks)

Unit Overview: Students continue to build an understanding of the number line in Unit 1 from their work in Grade 6. They learn to add, subtract, multiply, and

divide rational numbers. Unit 1 includes rational numbers as they appear in expressions and equations—work that is continued in Unit 2. This unit provides

opportunities for students to look for and make use of structure (MP7) as they apply their previous understandings of addition and subtraction to rational numbers.

Guiding Question: How is addition and subtraction with only positive numbers from elementary school both different as well as similar to addition and subtraction with rational

numbers?


Component Cluster 7.NS Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide

rational numbers.


addition and subtraction to add and subtract rational

numbers; represent addition and subtraction on a

horizontal or vertical number line diagram.

a. Describe situations in which opposite quantities

combine to make 0. For example, a hydrogen atom has 0

charge because its two constituents are oppositely charged.

b. Understand p + q as the number located a distance |q|

from p, in the positive or negative direction depending on

whether q is positive or negative. Show that a number and

its opposite have a sum of 0 (are additive inverses).

Interpret sums of rational numbers by describing real-

world contexts.

c. Understand subtraction of rational numbers as adding

the additive inverse, p – q = p + (–q). Show that the

distance between two rational numbers on the

number line is the absolute value of their difference, and

apply this principle in real world contexts.

d. Apply properties of operations as strategies to add and

subtract rational numbers.

Students add and subtract rational numbers. Visual representations may be helpful as students begin this work; they

become less necessary as students become more fluent with these operations. The expectation of the CCSS is to build

on student understanding of number lines developed in 6th grade.

In 6th grade, students found the distance of horizontal and vertical segments on the coordinate plane. In 7th

grade,

students build on this understanding to recognize subtraction is finding the distance between two numbers on a

number line.


multiplication and division and of fractions to multiply and

divide rational numbers.

a. Understand that multiplication is extended from

fractions to rational numbers by requiring that operations

continue to satisfy the properties of operations, particularly

the distributive property, leading to products such as (–

1)(–1) = 1 and the rules for multiplying signed numbers.

Students understand that multiplication and division of integers is an extension of multiplication and division of

whole numbers. Students recognize that when division of rational numbers is represented with a fraction bar, each

number can have a negative sign.

Using long division from elementary school, students understand the difference between terminating and repeating

decimals. This understanding is foundational for the work with rational and irrational numbers in

8th grade. Identify which fractions will terminate (the denominator of the fraction in reduced form only has factors

of 2 and/or 5)

Mathematics


Interpret products of rational numbers by describing real-

world contexts.

b. Understand that integers can be divided, provided that

the divisor is not zero, and every quotient of integers (with

non-zero divisor) is a rational number. If p and q are

integers, then –(p/q) = (–p)/q = p/(–

q). Interpret quotients of rational numbers by describing

real-world contexts.

c. Apply properties of operations as strategies to multiply

and divide rational numbers.

d. Convert a rational number to a decimal using long

division; know that the decimal form of a rational number

terminates in 0s or eventually repeats.

7.NS.3 Solve real-world and mathematical problems

involving the four operations with rational numbers.1

1Computations with rational numbers extend the rules for

manipulating fractions to complex fractions.*

*Computations with rational numbers extend the rules for

manipulating fractions to complex fractions.

Students use order of operations from 6th grade to write and solve problem with all rational numbers.

Component Cluster 7.EE Solve real-life and mathematical problems using numerical and algebraic expressions and equations.*

*The balance of this cluster is taught in Unit 2. 7.EE.4 Use variables to represent quantities in a real-









approach.



*In this unit the equations include negative rational

numbers.




7.EE.4a



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Grade 7 Accelerated Unit 2: The Number System and Properties of Exponents (~ 5 weeks) Unit Overview: In Unit 2, students extend the properties of exponents to integer exponents. They use the number line model to support their

understanding of the rational numbers and the number system. The number system is revisited throughout the year. This unit provides opportunities

for students to look for and make use of reasoning (MP 7) as they extend their understanding of the properties of exponents to integer exponents.

Guiding Question: What is the connection between exponents and scientific notation?


Component Cluster 8.EE Work with radicals and integer exponents.

8.EE.1. Know and apply the properties of integer

exponents to generate equivalent numerical expressions.

For example, 32 × 3

–5 = 3

–3 = 1/3

3 = 1/27.

In 6th

grade, students wrote and evaluated simple numerical expressions with whole number exponents

(ie. = 5 • 5 • 5 = 125).

Integer (positive and negative) exponents are further developed to generate equivalent numerical expressions when

multiplying, dividing or raising a power to a power. Using numerical bases and the laws of exponents, students

generate equivalent expressions.

Students understand:

• Bases must be the same before exponents can be added, subtracted or multiplied.

• Exponents are subtracted when like bases are being divided

• A number raised to the zero (0) power is equal to one.

• Negative exponents occur when there are more factors in the denominator. These exponents can be expressed as a

positive if left in the denominator.

• Exponents are added when like bases are being multiplied

• Exponents are multiplied when an exponents is raised to an exponent

• Several properties may be used to simplify an expression

8.EE.3. Use numbers expressed in the form of a single

digit times an integer power of 10 to estimate very large or

very small quantities, and to express how many times as

much one is than the other. For example, estimate the

population of the United States as 3 × 108 and the

population of the world as 7 × 109, and determine that the

world population is more than 20 times larger.

Students use scientific notation to express very large or very small numbers. Students compare and interpret scientific

notation quantities in the context of the situation, recognizing that if the exponent increases by one, the value

increases 10 times. Likewise, if the exponent decreases by one, the value decreases 10 times.

Students solve problems using addition, subtraction or multiplication, expressing the answer in scientific notation.

8.EE.4. Perform operations with numbers expressed in

scientific notation, including problems where both decimal

and scientific notation are used. Use scientific notation and

choose units of appropriate size for measurements of very

large or very small quantities (e.g., use millimeters per

year for seafloor spreading). Interpret scientific notation

that has been generated by technology.

Students understand scientific notation as generated on various calculators or other technology. Students enter

scientific notation using E or EE (scientific notation), * (multiplication), and ^ (exponent) symbols. Students use

laws of exponents to multiply or divide numbers written in scientific notation, writing the product or quotient in

proper scientific notation. Students understand the magnitude of the number being expressed in scientific notation

and choose an appropriate corresponding unit.

Component Cluster 8.NS Know that there are numbers that are not rational, and approximate them by rational numbers.

8.NS.1. Know that numbers that are not rational are called

irrational. Understand informally that every number has a

decimal expansion; for rational numbers show that the

Students understand that Real numbers are either rational or irrational. They distinguish between rational and

irrational numbers, recognizing that any number that can be expressed as a fraction is a rational number. The diagram

below illustrates the relationship between the subgroups of the real number system.

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decimal expansion repeats eventually, and convert a

decimal expansion which repeats eventually into a rational

number.

Students recognize that the decimal equivalent of a fraction will either terminate or repeat. Fractions that terminate

will have denominators containing only prime factors of 2 and/or 5. This understanding builds on work in 7th grade

when students used long division to distinguish between repeating and terminating decimals. Students convert

repeating decimals into their fraction equivalent using patterns or algebraic reasoning.

8.NS.2. Use rational approximations of irrational numbers

to compare the size of irrational numbers, locate them

approximately on a number line diagram, and estimate the

value of expressions (e.g., √2). For example, by truncating

the decimal expansion of √2, show that √2 is between 1

and 2, then between 1.4 and 1.5, and explain how to

continue on to get better approximations.

Students locate rational and irrational numbers on the number line. Students compare and order rational and irrational

numbers. Students also recognize that square roots may be negative and written as -√ . Additionally, students

understand that the value of a square root can be approximated between integers and that non-perfect square roots are

irrational.

Mathematics


Grade 7 Accelerated Unit 3: Expressions and Equations (~ 5 weeks)

Unit Overview: Unit 3 consolidates and expands students’ previous work with generating equivalent expressions and solving equations. Students solve real life

and mathematical problems using numerical and algebraic expressions and equations. Their work with expressions and equations is applied to finding unknown

angles and problems involving area, volume, and surface area, formulas learned in 6th grade. This unit provides the opportunity for students to reason abstractly

and quantitatively (MP2) as they move back and forth between the context of real life math problems and the algebraic representations they are using to solve the

problems.

Guiding Question: How does reflecting on the real-life context/initial math problem help to make sense of a problem?


Component Cluster 7.EE Use properties of operations to generate equivalent expressions.

7.EE.1 Apply properties of operations as strategies to add,

subtract, factor, and expand linear expressions with

rational coefficients.

This is a continuation of work from 6th grade using properties of operations (table 3, glossary) and combining like

terms. Students apply properties of operations and work with rational numbers (integers and positive / negative

fractions and decimals) to write equivalent expressions.

7.EE.2 Understand that rewriting an expression in

different forms in a problem context can shed light on the

problem and how the quantities in it are related. For

example, a + 0.05a = 1.05a means that “increase by 5%”

is the same as “multiply by 1.05.”

Students understand the reason for rewriting an expression in terms of a contextual situation. For example, students

understand that a 20% discount is the same as finding 80% of the cost, c (0.80c).






operations to calculate with numbers in any form; convert










computation.*

* Problems in this module take on any form but percent,

which is included in Unit 4.






















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about the quantities.






approach.



b. Solve word problems leading to inequalities of the form

px + q > r or px + q < r, where p, q, and r are specific

rational numbers. Graph the solution set of the inequality

and interpret it in the context of the problem. For example:

As a salesperson, you are paid $50 per week plus $3 per

sale. This week you want your pay to be at least $100.

Write an inequality for the number of sales you need to

make, and describe the solutions.

7.EE.4a



7.EE.4b Students solve and graph inequalities and make sense of the inequality in context. Inequalities may have negative

coefficients. Problems can be used to find a maximum or minimum value when in context.

Component Cluster 7.G Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.*

* Emphasis of 7.G.5 and 7.G.6 in this unit is on solving equations with formulas students have learned in previous grades. The standards are returned to in Unit 6. 7.G.5 Use facts about supplementary, complementary,



a figure.













grade, students





Mathematics


Grade 7 Accelerated Unit 4: Ratios and Proportional Relationships (~ 5 weeks)

Unit Overview: In Unit 4, students build on their Grade 6 experiences with ratios, unit rates, and fraction division to analyze proportional relationships. They

decide whether two quantities are in a proportional relationship, identify constants of proportionality, and represent the relationship by equations. These skills are

then applied to real‐world problems. Real-life problems include simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase

and decrease, and percent error. Additionally, this unit includes percent problems about populations, which prepare students for probability models about

populations covered in the next unit. Students have the opportunity to attend to precisions (MP6) as they solve real-world and mathematical problems.

Guiding Question: What are the strengths and weaknesses of the different representations of proportional relationships: tables, graphs, equations, diagrams, and verbal

descriptions?












between quantities.
















is the unit rate.


grade. Students


this standard.

Note: This standard focuses on the representations of proportions. Solving proportions is addressed in 7.RP.3.







proportionality.







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ratio and percent problems. Examples: simple interest, tax,

markups and markdowns, gratuities and commissions,

fees, percent increase and decrease, percent error









actual value













operations to calculate with numbe rs in any form; convert










computation.*




















7.EE.4a and b Students write an equation or inequality to model the situation. Students explain how they

determined whether to write an equation or inequality and the properties of the real number system that

you used to find a solution. In contextual problems, students define the variable and use appropriate units.

7.EE.4a

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approach.



* In this unit, the equations are derived from ratio

problems.

Students solve multi-step equations derived from word problems. Students use the arithmetic from the

problem to generalize an algebraic solution.

Component Cluster 7.G Draw, construct, and describe geometrical figures and describe the relationships between them.* *The balance of this cluster is taught in Unit 6.









Mathematics


Grade 7 Accelerated Unit 5: Linear Equations (~3 weeks)

Unit Overview: In Unit 5, students use ratios, proportions, and similar triangles to explain why the slope of a line is well-defined. Students learn the connection

between proportional relationships, lines, and linear equations as they develop ways to represent a line in different forms. They analyze and solve linear equations.

This unit provides the opportunity for students to use appropriate tools strategically (MP 6) as they use graphs, tables, triangles, and equivalent forms of equations

to prove their reasoning.

Guiding Question: What is the relationship between ratios and slope and how is an equation represented on a graph?


Component Cluster 8.EE Understand the connections between proportional relationships, lines, and linear equations.

8.EE.5 Graph proportional relationships, interpreting the

unit rate as the slope of the graph. Compare two different

proportional relationships represented in different ways.

For example, compare a distance-time graph to a

distance-time equation to determine which of two moving

objects has greater speed.

Students build on their work with unit rates from 6th grade and proportional relationships in 7th grade to compare

graphs, tables and equations of proportional relationships. Students identify the unit rate (or slope) in graphs, tables

and equations to compare two proportional relationships represented in different ways. Given an equation of a

proportional relationship, students draw a graph of the relationship. Students recognize that the unit rate is the

coefficient of x and that this value is also the slope of the line.

8.EE.6 Use similar triangles to explain why the slope m is

the same between any two distinct points on a non-vertical

line in the coordinate plane; derive the equation y = mx for

a line through the origin and the equation y = mx + b for a

line intercepting the vertical axis at b.

Triangles are similar when there is a constant rate of proportionality between them. Using a graph, students construct

triangles between two points on a line and compare the sides to understand that the slope (ratio of rise to run) is the

same between any two points on a line.

Given an equation in slope-intercept form, students graph the line represented.

Students write equations in the form y = mx for lines going through the origin, recognizing that m represents the

slope of the line. Students write equations in the form y = mx + b for lines not passing through the origin,

recognizing that m represents the slope and b represents the y-intercept.

Component Cluster 8.F Use functions to model relationships between quantities.

8.F.4 Construct a function to model a linear relationship

between two quantities. Determine the rate of change and

initial value of the function from a description of a

relationship or from two (x, y) values, including reading

these from a table or from a graph. Interpret the rate of

change and initial value of a linear function in terms of the

situation it models, and in terms of its graph or a table of

values.

Students identify the rate of change (slope) and initial value (y-intercept) from tables, graphs, equations or verbal

descriptions to write a function (linear equation). Students understand that the equation represents the relationship

between the x-value and the y-value; what math operations are performed with the x-value to give the y-value. Slopes

could be undefined slopes or zero slopes.

Tables:

Students recognize that in a table the y-intercept is the y-value when x is equal to 0. The slope can be determined by

finding the ratio

between the change in two y-values and the change between the two corresponding x-values.

Using graphs, students identify the y-intercept as the point where the line crosses the y-axis and the slope as the

.

Equations:

In a linear equation the coefficient of x is the slope and the constant is the y-intercept. Students need to be given the

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equations in formats other than y = mx + b, such as y = ax + b (format from graphing calculator), y = b + mx (often

the format from contextual situations), etc.

Point and Slope:

Students write equations to model lines that pass through a given point with the given slope.

Students also write equations given two ordered pairs. Note that point-slope form is not an expectation at this

level. Students use the slope and y-intercepts to write a linear function in the form y = mx +b.

Contextual Situations:

In contextual situations, the y-intercept is generally the starting value or the value in the situation when the

independent variable is 0. The slope is the rate of change that occurs in the problem. Rates of change can often occur

over years. In these situations it is helpful for the years to be “converted” to 0, 1, 2, etc. For example, the years of

1960, 1970, and 1980 could be represented as 0 (for 1960), 10 (for 1970) and 20 (for 1980). Students interpret the

rate of change and the y-intercept in the context of the problem. Classroom discussion about one-time fees vs.

recurrent fees will help students model contextual situations.

Mathematics


Grade 7 Accelerated Unit 6: Statistics and Probability (~ 5 weeks)

Unit Overview: In Unit 6, students learn to draw inferences about populations based on random samples. Through the study of chance processes, students learn to

develop, use and evaluate probability models. This unit provides an opportunity for students to construct viable arguments and critique the reasoning of others

(MP3) as they use math to reason about real-life events.

Guiding question: What is the relationship between theoretical and actual probability? Is there a connection between this idea and the idea of random sampling?


Component Cluster 7.SP Use random sampling to draw inferences about a population. 7.SP.1 Understand that statistics can be used to gain

information about a population by examining a sample of

the population; generalizations about a population from a

sample are valid only if the sample is representative of that

population. Understand that random sampling tends to

produce representative samples and support valid

inferences.

Students recognize that it is difficult to gather statistics on an entire population. Instead a random sample can be

representative of the total population and will generate valid predictions. Students use this information to draw

inferences from data. A random sample must be used in conjunction with the population to get accuracy. For

example, a random sample of elementary students cannot be used to give a survey about the prom.

7.SP.2 Use data from a random sample to draw inferences

about a population with an unknown characteristic of

interest. Generate multiple samples (or simulated samples)

of the same size to gauge the variation in estimates or

predictions. For example, estimate the mean word length

in a book by randomly sampling words from the book;

predict the winner of a school election based on randomly

sampled survey data. Gauge how far off the estimate or

prediction might be.

Students collect and use multiple samples of data to make generalizations about a population. Issues of variation in

the samples should be addressed.

Component Cluster 7.SP Draw informal comparative inferences about two populations. 7.SP.3 Informally assess the degree of visual overlap of

two numerical data distributions with similar variabilities,

measuring the difference between the centers by

expressing it as a multiple of a measure of variability. For

example, the mean height of players on the basketball team

is 10 cm greater than the mean height of players on the

soccer team, about twice the variability (mean absolute

deviation) on either team; on a dot plot, the separation

between the two distributions of heights is noticeable.

This is the students’ first experience with comparing two data sets. Students build on their understanding of graphs,

mean, median, Mean Absolute Deviation (MAD) and interquartile range from 6th grade. Students understand that

1. a full understanding of the data requires consideration of the measures of variability as well as mean or median,

2. variability is responsible for the overlap of two data sets and that an increase in variability can increase the overlap,

and

3. median is paired with the interquartile range and mean is paired with the mean absolute deviation .

The mean absolute deviation (MAD) is calculated by taking the mean of the absolute deviations for each data point.

The difference between each data point and the mean is recorded in the second column of the table. The difference

between each data point and the mean is recorded in the second column of the table. The absolute deviation, absolute

value of the deviation, is recorded in the third column. The absolute deviations are summed and divided by the

number of data points in the set.

7.SP.4 Use measures of center and measures of variability

for numerical data from random samples to draw informal

comparative inferences about two populations. For

Students compare two sets of data using measures of center (mean and median) and variability MAD and IQR).

Showing the two graphs vertically rather than side by side helps students make comparisons.

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example, decide whether the words in a chapter of a

seventh-grade science book are generally longer than the

words in a chapter of a fourth-grade science book.

Component Cluster 7.SP Investigate chance processes and develop, use, and evaluate probability models.

7.SP.5 Understand that the probability of a chance event is

a number between 0 and 1 that expresses the likelihood of

the event occurring. Larger numbers indicate greater

likelihood. A probability near 0 indicates an unlikely

event, a probability around 1/2 indicates an event that is

neither unlikely nor likely, and a probability near

1indicates a likely event.

This is the students’ first formal introduction to probability. Students recognize that the probability of any single

event can be can be expressed in terms such as impossible, unlikely, likely, or certain or as a number between 0 and

1, inclusive, as illustrated on the number line below.

The closer the fraction is to 1, the greater the probability the event will occur. Larger numbers indicate greater

likelihood. For example, if someone has 10 oranges and 3 apples, you have a greater likelihood of selecting an orange

at random. Students also recognize that the sum of all possible outcomes is 1.

7.SP.6 Approximate the probability of a chance event by

collecting data on the chance process that produces it and

observing its long-run relative frequency, and predict the

approximate relative frequency given the probability. For

example, when rolling a number cube 600 times, predict

that a 3 or 6 would be rolled roughly 200 times, but

probably not exactly 200 times.

Students collect data from a probability experiment, recognizing that as the number of trials increase, the

experimental probability approaches the theoretical probability. The focus of this standard is relative frequency --

The relative frequency is the observed number of successful events for a finite sample of trials. Relative frequency is

the observed proportion of successful event, expressed as the value calculated by dividing the number of times an

event occurs by the total number of times an experiment is carried out.

Students can collect data using physical objects or graphing calculator or web-based simulations. Students can

perform experiments multiple times, pool data with other groups, or increase the number of trials in a simulation to

look at the long-run relative frequencies. Students try the experiment and compare their predictions to the

experimental outcomes to continue to explore and refine conjectures about theoretical probability.

7.SP.7 Develop a probability model and use it to find

probabilities of events. Compare probabilities from a

model to observed frequencies; if the agreement is not

good, explain possible sources of the discrepancy.

a. Develop a uniform probability model by assigning equal

probability to all outcomes, and use the model to

determine probabilities of events. For example, if a student

is selected at random from a class, find the probability that

Jane will be selected and the probability that a girl will be

selected.

b. Develop a probability model (which may not be

uniform) by observing frequencies in data generated from

a chance process. For example, find the approximate

probability that a spinning penny will land heads up or

that a tossed paper cup will land open-end down. Do the

Probabilities are useful for predicting what will happen over the long run. Using theoretical probability, students

predict frequencies of outcomes. Students recognize an appropriate design to conduct an experiment with simple

probability events, understanding that the experimental data give realistic estimates of the probability of an event but

are affected by sample size.

Students need multiple opportunities to perform probability experiments and compare these results to theoretical

probabilities. Critical components of the experiment process are making predictions about the outcomes by applying

the principles of theoretical probability, comparing the predictions to the outcomes of the experiments, and

replicating the experiment to compare results. Experiments can be replicated by the same group or by compiling class

data. Experiments can be conducted using various random generation devices including, but not limited to, bag pulls,

spinners, number cubes, coin toss, and colored chips. Students can collect data using physical objects or graphing

calculator or web-based simulations. Students can also develop models for geometric probability (i.e. a target).

Mathematics


outcomes for the spinning penny appear to be equally

likely based on the observed frequencies?

7.SP.8 Find probabilities of compound events using

organized lists, tables, tree diagrams, and simulation.

a. Understand that, just as with simple events, the

probability of a compound event is the fraction of

outcomes in the sample space for which the compound

event occurs.

b. Represent for compound events using methods such as

organized lists, tables and tree diagrams.

For an event described in everyday language (e.g., “rolling

double sixes”), identify the outcomes in the sample space

which compose the event.

c. Design and use a simulation to generate frequencies for

compound events. For example,

use random digits as a simulation tool to approximate the

answer to the question: If 40% of donors

have type A blood, what is the probability that it will take

at least 4 donors to find one with type A blood?

Students use tree diagrams, frequency tables, and organized lists, and simulations to determine the probability of

compound events.

Mathematics


Grade 7 Accelerated Unit 7: Geometry (~ 5 weeks)

Unit Overview: The year concludes with students drawing and constructing geometrical figures and extending their knowledge to geometric formulas to include

the volumes of cones, cylinders, and spheres. Through real-world and mathematical problems, they also revisit unknown angle, area, volume, and surface area

problems, which now include problems involving percentages of areas or volumes. This unit provides the opportunity for students to use tools strategically (MP5)

as they choose tools ranging from actual concrete tools, technological support, or tools in the form of helpful formulas.

Guiding Question: What does it mean to “know” a formula in geometry?


Component Cluster 7.G Draw, construct, and describe geometrical figures and describe the relationships between them.

7.G.2 Draw (freehand, with ruler and protractor, and with

technology) geometric shapes with given conditions. Focus

on constructing triangles from three measures of angles or

sides, noticing when the conditions determine a unique

triangle, more than one triangle, or no triangle.

Students draw geometric shapes with given parameters. Parameters could include parallel lines, angles, perpendicular

lines, line segments, etc.

Students understand the characteristics of angles and side lengths that create a unique triangle, more than one triangle

or no triangle. Through exploration, students recognize that the sum of the angles of any triangle will be 180°.

7.G.3 Describe the two-dimensional figures that result

from slicing three-dimensional figures, as in plane sections

of right rectangular prisms and right rectangular pyramids.

Students need to describe the resulting face shape from cuts made parallel and perpendicular to the bases of right

rectangular prisms and pyramids. Cuts made parallel will take the shape of the base; cuts made perpendicular will

take the shape of the lateral (side) face. Cuts made at an angle through the right rectangular prism will produce a

parallelogram;

If the pyramid is cut with a plane (green) parallel to the base, the intersection of the pyramid and the plane is a square

cross section (red).

If the pyramid is cut with a plane (green) passing through the top vertex and perpendicular to the base, the

intersection of the pyramid and the plane is a triangular cross section (red).

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If the pyramid is cut with a plane (green) perpendicular to the base, but not through the top vertex, the intersection of

the pyramid and the plane is a trapezoidal cross section (red).


Component Cluster 7.G Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. 7.G.4 Know the formulas for the area and

circumference of a circle and use them to solve

problems; give an informal derivation of the

relationship between the circumference and area of a

circle.

Students understand the relationship between radius and diameter. Students also understand the ratio of

circumference to diameter can be expressed as pi. Building on these understandings, students generate the

formulas for circumference and area.

The illustration shows the relationship between the circumference and area. If a circle is cut into wedges

and laid out as shown, a parallelogram results. Half of an end wedge can be moved to the other end a

rectangle results. The height of the rectangle is the same as the radius of the circle. The base length is the

circumference (2Πr). The area of the rectangle (and therefore the circle) is found by the following

calculations:



Area = Πr x r

Area = Πr2


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Area = Πr x r

Area = Πr2


Students solve problems (mathematical and real-world) involving circles or semi-circles.

Note: Because pi is an irrational number that neither repeats nor terminates, the measurements are

approximate when 3.14 is used in place of Π.

Students build on their understanding of area from 6th grade to find the area of left-over materials when

circles are cut from squares and triangles or when squares and triangles are cut from circles. 7.G.5 Use facts about supplementary, complementary,



a figure.













grade, students





Component Cluster 8.G Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.


Mathematics


8.G.9. Know the formulas for the volumes of cones,

cylinders and spheres and use them to solve real-world and

mathematical problems.*

Students build on understandings of circles and volume from 7th grade to find the volume of cylinders, finding the

area of the base Π and multiplying by the number of layers (the height).

Students understand that the volume of a cylinder is 3 times the volume of a cone having the same base area and

height or that the volume of a cone is 1/3 the volume of a cylinder having the same base area and height.

A sphere can be enclosed with a cylinder, which has the same radius and height of the sphere (Note: the height of the

cylinder is twice the radius of the sphere). If the sphere is flattened, it will fill 2/3 of the cylinder. Based on this

model, students understand that the volume of a sphere is2/3 the volume of a cylinder with the same radius and

height. The height of the cylinder is the same as the diameter of the sphere or 2r. Using this information, the formula

for the volume of the sphere can be derived in the following way:

cylinder volume formula

multiply by 2/3 since the volume of a sphere is 2/3 the cylinder’s volume

substitute 2r for height since 2r is the height of the sphere

simplify

Students find the volume of cylinders, cones and spheres to solve real world and mathematical problems. Answers

could also be given in terms of Pi.


the formula works and how the formula relates to the measure (volume) and the figure. This understanding should be

for all students.

Note: At this level composite shapes will not be used and only volume will be calculated.

Mathematics


Grade 8 In Grade 8, instructional time should focus on three critical areas: (1) formulating and reasoning about expressions and equations, including modeling

an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a

function and using functions to describe quantitative relationships; (3) analyzing two- and three-dimensional space and figures using distance, angle,

similarity, and congruence, and understanding and applying the Pythagorean Theorem.

(1) Students use linear equations and systems of linear equations to represent, analyze, and solve a variety of problems. Students recognize

equations for proportions (y/x = m or y = mx) as special linear equations (y = mx + b), understanding that the constant of proportionality (m) is

the slope, and the graphs are lines through the origin. They understand that the slope (m) of a line is a constant rate of change, so that if the

input or x-coordinate changes by an amount A, the output or y-coordinate changes by the amount m·A. Students also use a linear equation to

describe the association between two quantities in bivariate data (such as arm span vs. height for students in a classroom). At this grade,

fitting the model, and assessing its fit to the data are done informally. Interpreting the model in the context of the data requires students to

express a relationship between the two quantities in question and to interpret components of the relationship (such as slope and y-intercept) in

terms of the situation.

Students strategically choose and efficiently implement procedures to solve linear equations in one variable, understanding that when they use

the properties of equality and the concept of logical equivalence, they maintain the solutions of the original equation. Students solve systems

of two linear equations in two variables and relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the same line.

Students use linear equations, systems of linear equations, linear functions, and their understanding of slope of a line to analyze situations and

solve problems.

(2) Students grasp the concept of a function as a rule that assigns to each input exactly one output. They understand that functions describe

situations where one quantity determines another. They can translate among representations and partial representations of functions (noting

that tabular and graphical representations may be partial representations), and they describe how aspects of the function are reflected in the

different representations.

(3) Students use ideas about distance and angles, how they behave under translations, rotations, reflections, and dilations, and ideas about

congruence and similarity to describe and analyze two-dimensional figures and to solve problems. Students show that the sum of the angles in

a triangle is the angle formed by a straight line, and that various configurations of lines give rise to similar triangles because of the angles

created when a transversal cuts parallel lines. Students understand the statement of the Pythagorean Theorem and its converse, and can

explain why the Pythagorean Theorem holds, for example, by decomposing a square in two different ways. They apply the Pythagorean

Theorem to find distances between points on the coordinate plane, to find lengths, and to analyze polygons. Students complete their work on

volume by solving problems involving cones, cylinders, and spheres.










Mathematics


Grade 8: Suggested Distribution of Units in Instructional Days Time # of Weeks

Unit 1: The Number System and Properties of Exponents 11% 5 weeks

Unit 2: Introduction to Irrational Numbers Using Geometry 20% 7 weeks


Unit 3: Congruence 9% 3 weeks

Unit 4: Similarity 11% 4 weeks


Unit 5: Linear Equations 25% 9 weeks

Unit 6: Linear Functions (Examples of Linear Functions from Geometry) 11% 4 weeks

State Test

Unit 7: Patterns and Data 5% 2 weeks

Unit 1: The Number System

and Properties of Exponents

11%

Unit 3: Congruence

14%

Unit 4: Similarity 14%

Unit 5: Linear Equations

25%

Unit 5: Examples of Linear

Functions from Geometry

25%

Unit 6: Linear Functions

11%

Unit 2: Introduction to

Irrational Numbers Using

Geometry 20%

Instructional Time

Mathematics


Grade 8 Unit 1: The Number System and Properties of Exponents (~ 4 weeks) Unit Overview: This year begins with students extending the properties of exponents to integer exponents in Unit 1. They use the number line model

to support their understanding of the rational numbers and the number system. The number system is revisited throughout the year.

Guiding Question: What is the connection between exponents and scientific notation?



8.EE.1. Know and apply the properties of integer

exponents to generate equivalent numerical

expressions. For example, 32 × 3

–5 = 3

–3 = 1/3

3

= 1/27.

In 6th

grade, students wrote and evaluated simple numerical expressions with whole number

exponents

(ie. = 5 • 5 • 5 = 125).

Integer (positive and negative) exponents are further developed to generate equivalent numerical

expressions when multiplying, dividing or raising a power to a power. Using numerical bases and

the laws of exponents, students generate equivalent expressions.

Students understand:

• Bases must be the same before exponents can be added, subtracted or multiplied.

• Exponents are subtracted when like bases are being divided

• A number raised to the zero (0) power is equal to one.

• Negative exponents occur when there are more factors in the denominator. These exponents can

be expressed as a positive if left in the denominator.

• Exponents are added when like bases are being multiplied

• Exponents are multiplied when an exponents is raised to an exponent

• Several properties may be used to simplify an expression

8.EE.3. Use numbers expressed in the form of a

single digit times an integer power of 10 to

estimate very large or very small quantities, and

to express how many times as much one is than

the other. For example, estimate the population

of the United States as 3 × 108 and the

population of the world as 7 × 109, and

determine that the world population is more than

20 times larger.

Students use scientific notation to express very large or very small numbers. Students compare

and interpret scientific notation quantities in the context of the situation, recognizing that if the

exponent increases by one, the value increases 10 times. Likewise, if the exponent decreases by

one, the value decreases 10 times.

Students solve problems using addition, subtraction or multiplication, expressing the answer in

scientific notation.

8.EE.4. Perform operations with numbers

expressed in scientific notation, including

problems where both decimal and scientific

notation are used. Use scientific notation and

choose units of appropriate size for

measurements of very large or very small

quantities (e.g., use millimeters per year for

Students understand scientific notation as generated on various calculators or other technology.

Students enter scientific notation using E or EE (scientific notation), * (multiplication), and ^

(exponent) symbols. Students use laws of exponents to multiply or divide numbers written in

scientific notation, writing the product or quotient in proper scientific notation. Students

understand the magnitude of the number being expressed in scientific notation and choose an

appropriate corresponding unit.

Mathematics


seafloor spreading). Interpret scientific notation

that has been generated by technology.

Component Cluster 8.NS Know that there are numbers that are not rational, and approximate them by rational numbers.

8.NS.1. Know that numbers that are not rational

are called irrational. Understand informally that

every number has a decimal expansion; for

rational numbers show that the decimal

expansion repeats eventually, and convert a

decimal expansion which repeats eventually into

a rational number.

Students understand that Real numbers are either rational or irrational. They distinguish between

rational and irrational numbers, recognizing that any number that can be expressed as a fraction is

a rational number. The diagram below illustrates the relationship between the subgroups of the

real number system.

Students recognize that the decimal equivalent of a fraction will either terminate or repeat.

Fractions that terminate will have denominators containing only prime factors of 2 and/or 5. This

understanding builds on work in 7th grade when students used long division to distinguish between

repeating and terminating decimals. Students convert repeating decimals into their fraction

equivalent using patterns or algebraic reasoning.

8.NS.2. Use rational approximations of irrational

numbers to compare the size of irrational

numbers, locate them approximately on a

number line diagram, and estimate the value of

expressions (e.g., √2). For example, by

truncating the decimal expansion of √2, show

that √2 is between 1 and 2, then between 1.4 and

1.5, and explain how to continue on to get better

approximations.

Students locate rational and irrational numbers on the number line. Students compare and order

rational and irrational numbers. Students also recognize that square roots may be negative and

written as -√ . Additionally, students understand that the value of a square root can be

approximated between integers and that non-perfect square roots are irrational.

Mathematics


Grade 8 Unit 2: Pythagorean Theorem (~ 7 weeks) Unit Overview: Students will learn and explain a proof of the Pythagorean Theorem on their own. The students will make connections between

rational and irrational numbers from Unit 1, and look at different ways to represent them (radicals, non-repeating decimal expansion). The

Pythagorean Theorem is also used to motivate a discussion of irrational square roots (irrational cube roots are introduced via volume of a sphere). Guiding Question: How are deductive reasoning and informal arguments used to prove the Pythagorean Theorem?



8.EE.2 Use square root and cube root symbols to

represent solutions to equations of the form x² =

p and x³ = p, where p is a positive rational

number. Evaluate square roots of small perfect

squares and cube roots of small perfect cubes.

Know that √2 is irrational.

Students recognize perfect squares and cubes, understanding that non-perfect squares and non-

perfect cubes are irrational.

Students recognize that squaring a number and taking the square root √ of a number are inverse

operations; likewise, cubing a number and taking the cube root √

are inverse operations.

Rational numbers would have perfect squares or perfect cubes for the numerator and

denominator. In the standard, the value of p for square root and cube root equations must be

positive.

Students understand that in geometry the square root of the area is the length of the side of a

square and a cube root of the volume is the length of the side of a cube. Students use this

information to solve problems, such as finding the perimeter.

Component Cluster 8.G Understand and apply the Pythagorean Theorem.

8.G.6 Explain a proof of the Pythagorean

Theorem and its converse.

Using models, students explain the Pythagorean Theorem, understanding that the sum of the

squares of the legs is equal to the square of the hypotenuse in a right triangle.

Students also understand that given three side lengths with this relationship forms a right triangle.

8.G.7 Apply the Pythagorean Theorem to

determine unknown side lengths in right

triangles in real-world and mathematical

problems in two and three dimensions.

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-

world and mathematical problems in two and three dimensions.

Based on this work, students could then find the volume or surface area.

8.G.8 Apply the Pythagorean Theorem to find

the distance between two points in a coordinate

system.

One application of the Pythagorean Theorem is finding the distance between two points on the

coordinate plane. Students build on work from 6th grade (finding vertical and horizontal distances

on the coordinate plane) to determine the lengths of the legs of the right triangle drawn

connecting the points. Students understand that the line segment between the two points is the

length of the hypotenuse.

NOTE: The use of the distance formula is not an expectation.

Mathematics


Students find area and perimeter of two-dimensional figures on the coordinate plane, finding the

distance between each segment of the figure. (Limit one diagonal line, such as a right trapezoid or

parallelogram)

Mathematics


Grade 8 Unit 3: Congruence (~ 3 weeks)

Unit Overview: In Unit 3, students study congruence by experimenting with rotations, reflections, and translations of geometrical figures. This unit

concludes by using deductive reasoning and informal arguments to establish the angle sum theorem and parallel lines cut by transversals.

Guiding Questions: How are congruent figures represented in transformations? How are congruent triangles used to determine relationships that exist in triangles and parallel

lines?


Component Cluster 8.G Understand congruence and similarity using physical models, transparencies, or geometry software.

8.G.1 Verify experimentally the properties of

rotations, reflections, and translations:

a. Lines are taken to lines, and line segments to

line segments of the same length.

b. Angles are taken to angles of the same

measure.

c. Parallel lines are taken to parallel lines.

Students use compasses, protractors and rulers or technology to explore figures created from

translations, reflections and rotations. Characteristics of figures, such as lengths of line segments,

angle measures and parallel lines, are explored before the transformation (pre-image) and after

the transformation (image). Students understand that these transformations produce images of

exactly the same size and shape as the pre-image and are known as rigid transformations.

8.G.2 Understand that a two-dimensional figure

is congruent to another if the second can be

obtained from the first by a sequence of

rotations, reflections and translations; given two

congruent figures, describe a sequence that

exhibits the congruence between them.

This standard is the students’ introduction to congruency. Congruent figures have the same shape

and size. Translations, reflections and rotations are examples of rigid transformations. A rigid

transformation is one in which the pre-image and the image both have exactly the same size and

shape since the measures of the corresponding angles and corresponding line segments remain

equal (are congruent).

Students examine two figures to determine congruency by identifying the rigid transformation(s)

that produced the figures. Students recognize the symbol for congruency (≅) and write statements

of congruency.

8.G.3 Describe the effect of translations,

rotations, and reflections on two-dimensional

figures using coordinates.

Students identify resulting coordinates from translations, reflections, and rotations (90º, 180º and

270º both clockwise and counterclockwise), recognizing the relationship between the coordinates

and the transformation.

Translations

Translations move the object so that every point of the object moves in the same direction as well

as the same distance. In a translation, the translated object is congruent to its pre-image.

Reflections

A reflection is the “flipping” of an object over a line, known as the “line of reflection”. In the 8th

grade, the line of reflection will be the x-axis and the y-axis. Students recognize that when an

object is reflected across the y-axis, the reflected x-coordinate is the opposite of the pre-image x-

coordinate.

Mathematics


Rotations

A rotation is a transformation performed by “spinning” the figure around a fixed point known as

the center of rotation. The figure may be rotated clockwise or counterclockwise up to 360º (at 8th

grade, rotations will be around the origin and a multiple of 90º). In a rotation, the rotated object is

congruent to its pre-image.

Students recognize the relationship between the coordinates of the pre-image and the image.

Students identify the transformations based on the given coordinates.

Mathematics


Grade 8 Unit 4: Similarity (~ 4 weeks) Unit Overview: The experimental study of rotations, reflections, and translations in Unit 2 prepares students for the more complex work of

understanding the effects of dilations on geometrical figures in their study of similarity in Unit 3. They use similar triangles to solve unknown angle,

side length and area problems. Unit 3 concludes with revisiting a proof of the Pythagorean Theorem from the perspective of similar triangles. Guiding Question: How are similar figures represented in transformations and what is the relationship between congruent and similar figures?



8.G.3 Describe the effect of dilations on two-

dimensional figures using coordinates.




Dilations

A dilation is a non-rigid transformation that moves each point along a ray which starts from a

fixed center, and multiplies distances from this center by a common scale factor. Dilations

enlarge (scale factors greater than one) or reduce (scale factors less than one) the size of a figure

by the scale factor. In 8th grade, dilations will be from the origin. The dilated figure is similar to

its pre-image.

Students recognize the relationship between the coordinates of the pre-image, the image and the

scale factor for a dilation from the origin. Using the coordinates, students are able to identify the

scale factor (image/pre-image).

Students identify the transformation based on given coordinates.


is similar to another if the second can be


rotations, reflections, translations, and dilations;

given two similar two-dimensional figures,

describe a sequence that exhibits the similarity

between them.

Similar figures and similarity are first introduced in the 8th

grade. Students understand similar

figures have congruent angles and sides that are proportional. Similar figures are produced from

dilations. Students describe the sequence that would produce similar figures, including the scale

factors. Students understand that a scale factor greater than one will produce an enlargement in

the figure, while a scale factor less than one will produce a reduction in size.

Students need to be able to identify that triangles are similar or congruent based on given

information.

8.G.5 Use informal arguments to establish facts

about the angle sum and exterior angle of

triangles, about the angles created when parallel

lines are cut by a transversal, and the angle-angle

criterion for similarity of triangles. For example,

arrange three copies of the same triangle so that

the sum of the three angles appears to form a

line, and give an argument in terms of

Students use exploration and deductive reasoning to determine relationships that exist between

the following:

a) angle sums and exterior angle sums of triangles, b) angles created when parallel lines are cut

by a transversal, and c) the angle-angle criterion for similarity of triangle.

Students construct various triangles and find the measures of the interior and exterior angles.

Students make conjectures about the relationship between the measure of an exterior angle and

the other two angles of a triangle.

(the measure of an exterior angle of a triangle is equal to the sum of the measures of the other two

Mathematics


transversals why this is so. interior angles) and the sum of the exterior angles (360º). Using these relationships, students use

deductive reasoning to find the measure of missing angles.

Students construct parallel lines and a transversal to examine the relationships between the

created angles. Students recognize vertical angles, adjacent angles and supplementary angles from

7th grade and build on these relationships to identify other pairs of congruent angles. Using these

relationships, students use deductive reasoning to find the measure of missing angles.

Students can informally conclude that the sum of the angles in a triangle is 180º (the angle-sum

theorem) by applying their understanding of lines and alternate interior angles.

Students construct various triangles having line segments of different lengths but with two

corresponding congruent angles. Comparing ratios of sides will produce a constant scale factor,

meaning the triangles are similar. Students solve problems with similar triangles.

Mathematics


Grade 8 Unit 5: Linear Equations (~ 8 weeks)

Unit Overview: In Unit 5, students use similar triangles learned in Unit 4 to explain why the slope of a line is well‐defined. Students learn the

connection between proportional relationships, lines, and linear equations as they develop ways to represent a line by different equations (y = mx + b,

y – = m (x – ), etc.). They analyze and solve linear equations and pairs of simultaneous linear equations. The equation of a line provides a

natural transition into the idea of a function explored in the next units.

Guiding Question: What is the relationship between an equation and a graphical representation? How is this relationship used to solve a system of equations?


Component Cluster 8.EE Understand the connections between proportional relationships, lines, and linear equations.

8.EE.5 Graph proportional relationships,

interpreting the unit rate as the slope of the

graph. Compare two different proportional

relationships represented in different ways. For

example, compare a distance-time graph to a

distance-time equation to determine which of two

moving objects has greater speed.

Students build on their work with unit rates from 6th grade and proportional relationships in 7th

grade to compare graphs, tables and equations of proportional relationships. Students identify the

unit rate (or slope) in graphs, tables and equations to compare two proportional relationships

represented in different ways. Given an equation of a proportional relationship, students draw a

graph of the relationship. Students recognize that the unit rate is the coefficient of x and that this

value is also the slope of the line.

8.EE.6 Use similar triangles to explain why the

slope m is the same between any two distinct

points on a non-vertical line in the coordinate

plane; derive the equation y = mx for a line

through the origin and the equation y = mx + b

for a line intercepting the vertical axis at b..

Triangles are similar when there is a constant rate of proportionality between them. Using a

graph, students construct triangles between two points on a line and compare the sides to

understand that the slope (ratio of rise to run) is the same between any two points on a line.

Given an equation in slope-intercept form, students graph the line represented.

Students write equations in the form y = mx for lines going through the origin, recognizing that m

represents the slope of the line. Students write equations in the form y = mx + b for lines not

passing through the origin, recognizing that m represents the slope and b represents the y-

intercept.

Component Cluster 8.EE Analyze and solve linear equations and pairs of simultaneous linear equations.

8.EE.7. Solve linear equations in one variable.

a. Give examples of linear equations in one

variable with one solution, infinitely many

solutions, or no solutions. Show which of

these possibilities is the case by successively

transforming the given equation into simpler

forms, until an equivalent equation of the

form x = a, a = a, or a = b results (where a

and b are different numbers).

Students solve one-variable equations including those with the variables being on both sides of

the equals sign. Students recognize that the solution to the equation is the value(s) of the variable,

which make a true equality when substituted back into the equation. Equations shall include

rational numbers, distributive property and combining like terms.

Equations have one solution when the variables do not cancel out. If each side of the equation

were treated as a linear equation and graphed, the solution of the equation represents the

coordinates of the point where the two lines would intersect.

Mathematics


b. Solve linear equations with rational number

coefficients, including equations whose

solutions require expanding expressions

using the distributive property and collecting

like terms.

Equations having no solution have variables that will cancel out and constants that are not equal.

This means that there is not a value that can be substituted for x that will make the sides equal.

This solution means that no matter what value is substituted for x the final result will never be

equal to each other.

If each side of the equation were treated as a linear equation and graphed, the lines would be

parallel.

An equation with infinitely many solutions occurs when both sides of the equation are the same.

Any value of x will produce a valid equation. If each side of the equation were treated as a linear

equation and graphed, the graph would be the same line.

Students write equations from verbal descriptions and solve.

8.EE.8. Analyze and solve pairs of simultaneous

linear equations.

a. Understand that solutions to a system of

two linear equations in two variables

correspond to points of intersection of their

graphs, because points of intersection

satisfy both equations simultaneously.

b. Solve systems of two linear equations in

two variables algebraically, and estimate

solutions by graphing the equations. Solve

simple cases by inspection. For example,

3x + 2y = 5 and 3x + 2y = 6 have no

solution because 3x + 2y cannot

simultaneously be 5 and 6.

c. Solve real-world and mathematical

problems leading to two linear equations in

two variables. For example, given

coordinates for two pairs of points,

determine whether the line through the

first pair of points intersects the line

through the second pair.

Systems of linear equations can also have one solution, infinitely many solutions or no solutions.

Students will discover these cases as they graph systems of linear equations and solve them

algebraically.

Students graph a system of two linear equations, recognizing that the ordered pair for the point of

intersection is the x-value that will generate the given y-value for both equations. Students

recognize that graphed lines with one point of intersection (different slopes) will have one

solution, parallel lines (same slope, different y-intercepts) have no solutions, and lines that are the

same (same slope, same y-intercept) will have infinitely many solutions.

By making connections between algebraic and graphical solutions and the context of the system

of linear equations, students are able to make sense of their solutions. Students need opportunities

to work with equations and context that include whole number and/or decimals/fractions.

Students define variables and create a system of linear equations in two variables.

Note: Students are not expected to change linear equations written in standard form to slope-

intercept form or solve systems using elimination.

For many real world contexts, equations may be written in standard form. Students are not

expected to change the standard form to slope-intercept form. However, students may generate

ordered pairs recognizing that the values of the ordered pairs would be solutions for the equation.

Mathematics


Grade 8 Unit 6: Linear Functions (~ 4 weeks) and Examples of Functions from Geometry (~ 3 weeks)

Unit Overview: In this unit, students will build upon informal knowledge of one-variable linear equations, to solve equations, which include cases with infinitely

many solutions or no solutions as well as cases requiring algebraic manipulation. Solving an equation requires students to see and make use of structure (MP7).

Students will use the equation of a linear model to solve problems in the context of interpreting the slope and intercept. This unit introduces two-way tables as a

way to organize categorical data and calculate relative frequencies. Also in this unit, students will extend their previous work with geometric measurement to

include solving problems involving volumes of cones, cylinders and spheres.

Guiding Question: How are applications of linear equations used to solve contextual problems?



8.F.4 Construct a function to model a linear

relationship between two quantities. Determine

the rate of change and initial value of the

function from a description of a relationship or

from two (x, y) values, including reading these

from a table or from a graph. Interpret the rate of

change and initial value of a linear function in

terms of the situation it models, and in terms of

its graph or a table of values.

Students identify the rate of change (slope) and initial value (y-intercept) from tables, graphs,

equations or verbal descriptions to write a function (linear equation). Students understand that the

equation represents the relationship between the x-value and the y-value; what math operations

are performed with the x-value to give the y-value. Slopes could be undefined slopes or zero

slopes.

Tables:

Students recognize that in a table the y-intercept is the y-value when x is equal to 0. The slope can

be determined by finding the ratio

between the change in two y-values and the change between

the two corresponding x-values.

Using graphs, students identify the y-intercept as the point where the line crosses the y-axis and

the slope as the

.

Equations:

In a linear equation the coefficient of x is the slope and the constant is the y-intercept. Students

need to be given the equations in formats other than y = mx + b, such as y = ax + b (format from

graphing calculator), y = b + mx (often the format from contextual situations), etc.

Point and Slope:

Students write equations to model lines that pass through a given point with the given slope.

Students also write equations given two ordered pairs. Note that point-slope form is not an

expectation at this level. Students use the slope and y-intercepts to write a linear function in the

form y = mx +b.

Contextual Situations:

In contextual situations, the y-intercept is generally the starting value or the value in the situation

when the independent variable is 0. The slope is the rate of change that occurs in the problem.

Mathematics


Rates of change can often occur over years. In these situations it is helpful for the years to be

“converted” to 0, 1, 2, etc. For example, the years of 1960, 1970, and 1980 could be represented

as 0 (for 1960), 10 (for 1970) and 20 (for 1980). Students interpret the rate of change and the y-

intercept in the context of the problem. Classroom discussion about one-time fees vs. recurrent

fees will help students model contextual situations.

8.F.5 Describe qualitatively the functional

relationship between two quantities by analyzing

a graph, (e.g. where the function is increasing or

decreasing, linear or nonlinear). Sketch a graph

that exhibits the qualitative features of a function

that has been described verbally.

Given a verbal description of a situation, students sketch a graph to model that situation. Given a

graph of a situation, students provide a verbal description of the situation.

Component Cluster 8.F Define, evaluate, and compare functions.*

* Linear and non-linear functions are compared in this module using linear equations and area/volume formulas as examples.

8.F.1 Understand that a function is a rule that

assigns to each input exactly one output. The

graph of a function is the set of ordered pairs

consisting of an input and the corresponding

output.*

*Function notation is not required in

Grade 8.

Students understand rules that take x as input and gives y as output is a function. Functions occur

when there is exactly one y-value is associated with any x-value. Using y to represent the output

we can represent this function with the equations y = x2+ 5x + 4. Students are not expected to use

the function notation f(x) at this level. Students identify functions from equations, graphs, and

tables/ordered pairs.

Graphs

Students recognize graphs such as the one below is a function using the vertical line test, showing

that each x-value

has only one y-value;

whereas, graphs are not functions if there are 2 y-values for multiple x-value.

Tables or Ordered Pairs

Mathematics


Students read tables or look at a set of ordered pairs to determine functions and identify equations

where there is only one output (y-value) for each input (x-value).

Equations

Students recognize equations such as y = x or y = + 3x + 4 as functions; whereas, equations

such as +

= 25 are not functions.

8.F.2 Compare properties of two functions each

represented in a different way (algebraically,

graphically, numerically in tables, or by verbal

descriptions). For example, given a linear

function represented by a table of values and a

linear function represented by an algebraic

expression, determine which function has the

greater rate of change.

Students compare two functions from different representations.

NOTE: Functions could be expressed in standard form. However, the intent is not to change from

standard form to slope-intercept form but to use the standard form to generate ordered pairs.

Substituting a zero (0) for x and y will generate two ordered pairs. From these ordered pairs, the

slope could be determined.

8.F.3 Interpret the equation y = mx + b as

defining a linear function, whose graph is a

straight line; give examples of functions that are

not linear. For example, the function A = s²

giving the area of a square as a function of its

side length is not linear because its graph

contains the points (1,1), (2,4) and (3,9), which

are not on a straight line.

Students understand that linear functions have a constant rate of change between any two points.

Students use equations, graphs and tables to categorize functions as linear or non-linear.

Component Cluster 8.G Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

8.G.9. Know the formulas for the volumes of

cones, cylinders and spheres and use them to

solve real-world and mathematical problems.*

* Solutions that introduce rational numbers are

not introduced until Unit 7.

Students build on understandings of circles and volume from 7th grade to find the volume of

cylinders, finding the area of the base Π and multiplying by the number of layers (the height).

Students understand that the volume of a cylinder is 3 times the volume of a cone having the

same base area and

height or that the volume of a cone is 1/3 the volume of a cylinder having the same base area and

height.

A sphere can be enclosed with a cylinder, which has the same radius and height of the sphere

(Note: the height of the cylinder is twice the radius of the sphere). If the sphere is flattened, it will

Mathematics


fill 2/3 of the cylinder. Based on this model, students understand that the volume of a sphere is2/3

the volume of a cylinder with the same radius and height. The height of the cylinder is the same

as the diameter of the sphere or 2r. Using this information, the formula for the volume of the

sphere can be derived in the following way:

cylinder volume formula

multiply by 2/3 since the volume of a sphere is 2/3 the cylinder’s volume

substitute 2r for height since 2r is the height of the sphere

simplify

Students find the volume of cylinders, cones and spheres to solve real world and mathematical

problems. Answers could also be given in terms of Pi.

“Know the formula” does not mean memorization of the formula. To “know” means to have an

understanding of why the formula works and how the formula relates to the measure (volume)

and the figure. This understanding should be for all students.

Note: At this level composite shapes will not be used and only volume will be calculated.

Mathematics


Grade 8 Unit 7: Patterns and Data

Unit Overview: In Unit 7, students return to linear functions in the context of statistics and probability as bivariate data provides support in the use of

linear functions.

Guiding Questions: How are linear functions used in the context of statistics and probability?


Component Cluster 8.SP Investigate patterns of association in bivariate data.

8.SP.1 Construct and interpret scatter plots for

bivariate measurement data to investigate

patterns of association between two quantities.

Describe patterns such as clustering, outliers,

positive or negative association, linear

association, and nonlinear association.

Bivariate data refers to two-variable data, one to be graphed on the x-axis and the other on the y-

axis.

Students represent numerical data on a scatter plot, to examine relationships between variables.

They analyze scatter plots to determine if the relationship is linear (positive, negative association

or no association) or nonlinear. Students can use tools such as those at the National Center for

Educational Statistics to create a graph or generate data sets.

(http://nces.ed.gov/nceskids/createagraph/default.aspx)

Data can be expressed in years. In these situations it is helpful for the years to be “converted” to

0, 1, 2, etc. For example, the years of 1960, 1970, and 1980 could be represented as 0 (for 1960),

10 (for 1970) and 20 (for 1980).

Students recognize that points may be away from the other points (outliers) and have an effect on

the linear model.

NOTE: Use of the formula to identify outliers is not expected at this level.

Students recognize that not all data will have a linear association. Some associations will be non-

linear as in the

example below:

8.SP.2 Know that straight lines are widely used

to model relationships between two quantitative

variables. For scatter plots that suggest a linear

association, informally fit a straight line, and

informally assess the model fit by judging the

Students understand that a straight line can represent a scatter plot with linear association. The

most appropriate linear model is the line that comes closest to most data points. The use of linear

regression is not expected. If there is a linear relationship, students draw a linear model. Given a

linear model, students write an equation.

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closeness of the data points to the line.

8.SP.3 Use the equation of a linear model to

solve problems in the context of bivariate

measurement data, interpreting the slope and

intercept. For example, in a linear model for a

biology experiment, interpret a slope of 1.5

cm/hr as meaning that an additional hour of

sunlight each day is associated with an

additional 1.5 cm in mature plant height.

Linear models can be represented with a linear equation. Students interpret the slope and y-

intercept of the line in the context of the problem.

8.SP.4 Understand that patterns of association

can also be seen in bivariate categorical data by

displaying frequencies and relative frequencies

in a two-way table. Construct and interpret a

two-way table summarizing data on two

categorical variables collected from the same

subjects. Use relative frequencies calculated for

rows or columns to describe possible association

between the two variables. For example, collect

data from students in your class on whether or

not they have a curfew on school nights and

whether or not they have assigned chores at

home. Is there evidence that those who have a

curfew also tend to have chores?

Students understand that a two-way table provides a way to organize data between two

categorical variables. Data for both categories needs to be collected from each subject. Students

calculate the relative frequencies to describe associations.

Mathematics


NQ = Number and Quantity A = Algebra F = Functions M = Modeling G= Geometry S= Statistics and Probability

8th

Grade Algebra I The fundamental purpose of this course is to complete, formalize and extend the mathematics that students have learned in the sixth and seventh grades. Because this accelerated

course will be completing, incorporating and building off of the totality of the middle grades standards, this is an even more ambitious version of Algebra I than has generally been

offered. The critical areas, called units, deepen and extend understanding of linear and exponential relationships by contrasting them with each other and by applying linear models

to data that exhibit a linear trend, and students engage in methods for analyzing, solving, and using quadratic functions. The Mathematical Practice Standards apply throughout

each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to

make sense of problem situations.

Through the review and completion of the middle grades work, students will have learned to solve linear equations in one variable and how to apply graphical and algebraic

methods to analyze and solve systems of linear equations in two variables. Then, students analyze and explain the process of solving an equation. Students develop fluency writing,

interpreting, and translating between various forms of linear equations and inequalities, and using the different modalities to solve problems. They master the solution of linear

equations and apply related solution techniques and the laws of exponents to the creation and solution of simple exponential equations.

Students also complete their understanding of how to define, evaluate, and compare functions, and use them to model relationships between quantities. Then students will learn

function notation and develop the concepts of domain and range. They explore many examples of functions, including sequences; they interpret functions given graphically,

numerically, symbolically, and verbally, translate between representations, and understand the limitations of various representations. Students build on and informally extend their

understanding of integer exponents to consider exponential functions. They compare and contrast linear and exponential functions, distinguishing between additive and

multiplicative change. Students explore systems of equations and inequalities, and they find and interpret their solutions. They interpret arithmetic sequences as linear functions

and geometric sequences as exponential functions.

Building upon students’ prior experiences with data, students are now provided with more formal means of assessing how a model fits data. Students use regression techniques to

describe approximately linear relationships between quantities. They use graphical representations and knowledge of the context to make judgments about the appropriateness of

linear models. With linear models, they look at residuals to analyze the goodness of fit.

Students apply this new understanding of number and rules of exponents to strengthen their ability to see structure in and create quadratic and exponential expressions. They create

and solve equations, inequalities, and systems of equations involving quadratic expressions.

Students will also consider quadratic functions, comparing the key characteristics of quadratic functions to those of linear and exponential functions. They select from among these

functions to model phenomena. Students learn to anticipate the graph of a quadratic function by interpreting various forms of quadratic expressions. In particular, they identify the

real solutions of a quadratic equation as the zeros of a related quadratic function. Students expand their experience with functions to include more specialized functions—absolute

value, step, and those that are piecewise-defined.










Mathematics



8th

Grade Algebra I: Suggested Distribution of Units in Instructional Days Time # of Weeks

Unit 0: Geometry 5 weeks

Unit 1: Patterns 3 weeks

Unit 2: Linear Equations and Inequalities 5 weeks

Unit 3: Functions 3 weeks

Unit 4: Linear Functions 5 weeks

Unit 5: Scatter Plots and Trend Lines 3 weeks

Unit 6: Systems of Equations 3 weeks

Unit 7: Introduction to Exponential Functions 3 weeks

Unit 8: Quadratic Functions and Equations 4 weeks

Performance Task 2 weeks


Unit 1: Patterns 8%

Unit 2: Linear Equations and

Inequalities 15%

Unit 3: Functions 8%


15%

Unit 5: Scatter Plots and Trend

Lines 9%

Unit 6: Systems of Equations

9%

Unit 7: Introduction to

Exponential Functions

9%

Unit 8: Quadratic Functions and

Equations 12%

Instructional Time

Mathematics



8th

Grade Algebra I Unit 0: Geometry (~ 3 weeks)

Unit Overview: The 8th

grade year begins with a mini-geometry unit that completes the foundation for high school geometry before diving deeply

into the Algebra curriculum. Students will learn and explain a proof of the Pythagorean Theorem on their own. The Pythagorean Theorem is used to

motivate a discussion and review of irrational square roots. This is followed by an experimental study of rotations, reflections, and translations which

prepares students for the more complex later work of understanding the effects of dilations on geometrical figures in their study of similarity. They

also use informal arguments to establish ideas about angle sums, parallel lines cut by transverals, and similar triangles. Unit 0 concludes with

revisiting a proof of the Pythagorean Theorem from the perspective of similar triangles.

Guiding Questions:

How are deductive reasoning and informal arguments used to prove the Pythagorean Theorem?

How are congruent figures represented in transformations? How are congruent triangles used to determine relationships that exist in triangles and parallel lines?

How are similar figures represented in transformations and what is the relationship between congruent and similar figures?



8.G.1 Verify experimentally the properties of

rotations, reflections, and translations:

d. Lines are taken to lines, and line segments to

line segments of the same length.

e. Angles are taken to angles of the same

measure.

f. Parallel lines are taken to parallel lines.

Students use compasses, protractors and rulers or technology to explore figures created from

translations, reflections and rotations. Characteristics of figures, such as lengths of line segments,

angle measures and parallel lines, are explored before the transformation (pre-image) and after

the transformation (image). Students understand that these transformations produce images of

exactly the same size and shape as the pre-image and are known as rigid transformations.


is congruent to another if the second can be


rotations, reflections and translations; given two

congruent figures, describe a sequence that

exhibits the congruence between them.

This standard is the students’ introduction to congruency. Congruent figures have the same shape

and size. Translations, reflections and rotations are examples of rigid transformations. A rigid

transformation is one in which the pre-image and the image both have exactly the same size and

shape since the measures of the corresponding angles and corresponding line segments remain

equal (are congruent).

Students examine two figures to determine congruency by identifying the rigid transformation(s)

that produced the figures. Students recognize the symbol for congruency (≅) and write statements

of congruency.

8.G.3 Describe the effect of translations,

rotations, and reflections on two-dimensional

figures using coordinates.




Translations

Translations move the object so that every point of the object moves in the same direction as well

as the same distance. In a translation, the translated object is congruent to its pre-image.

Mathematics



Reflections

A reflection is the “flipping” of an object over a line, known as the “line of reflection”. In the 8th

grade, the line of reflection will be the x-axis and the y-axis. Students recognize that when an

object is reflected across the y-axis, the reflected x-coordinate is the opposite of the pre-image x-

coordinate.

Rotations

A rotation is a transformation performed by “spinning” the figure around a fixed point known as

the center of rotation. The figure may be rotated clockwise or counterclockwise up to 360º (at 8th

grade, rotations will be around the origin and a multiple of 90º). In a rotation, the rotated object is

congruent to its pre-image.

Students recognize the relationship between the coordinates of the pre-image and the image.

Students identify the transformations based on the given coordinates.


8.G.3 Describe the effect of dilations on two-

dimensional figures using coordinates.




Dilations

A dilation is a non-rigid transformation that moves each point along a ray which starts from a

fixed center, and multiplies distances from this center by a common scale factor. Dilations

enlarge (scale factors greater than one) or reduce (scale factors less than one) the size of a figure

by the scale factor. In 8th grade, dilations will be from the origin. The dilated figure is similar to

its pre-image.

Students recognize the relationship between the coordinates of the pre-image, the image and the

scale factor for a dilation from the origin. Using the coordinates, students are able to identify the

scale factor (image/pre-image).

Students identify the transformation based on given coordinates.


is similar to another if the second can be


rotations, reflections, translations, and dilations;

given two similar two-dimensional figures,

describe a sequence that exhibits the similarity

between them.

Similar figures and similarity are first introduced in the 8th

grade. Students understand similar

figures have congruent angles and sides that are proportional. Similar figures are produced from

dilations. Students describe the sequence that would produce similar figures, including the scale

factors. Students understand that a scale factor greater than one will produce an enlargement in

the figure, while a scale factor less than one will produce a reduction in size.

Students need to be able to identify that triangles are similar or congruent based on given

information.

Mathematics



Component Cluster 8.G Understand and apply the Pythagorean Theorem.

8.G.6 Explain a proof of the Pythagorean

Theorem and its converse.

Using models, students explain the Pythagorean Theorem, understanding that the sum of the

squares of the legs is equal to the square of the hypotenuse in a right triangle.

Students also understand that given three side lengths with this relationship forms a right triangle.

8.G.7 Apply the Pythagorean Theorem to

determine unknown side lengths in right

triangles in real-world and mathematical

problems in two and three dimensions.

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-

world and mathematical problems in two and three dimensions.

Based on this work, students could then find the volume or surface area.

8.G.8 Apply the Pythagorean Theorem to find

the distance between two points in a coordinate

system.

One application of the Pythagorean Theorem is finding the distance between two points on the

coordinate plane. Students build on work from 6th grade (finding vertical and horizontal distances

on the coordinate plane) to determine the lengths of the legs of the right triangle drawn

connecting the points. Students understand that the line segment between the two points is the

length of the hypotenuse.

8.G.5 Use informal arguments to establish facts

about the angle sum and exterior angle of

triangles, about the angles created when parallel

lines are cut by a transversal, and the angle-angle

criterion for similarity of triangles. For example,

arrange three copies of the same triangle so that

the sum of the three angles appears to form a

line, and give an argument in terms of

transversals why this is so.

Students use exploration and deductive reasoning to determine relationships that exist between

the following:

a) angle sums and exterior angle sums of triangles, b) angles created when parallel lines are cut

by a transversal, and c) the angle-angle criterion for similarity of triangle.

Students construct various triangles and find the measures of the interior and exterior angles.

Students make conjectures about the relationship between the measure of an exterior angle and

the other two angles of a triangle.

(the measure of an exterior angle of a triangle is equal to the sum of the measures of the other two

interior angles) and the sum of the exterior angles (360º). Using these relationships, students use

deductive reasoning to find the measure of missing angles.

Students construct parallel lines and a transversal to examine the relationships between the

created angles. Students recognize vertical angles, adjacent angles and supplementary angles from

7th grade and build on these relationships to identify other pairs of congruent angles. Using these

relationships, students use deductive reasoning to find the measure of missing angles.

Students can informally conclude that the sum of the angles in a triangle is 180º (the angle-sum

theorem) by applying their understanding of lines and alternate interior angles.

Students construct various triangles having line segments of different lengths but with two

corresponding congruent angles. Comparing ratios of sides will produce a constant scale factor,

meaning the triangles are similar. Students solve problems with similar triangles.

Mathematics



NOTE: The use of the distance formula is not an expectation.

Students find area and perimeter of two-dimensional figures on the coordinate plane, finding the

distance between each segment of the figure. (Limit one diagonal line, such as a right trapezoid or

parallelogram)

Mathematics



8th

Grade Algebra I Unit 1: Patterns (~ 4 weeks) Unit Overview: In Unit 1, students will have an opportunity to investigate patterns including arithmetic and geometric sequences for the purpose of

writing equations of various types.

Guiding Question: How can patterns describe real world phenomena?


Component Cluster 9-12.F.IF Understand the concept of a function and use function notation

9-12.F.IF.3 Recognize that sequences are

functions, sometimes defined recursively, whose

domain is a subset of the integers.

Patterns are developed using tables which are then used to develop the equations.

Component Cluster 9-12.F.BF Build a function that models a relationship between two quantities

9-12.F.BF.1 Write a function that describes a

relationship between two quantities.

a. Determine an explicit expression, a

recursive process, or steps for calculation

from a context.

Multiple activities build this standard using experiments and other real life situations.

9-12.F.BF.2 Write arithmetic and geometric

sequences both recursively and with an explicit

formula, use them to model situations, and

translate between the two forms.

Have students think about the starting point and how to get to the next level as they begin to

develop their function equations.

Mathematics



8th

Grade Algebra I Unit 2: Linear Equations and Inequalities (~ 5 weeks) Unit Overview: In Unit 2, students will write, simplify, evaluate and model situations with linear expressions.

Guiding Question: How can we use linear equations and inequalities to solve real world problems?



8.EE.7. Solve linear equations in one variable.

c. Give examples of linear equations in one

variable with one solution, infinitely many

solutions, or no solutions. Show which of

these possibilities is the case by successively

transforming the given equation into simpler

forms, until an equivalent equation of the

form x = a, a = a, or a = b results (where a

and b are different numbers).

d. Solve linear equations with rational number

coefficients, including equations whose

solutions require expanding expressions

using the distributive property and collecting

like terms.

Students solve one-variable equations including those with the variables being on both sides of

the equals sign. Students recognize that the solution to the equation is the value(s) of the variable,

which make a true equality when substituted back into the equation. Equations shall include

rational numbers, distributive property and combining like terms.

Equations have one solution when the variables do not cancel out. If each side of the equation

were treated as a linear equation and graphed, the solution of the equation represents the

coordinates of the point where the two lines would intersect.

Equations having no solution have variables that will cancel out and constants that are not equal.

This means that there is not a value that can be substituted for x that will make the sides equal.

This solution means that no matter what value is substituted for x the final result will never be

equal to each other.

If each side of the equation were treated as a linear equation and graphed, the lines would be

parallel.

An equation with infinitely many solutions occurs when both sides of the equation are the same.

Any value of x will produce a valid equation. If each side of the equation were treated as a linear

equation and graphed, the graph would be the same line.

Students write equations from verbal descriptions and solve.

Component Cluster 9-12.A.SSE Interpret the structure of expressions

9-12.A.SSE.1 Interpret expressions that

represent a quantity in terms of its context.

a. Interpret parts of an expression, such as

terms, factors, and coefficients

b. Interpret complicated expressions by

viewing one or more of their parts as a

single entity.

For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-

1) for n > 1. Continue to focus on the terms in the context: what is the starting point and what is

the change? Repeated addition or repeated multiplication?

Component Cluster 9-12.A.SSE Write expressions in equivalent forms to solve problems

Mathematics



9-12.A.SSE.3 Choose and produce an

equivalent form of an expression to reveal and

explain properties of quantity represented by the

expression.

Simplifying the initial equations can make them easier to understand.

Component Cluster 9-12.A.CED Create equations that describe numbers or relationships

9-12.A.CED.1 Create equations and inequalities

in one variable and use them to solve problems.

Once the equation is developed it can be used to predict what will happen in a much larger

situation or one where the data was not given.

9-12.A.CED.4 Rearrange formulas to highlight a

quantity of interest, using the same reasoning as

in solving equations

Solving for y might allow it to be easier to graph given their graphing background in earlier

grades.

Component Cluster 9-12.A.REI Understanding solving equations as a process of reasoning and explain the reasoning

9-12.A.REI.1 Explain each step in solving a

simple equation as following from the equality

of numbers asserted at the previous step, starting

from the assumption that the original equation

has a solution. Construct a viable argument to

justify a solution method.

Use the addition property of equality, the multiplication property of equality, etc. and refer to

them by name so students become familiar with them.

Component Cluster 9-12.A.REI Solve equations and inequalities in one variable

9-12.A.REI.3 Solve linear equations and

inequalities in one variable, including equations

with coefficients represented by letters.

Continue to reinforce solving skills developed in earlier grades. Solving of literal equations is an

extension of solving regular equations.

Component Cluster 9-12.N-Q.1 Reason quantitatively and use units to solve problems

9-12.N-Q.1 Use units as a way to understand

problems and to guide the solution of multi-step

problems; choose and interpret units consistently

in formulas.

Have students use units as a way of connecting back to the concrete of the real world context.

9-12.N-Q.2 Define appropriate quantities for the

purpose of descriptive modeling.

Students should be able to determine what variable is being represented.

9-12.N-Q.3 Choose a level of accuracy

appropriate to limitations on measurements when

reporting quantities.

It helps to keep the context in mind when determining values. For example, does it make sense to

have a negative number of people? Or does zero of something make sense in number of steps?

Mathematics



8th

Grade Algebra I Unit 3: Functions (~ 3 weeks) Unit Overview: In Unit 3, students will develop a definition for relations and, subsequently, functions and determine if relations are functions.

Multiple representations are used (tables, mappings, graphs, ordered pairs, verbal descriptions and equations).

Guiding Question: What is a function and how can it be represented?


Component Cluster 8.F Define, evaluate, and compare functions.*

* Linear and non-linear functions are compared in this module using linear equations and area/volume formulas as examples.

8.F.1 Understand that a function is a rule that

assigns to each input exactly one output. The

graph of a function is the set of ordered pairs

consisting of an input and the corresponding

output.*

*Function notation is not required in

Grade 8.

Students understand rules that take x as input and gives y as output is a function. Functions occur

when there is exactly one y-value is associated with any x-value. Using y to represent the output

we can represent this function with the equations y = x2+ 5x + 4. Students are not expected to use

the function notation f(x) at this level. Students identify functions from equations, graphs, and

tables/ordered pairs.

Graphs

Students recognize graphs such as the one below is a function using the vertical line test, showing

that each x-value

has only one y-value;

whereas, graphs are not functions if there are 2 y-values for multiple x-value.

Tables or Ordered Pairs

Students read tables or look at a set of ordered pairs to determine functions and identify equations

where there is only one output (y-value) for each input (x-value).

Equations

Students recognize equations such as y = x or y = + 3x + 4 as functions; whereas, equations

such as +

= 25 are not functions.

Mathematics




8.F.5 Describe qualitatively the functional

relationship between two quantities by analyzing

a graph, (e.g. where the function is increasing or

decreasing, linear or nonlinear). Sketch a graph

that exhibits the qualitative features of a function

that has been described verbally.

Given a verbal description of a situation, students sketch a graph to model that situation. Given a

graph of a situation, students provide a verbal description of the situation.


9-12.A.CED.2 Create equations in two or more

variables to represent relationships between

quantities; graph equations on coordinate axes

with labels and scales.

The standard is an extension of those developed in earlier grades. Students should be reaching

this standard more consistently.

Component Cluster 9-12.A.REI Represent and solve equations and inequalities graphically

9-12.A.REI.10 Understand that the graph of an

equation in two variables is the set of all its

solutions plotted in the coordinate plane, often

forming a curve (which could be a line).

Context may also dictate that graphs are connected or not connected. For example, number of

people cannot be partial so the graph would consist of points. Explain how one might want to

connect the points anyway to see a “model” that could predict future trends.


9-12.F.IF.1 Understand that a function from one

set (called the domain) to another set (called the

range) assigns to each element of the domain

exactly one element of the range. If f is a

function and x is an element of its domain, then

f(x) denotes the output of f corresponding to the

input x. The graph of f is the graph of the

equation y=f(x).

The definition of a function is key to future mathematics. This concept will continue to thread

through all course work.

8.F.2 Compare properties of two functions each

represented in a different way (algebraically,

graphically, numerically in tables, or by verbal

descriptions). For example, given a linear

function represented by a table of values and a

linear function represented by an algebraic

expression, determine which function has the

greater rate of change.

Students compare two functions from different representations.

NOTE: Functions could be expressed in standard form. However, the intent is not to change from

standard form to slope-intercept form but to use the standard form to generate ordered pairs.

Substituting a zero (0) for x and y will generate two ordered pairs. From these ordered pairs, the

slope could be determined.

Mathematics



9-12.F.IF.2 Use function notation, evaluate

functions for inputs in their domains, and

interpret statements that use function notation in

terms of context.

F(x) notation is used and must be distinguished between f x, a common misconception. The use

of words written out “f of x” can help this misunderstanding. Also, while f is often the function

name, other letters are used depending on the context.

Component Cluster 9-12.F.IF Interpret functions that arise in applications in terms of the context

9-12.F.IF.4 For a function that models a

relationship between two quantities, interpret key

features of graphs and tables in terms of the

quantities and sketch graphs showing key

features given a verbal description of the

relationship.

Connecting to the context is important.

9-12.F.IF.5 Relate the domain of a function to

its graph, and where applicable, to the

quantitative relationship it describes.

Positive domains produce graphs that are in the first quadrant for most of the examples here.

Component Cluster 9-12.F.IF Analyze functions using different representations

9-12.F.IF.7 Graph functions expressed

symbolically and show key features of the graph,

by hand in simple cases and using technology for

more complicated cases.

b.Graph square root, cube root, and

piece-wise functions, including step

functions and absolute value functions.

Use of the graphing calculator and analysis to help determine an appropriate window needs to be

stressed. By hand, scale and labeling appropriately is important. (b) identifies functions that may

be included in the parent function discussion.

9-12.F.IF.9 Compare properties of two

functions each represented in a different way

(algebraically, graphically, numerically in tables,

or by verbal description).

Comparison helps identify key similarities and differences between two different functions or

between two functions of the same family.

Mathematics



8th

Grade Algebra I Unit 4: Linear Functions (~ 6 weeks) Unit Overview: In Unit 4, students thoroughly explore linear functions, deriving linear models to describe behavior, talking about rates of change,

recognizing linear relationships from tables and graphs, followed by the development of equations of lines using slope intercept and point slope form.

Guiding Question: How may linear functions help us analyze real world situations and solve practical problems?



9-12.F.IF.6 Calculate and interpret the average

rate of change of a function (presented

symbolically or as a table) over a specified

interval. Estimate the rate of change from a

graph.

In linear functions this average rate of change is going to connect to the slope of the function and

the graph.






a. Graph linear functions and show

intercepts.

Linear functions only are emphasized in this unit.

9-12.F.IF.8 Write a function defined by an

expression in different by equivalent forms to

reveal and explain different properties of the

function.

Manipulation between different forms of a linear function will be helpful in graphing different

forms.

Component Cluster 9-12.F.LE Construct and compare linear, quadratic, and exponential models and solve problems

9-12.F.LE.1 Distinguish between situations that

can be modeled with linear functions and with

exponential functions.

a. Prove that linear functions grow by equal

differences over interval.

b. Recognize situations in which one

quantity changes at a constant rate per

unit interval relative to another.

This goes back to the first unit when linear and exponential functions were developed in the

patterns. Linear functions will have a common addition term while exponential functions will be

repeated multiplications.

9-12.F.LE.2 Construct linear and exponential

functions, including arithmetic and geometric

sequences, given a graph, a description of a

relationship, or two input-output pairs (including

reading these from a table).

Connections are made to the first unit pattern development and now identifying those functions as

linear and exponential.

Mathematics



Component Cluster 9-12.F.LE

9-12.F.LE.5 Interpret the parameters in a linear

… function in terms of context.

Context can help determine intercepts and slope as well as domain if there are any restrictions.

Mathematics



8th

Grade Algebra I Unit 5: Scatter Plots and Trend Lines (~ 4 weeks) Unit Overview: In Unit 5, students will use data and regression capabilities to find the best fit line.

Guiding Question: How do we make predictions and informed decisions based on current numerical information?


8.SP.3 Use the equation of a linear model to

solve problems in the context of bivariate

Linear models can be represented with a linear equation. Students interpret the slope and y-

intercept of the line in the context of the problem.

Component Cluster 8.SP Investigate patterns of association in bivariate data.

8.SP.1 Construct and interpret scatter plots for

bivariate measurement data to investigate

patterns of association between two quantities.

Describe patterns such as clustering, outliers,

positive or negative association, linear

association, and nonlinear association.

Bivariate data refers to two-variable data, one to be graphed on the x-axis and the other on the y-

axis.

Students represent numerical data on a scatter plot, to examine relationships between variables.

They analyze scatter plots to determine if the relationship is linear (positive, negative association

or no association) or nonlinear. Students can use tools such as those at the National Center for

Educational Statistics to create a graph or generate data sets.

(http://nces.ed.gov/nceskids/createagraph/default.aspx)

Data can be expressed in years. In these situations it is helpful for the years to be “converted” to

0, 1, 2, etc. For example, the years of 1960, 1970, and 1980 could be represented as 0 (for 1960),

10 (for 1970) and 20 (for 1980).

Students recognize that points may be away from the other points (outliers) and have an effect on

the linear model.

NOTE: Use of the formula to identify outliers is not expected at this level.

Students recognize that not all data will have a linear association. Some associations will be non-

linear as in the

example below:

8.SP.2 Know that straight lines are widely used

to model relationships between two quantitative

variables. For scatter plots that suggest a linear

association, informally fit a straight line, and

informally assess the model fit by judging the

closeness of the data points to the line.

Students understand that a straight line can represent a scatter plot with linear association. The

most appropriate linear model is the line that comes closest to most data points. The use of linear

regression is not expected. If there is a linear relationship, students draw a linear model. Given a

linear model, students write an equation.

Mathematics



measurement data, interpreting the slope and

intercept. For example, in a linear model for a

biology experiment, interpret a slope of 1.5

cm/hr as meaning that an additional hour of

sunlight each day is associated with an

additional 1.5 cm in mature plant height.

8.SP.4 Understand that patterns of association

can also be seen in bivariate categorical data by

displaying frequencies and relative frequencies

in a two-way table. Construct and interpret a

two-way table summarizing data on two

categorical variables collected from the same

subjects. Use relative frequencies calculated for

rows or columns to describe possible association

between the two variables. For example, collect

data from students in your class on whether or

not they have a curfew on school nights and

whether or not they have assigned chores at

home. Is there evidence that those who have a

curfew also tend to have chores?

Students understand that a two-way table provides a way to organize data between two

categorical variables. Data for both categories needs to be collected from each subject. Students

calculate the relative frequencies to describe associations.

Component Cluster 9-12.S.ID Summarize, represent, and interpret data on a single count or measurement variable

9-12.S.ID 2 Use statistics appropriate to the

shape of the data distribution to compare center

(median, mean) and spread (interquartile range,

standard deviation) of two or more different data

sets.

Graphing calculator statistical features are used. Analysis of how to mathematically determine

outliers is introduced.

9-12.S.ID 3 Interpret differences in shape,

center, and spread in the context of the data sets,

accounting for possible effects of extreme data

points (outliers).

Component Cluster 9-12.S.ID Summarize, represent, and interpret data on two categorical and quantitative variables

9-12.S.ID 6 Represent data on two quantitative

variables on a scatter plot, and describe how the

variables are related.

a. Fit a function to the data; use functions

fitted to data to solve problems in the

context of the data.

Best fit lines are done by hand and using technology.

Mathematics



c. Fit a linear function for a scatter plot

that suggests a linear association.

Component Cluster 9-12.S.ID Interpret linear models

9-12.S.ID.7 Interpret the slope (rate of change)

and the intercept (constant term) of a linear

model in the context of the data.

Students may use spreadsheets or graphing calculators to create representations of data sets and

create linear models.

Mathematics



8th

Grade Algebra I Unit 6: Systems of Linear Equations (~ 3 weeks) Unit Overview: In Unit 6, students will represent, compare and analyze two linear equations, look for common solutions and use this information to

make choices between competing situations in real world contexts.

Guiding Question: What does the number of solutions (none, one or infinite) of a system of linear equations represent? What are the advantages and disadvantages of solving a

system of linear equations graphically versus algebraically?



9-12.A.CED.3 Represent constraints by

equations or inequalities, and by systems of

equations and/or inequalities, and interpret

solutions as viable or nonviable options in a

modeling context.

Development of equations from context is one of the key focuses of this problem solving related

unit.


8.EE.8. Analyze and solve pairs of simultaneous

linear equations.

a. Understand that solutions to a system of

two linear equations in two variables

correspond to points of intersection of their

graphs, because points of intersection

satisfy both equations simultaneously.

b. Solve systems of two linear equations in

two variables algebraically, and estimate

solutions by graphing the equations. Solve

simple cases by inspection. For example,

3x + 2y = 5 and 3x + 2y = 6 have no

solution because 3x + 2y cannot

simultaneously be 5 and 6.

c. Solve real-world and mathematical

problems leading to two linear equations in

two variables. For example, given

coordinates for two pairs of points,

determine whether the line through the

first pair of points intersects the line

through the second pair.

Systems of linear equations can also have one solution, infinitely many solutions or no solutions.

Students will discover these cases as they graph systems of linear equations and solve them

algebraically.

Students graph a system of two linear equations, recognizing that the ordered pair for the point of

intersection is the x-value that will generate the given y-value for both equations. Students

recognize that graphed lines with one point of intersection (different slopes) will have one

solution, parallel lines (same slope, different y-intercepts) have no solutions, and lines that are the

same (same slope, same y-intercept) will have infinitely many solutions.

By making connections between algebraic and graphical solutions and the context of the system

of linear equations, students are able to make sense of their solutions. Students need opportunities

to work with equations and context that include whole number and/or decimals/fractions.

Students define variables and create a system of linear equations in two variables.

Note: Students are not expected to change linear equations written in standard form to slope-

intercept form or solve systems using elimination.

For many real world contexts, equations may be written in standard form. Students are not

expected to change the standard form to slope-intercept form. However, students may generate

ordered pairs recognizing that the values of the ordered pairs would be solutions for the equation.

Component Cluster 9-12.A.REI Solve systems of equations

Mathematics



9-12.A.REI.5 Prove that, given a system of two

equations in two variables, replacing one

equation by the sum of that equation and a

multiple of the other produces a system with the

same solutions.

This standard helps develop the method of elimination.

9-12.A.REI.6 Solve systems of linear equations

exactly and approximately (e.g., with graphs),

focusing on pairs of linear equations in two

variables.

Graphical solutions are limited in larger scale or in decimal solutions. This leads to need for

additional solution methods.

Component Cluster 9-12.REI Represent and solve equations and inequalities graphically

9-12.A.REI.11 Explain why the x-coordinates

of the points where the graphs of the equations

y= f(x) and y =g(x) intersect are the solutions of

the equation f(x) = g(x); find the solutions

approximately, e.g., using technology to graph

the functions, make tables of values, or find

successive approximations. Include cases where

f(x) and/or g(x) are linear functions.*

Graphing technology can help produce graphical and table solutions, or at least approximations of

those solutions.

Mathematics



8th

Grade Algebra I Unit 7: Introduction to Exponential Functions (~ 4 weeks) Unit Overview: In Unit 7, students will explore relationships that grow exponentially.

Guiding Question: How can exponential functions be used to model real world situations?
















positive.




Component Cluster 9-12.N.RN Extend the properties of exponents to rational numbers

9-12.N.RN.1 Explain how the definition of the

meaning of rational exponents follows from

extending the properties of integer exponents to

those values, allowing for a notation of radicals

in terms of rational exponents.

Properties of exponents are developed in earlier grades and extended to rational exponents in this

unit.

9-12.N.RN.2 Rewrite expression involving

radicals and rational exponents using the

properties of exponents.

3 23

2

bb


9-12.A.SSE.1b Interpret complicated

expressions by viewing one or more of their

parts as a single entity.

Focus on base, exponent, root, etc.


9-12.A.SSE.3c Complete the square in a

quadratic expression to reveal the maximum or

minimum value of the function it defines.

Process of completing the square to solve is used in the next unit.


Mathematics



9-12.F.IF.7e Graph exponential functions,

showing intercepts and end behavior.

Graph by hand and using technology.

9-12.F.IF.8b Use the properties of exponents to

interpret expressions for exponential functions.






Connections back to the first unit on patterns make the concept clearer to students.





a. Prove that linear functions grow by equal

differences over equal intervals, and that

exponential functions grow by equal

factors over equal intervals.

b. Recognize situations in which one



c. Recognize situations in which a quantity

grows or decays by a constant percent

rate per unit interval relative to another.

Linear functions are compared and contrasted against exponential functions.

9-12.F.LE.3 Observe using graphs and tables

that a quantity increasing exponentially

eventually exceeds a quantity increasing linearly,

quadratically, or (more generally) as a

polynomial function.

This may require extending the domain past any local linearity.

Component Cluster 9-12.F.LE Interpret expressions for functions in terms of the situation they model


or exponential function in terms of context.

Identify beginning, change and type of change.




Mathematics


















9-12.F.IF.7e Graph exponential and logarithmic

functions, showing intercepts and end behavior.












d. Prove that linear functions grow by equal




e. Recognize situations in which one



Mathematics



f. Recognize situations in which a quantity











Mathematics



8th

Grade Algebra I Unit 8: Quadratic Functions and Equations (~ 5 weeks) Unit Overview: In Unit 8, students will model situations with quadratic functions. They will find and interpret intercepts, maxima and minima, and

determine symmetries. Students will solve quadratic equations by factoring, completing the square and the quadratic equation.

Guiding Question: What can the zeros, intercepts, vertex, maximum, minimum and other features of a quadratic function tell you about real world relationships?
















positive.




Component Cluster 9-12.A.SSE Write expression in equivalent forms to solve problems

9-12.A.SSE 3 Choose and produce an


explain properties of the quantity represented by

the expression.

a. Factor a quadratic expression to reveal

the zeros of the function it defines.

b. Complete the square in a quadratic

expression to reveal the maximum or


Rewriting quadratic functions can help solve them in different ways.


9-12.A.REI.4 Solve quadratic functions in one

variable.

a. Use the method of completing the square to

transform any quadratic equation in x into an

equation of the form that has the

same solutions. Derive the quadratic formula

from this form. Solve quadratic equations by

All methods of solving are used and analyzed for when one is more appropriate than another.

Mathematics



inspection (e.g., for ), taking square

roots, completing the square, the quadratic

formula and factoring, as appropriate to the

initial form of the equation.

Component Cluster 9-12.A.APR Perform arithmetic operations on polynomials

9-12.A.APR.1 Understand that polynomials

form a system analogous to the integers, namely,

they are closed under the operations of addition,

subtraction, and multiplication; add, subtract,

and multiply polynomials.

Polynomial operations are seen as an extension of the arithmetic operations on number systems.


9-12. A.CED.1 Create equations and inequalities


Focus in this unit is on quadratic functions.

9-12.A.CED. 2 Create equations in two or more




Graphing continues to emphasize using appropriate scales.


9-12.F.IF 4 For a function that models a



quantities, and sketch graphs showing key


relationship. Key features include: intercepts;

intervals where the function is increasing,

decreasing, positive, or

negative; relative maxima and minima;

symmetries...

Features are dependent on the type of function being used. In this unit quadratic functions are

emphasized.


9-12.F.IF. 7a. Graph ... quadratic functions and

show intercepts, maxima, and minima.

Graphing will be both by hand and using technology.

9-12.F.IF.8a Use the process of factoring and

completing the square in a quadratic function to

show zeros, extreme values, and symmetry of the

The connection between the symbolic and graphical representations is important to show key

features in the symbolic function.

Mathematics



graph, and interpret these in terms of a context.

Component Cluster 9-12.F.BF Build new functions from existing functions

9-12.F.BF.3 Identify the effect on the graph of

replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k)

for specific values of k (both positive and

negative); find the value of k given the graphs.

Experiment with cases and illustrate an

explanation of the effects on the graph using

technology...

Transformations of basic functions from the parent function are explored. This can be done using

graphing calculators to show the effect of changing parameters.

Mathematics


Algebra 1 The fundamental purpose of this course is to formalize and extend the mathematics that students learned in the middle grades. Because it is built on the middle grades standards,

this is a more ambitious version of Algebra I than has generally been offered. The critical areas, called units, deepen and extend understanding of linear and exponential

relationships by contrasting them with each other and by applying linear models to data that exhibit a linear trend, and students engage in methods for analyzing, solving, and using

quadratic functions. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as

a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations.

By the end of eighth grade, students have learned to solve linear equations in one variable and have applied graphical and algebraic methods to analyze and solve systems of linear

equations in two variables. Now, students analyze and explain the process of solving an equation. Students develop fluency writing, interpreting, and translating between various

forms of linear equations and inequalities, and using them to solve problems. They master the solution of linear equations and apply related solution techniques and the laws of

exponents to the creation and solution of simple exponential equations.

In earlier grades, students define, evaluate, and compare functions, and use them to model relationships between quantities. Now students will learn function notation and develop

the concepts of domain and range. They explore many examples of functions, including sequences; they interpret functions given graphically, numerically, symbolically, and

verbally, translate between representations, and understand the limitations of various representations. Students build on and informally extend their understanding of integer

exponents to consider exponential functions. They compare and contrast linear and exponential functions, distinguishing between additive and multiplicative change. Students

explore systems of equations and inequalities, and they find and interpret their solutions. They interpret arithmetic sequences as linear functions and geometric sequences as


Building upon prior students’ prior experiences with data, students are now provided with more formal means of assessing how a model fits data. Students use regression

techniques to describe approximately linear relationships between quantities. They use graphical representations and knowledge of the context to make judgments about the

appropriateness of linear models. With linear models, they look at residuals to analyze the goodness of fit.

Students apply this new understanding of number and rules of exponents to strengthen their ability to see structure in and create quadratic and exponential expressions. They create

and solve equations, inequalities, and systems of equations involving quadratic expressions.

Students will also consider quadratic functions, comparing the key characteristics of quadratic functions to those of linear and exponential functions. They select from among these

functions to model phenomena. Students learn to anticipate the graph of a quadratic function by interpreting various forms of quadratic expressions. In particular, they identify the

real solutions of a quadratic equation as the zeros of a related quadratic function. Students expand their experience with functions to include more specialized functions—absolute

value, step, and those that are piecewise-defined.










Mathematics


Algebra 1: Suggested Distribution of Units in Instructional Weeks Time # of Weeks

Unit 1: Patterns 11% 4 weeks

Unit 2: Linear Equations and Inequalities 14% 5 weeks

Unit 3: Functions 8% 3 weeks

Unit 4: Linear Functions 17% 6 weeks

Unit 5: Scatter Plots and Trend Lines 11% 4 weeks

Unit 6: Systems of Equations 8% 3 weeks

Unit 7: Introduction to Exponential Functions 11% 4 weeks

Unit 8: Quadratic Functions and Equations 15% 5 weeks

Performance Task 2 weeks

Unit 1: Patterns 11%

Unit 2: Linear Equations and

Inequalities 14%

Unit 3: Functions 8%


17%

Unit 5: Scatter Plots and Trend

Lines 11%

Unit 6: Systems of Equations

8%

Unit 7: Introduction to Exponoential

Functions 11%

Unit 8: Quadratic Functions and

Equations 15%

Instructional Time

Mathematics


Algebra 1 Unit 1: Patterns (~ 4 weeks) Unit Overview: In Unit 1, students will have an opportunity to investigate patterns including arithmetic and geometric sequences for the purpose of

writing equations of various types.

Guiding Question: How can patterns describe real world phenomena?



9-12.F.IF.3 Recognize that sequences are

functions, sometimes defined recursively, whose

domain is a subset of the integers.

Patterns are developed using tables which are then used to develop the equations.




b. Determine an explicit expression, a

recursive process, or steps for calculation

from a context.

Multiple activities build this standard using experiments and other real life situations.





Have students think about the starting point and how to get to the next level as they begin to

develop their function equations.

Mathematics


Algebra 1 Unit 2: Linear Equations and Inequalities (~ 5 weeks) Unit Overview: In Unit 2, students will write, simplify, evaluate and model situations with linear expressions.

Guiding Question: How can we use linear equations and inequalities to solve real world problems?





c. Interpret parts of an expression, such as

terms, factors, and coefficients

d. Interpret complicated expressions by

viewing one or more of their parts as a

single entity.

For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-

1) for n > 1. Continue to focus on the terms in the context: what is the starting point and what is

the change? Repeated addition or repeated multiplication?


9-12.A.SSE.3 Choose and produce an


explain properties of quantity represented by the

expression.

Simplifying the initial equations can make them easier to understand.


9-12.A.CED.1 Create equations and inequalities


Once the equation is developed it can be used to predict what will happen in a much larger

situation or one where the data was not given.

9-12.A.CED.4 Rearrange formulas to highlight a

quantity of interest, using the same reasoning as

in solving equations

Solving for y might allow it to be easier to graph given their graphing background in earlier

grades.

Component Cluster 9-12.A.REI Understanding solving equations as a process of reasoning and explain the reasoning

9-12.A.REI.1 Explain each step in solving a

simple equation as following from the equality

of numbers asserted at the previous step, starting

from the assumption that the original equation

has a solution. Construct a viable argument to

justify a solution method.

Use the addition property of equality, the multiplication property of equality, etc. and refer to

them by name so students become familiar with them.


9-12.A.REI.3 Solve linear equations and

inequalities in one variable, including equations

with coefficients represented by letters.

Continue to reinforce solving skills developed in earlier grades. Solving of literal equations is an

extension of solving regular equations.

Component Cluster 9-12.N-Q.1 Reason quantitatively and use units to solve problems

Mathematics


9-12.N-Q.1 Use units as a way to understand

problems and to guide the solution of multi-step

problems; choose and interpret units consistently

in formulas.

Have students use units as a way of connecting back to the concrete of the real world context.

9-12.N-Q.2 Define appropriate quantities for the

purpose of descriptive modeling.

Students should be able to determine what variable is being represented.

9-12.N-Q.3 Choose a level of accuracy

appropriate to limitations on measurements when

reporting quantities.

It helps to keep the context in mind when determining values. For example, does it make sense to

have a negative number of people? Or does zero of something make sense in number of steps?

Mathematics


Algebra 1 Unit 3: Functions (~ 3 weeks) Unit Overview: In Unit 3, students will develop a definition for relations and, subsequently, functions and determine if relations are functions.

Multiple representations are used (tables, mappings, graphs, ordered pairs, verbal descriptions and equations).








The standard is an extension of those developed in earlier grades. Students should be reaching

this standard more consistently.


9-12.A.REI.10 Understand that the graph of an

equation in two variables is the set of all its

solutions plotted in the coordinate plane, often

forming a curve (which could be a line).

Context may also dictate that graphs are connected or not connected. For example, number of

people cannot be partial so the graph would consist of points. Explain how one might want to

connect the points anyway to see a “model” that could predict future trends.


9-12.F.IF.1 Understand that a function from one

set (called the domain) to another set (called the

range) assigns to each element of the domain

exactly one element of the range. If f is a

function and x is an element of its domain, then

f(x) denotes the output of f corresponding to the

input x. The graph of f is the graph of the

equation y=f(x).

The definition of a function is key to future mathematics. This concept will continue to thread

through all course work.

9-12.F.IF.2 Use function notation, evaluate

functions for inputs in their domains, and

interpret statements that use function notation in

terms of context.

F(x) notation is used and must be distinguished between f x, a common misconception. The use

of words written out “f of x” can help this misunderstanding. Also, while f is often the function

name, other letters are used depending on the context.





Connecting to the context is important.

Mathematics


quantities and sketch graphs showing key


relationship.


its graph, and where applicable, to the


Positive domains produce graphs that are in the first quadrant for most of the examples here.






b.Graph square root, cube root, and

piece-wise functions, including step

functions and absolute value functions.

Use of the graphing calculator and analysis to help determine an appropriate window needs to be

stressed. By hand, scale and labeling appropriately is important. (b) identifies functions that may

be included in the parent function discussion.





Comparison helps identify key similarities and differences between two different functions or

between two functions of the same family.

Mathematics


Algebra 1 Unit 4: Linear Functions (~ 6 weeks) Unit Overview: In Unit 4, students thoroughly explore linear functions, deriving linear models to describe behavior, talking about rates of change,

recognizing linear relationships from tables and graphs, followed by the development of equations of lines using slope intercept and point slope form.

Guiding Question: How may linear functions help us analyze real world situations and solve practical problems?



9-12.F.IF.6 Calculate and interpret the average

rate of change of a function (presented

symbolically or as a table) over a specified

interval. Estimate the rate of change from a

graph.

In linear functions this average rate of change is going to connect to the slope of the function and

the graph.






b. Graph linear functions and show

intercepts.

Linear functions only are emphasized in this unit.


expression in different by equivalent forms to


function.

Manipulation between different forms of a linear function will be helpful in graphing different

forms.





c. Prove that linear functions grow by equal

differences over interval.

d. Recognize situations in which one



This goes back to the first unit when linear and exponential functions were developed in the

patterns. Linear functions will have a common addition term while exponential functions will be

repeated multiplications.

9-12.F.LE.2 Construct linear and exponential

functions, including arithmetic and geometric

sequences, given a graph, a description of a

relationship, or two input-output pairs (including

Connections are made to the first unit pattern development and now identifying those functions as

linear and exponential.

Mathematics


reading these from a table).

Component Cluster 9-12.F.LE


… function in terms of context.

Context can help determine intercepts and slope as well as domain if there are any restrictions.

Mathematics


Algebra 1 Unit 5: Scatter Plots and Trend Lines (~ 4 weeks) Unit Overview: In Unit 5, students will use data and regression capabilities to find the best fit line.

Guiding Question: How do we make predictions and informed decisions based on current numerical information?



9-12.S.ID 2 Use statistics appropriate to the

shape of the data distribution to compare center

(median, mean) and spread (interquartile range,

standard deviation) of two or more different data

sets.

Graphing calculator statistical features are used. Analysis of how to mathematically determine

outliers is introduced.

9-12.S.ID 3 Interpret differences in shape,

center, and spread in the context of the data sets,

accounting for possible effects of extreme data

points (outliers).

Component Cluster 9-12.S.ID Summarize, represent, and interpret data on two categorical and quantitative variables

9-12.S.ID 6 Represent data on two quantitative

variables on a scatter plot, and describe how the

variables are related.

a. Fit a function to the data; use functions

fitted to data to solve problems in the

context of the data.

c. Fit a linear function for a scatter plot

that suggests a linear association.

Best fit lines are done by hand and using technology.

Component Cluster 9-12.S.ID Interpret linear models

9-12.S.ID.7 Interpret the slope (rate of change)

and the intercept (constant term) of a linear

model in the context of the data.

Students may use spreadsheets or graphing calculators to create representations of data sets and

create linear models.

9-12.S.ID 8 Compute (using technology) and

interpret the correlation coefficient of a linear fit.

9-12.S.ID 9 Distinguish between correlation and

causation.

Mathematics


Algebra 1 Unit 6: Systems of Linear Equations (~ 3 weeks) Unit Overview: In Unit 6, students will represent, compare and analyze two linear equations, look for common solutions and use this information to

make choices between competing situations in real world contexts.

Guiding Question: What does the number of solutions (none, one or infinite) of a system of linear equations represent? What are the advantages and disadvantages of solving a

system of linear equations graphically versus algebraically?




equations or inequalities, and by systems of

equations and/or inequalities, and interpret

solutions as viable or nonviable options in a

modeling context.

Development of equations from context is one of the key focuses of this problem solving related

unit.

Component Cluster 9-12.A.REI Solve systems of equations

9-12.A.REI.5 Prove that, given a system of two

equations in two variables, replacing one

equation by the sum of that equation and a

multiple of the other produces a system with the

same solutions.

This standard helps develop the method of elimination.

9-12.A.REI.6 Solve systems of linear equations

exactly and approximately (e.g., with graphs),

focusing on pairs of linear equations in two

variables.

Graphical solutions are limited in larger scale or in decimal solutions. This leads to need for

additional solution methods.

Component Cluster 9-12.REI Represent and solve equations and inequalities graphically


of the points where the graphs of the equations

y= f(x) and y =g(x) intersect are the solutions of

the equation f(x) = g(x); find the solutions

approximately, e.g., using technology to graph

the functions, make tables of values, or find

successive approximations. Include cases where

f(x) and/or g(x) are linear functions.*

Graphing technology can help produce graphical and table solutions, or at least approximations of

those solutions.

Mathematics


Algebra 1 Unit 7: Introduction to Exponential Functions (~ 4 weeks) Unit Overview: In Unit 7, students will explore relationships that grow exponentially.

Guiding Question: How can exponential functions be used to model real world situations?








Properties of exponents are developed in earlier grades and extended to rational exponents in this

unit.




3 23

2

bb





Focus on base, exponent, root, etc.





Process of completing the square to solve is used in the next unit.


9-12.F.IF.7e Graph exponential functions,

showing intercepts and end behavior.

Graph by hand and using technology.








Connections back to the first unit on patterns make the concept clearer to students.


Mathematics





g. Prove that linear functions grow by equal




h. Recognize situations in which one



i. Recognize situations in which a quantity



Linear functions are compared and contrasted against exponential functions.






This may require extending the domain past any local linearity.




Identify beginning, change and type of change.

Mathematics


Algebra 1 Unit 8: Quadratic Functions and Equations (~ 5 weeks) Unit Overview: In Unit 8, students will model situations with quadratic functions. They will find and interpret intercepts, maxima and minima, and

determine symmetries. Students will solve quadratic equations by factoring, completing the square and the quadratic equation.

Guiding Question: What can the zeros, intercepts, vertex, maximum, minimum and other features of a quadratic function tell you about real world relationships?



9-12.A.SSE 3 Choose and produce an


explain properties of the quantity represented by

the expression.

c. Factor a quadratic expression to reveal

the zeros of the function it defines.

d. Complete the square in a quadratic

expression to reveal the maximum or


Rewriting quadratic functions can help solve them in different ways.


9-12.A.REI.4 Solve quadratic functions in one

variable.

b. Use the method of completing the square to

transform any quadratic equation in x into an

equation of the form that has the

same solutions. Derive the quadratic formula

from this form. Solve quadratic equations by

inspection (e.g., for ), taking square

roots, completing the square, the quadratic

formula and factoring, as appropriate to the

initial form of the equation.

All methods of solving are used and analyzed for when one is more appropriate than another.







Polynomial operations are seen as an extension of the arithmetic operations on number systems.


Mathematics


9-12. A.CED.1 Create equations and inequalities


Focus in this unit is on quadratic functions.

9-12.A.CED. 2 Create equations in two or more




Graphing continues to emphasize using appropriate scales.


9-12.F.IF 4 For a function that models a





relationship. Key features include: intercepts;

intervals where the function is increasing,

decreasing, positive, or

negative; relative maxima and minima;

symmetries...

Features are dependent on the type of function being used. In this unit quadratic functions are

emphasized.


9-12.F.IF. 7a. Graph ... quadratic functions and

show intercepts, maxima, and minima.

Graphing will be both by hand and using technology.

9-12.F.IF.8a Use the process of factoring and

completing the square in a quadratic function to

show zeros, extreme values, and symmetry of the

graph, and interpret these in terms of a context.

The connection between the symbolic and graphical representations is important to show key

features in the symbolic function.



replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k)





technology...

Transformations of basic functions from the parent function are explored. This can be done using

graphing calculators to show the effect of changing parameters.

Mathematics


Geometry Geometry is devoted primarily to plane Euclidean geometry, studied both synthetically (without coordinates) and analytically (with coordinates).

During high school, students begin to formalize their geometry experiences from elementary and middle school, using more precise definitions and

developing careful proofs.

The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation. Fundamental are the

rigid motions: translations, rotations, reflections, and combinations of these.

Three dimensional geometry is analyzed using two dimensional cross sections to inform new formulas for volume and surface area.

Circles are analyzed and properties developed, including an analytical approach to developing the equation for a circle.

Two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries one onto the other. For triangles, congruence

means the equality of all corresponding pairs of sides and all corresponding pairs of angles. Once these triangle congruence criteria (ASA, SAS, and

SSS) are established using rigid motions, they can be used to prove theorems about triangles, quadrilaterals, and other geometric figures.

Similarity transformations (rigid motions followed by dilations) define similarity in the same way that rigid motions define congruence, thereby

formalizing the similarity ideas of "same shape" and "scale factor”. These transformations lead to the criterion for triangle similarity that two pairs of

corresponding angles are congruent. The definitions of sine, cosine, and tangent for acute angles are founded on right triangles and similarity, and,

with the Pythagorean Theorem, are fundamental in many real-world and theoretical situations.

Analytic geometry connects algebra and geometry, resulting in powerful methods of analysis and problem solving. Geometric shapes can be

described by equations, making algebraic manipulation into a tool for geometric understanding, modeling, and proof.

Dynamic geometry environments provide students with experimental and modeling tools that allow them to investigate geometric phenomena.

Probability is expanded upon, developing ways to describe the probability in independent or conditional events.










Mathematics


Geometry: Suggested Distribution of Units in Instructional Weeks Time # of Weeks

Unit 1: Transformations and the Coordinate Plane 16% ~ 5 weeks

Unit 2: Congruence, Proof and Constructions 19% ~ 6 weeks

Unit 3: Three Dimensional Geometry 16% ~ 5 weeks

Unit 4: Similarity, Proof and Trigonometry 19% ~ 6 weeks

Unit 5: Circles and Other Conic Sections 16% ~ 5 weeks

Unit 6: Applications of Probability 13% ~ 4 weeks

Unit 1: Transformations

and the Coordinate Plane

11%

Unit 2: Congruence,

Proof and Constructions

14% Unit 3: Three Dimensional

Geometry 14%

Unit 4: Similarity, Proof

and Trigonometry

22%

Unit 5: Circles and Other Conic

Sections 8%

Unit 6: Applications of

Probability 11%

Instructional Time

Mathematics


Geometry Unit 1: Transformations and the Coordinate Plane (~ 5 weeks) Unit Overview: In Unit 1, students will experiment with different transformations to build knowledge of geometric terms and set the foundation for

similarity and congruence development later.

Guiding Questions: How can transformations describe geometric events in the real world?


Component Cluster 9-12.G.CO Experiment with transformations in the plane

9-12.G.CO.1 Know precise definitions of angle,

circle, perpendicular line, parallel line, and line

segment, based on the undefined notions of point,

line, distance along a line, and distance around a

circular arc.

Build on student experience with rigid motion from earlier grades. Point out the basis of rigid

motions in geometric concepts, e.g., translations move points a specified distance along a line

parallel to a specified line; rotations move objects along a circular arc with a specified center

through a specified angle.

Students may use geometry software and/or manipulatives to model transformations and

demonstrate a sequence of transformations that will carry a given figure onto another. 9-12.G.CO.2 Represent transformations in the

plane using, e.g., transparencies and geometry

software; describe transformations as functions that

take points in the plane as inputs and give other

points as outputs. Compare transformations that

preserve distance and angle to those that do not

(e.g., translations versus horizontal stretch).

9-12.G.CO.3 Given a rectangle, parallelogram,

trapezoid, or regular polygon, describe the rotations

and reflections that carry it onto itself.

9-12.G.CO.4 Develop definitions of rotations,

reflections, and translations in terms of angles,

circles, perpendicular lines, parallel lines, and line

segments.

9-12.G.CO.5 Given a geometric figure and a

rotation, reflection, or translation, draw the

transformed figure using, e.g., graph paper, tracing

paper, or geometry software. Specify a sequence of

transformations that will carry a given figure onto

another.

Component Cluster 9-12.G.GPE Use coordinates to prove simple geometric theorems algebraically

9-12.G.GPE.4 Use coordinates to prove simple

geometric theorems algebraically. For example,

Students may use geometric simulation software to model figures and prove simple geometric

theorems.

Mathematics


prove or disprove that a figure defined by four

given points in the coordinate plane is a rectangle;

prove or disprove that the point (1, √ ) lies on the

circle centered at the origin and containing the

point (0,2).

Example: Use slope and distance formula to verify the polygon formed by connecting the points

(-3,-2), (5,3), (9,9), (1,4) is a parallelogram.

9-12.G.GPE.5 Prove the slope criteria for parallel

and perpendicular lines and use them to solve

geometric problems (e.g., find the equation of a

line parallel or perpendicular to a given line that

passes through a given point).

Lines can be horizontal, vertical or neither.

Students may use a variety of different methods to construct a parallel or perpendicular line to a

given line and calculate the slopes to compare the relationships.

Relate work on parallel lines to work in Algebra 1 involving systems of equations having no

solutions or infinitely many solutions.

9-12.G.GPE.6 Find the point on a directed line

segment between two given points that partitions

the segment in a given ratio.

Students may use geometric simulation software to model figures or line segments.

Example: given A(3,2) and B(6,11)

- Find the point that divides the line segment AB two thirds of the way from A to B. The

point two-thirds of the way from A to B has x-coordinate two-thirds of the way from 3 to

6 and y coordinate two-thirds of the way from 2 to 11. So, (5,8) is the point that is two-

thirds from point A to point B.

- Find the midpoint of line segment AB.

9-12.G.GPE.7 Use coordinates to compute

perimeters of polygons and areas of triangles and

rectangles, e.g., using the distance formula.

Practice with the distance formula and its connection with the Pythagorean theorem.

Students may use geometric simulation software to model figures.

Mathematics


Geometry Unit 2: Congruence, Proof and Constructions (~ 6 weeks) Unit Overview: In Unit 2, students will use transformations to establish congruence related to parallel lines, theorems about triangles, and between

two triangles (SSS, ASA, SAS, AAS).

Guiding Question: How can transformations describe congruence?


Component Cluster 9-12.G.CO Understanding congruence in terms of rigid motion

9-12.G.CO.6 Use geometric descriptions of rigid

motions to transform figures and to predict the

effect of a given rigid motion on a given figure;

given two figures, use the definition of congruence

in terms of rigid motions to decide if they are

congruent.

Rigid motions are at the foundation of the definition of congruence. Students reason from the

basic properties of rigid motions (that they preserve distance and angle), which are assumed

without proof. Rigid motions and their assumed properties can be used to establish the usual

triangle congruence criteria, which can then be used to prove other theorems.

9-12.G.CO.7 Use the definition of congruence in

terms of rigid motions to show that two triangles

are congruent if and only if corresponding pairs of

sides and corresponding pairs of angles are

congruent.

9-12.G.CO.8 Explain how the criteria for triangle

congruence (ASA, SAS, and SSS) follow from the

definitions of congruence in terms of rigid motion.

Component Cluster 9-12.G.CO Prove geometric theorems

9-12.G.CO.9 Prove theorems about lines and

angles. Theorems include: vertical angles are

congruent; when a transversal crosses parallel lines,

alternate interior angles are congruent and

corresponding angles are congruent; points on a

perpendicular bisector of a line segment are

equidistant from the segment’s endpoints.

Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams,

in two-column format, and using diagrams without words. Students should be encouraged to

focus on the validity of the underlying reasoning while exploring a variety of formats for

expressing that reasoning.

9-12.G.CO.10 Prove theorems about triangles.

Theorems include: measures of interior angles of a

triangle sum to 180°; base angles of isosceles

triangles are congruent; the segment joining

midpoints of two sides of a triangle is parallel to

the third side and half the length; the medians of a

triangle meet at a point.

9-12.G.CO.11 Prove theorems about

Mathematics


parallelograms. Theorems include: opposite sides

are congruent, opposite angles are congruent, the

diagonals of a parallelogram bisect each other, and

conversely, rectangles are parallelograms with

congruent diagonals.

Component Cluster 9-12.G.CO Make geometric constructions

9-12.G.CO.12 Make formal geometric

constructions with a variety of tools and methods

(compass and straightedge, string, reflective

devices, paper folding, dynamic geometric

software, etc.). Copying a segment; copying an

angle; bisecting a segment; bisecting an angle;

constructing perpendicular lines, including the

perpendicular bisector of a line segment; and

constructing a line parallel to a given line through a

point not on the line.

Build on prior student experience with simple constructions. Emphasize the ability to formalize

and explain how these constructions result in the desired objects.

Students may use geometric software to make geometric constructions.

Mathematics


Geometry Unit 3: Three Dimensional Geometry (~ 5 weeks) Unit Overview: In Unit 3, students will build on prior knowledge of two dimensional shapes and apply volumes to real world problems.

Guiding Question: How can properties of two dimensional figures inform our understanding of three dimensional figures?


Component Cluster 9-12.G.GMD Explain volume formulas and use them to solve problems.

9-12.G.GMD.1 Given an informal argument for

the formulas for the circumference of a circle, area

of a circle, volume of a cylinder, pyramid and cone.

Cavalieri’s principle is if two solids have the same height and the same cross-sectional area at

every level, then they have the same volume.

9-12.G.GMD.3 Use volume formulas for cylinders,

pyramids, cones, and spheres to solve problems.

Missing measures can include but are not limited to slant height, altitude, height, diagonal of a

prism, edge length, and radius.

Component Cluster 9-12.G.GMD Visualize relationships between two-dimensional and three-dimensional objects

9-12.G.GMD.4 Identify the shapes of two-

dimensional cross-sections of three-dimensional

objects, and identify three-dimensional objects

generated by rotations of two-dimensional objects

Students may use geometric simulation software to model figures and create cross sectional

views.

Example: Identify the shape of the vertical, horizontal, and other cross sections of a cylinder.

Component Cluster 9-12.G.MG Apply geometric concepts in modeling situations

9-12.G.MG.1 Use geometric shapes, their

measures, and their properties to describe objects

(e.g., modeling a tree trunk or a human torso as a

cylinder).

Students may use simulation software and modeling software to explore which model best

describes a set of data or situation.

9-12.G.MG.2 Apply concepts of density based on

area and volume in modeling situations (e.g.,

persons per square mile, BTU’s per cubic foot).

Mathematics


Geometry Unit 4: Similarity, Proof and Trigonometry (~ 6 weeks) Unit Overview: In Unit 4, students will take their knowledge of transformations and develop an understanding of similarity. The similarity will

then be used to define trigonometric ratios and other properties.

Guiding Question: How can transformations describe similarity in the real world?


Component Cluster 9-12.G.SRT Understand similarity in terms of similarity transformations

9-12.G.SRT.1 Verify experimentally the properties

of dilations given by a center and a scale factor:

a. A dilation takes a line not passing through

the center of the dilation to a parallel line,

and leaves a line passing through the center

unchanged.

b. The dilation of a line segment is longer or

shorter in the ratio given by the scale factor.

A dilation is a transformation that moves each point along the ray through the point emanating

from a fixed center, and multiplies distances from the center by a common scale factor.

Students may use geometric simulation software to model transformations. Students may observe

patterns and verify experimentally the properties of dilations.

9-12.G.SRT.2 Given two figures, use the definition

of similarity in terms of similarity transformations

to decide if they are similar; explain using

similarity transformations the meaning of similarity

for triangles as the equality of all corresponding

pairs of angles and the proportionality of all

corresponding pairs of sides.

A similarity transformation is a rigid motion followed by a dilation.

Students may use geometric simulation software to model transformations and demonstrate a

sequence of transformations to show congruence or similarity of figures.

9-12.G.SRT.3 Use the properties of similarity

transformations to establish the AA criterion for

two triangles to be similar.

Component Cluster 9-12.G.SRT Prove theorems involving similarity

9-12.G.SRT.4 Prove theorems about triangles.

Theorems include: a line parallel to one side of a

triangle divides the other two proportionally, and

conversely; the Pythagorean theorem proved using

triangle similarity.



9-12.G.SRT.5 Use congruence and similarity

criteria for triangles to solve problems and to prove

relationships in geometric figures.

Similarity postulates include SSS, SAS, and AA.

Congruence postulates include SSS, SAS, ASA, AAS, and HL.


Mathematics



Component Cluster 9-12.G.SRT Define trigonometric ratios and solve problems involving right triangles

9-12.G.SRT.6 Understand that by similarity, side

ratios in right triangles are properties of the angles

in the triangle, leading to definitions of

trigonometric ratios for acute angles.

Students may use applets to explore the range of values of the trigonometric ratios as ɵ ranges

from 0 to 90 degrees.

9-12.G.SRT.7 Explain and use the relationship

between the sine and cosine of complementary

angles.

Geometric simulation software, applets, and graphing calculators can be used to explore the

relationship between sine and cosine.

9-12.G.SRT.8 Use trigonometric ratios and the

Pythagorean Theorem to solve right triangles in

applied problems.

Students may use graphing calculators or programs, tables, spreadsheets, or computer algebra systems to

solve right triangle problems.

Example: find the height of a tree to the nearest tenth if the angle of elevation of the sun is 28° and the

shadow of the tree is 50 ft.

Component Cluster 9-12.G.MG Apply geometric concepts in modeling situations

9-12.G.MG.3 Apply geometric methods to solve

design problems (e.g., designing an object or

structure to satisfy physical constraints or minimize

cost; working with typographic grid systems based

on ratios).

Focus on situations well modeled by trigonometric ratios for acute angles.

Mathematics


Geometry Unit 5: Circles and Other Conic Sections (~ 5 weeks) Unit Overview: In Unit 5, students will develop a thorough understanding of circles and their properties. This will include an analytic geometry

approach writing the equation of circles.

Guiding Question: How can a circle be modeled analytically?


Component Cluster 9-12.G.C Understand and apply theorems about circles

9-12.G.C.1 Prove that all circles are similar. Students may use geometric simulation software to model transformations and demonstrate a


9-12.G.C.2 Identify and describe relationships

among inscribed angles, radii, and chords. Include

the relationship between central, inscribed, and

circumscribed angles; inscribed angles on a

diameter are right angles; the radius of a circle is

perpendicular to the tangent where the radius

intersects the circle.

Examples:

Given the circle with radius of 10 and chord length of 12, find the distance from the chord to the

center of the circle.

Find the unknown length (tangent length given radius and secant)

9-12.G.C.3 Construct the inscribed and

circumscribed circles of a triangle, and prove

properties of angles for a quadrilateral inscribed in

a circle.

Students may use geometric simulation software to make geometric constructions.

Component Cluster 9-12.G.C Find arc lengths and areas of sectors of circles

9-12.G.C.5 Derive using similarity the fact that the

length of the arc intercepted by an angle is

proportional to the radius, and define the radian

measure of the angle as the constant of

proportionality; derive the formula for the area of a

sector.

Emphasis the similarity of all circles. Use this as a basis for introducing radian as a unit of

measure. It is not intended that it be applied to the development of circular trigonometry in this

course.

Component Cluster 9-12.G.GPE Translate between the geometric description and the equation for a conic section

9-12.G.GPE.1 Derive the equation of a circle of

given center and radius using the Pythagorean

Theorem; complete the square to find the center

and radius of a circle given by an equation.

Students may use geometric simulation software to explore the connection between circles and

the Pythagorean Theorem.

Examples:

- Write an equation for a circle with a radius of 2 units and center at (1,3)

- Write an equation for a circle given that the endpoints of the diameter are (-2,7) and (4,-8)

Mathematics


- Find the center and radius of the circle 012444 22 yxyx

9-12.G.GPE.2 Derive the equation of a parabola

given a focus and directrix.

Students may use geometric simulation software to explore parabolas.

Example: Write and graph an equation for a parabola with focus (2,3) and directrix y=1.

Component Cluster 9-12.G.GPE Use coordinates to prove simple geometric theorems algebraically

9-12.G.GPE.4 Use coordinates to prove simple

geometric theorems algebraically. For example,

prove or disprove that a figure defined by four

given points in the coordinate plane is a rectangle;

prove or disprove that the point (1, √ ) lies on the

circle centered at the origin and containing the

point (0,2).

Include simple proofs involving circles. Students may use geometric simulation software to

model figures and prove simple geometric theorems.

Example:

Prove the diameter is twice the radius.

Mathematics


Geometry Unit 6: Applications of Probability (~ 4 weeks) Unit Overview: In Unit 6, students will take their understanding of simple probabilities and expand that to include conditional probabilities and

dependent relationships.

Guiding Question: How can probability be used to make decisions?


Component Cluster 9-12.S.CP Understand independence and conditional probability and use them to interpret data

9-12.S.CP.1 Describe events as subsets of a sample

space (the set of outcomes) using characteristics (or

categories) of the outcomes, or as unions,

intersections, or complements of other events (“or”,

“and”, “not”).

Intersection: The intersection of two sets A and B is the set of elements that are common to both

set A and set B. It is denoted by A ∩ B and is read ‘A intersection B’.

A ∩ B in the diagram is {1, 5}

this means: BOTH/AND

Union: The union of two sets A and B is the set of elements, which are in A or in B or in both. It

is

denoted by A ∪ B and is read ‘A union B’.

A ∪ B in the diagram is {1, 2, 3, 4, 5, 7}

this means: EITHER/OR/ANY

could be both

Complement: The complement of the set A ∪B is the set of elements that are members of the

universal set U but are not in A ∪B. It is denoted by (A ∪ B )’

U

B A

7 5

4 3

2 1

8

Mathematics


(A ∪ B )’ in the diagram is {8}

9-12.S.CP.2 Understand that two events A and B

are independent if the probability of A and B

occurring together is the product of their

probabilities, and use this characterization to

determine if they are independent.

9-12.S.CP.3 Understand the conditional probability

of A given B as P(A and B)/P(B), and interpret

independence of A and B as saying the conditional

probability of A given B is the same as the

probability of A, and the conditional probability of

B given A is the same as the probability of B.

9-12.S.CP.4 Construct and interpret two-way

frequency tables of data when two categories are

associated with each object being classified. Use

the two-way table as a sample space to decide if

events are independent and to approximate

conditional probabilities. For example, collect data

from a random sample of students in your school

on their favorite subject among math, science, and

English. Estimate the probability that a randomly

selected student from your school will favor

science given that the student is in tenth grade. Do

the same for other subjects.

Students may use spreadsheets, graphing calculators, and simulations to create frequency tables

and conduct analyses to determine if events are independent or determine approximate

conditional probabilities.

9-12.S.CP.5 Recognize and explain the concepts of

conditional probability and independence in

everyday language and everyday situations. For

example, compare the chance of having lung cancer

if you are a smoker with the chance being a smoker

if you have lung cancer.

Examples:

What is the probability of drawing a heart from a standard deck of cards on a second

draw, given that a heart was drawn on the first draw and not replaced? Are these events

independent or dependent?

At Johnson Middle School, the probability that a student takes computer science and

French is 0.062. The probability that a student takes computer science is 0.43. What is the

probability that a student takes French given that the student is taking computer science?

Component Cluster 9-12.S.CP Use the rules of probability to compute probabilities of compound events in a uniform probability model

9-12.S.CP.6 Find the conditional probability of A

given B as the fraction of B’s outcomes that also

belong to A, and interpret the answer in terms of

Students could use graphing calculators, simulations, or applets to model probability experiments

and interpret outcomes.

Mathematics


the model.

9-12.S.CP.7 Apply the Addition Rule, P(A or B) =

P(A) + P(B) = P(A and B), and interpret the answer

in terms of the model.

Students could use graphing calculators, simulations, or applets to model probability experiments

and interpret outcomes.

Mathematics


Algebra 2 In Algebra 2, building on their work with linear, quadratic, and exponential functions, students extend their repertoire of functions to include

polynomial, rational, and radical functions. Students work closely with the expressions that define the functions, and continue to expand and hone

their abilities to model situations and to solve equations. The Mathematical Practice Standards apply throughout the course and, together with the

content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make

sense of problem situations.

Students develop the structural similarities between the system of polynomials and the system of integers. Students draw on analogies between

polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property. Students connect

multiplication of polynomials with multiplication of multi-digit integers, and division of polynomials with long division of integers. Students identify

zeros of polynomials, including complex zeros of quadratic polynomials, and make connections between zeros of polynomials and solutions of

polynomial equations

Building on their previous work with functions, and on their work with trigonometric ratios and circles in Geometry, students now use the coordinate

plane to extend trigonometry to model periodic phenomena.

Students synthesize and generalize what they have learned about a variety of function families. They extend their work with exponential functions to

include solving exponential equations with logarithms. They explore the effects of transformations on graphs of diverse functions, including

functions arising in an application, in order to abstract the general principle that transformations on a graph always have the same effect regardless of

the type of the underlying function. They identify appropriate types of functions to model a situation, they adjust parameters to improve the model,

and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit. The narrative

discussion and diagram of the modeling cycle should be considered when knowledge of functions, statistics, and geometry is applied in a modeling

context.

Students see how the visual displays and summary statistics they learned in earlier grades relate to different types of data and to probability

distributions. They identify different ways of collecting data— including sample surveys, experiments, and simulations—and the role that

randomness and careful design play in the conclusions that can be drawn.










Mathematics


Algebra 2: Suggested Distribution of Units in Instructional Weeks Time # of Weeks

Unit 1: Functions and Inverses 16% ~ 5 weeks

Unit 2: Polynomial Functions 19% ~ 6 weeks

Unit 3: Rational Expressions and Functions 13% ~ 4 weeks

Unit 4: Exponential and Logarithmic Functions 19% ~ 6 weeks

Unit 5: Trigonometric Functions 16% ~ 5 weeks

Unit 6: Statistics 19% ~ 6 weeks

Unit 1: Functions and Inverses

16%

Unit 2: Polynomial Functions

19%

Unit 3: Rational Expressions and

Functions 13%

Unit 4: Exponential and

Logarithmic Functions

19%

Unit 5: Trigonometric

Functions 16%

Unit 6: Statistics 19%

Instructional Time

Mathematics


Algebra 2 Unit 1: Functions and Inverses (~ 5 weeks) Unit Overview: In Unit 1, students expand their work with functions from Algebra 1 to include new functions as well as previously learned

functions in more depth. New functions are created by transforming functions and by creating inverses. Domain and range continue to be identified

and used to explain real world restrictions. Functions are analyzed using multiple representations: tables, graphs, ordered pairs, equations, and

verbal descriptions.



Component Cluster 9-12.F.IF Interpret functions that arise in applications in terms of context


its graph and, where applicable, to the


Students will examine all functions studied thus far with respect to domain. Restrictions will be

examined as well.






Focus on applications and how key features relate to characteristics of a situation, making

selection of a particular type of function model appropriate.








c. (+) Compose functions.

Build a function that models a relationship between two quantities. Develop models for more

complex or sophisticated situations than in previous courses.



replacing f(x) by f(x) + k, k f(x), f(kx), and f(x+k)





technology.

Build new functions from existing functions. Note the effect of multiple transformations on a

single graph and the common effect of each transformation across function types.

9-12.F.BF.4 Find inverse functions.

a. Solve an equation of the form f(x)=c for a

Find inverses algebraically and graphically.

Mathematics


simple function f that has an inverse and

write an expression for the inverse.

b. (+) Verify by composition that one

function is the inverse of another.

c. (+) Read values of an inverse function

from a graph or table, given that the

function has an inverse.



variables to represent relationships between two



Create equations that describe numbers or relationships. Problems should extend the functions

used in Algebra 1.

Mathematics


Algebra 2 Unit 2: Polynomial Functions (~ 6 weeks) Unit Overview: In Unit 2, students develop an understanding of the structural similarities between the system of polynomials and the system of

integers, including operations. Students identify zeros of polynomials, including complex zeros, and make connections between zeros of polynomials

and solutions of polynomial equations. Students are introduced to the complex numbers and learn to perform arithmetic operations with complex

numbers. Polynomial functions are analyzed in multiple representations and used to model real world situations.

Guiding Question: What can the characteristics of polynomial functions tell you about real world relationships?


Component Cluster 9-12.N.CN Perform arithmetic operations with complex numbers

9-12.N.CN.1 Know there is a complex number i

such that 12 i , and every complex number

has the form a+bi where a and b are real.

Understand that there is no real solution to taking the square root of a negative number; taking the

square root of a negative number results in an imaginary number. However, when you square an

imaginary number the result is a real number. Students will examine operations in the complex

number system. When dealing with powers of i, students will focus on the four basic powers of i.

For example, .,,, 3210 iiii 9-12.N.CN.2 Use the relation 12 i and the

commutative, associative, and distributive

properties to add, subtract, and multiply complex

numbers.

Component Cluster 9-12.N.CN Use complex numbers in polynomial identities and equations

9-12.N.CN.7 Solve quadratic equations with

real coefficients that have complex solutions.

Limit to polynomials with real coefficients. Review methods for solving quadratic equations

from Algebra 1 (graphing, factoring, completing the square, square rooting, and quadratic

formula) now including complex solutions. Note complex solutions will not show up on graph.

Only real solutions can be found graphically.

9-12.N.CN.8 (+) Extend polynomial identities

to the complex numbers.

This will help convert factored forms into polynomial functions, but should be limited to only

pure imaginary not complex roots. For example: 4)2)(2( 2 xixix

9-12.N.CN.9 (+) Know the Fundamental

Theorem of Algebra; show that it is true for

quadratic polynomials.

Examples:

● How many zeros does -2x2 + 3x – 8 have? Find all the zeros and explain, orally or in written

format, your answer in terms of the Fundamental Theorem of Algebra.

● How many complex zeros does the following polynomial have? How do you know?

p(x) = (x2 -3) (x

2 +2)(x - 3)(2x – 1)


9-12.A.REI.4 Solve quadratic equations in one

variable.

Extend Algebra 1 solution of quadratics to include complex solutions.

9-12.A.REI.4b Solve quadratic equations by

inspecting, taking square roots, completing the

square, the quadratic formula and factoring, as

Mathematics


appropriate to the initial form of the equation.

Recognize when the quadratic formula give

complex solutions and write them as bia for

real numbers a and b.


9-12.A.CED.1 Create equations in one variable

and use them to solve problems.

Equations can represent real world and mathematical problems. Include equations that arise when

comparing the values of two different functions.

Examples:

Given that the following trapezoid has area 54 cm2, set up an equation to find the length of

the base, and solve the equation.

Lava coming from the eruption of a volcano follows a parabolic path. The height h in feet

of a piece of lava t seconds after it is ejected from the volcano is given by h(t)= -t2 + 16t

+ 936. After how many seconds does the lava reach its maximum height of 1000 feet?











relationship.

Key features include: intercepts; intervals where the function is increasing, decreasing, positive,

or negative; relative maximums and minimums; symmetries; and end behavior.






Build understanding of graphs by graphing by hand and graphing calculator. Note that real zeros

will appear on the graph as x-intercepts. Key features include: intercepts; intervals where the

function is increasing, decreasing, positive, or negative; relative maximums and minimums;

symmetries; and end behavior.

9-12.F.IF.7c Graph polynomial functions,

identifying zeros when suitable factorizations are

available, and showing end behavior.


9-12.A.SSE.2 Use the structure of an expression

to identify ways to rewrite it. For example, recognize 44 yx as a difference of two squares that can be factored as

Mathematics


))(( 2222 yxyx .







Extend beyond the quadratic polynomials found in Algebra 1.

Component Cluster 9-12.A.APR Understand the relationship between the zeros and factors of polynomials

9-12.A.APR.2 Know and apply the Remainder

Theorem: For a polynomial p(x) and a number

a, the remainder on division by x-a is p(a), so

p(a)=0 if and only if (x-a) is a factor of p(x).

Show that when you evaluate the function at the root then the result is zero. Students may be

introduced to synthetic division here. Build on the student’s understanding of integer division to

develop polynomial division and an understanding of remainders and factors.

9-12.A.APR.3 Identify zeros of polynomials

when suitable factorizations are available, and

use the zeros to construct a rough graph of the

function defined by the polynomial.

Make connections between factors, zeros and constructing graphs. Note how zeros of multiplicity

will be graphed.

Component Cluster 9-12.A.APR Use polynomial identities to solve problems

9-12.A.APR.4 Prove polynomial identities and

use them to describe numerical relationships

Be able to recognize different forms of polynomials and rewrite them to factor, solve or graph.

Mathematics


Algebra 2 Unit 3: Rational Expressions and Functions (~ 4 weeks) Unit Overview: In Unit 3, students take their understanding of rational numbers and extend to rational expressions. Rational numbers extend the

arithmetic of integers by allowing division by all numbers except 0. Similarly, rational expressions extend the arithmetic of polynomials by allowing

division of polynomials except the zero polynomial. Graphical representations are explored and critical pieces, like asymptotes, identified to get a

clearer understanding of the functions’ properties.

Guiding Question: What can the properties of rational functions tell you about the real world relationships they describe?


Component Cluster 9-12.A.REI Understand solving equations as a process of reasoning and explain the reasoning

9-12.A.REI.2 Solve simple rational equations in

one variable, and give examples showing how

extraneous solutions may arise.

Note that potential solutions must be checked in the original equation to make sure that these

values do not cause the expression to become undefined; such extraneous solutions must be

rejected.




Students should understand the vocabulary for the parts that make up the whole expression and be

able to identify those parts and interpret their meaning in terms of a context. For example, use

rational expressions to represent Average Speed as (Total Distance) / (Total Time), based on d = r

x t.




Show how complex fractions can be viewed as division, a/b.

Component Cluster 9-12.A.APR Rewrite rational expressions

9-12.A.APR.6 Rewrite simple rational

expressions in different forms; write a(x)/b(x) in

the form q(x) + r(x)/b(x), where a(x), b(x), q(x),

and r(x) are polynomials with degree of r(x) less

than the degree of b(x), using inspection, long

division, or, for more complicated examples, a

computer algebra system.

The polynomial q(x) is called the quotient and the polynomial r(x) is called the remainder.

Example: Find the quotient and remainder for the rational expression

and use them

to write the expression in a different form.

9-12.A.APR.7(+) Understand that rational

expressions form a system analogous to the

rational numbers, closed under addition,

subtraction, multiplication, and division by a

nonzero rational expression; add, subtract,

multiply, and divide rational expressions.

Show how students’ prior knowledge of working with fractions applies to performing operations

(+, -, x, /) with rational expressions.

Example: Express

in the form , where a(x) and b(x) are polynomials.

Mathematics




and use them to solve problems.

Equations can represent real world and mathematical problems. Include equations that arise when

comparing the values of two different functions.


equations, and by systems of equations, and

interpret solutions as viable or non-viable

options in a modeling context.

Consider issues such as where a solution of 0 or a negative number may not be appropriate in the

real-life context.






Students should be very familiar with the parent graph x

y1

and be able to graph it by hand,

showing key features.

Students may use graphing calculators to graph more difficult rational functions. When using the

graphing calculator, point out how the table may show an error where the function is undefined

and how that corresponds to the asymptotes or holes that exist in the function.

9-12.F.IF.7d(+) Graph rational functions,

identifying zeros and asymptotes when suitable

factorizations are available, and showing end

behavior.

Find vertical and horizontal asymptotes. Show how factoring is used in determining features of

rational functions, such as zeros and multiple vertical asymptotes.

Mathematics


Algebra 2 Unit 4: Exponential and Logarithmic Functions (~ 6 weeks) Unit Overview: In Unit 4, students extend their basic understanding of exponential growth functions to gain a deeper understanding of the function

and all its properties. Students will use more precise graphical representations to better define the relationships. Then students will utilize inverse

relationships to develop the logarithmic function and explore its properties. Both exponential and logarithmic functions are used to model real world

relationships and solve real world problems.

Guiding Question: How do exponential and logarithmic functions explain real world relationships?




relationship between two quantities, interpret

key features of graphs and tables in terms of the



relationship.


or negative; symmetries; asymptotes; and end behavior.



symbolically and show key features of the

graph, by hand in simple cases and using

technology for more complicated cases.

Focus on applications and how key features relate to characteristics of a situation, making

selection of a particular type of function model appropriate.

9-12.F.IF.7e Graph exponential and

logarithmic functions, showing intercepts and

end behavior.

Note that for exponential functions in the form xbay , the y-intercept is ),0( a and if 1b ,

the function is increasing, whereas if 10 b , the function is decreasing. For xy blog , the

x-intercept is )0,1( .


expression in different but equivalent forms to


function.

Show how to go from exponential to logarithmic form and from logarithmic to exponential

form. For example:

xyby b

x log



For example, identify percent rate of change in functions such as 10/12 )2.1(,)02.1(,)97.0(,)02.1( tttt yyyy and classify them as representing exponential

growth or decay.




For example, build a function that models appreciation/depreciation, radioactive decay or

population growth.

9-12.F.BF.1b Combine standard function types

using arithmetic operations.

Can be demonstrated with the properties of logs. Models of growth and decay that include

constant functions are also included, combining the log or exponential with a constant function.

Mathematics




replacing f(x) by f(x)+k, kf(x), f(kx), and f(x+k)



Show transformations of the parent function depending on the model criteria. Include vertical

and horizontal shifts, reflections, compressions and stretches.

9-12.F.BF.5(+) Understand the inverse

relationship between exponents and logarithms

and use this relationship to solve problems

involving logarithms and exponents.

For example: xby and xy blog are inverses. Note the usefulness of the inverse properties:

xbxb

log and xb x

b log .


9-12.F.LE.4 For exponential models, express

as a logarithm the solution to dabct where

a, c, and d are numbers and the base b is 2, 10,

or e; evaluate the logarithm using technology.

Show how to use logarithms to solve exponential equations.

Note that when evaluating logarithms on a graphing calculator, log uses base 10 and ln uses

base e.




Show how exponential and logarithmic expressions can model real-life situations.




For example, show how the principle yx bb can be used to understand more complicated

problems such as 523 2 xxx bb .


9-12.A.SSE.4 Derive the formula for the sum

of a finite geometric series (when the common

ratio is not 1), and use the formula to solve

problems.

For example, calculate mortgage payments.



and use them to solve problems. Include

equations arising from exponential functions.

Equations can represent real world and mathematical problems. Include equations that arise

when comparing the values of two different functions.





Create equations that describe numbers or relationships.

Show the connection between the numeric and graphic representations.

Emphasize the importance of using appropriate labels and scales.


Mathematics



of the points where the graphs of the equations y

= f(x) and y = g(x) intersect are the solutions of

the equation f(x)=g(x); find the solutions

approximately.

Solve systems of equations graphically, using the intersect feature on the graphing calculator.

Mathematics


Algebra 2 Unit 5: Trigonometric Functions (~ 5 weeks) Unit Overview: In Unit 5, students build on their knowledge of trigonometry in right triangles to extend to the trigonometric functions based on a

unit circle. Students explore the basic trigonometric functions, including their graphs, to gain a better understanding of them.

Guiding Question: What can the properties of trigonometric functions tell you about the real world relationships they describe?








relationship.


or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.





9-12.F.IF.7e Graph trigonometric functions,

showing period, midline, and amplitude.



replacing f(x) by f(x)+k, kf(x), f(kx), and f(x+k)



Experiment with cases and illustrate an explanation of the effects on the graph using technology.

Note the common effect of transformations across function types.

Component Cluster 9-12.F.TF Extend the domain of trigonometric functions using the unit circle

9-12.F.TF.1 Understand radian measure of an

angle as the length of the arc on the unit circle

subtended by the angle.

Consider relating it to a special case of r

s where s is the arc length and r = 1 for the unit circle.

9-12.F.TF.2 Explain how the unit circle in the

coordinate plane enables the extension of

trigonometric functions to all real numbers,

interpreted as radian measures of angles

traversed counterclockwise around the unit

circle.

Note that every coordinate on the circle (x,y) represents )sin,(cos .

9-12.F.TF.3(+) Use special triangles to

determine geometrically the values of sine,

Visual representations of the unit circle, with reflections and rotations, can prove helpful in

demonstrating the angles outside of quadrant I.

Mathematics


cosine, tangent for π/3, π/4, and π/6, and use the

unit circle to express the values of sine, cosine,

and tangent for π-x, π+x, and 2π-x in terms of

their values for x, where x is any real number.

Component Cluster 9-12.F.TF Model periodic phenomena with trigonometric functions

9-12.F.TF.5 Choose trigonometric functions to

model periodic phenomena with specified

amplitude, frequency, and midline.

Show how trigonometric functions can model real life situations (for example: tides, sunrise,

temperature change). Students should be able to interpret what amplitude, frequency and midline

mean in context.

Component Cluster 9-12.F.TF Prove and apply trigonometric identities

9-12.F.TF.8 Prove the Pythagorean identity (sin

A)² + (cos A)² = 1 and use it to calculate

trigonometric ratios.

Use to find additional trigonometric ratios when one ratio is given.

Mathematics


Algebra 2 Unit 6: Statistics (~ 6 weeks) Unit Overview: In working with statistics, students will understand all the essential processes involved in dealing with data: collecting, organizing,

representing, analyzing, making predictions/conclusions. Calculations of measures of central tendency and dispersion are used to analyze data.

Inferences and conclusions will help interpret real world situations.

Guiding Question: How can statistics be used to understand real world phenomena and help us draw appropriate conclusions?



9-12.S.ID.4 Use the mean and standard

deviation of a data set to fit it to a normal

distribution and to estimate population

percentages. Recognize that there are data sets

for which such a procedure is not appropriate.

Use calculators, spreadsheets, and tables to

estimate areas under the normal curve.

Note that the Normal Distribution is also referred to as the Bell Curve. Show how Measures of

Central Tendency, Measures of Dispersion, and percentiles are represented and can be interpreted

in a Normal Distribution. Point out that a special characteristic of a normal distribution is that

mean = median = mode.

While students may have heard of the normal distribution, it is unlikely that they will have prior

experience using it to make specific estimates. Build on students’ understanding of data

distributions to help them see how the normal distribution uses area to make estimates of

frequencies (which can be expressed as probabilities).

Emphasize that only some data are well described by a normal distribution (i.e., not all data are

normally distributed). Use percentiles, standard deviation, and the empirical rule to verify if a

data distribution is normal.

Component Cluster 9-12.S.IC Understand and evaluate random processes underlying statistical experiments

9-12.S.IC.1 Understand statistics as a process

for making inferences about population

parameters based on a random sample from that

population.

Use simulated results to informally compare a hypothesized value of a population parameter to

the collected sample statistic. Judge the likelihood of obtaining a sample statistic using

simulations and/or probability models.

9-12.S.IC.2 Decide if a specified model is

consistent with results from a given data-

generating process.

Include comparing theoretical and empirical results to evaluate the effectiveness of a treatment.

Component Cluster 9-12.S.IC Make inferences and justify conclusions from sample surveys, experiments, and observational studies

9-12.S.IC.3 Recognize the purposes of and

differences among sample surveys, experiments,

and observational studies; explain how

randomization relates to each.

In earlier grades, students are introduced to different ways of collecting data and use graphical

displays and summary statistics to make comparisons. These ideas are revisited with a focus on

how the way in which data are collected determines the scope and nature of the conclusions that

can be drawn from that data. The concept of causation should be stressed.

9-12.S.IC.4 Use data from a sample survey to

estimate a population mean or proportion;

develop a margin of error through the use of

The concept of statistical significance is developed informally through simulation as meaning a

result that is unlikely to have occurred solely as a result of random selection in sampling or

random assignment in an experiment.

Mathematics


simulation models from random sampling.

Focus on the variability of results from experiments—that is, focus on statistics as a way of

dealing with, not eliminating, inherent randomness. 9-12.S.IC.5 Use data from a randomized

experiment to compare two treatments; use

simulations to decide if differences between two

parameters are significant.

9-12.S.IC.6 Evaluate reports based on data. Demonstrate how to both write a report as well as critique a report already written. Investigate

how data could be used to present a misleading conclusion, for example using the mean to

summarize a set of data that is skewed or contains outliers. Emphasis should be placed on

discussion of the shape, center, and spread of quantitative data or proportion of each category in

categorical data.

Mathematics

Glossary

Addition and subtraction within 5, 10, 20, 100, or 1000. Addition or subtraction of two whole numbers with whole number answers, and with sum or minuend in the range 0-5, 0-10, 0-20, or 0-100,

respectively. Example: 8 + 2 = 10 is an addition within 10, 14 – 5 = 9 is a subtraction within 20, and 55 – 18 = 37 is a subtraction within 100.

Additive inverses. Two numbers whose sum is 0 are additive inverses of one another. Example: 3/4 and – 3/4 are additive inverses of one another because 3/4 + (– 3/4) = (– 3/4) + 3/4 = 0.

Associative property of addition. See Table 3 in this Glossary.

Associative property of multiplication. See Table 3 in this Glossary.

Bivariate data. Pairs of linked numerical observations. Example: a list of heights and weights for each player on a football team.

Box plot. A method of visually displaying a distribution of data values by using the median, quartiles, and extremes of the data set. A box shows the middle 50% of the data.1

Commutative property. See Table 3 in this Glossary.

Complex fraction. A fraction A/B where A and/or B are fractions (B nonzero).

Computation algorithm. A set of predefined steps applicable to a class of problems that gives the correct result in every case when the steps are carried out correctly. See also: computation strategy.

Computation strategy. Purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at converting one problem into another. See also:

computation algorithm.

Congruent. Two plane or solid figures are congruent if one can be obtained from the other by rigid motion (a sequence of rotations, reflections, and translations).

Counting on. A strategy for finding the number of objects in a group without having to count every member of the group. For example, if a stack of books is known to have 8 books and 3 more books

are added to the top, it is not necessary to count the stack all over again. One can find the total by counting on—pointing to the top book and saying “eight,” following this with “nine, ten, eleven.

There are eleven books now.”

Dot plot. See: line plot.

Dilation. A transformation that moves each point along the ray through the point emanating from a fixed center, and multiplies distances from the center by a common scale factor.

Expanded form. A multi-digit number is expressed in expanded form when it is written as a sum of single-digit multiples of powers of ten. For example, 643 = 600 + 40 + 3.

Expected value. For a random variable, the weighted average of its possible values, with weights given by their respective probabilities.

First quartile. For a data set with median M, the first quartile is the median of the data values less than M. Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the first quartile is 6.2 See

also: median, third quartile, interquartile range.

Fraction. A number expressible in the form a/b where a is a whole number and b is a positive whole number. (The word fraction in these standards always refers to a non-negative number.) See also:

rational number.

Identity property of 0. See Table 3 in this Glossary.

Independently combined probability models. Two probability models are said to be combined independently if the probability of each ordered pair in the combined model equals the product of the

original probabilities of the two individual outcomes in the ordered pair.

Integer. A number expressible in the form a or –a for some whole number a.

Interquartile Range. A measure of variation in a set of numerical data, the interquartile range is the distance between the first and third quartiles of the data set. Example: For the data set {1, 3, 6, 7,

10, 12, 14, 15, 22, 120}, the interquartile range is 15 – 6 = 9. See also: first quartile, third quartile.

Line plot. A method of visually displaying a distribution of data values where each data value is shown as a dot or mark above a number line. Also known as a dot plot.3

Mean. A measure of center in a set of numerical data, computed by adding the values in a list and then dividing by the number of values in the list. (To be more precise, this defines the arithmetic

mean.)Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the mean is 21.

Mean absolute deviation. A measure of variation in a set of numerical data, computed by adding the distances between each data value and the mean, then dividing by the number of data values.

Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the mean absolute deviation is 20.

Median. A measure of center in a set of numerical data. The median of a list of values is the value appearing at the center of a sorted version of the list—or the mean of the two central values, if the

list contains an even number of values. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 90}, the median is 11.

Midline. In the graph of a trigonometric function, the horizontal line halfway between its maximum and minimum values.

1Adapted from Wisconsin Department of Public Instruction, http://dpi.wi.gov/standards/mathglos.html , accessed March 2, 2010.

http://dpi.wi.gov/standards/mathglos.html

Mathematics

2Many different methods for computing quartiles are in use. The method defined here is sometimes called the Moore and McCabe method. See Langford, E., “Quartiles in Elementary Statistics,” Journal of

Statistics Education Volume 14, Number 3 (2006). 3Adapted from Wisconsin Department of Public Instruction, op. cit.

Multiplication and division within 100. Multiplication or division of two whole numbers with whole number answers, and with product or dividend in the range 0-100. Example: 72 ÷ 8 = 9.

Multiplicative inverses. Two numbers whose product is 1 are multiplicative inverses of one another. Example: 3/4 and 4/3 are multiplicative inverses of one another because 3/4 × 4/3 = 4/3 × 3/4 = 1

Number line diagram. A diagram of the number line used to represent numbers and support reasoning about them. In a number line diagram for measurement quantities, the interval from 0 to 1 on

the diagram represents the unit of measure for the quantity.

Percent rate of change. A rate of change expressed as a percent. Example: if a population grows from 50 to 55 in a year, it grows by 5/50 = 10% per year.

Probability distribution. The set of possible values of a random variable with a probability assigned to each.

Properties of operations. See Table 3 in this Glossary.

Properties of equality. See Table 4 in this Glossary.

Properties of inequality. See Table 5 in this Glossary.

Properties of operations. See Table 3 in this Glossary.

Probability. A number between 0 and 1 used to quantify likelihood for processes that have uncertain outcomes (such as tossing a coin, selecting a person at random from a group of people, tossing a

ball at a target, or testing for a medical condition).

Probability model. A probability model is used to assign probabilities to outcomes of a chance process by examining the nature of the process. The set of all outcomes is called the sample space, and

their probabilities sum to 1. See also: uniform probability model.

Random variable. An assignment of a numerical value to each outcome in a sample space.

Rational expression. A quotient of two polynomials with a non-zero denominator.

Rational number. A number expressible in the form a/b or – a/b for some fraction a/b. The rational numbers include the integers.

Rectilinear figure. A polygon all angles of which are right angles.

Rigid motion. A transformation of points in space consisting of a sequence of one or more translations, reflections, and/or rotations. Rigid motions are here assumed to preserve distances and angle

measures.

Repeating decimal. The decimal form of a rational number. See also: terminating decimal.

Sample space. In a probability model for a random process, a list of the individual outcomes that are to be considered.

Scatter plot. A graph in the coordinate plane representing a set of bivariate data. For example, the heights and weights of a group of people could be displayed on a scatter plot.5

Similarity transformation. A rigid motion followed by a dilation

Tape diagram. A drawing that looks like a segment of tape, used to illustrate number relationships. Also known as a strip diagram, bar model, fraction strip, or length model.

Terminating decimal. A decimal is called terminating if its repeating digit is 0.

Third quartile. For a data set with median M, the third quartile is the median of the data values greater than M. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the third quartile is 15.

See also: median, first quartile, interquartile range.

Transitivity principle for indirect measurement. If the length of object A is greater than the length of object B, and the length of object B is greater than the length of object C, then the length of

object A is greater than the length of object C. This principle applies to measurement of other quantities as well.

Uniform probability model. A probability model which assigns equal probability to all outcomes. See also: probability model.

Vector. A quantity with magnitude and direction in the plane or in space, defined by an ordered pair or triple of real numbers.

Visual fraction model. A tape diagram, number line diagram, or area model.

Whole numbers. The numbers 0, 1, 2, 3, ….

5Adapted from Wisconsin Department of Public Instruction, op. cit.

Mathematics

Standards for Mathematical Practice

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on

important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning

and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It

Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying

out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled

with a belief in diligence and one’s own efficacy).


Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints,

relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They

consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and

change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing

calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw

diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help

conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this

make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.


Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving

quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life

of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the

referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to

the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.


Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and

build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use

counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible

arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments,

distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using

concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later

grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense,

and ask useful questions to clarify or improve the arguments.


Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as

simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the

community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically

proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need

revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and

Mathematics

formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on

whether the results make sense, possibly improving the model if it has not served its purpose.


Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a

protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools

appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For

example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by

strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of

varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external

mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their

understanding of concepts.


Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the

meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify

the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem

context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make

explicit use of definitions.


Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven

and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in

preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of

an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective.

They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus

a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.


Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing

25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they

repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way

terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x

3 + x

2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work

to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their

intermediate results.

Mathematics

Table 1. Common addition and subtraction situations.6

Result Unknown Change Unknown Start Unknown

Add to

Two bunnies sat on the grass. Three more

bunnies hopped there. How many bunnies are

on the grass now?

2 + 3 = ?

Two bunnies were sitting on the grass.

Some more bunnies hopped there. Then

there were five bunnies. How many

bunnies hopped over to the first two?

2 + ? = 5

Some bunnies were sitting on the grass.

Three more bunnies hopped there. Then

there were five bunnies. How many bunnies

were on the grass before?

? + 3 = 5

Take from

Five apples were on the table. I ate two apples.

How many apples are on the table now?

5 – 2 = ?

Five apples were on the table. I ate some

apples. Then there were three apples. How

many apples did I eat?

5 – ? = 3

Some apples were on the table. I ate two

apples. Then there were three apples. How

many apples were on the table before?

? – 2 = 3

Total Unknown Addend Unknown Both Addends Unknown1

Put Together/

Take Apart2

Three red apples and two green apples are on

the table. How many apples are on the table?

3 + 2 = ?

Five apples are on the table. Three are red

and the rest are green. How many apples

are green?

3 + ? = 5, 5 – 3 = ?

Grandma has five flowers. How many can

she put in her red vase and how many in her

blue vase?

5 = 0 + 5, 5 = 5 + 0

5 = 1 + 4, 5 = 4 + 1

5 = 2 + 3, 5 = 3 + 2

Difference Unknown Bigger Unknown Smaller Unknown

Compare3

(“How many more?” version):

Lucy has two apples. Julie has five apples.

How many more apples does Julie have than

Lucy?

(“How many fewer?” version):

Lucy has two apples. Julie has five apples.

How many fewer apples does Lucy have than

Julie?

2 + ? = 5, 5 – 2 = ?

(Version with “more”):

Julie has three more apples than Lucy.

Lucy has two apples. How many apples

does Julie have?

(Version with “fewer”):

Lucy has 3 fewer apples than Julie. Lucy

has two apples. How many apples does

Julie have?

2 + 3 = ?, 3 + 2 = ?

(Version with “more”):

Julie has three more apples than Lucy. Julie

has five apples. How many apples does

Lucy have?

(Version with “fewer”):

Lucy has 3 fewer apples than Julie. Julie

has five apples. How many apples does

Lucy have?

5 – 3 = ?, ? + 3 = 5 1These take apart situations can be used to show all the decompositions of a given number. The associated equations, which have the total on the left of the equal sign, help children understand that the = sign does not always mean makes or

results in but always does mean is the same number as. 2Either addend can be unknown, so there are three variations of these problem situations. Both Addends Unknown is a productive extension of this basic situation, especially for small numbers less than or equal to 10. 3For the Bigger Unknown or Smaller Unknown situations, one version directs the correct operation (the version using more for the bigger unknown and using less for the smaller unknown). The other versions are more difficult.

6Adapted from Box 2-4 of National Research Council (2009, op. cit., pp. 32, 33).

Mathematics

Table 2. Common multiplication and division situations.7

Unknown Product

Group Size Unknown

(“How many in each group?”

Division)

Number of Groups Unknown

(“How many groups?” Division)

3 x 6 = ? 3 x ? = 18, and 18 3 = ? ? x 6 = 18, and 18 6 = ?

Equal Groups

There are 3 bags with 6 plums in each

bag. How many plums are there in all?

Measurement example. You need 3

lengths of string, each 6 inches long.

How much string will you need

altogether?

If 18 plums are shared equally into 3

bags, then how many plums will be

in each bag?

Measurement example. You have 18

inches of string, which you will cut

into 3 equal pieces. How long will

each piece of string be?

If 18 plums are to be packed 6

to a bag, then how many bags

are needed?

Measurement example. You

have 18 inches of string, which

you will cut into pieces that are

6 inches long. How many pieces

of string will you have?

Arrays,4

Area5

There are 3 rows of apples with 6

apples in each row. How many apples

are there?

Area example. What is the area

of a 3 cm by 6 cm rectangle?

If 18 apples are arranged into 3

equal rows, how many apples will

be in each row?

Area example. A rectangle has area

18 square centimeters. If one side is

3 cm long, how long is a side next to

it?

If 18 apples are arranged into

equal rows of 6 apples, how

many rows will there be?

Area example. A rectangle has

area 18 square centimeters. If

one side is 6 cm long, how long

is a side next to it?

Compare

A blue hat costs $6. A red hat costs 3

times as much as the blue hat. How

much does the red hat cost?

Measurement example. A rubber band

is 6 cm long. How long will the rubber

band be when it is stretched to be 3

times as long?

A red hat costs $18 and that is 3

times as much as a blue hat costs.

How much does a blue hat cost?

Measurement example. A rubber

band is stretched to be 18 cm long

and that is 3 times as long as it was

at first. How long was the rubber

band at first?

A red hat costs $18 and a blue hat

costs $6. How many times as much

does the red hat cost as the blue hat?

Measurement example. A rubber

band was 6 cm long at first. Now it

is stretched to be 18 cm long. How

many times as long is the rubber

band now as it was at first?

General a × b = ? a × ? = p, and p ÷ a = ? ? × b = p, and p ÷ b = ?

4The language in the array examples shows the easiest form of array problems. A harder form is to use the terms rows and columns: The apples in the grocery window are in 3 rows and 6 columns. How many apples are in

there? Both forms are valuable. 5Area involves arrays of squares that have been pushed together so that there are no gaps or overlaps, so array problems include these especially important measurement situations. 7The first examples in each cell are examples of discrete things. These are easier for students and should be given

before the measurement examples.

Mathematics

Table 3. The properties of operations. Here a, b and c stand for arbitrary numbers in a given number system. The properties of operations apply to

the rational number system, the real number system, and the complex number system.

Associative property of addition (a + b) + c = a + (b + c)

Commutative property of addition a + b = b + a

Additive identity property of 0 a + 0 = 0 + a = a

Existence of additive inverses For every a there exists –a so that a + (–a) = (–a) + a = 0.

Associative property of multiplication (a x b) x c = a x (b x c)

Commutative property of multiplication a x b = b x a

Multiplicative identity property of 1 a x 1 = 1 x a = a

Existence of multiplicative inverses For every a ≠ 0 there exists 1/a so that a x 1/a = 1/a x a = 1.

Distributive property of multiplication over addition a x (b + c) = a x b + a x c

Table 4. The properties of equality. Here a, b and c stand for arbitrary numbers in the rational, real, or complex number systems.

Reflexive property of equality a = a

Symmetric property of equality If a = b, then b = a.

Transitive property of equality If a = b and b = c, then a = c.

Addition property of equality If a = b, then a + c = b + c.

Subtraction property of equality If a = b, then a – c = b – c.

Multiplication property of equality If a = b, then a x c = b x c.

Division property of equality If a = b and c ≠ 0, then a c = b c.

Substitution property of equality If a = b, then b may be substituted for a

in any expression containing a.

Table 5. The properties of inequality. Here a, b and c stand for arbitrary numbers in the rational or real number systems.

Exactly one of the following is true: a < b, a = b, a > b.

If a > b and b > c then a > c.

If a > b, then b < a.

If a > b, then –a < –b.

If a > b, then a ± c > b ± c.

If a > b and c > 0, then a x c > b x c.

If a > b and c < 0, then a x c < b x c.

If a > b and c > 0, then a c > b c.

If a > b and c < 0, then a c < b c.

Mathematics

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Mathematics


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