2015 - 2016 - elementarypgms.sp.brevardschools.orgelementarypgms.sp.brevardschools.org/home... ·...

A Teacher’s Guide to

Marching Ahead

with the

Mathematics Florida Standards

2015 - 2016Grade 4

Getting the Facts about Mathematics Florida Standards

Fourth Grade Table of Contents

I. Planning

Introduction to Pacing and Sequencing

Pacing and Sequencing Chart

Test Item Specifications

Operations and Properties Tables

II. Standards for Mathematical Practice

What Do Good Problem Solvers Do?

What Constitutes a Cognitively Demanding Task?

Key Ideas in Mathematics

Standards for Mathematical Practice Descriptions

Standards for Mathematical Practice – Student Behaviors

Standards for Mathematical Practice – Student Friendly Language

Standards for Mathematical Practice – Sample Questions for Teachers to Ask

Standards for Mathematical Practice in Action

Standards for Mathematical Practice in 4th Grade

Standards for Mathematical Practice Posters

III. Getting to know the Mathematics Florida Standards (MAFS)

Breaking the Code

MAFS by Grade Level at a Glance

Mathematics Florida Standards Changes

CCSS Domains, Clusters, and Critical Areas of Focus

Domain Progression

Fourth Grade Domain/Cluster Descriptors and Clarifications

NOTE: While some of the documents in this section were written based on Common Core Standards, they still contain information that can be used with Mathematics Florida Standards (MAFS). The changes as listed on the chart titled Mathematics Florida Standards Changes must be considered when using these documents.

IV. Additional Resources

Addition and Subtraction Strategies

Basic Multiplication and Division Strategies

Four Corners and Rhombus Math Graphic Organizer

Depth of Knowledge Levels/ Cognitive Complexity of Mathematics Items

Planning

INTRODUCTION TO PACING AND SEQUENCING- GRADE 4

INSTRUCTION:

All instruction must be standards-based. The textbook is a resource and textbook

lessons must be carefully chosen and aligned with the standards targeted for instruction.

It is critical that the Pacing and Sequencing Chart and the FSA Test Item Specifications are used for planning and implementing lessons.

The entire Pacing and Sequencing Chart should be previewed in order to begin with the end in mind and understand how the mathematical concepts grow throughout the year.

MAFS.4.OA.1a and MAFS.4.OA.1b are not addressed in the enVision program. This content can be taught through number talks.

CONNECTIONS BETWEEN THE DOMAINS:

Standards are not meant to be taught in isolation.

Each standard supports other standards and will continue to be developed throughout the year.

PROBLEM-SOLVING:

Emphasis should be on engaging students in deeper levels of thinking and analyzing.

Students must have many opportunities to explore the content of the standards through real-world problem-solving tasks.

Mathematical discourse must be an integral part of instruction.

MEASUREMENT:

Hands-on opportunities for students to be engaged in measurement are critical.

Hands-on measurement tasks may be taught within the science and social studies curricula.

VOCABULARY:

Correct mathematical vocabulary MUST be used. For example, students are expected to use terms such as addend, sum, factor, product, and so on.

ALGORITHMS AND FORMULAS:

Standard algorithms for addition and subtraction ONLY are introduced. The intent is that these are not just “rules” to be followed, but that students understand why they work.

Standard algorithms for multiplication and division are NOT to be introduced. Students are expected to solve these types of problems using strategies based on place value and properties. These algorithms are in standards in later grades.

Formulas for area (A = lw) and perimeter (P = 2l + 2w) are first introduced in 4th grade. The intent is that students understand why these work, not just memorize and use them. See your FSA Mathematics Reference Sheet for more information.

POST-FSA IDEAS:

Students should continue to work on critical areas within the grade level standards.

Project-based lessons and activities are encouraged.

Possible resources to use are:

AIMS Solve It! Navigating Through Numbers and

Operations in Grades 3-5, NCTM EnVision Math Worldscapes Literature

Library The Super Source Series, ETA/Cuisenaire Teaching Student-Centered Mathematics,

Vol.1, J.A.Van de Walle and L.H. Lovin Good Questions for Math Teaching, by

Peter Sullivan and Pat Lilburn

Pacing and Sequencing Chart Fourth Grade Mathematics Florida Standards

2015–2016

Fourth Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 1 of 23, Brevard Public Schools, 2015 – 2016

*Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction documents, and FSA Test Item Specifications.

Standards for Mathematical Practice

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and

critique the reasoning of others.

Model with mathematics.

Use appropriate tools

strategically.

Attend to precision.

Look for and make use of

structure.

Look for and express regularity in repeated

reasoning.

First Nine Weeks

Mathematics Florida Standards (MAFS) Explanations and Examples *

MAFS.4.NBT.1.1:

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.

Example:

How is the 2 in the number 582 similar to and different from the 2 in the number 528?

How many times greater is the value of the 4 in 640,700 than the value of the 4 in 64,070?

MAFS.4.NBT.1.2:

Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

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 and 5 ones; 1 hundred, 18 tens, and 5 ones; or 285 ones.

To read numerals, students need to understand the role of commas to separate the sequence of digits that makes up a period. Note: The word “and” signifies a decimal point. It should only be used at that point when reading a numeral. So, 457,328 is read, “Four hundred fifty-seven, three hundred twenty-eight.” Example:


2015–2016



MAFS.4.NBT.1.3:

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.

Example:

Round 590,340 to the nearest hundred thousand.

Example:

Original numbers are rounded to the nearest hundred and to the nearest thousand. The original numbers are different from all the rounded numbers in the table. Complete the table with possible original numbers.

Example:

Complete the table to show how each original number was rounded to make the new number.

Original New Nearest 100 Nearest 1,000

3,545 3,500

14,675 15,000

16,789 16,800


2015–2016



MAFS.4.NBT.2.4:

Fluently add and subtract multi-digit whole numbers using the standard algorithm.

Fluency involves accuracy (correct answer), efficiency (a reasonable amount of steps), and flexibility (using strategies that demonstrate number sense).

Example: An addition problem is shown. Calculate the sum.

63,829 24,343 + 1,424 Example: Enter the missing digit to complete the subtraction statement.

4 0 9, 8 4 5 – 1 □ 6, 6 7 5 2 1 3, 1 7 0

Example: What do each of the boxed digits stand for in this problem?

Students also need to be able to look at a problem written in the standard algorithm and explain what each number means.

Example: What do each of the boxed digits stand for in this problem?

1 1

739 + 556 1295

Student answer: This digit represents the number of tens in the sum of 9 and 6. 9 + 6 = 15, which is 1 ten and 5 ones. You keep the 5 ones in the ones place and move the 1 ten to the tens place.

Student answer: This number is the thousands from the sum of 7 hundreds and 5 hundreds, which is 12 hundreds. You regroup the 10 hundreds into 1 thousand and move it to the thousands place. This leaves 2 hundreds in the hundreds place.

1 14

524 - 318 206

Student answer: In order to subtract the ones place, I had to regroup the 24 as 1 ten and 14 ones.


2015–2016



MAFS.4.OA.1.1:

Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 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.

Additive comparison: How many more? or How many fewer? (quantities are different) Multiplicative comparison: How many times more? or How many times fewer? ( equal groups of quantities)

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.

Example:

5 x 8 = 40. Sally is five years old. Her mom is eight times older. How old is Sally’s Mom? 5 x 5 = 25 Sally has five times as many pencils as Mary. If Sally has 5 pencils, how many does Mary have? Example: Reggie has 12 times as many model cars as Jackson. Jackson has 5 model cars. Select all the equations below that show how many cars Reggie has.

A. 5 x 12 = ?

B. 5 + 12 = ?

C. 12 + 5 = ?

D. 12(5) = ?

E. 12(12 + 5) = ?

MAFS.4.OA.1.2:

Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

This standard calls for students to translate comparative situations into equations with an unknown and solve them.

Unknown Product: A blue hat costs $6. A red hat costs 3 times as much as the blue hat. How much does the red hat cost? (3 x 6 = p).

Group Size Unknown: A red hat costs $18. That is 3 times as much as a blue hat. How much does a blue hat cost? (18 ÷ a = 3 or 3 x a = 18).

Number of Groups Unknown: A red hat costs $18. A blue hat costs $6. How many times as much does the red hat cost compared to the blue hat? (18 ÷ 6 = b or 6 x b = 18).

*NOTE: See the Operations and Properties Tables in this section for common problem situations.


2015–2016



MAFS.4.OA.1.3:

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 reasonableness of answers using mental computation and estimation strategies including rounding. *Note: This standard should be developed throughout the year as new operations are introduced.

This standard references interpreting remainders. Remainders should be put into context for interpretation:

A. discard leaving only the whole number answer

B. remain as a left over

C. increase the whole number answer up one

D. partition into fractions or decimals (addressed in a later grade)

Possible problems for A – C above:

• Problem A: Mary had 44 pencils. Six pencils fit into each of her pencil pouches. How many pouches

did she fill? 44 ÷ 6 = p; p = 7 r 2. Mary can fill 7 pouches completely.

• Problem B: Mary had 44 pencils. Six pencils fit into each of her pencil pouches. How many pouches

could she fill and how many pencils would she have left? 44 ÷ 6 = p; p = 7 r 2; Mary can fill 7 pouches

and have 2 left over.

• Problem C: Mary had 44 pencils. Six pencils fit into each of her pencil pouches. What would be the

fewest number of pouches she would need in order to hold all of her pencils? 44 ÷ 6 = p; p = 7 r 2;

Mary needs 8 pouches to hold all the pencils.

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 may include, but are not limited to:

• using friendly or compatible numbers such as factors (students seek to fit numbers together - e.g.,

rounding to factors and grouping numbers together that have round sums like 100 or 1000),

• using benchmark numbers that are easy to compute (students select close whole numbers for fractions

or decimals to determine an estimate).


2015–2016



MAFS.4.OA.1.3: (Continued)

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 reasonableness of answers using mental computation and estimation strategies including rounding. *Note: Should be taught throughout the year as new skills are introduced.

Example:

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 miles did they travel total? Some typical estimation strategies for this problem:

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, 1 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.

MAFS.4.OA.3.5:

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 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 explain the reason for those features.

* NOTE: Rules can contain up to TWO procedural operations.

Example:

Pattern Rule Feature(s)

3, 8, 13, 18, 23, 28, … Start with 3, add 5 The numbers alternately end with a 3 or 8

5, 10, 15, 20 … Start with 5, add 5

The numbers are multiples of 5 and end with either 0 or 5. The numbers that end with 5 are products of 5 and an odd number. The numbers that end in 0 are products of 5 and an even number.


2015–2016



MAFS.4.OA.1.a:

Determine whether an equation is true or false by

using comparative relational thinking. For example,

without adding 60 and 24, determine whether the

equation 60 + 24 = 57 + 27 is true or false.

The equal sign can be defined in two ways – operational and relational. Many students express the equal sign as part of an

operation: “The equal sign means do the operation or give the answer.” This standard asks students to think of the equal sign

as showing a relationship between the values on the left side and the right side of the equal sign. This relational

understanding points at the equal sign as a balance—do the two sides balance? This understanding allows students to think

algebraically. Students need multiple opportunities to see the equal sign as a relationship between different quantities, e.g.,

6 + 4 = 7 + 3 (true) or 6 + 5 = 7 + 3 (false).

MAFS.4.OA.1.b:

Determine the unknown whole number in an equation relating four whole numbers using comparative relational thinking. For example, solve 76 + 9 = n + 5 for n by arguing that nine is four more than five, so the unknown number must be four greater than 76.

This standard carries comparative relational thinking into the use of unknowns. It emphasizes the relationship between the

two sides of the equation:

Example:

17 + 5 = 20 + ?

Student: “17 is 3 less than 20, so the unknown should be 3 less than 5.”


2015–2016



14

x 12

8 ( 2 x 4)

20 (2 x 10)

40 (10 x 4)

+ 100 (10 x 10)

168

MAFS.4.NBT.2.5:

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 5th grade, not fourth.

Example:

There are 14 dozen cookies in the bakery. What is the total number of cookies at the bakery? STUDENT 1: STUDENT 2: 14 x 12 14 x 12 I broke 12 up into 10 and 2 I broke 14 up into 2 groups 14 x 10 = 140 of 7: 7 x 12 = 84 14 x 2 = 28 I have 2 groups in 84, so 140 + 28 = 168 2 x 84 = 168

Example: What would an array area model of 14 x 12 look like?

Example:

Students should use base 10 blocks or drawings in order to develop understanding of the distributive property.

10 x 10 = 100

4 x 10 = 40

70

10 x 2 = 20

4 x 2 = 8

10 2

10

4

14 x 12

(10 + 4) x (10 + 2)

(10 x 10) + (4 x 10) + (10 x 2) + (4 x 2)

100 + 20 + 40 + 8 = 168


2015–2016










strategically.



structure.


reasoning.

Second Nine Weeks


MAFS.4.OA.2.4:

Investigate factors and multiples.

a. Find all factor pairs for a whole number in therange 1-100.

b. Recognize that a whole number is a multiple ofeach of its factors. Determine whether a givenwhole number in the range 1-100 is a multiple ofa given one-digit number.

c. Determine whether a given whole number in therange 1-100 is prime or composite.

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 factor pairs of the number, e.g., 96: (1, 96), (2, 48), (3, 32), (4, 24), (6, 16), (8, 12)Example:

Which factors do 36 and 42 have in common? Select all that apply. A. 1

B. 2

C. 3

D. 4

E. 6

F. 7

Example: Sarah is arranging 16 chairs for a recital. She is going to put them into a rectangular array. Complete the table below to show three ways Sarah can arrange the chairs.

Number of Rows

Number of

Chairs in Each Row

Arrangement 1

Arrangement 2

Arrangement 3


2015–2016



MAFS.4.NBT.2.6:

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.

Example:

There are 592 students participating in Field Day. They are put into teams of 8 for the competition. How many teams get

created?

Student 1 592 divided by 8 There are 70 8’s in 560 592 - 560 = 32 There are 4 8’s in 32 70 + 4 = 74

Student 2 592 divided by 8 I know that 10 8’s is 80 If I take out 50 8’s that is 400 592 - 400 = 192 I take out 20 more 8’s which is 160. 192 - 160 = 32 8 goes into 32 4 times with none left I took out 50, then 20 more, then

4 more. That’s 74.

592

- 400 50 192 - 160 20 32 - 32 4 0

Student 3 I want to get to 592 8 x 25 = 200 8 x 25 = 200 8 x 25 = 200 200 + 200 + 200 = 600 600 - 8 = 592 I had 75 groups of 8 and took one away, so there are 74 teams

MAFS.4.NF.1.1:

Explain why a fraction b

a is equivalent to a fraction

bn

an

×

× 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 addresses equivalent fractions by using visual fraction models to examine 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. Using an area model to show that 3

2 =

34

24

×

×

The whole is the square, measured by area. On the left it is divided horizontally into 3 rectangles of equal area, and the

shaded region is 2 of these and so represents 3

2. On the right it is divided into 4 x 3 small rectangles of equal area, and the

shaded area comprises 4 x 2 of these, and so it represents 34

24

×

×.

*NOTE: Fraction models are limited to number lines, rectangles, squares, and circles.


2015–2016



MAFS.4.NF.1.2:

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 2

1 . 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.

Possible student thinking by using benchmarks:

• 8

1 is smaller than

2

1 because when 1 whole is cut into 8 pieces, the pieces are much smaller than when 1 whole is

cut into 2 pieces

• 4

3 is more than

2

1 and

5

2 is less than

2

1, so

4

3>

5

2

• 8

1 is closer to 0 and

6

5 is closer to 1, so

8

1<

6

5

Possible student thinking by creating common denominators:

• 6

5>

2

1 because

6

3 =

2

1 and

6

5 >

6

3

*NOTE: Benchmark fractions are limited to;2

1,

4

1,

4

3

Example: Select >, <, or = to make the inequalities below true.

Example:

Kari has two fraction models, each divided into equal‐sized sections. The fraction represented by Model A is greater than the fraction represented by Model B. Model A is divided into 8 sections, and 2 sections are shaded. Model B is divided into 12 sections. What do you know about the number of sections shaded in Model B? Explain your answer.


2015–2016



MAFS.4.NF.2.3:

Understand a fraction b

a with a > 1 as a sum of

fractions b

1.

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:

8

3=

8

1+

8

1 +

8

1;

8

3 =

8

1 +

8

2;

28

1 = 1 + 1 +

8

1 =

8

8 +

8

8 +

8

1

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.

A fraction with a numerator of one is called a unit fraction. When students investigate fractions other than unit fractions, such

as 3

2, they should be able to decompose the non-unit fraction into a combination of several unit fractions.

Example:

3

2 =

3

1 +

3

1

Being able to visualize this decomposition into unit fractions helps students when adding or subtracting fractions. 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. Example:

14

1 -

4

3 = _____

4

4 +

4

1 =

4

5

4

5 –

4

3 =

4

2 or

2

1


2015–2016



MAFS.4.NF.2.3: (Continued)


a with a > 1 as a sum of

fractions b

1.

a. Understand addition and subtraction of fractionsas joining and separating parts referring to thesame whole.

b. Decompose a fraction into a sum of fractions withthe same denominator in more than one way,recording each decomposition by an equation.Justify decompositions, e.g. by using a visualfraction model. Examples:

8

3=

8

1+

8

1 +

8

1;

8

3 =

8

1 +

8

2;

28

1 = 1 + 1 +

8

1 =

8

8 +

8

8 +

8

1

c. Add and subtract mixed numbers with likedenominators, e.g., by replacing each mixednumber with an equivalent fraction, and/or byusing properties of operations and therelationship between addition and subtraction.

d. Solve word problems involving addition andsubtraction of fractions referring to the samewhole and having like denominators, e.g., byusing visual fraction models and equations torepresent the problem.

Example:

What is the sum of 23

2 and 1

3

2?

A. Give your answer as a mixed number.

B. Give your answer as a fraction.

Example:

Susan and Maria need 8 8

3 feet of ribbon to package gift baskets. Susan has 3

8

1 feet of ribbon and Maria has 5

8

3 feet of

ribbon. How much ribbon do they have altogether? Will it be enough to complete the project?

Explain why or why not.

The student thinks: I can add the ribbon Susan has to the ribbon Maria has to find out how much ribbon they have altogether.

Susan has 3 8

1 feet of ribbon and Maria has 5

8

3 feet of ribbon. I can write this as 3

8

1 + 5

8

3. I know they have 8 feet of

ribbon by adding the 3 and 5. They also have 8

1 and

8

3 which makes a total of

8

4 more. Altogether they have 8

8

4 feet of

ribbon. 8 8

4 is larger than 8

8

3 so they will have enough ribbon to complete the project. They will even have a little extra

ribbon left, 8

1 foot.


2015–2016



2

1

2

1

2

1

2

1 jump

MAFS.4.NF.2.4:

Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

a. Understand a fraction b

a as a multiple of

b

1. For

example, use a visual fraction model to represent

4

5 as the product 5

4

1, recording the

conclusion by the equation 4

5= 5

4

1.

b. Understand a multiple of b

a as a multiple of

b

1,

and use this understanding to multiply a fraction by a whole number. For example, use a visual

fraction model to express 3 5

2 as 6

5

1,

recognizing this product as 5

6.

(In general, n b

a =

b

an ×.)

c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party

will eat 8

3 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?

This standard builds on students’ work of adding fractions and extending that work into multiplication.

Example:

In a relay race, each runner runs 2

1 of a lap. If there are 4 team members how long is the race?

Number line:

0 2

1 1 1

2

1 2 2

2

1 3

Answer:

4 x 2

1

2

1 +

2

1 +

2

1 +

2

1 =

2

4

2

4 = 2


2015–2016



MAFS.4.MD.2.4:

Make a line plot to display a data set of

measurements in fractions of a unit (2

1 ,4

1 ,8

1 ). 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.

Questions based on the data must require students to add or subtract fractions with like denominators only. Students are not

expected to add or subtract fractions with unlike denominators until 5th grade.

Example:

Ten students in Room 31 measured their pencils (in inches). They recorded their results on the line plot below. X X X X X X X X X X __________________________________________________

32

1” 4” 4

4

1” 5

8

1” 5

2

1”

Possible questions:

• What is the difference in length from the longest to the shortest pencil? Answer: 52

1 - 3

2

1 = 2

• If the 58

1” pencils are placed end to end, what would be their total length? Answer: 5

8

1+ 5

8

1= 10

8

2

Example:

Benny recorded the results for his top four long jumps. The total length of all his jumps was 57 feet. The first two jumps

are shown on the number line. Record with an “x” the possible lengths of Benny’s last two jumps.


2015–2016










strategically.



structure.


reasoning.

Third Nine Weeks


MAFS.4.NF.3.5:

Express a fraction with a 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 10

3

as 100

30, and add

10

3 +

100

4 =

100

34.

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. Students should have many opportunities to shade decimal grids (10 x 10). 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 use the representations explored to understand 100

32 can be expanded to

10

3 and

100

2.

MAFS.4.NF.3.6:

Use decimal notation for fractions with denominators

10 or 100. For example, rewrite 0.62 as 100

62;

describe a length as 0.62 meters; locate 0.62 on a number line diagram.

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.

*NOTE: Items may contain decimals or fractions greater than 1 and/or mixed numbers.


2015–2016



MAFS.4.NF.3.7:

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 build area and other models to compare decimals. Through these experiences and their work with fraction models, they build the understanding that comparisons between decimals or fractions are only valid when the whole is the same for both cases.

Example:

Example:

Mr. Shelby bought a new plant. The plant grew 2.6 centimeters in the first week and 3.42 centimeters the second week.

Select all the comparisons below that correctly compare the two plant growths.

A. 2.6 > 3.42

B. 3.42 > 2.6

C. 2.6 < 3.42

D. ’3.42 < 2.6

E. 2.6 = 3.42


2015–2016



MAFS.4.G.1.1:

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:

Draw two different types of quadrilaterals that have two pairs of parallel sides.

Is it possible to have an acute right triangle? Justify your reasoning using pictures and words.

How many acute, obtuse and right angles are in a trapezoid?

Draw a parallelogram and list its properties. Do the same with a rectangle. How are your drawings and lists alike? How

are they different?

Example:

Put an X on all the acute angles in the shape to the right.

MAFS.4.G.1.2:

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. Angle Measurement: This expectation is closely connected to 4.MD.3.5, 4.MD.3.6, and 4.G.1.1. Students’ experiences with drawing

and identifying right, acute, and obtuse angles support them in classifying two-dimensional figures based on specified angle

measurements.

Example:

Select all the properties that always belong to each shape.

Has a right

angle

Has perpendicular

lines

Has parallel lines

right triangle

rhombus

rectangle

Example

Select all the shapes that always contain perpendicular sides. A. obtuse triangle B. acute triangle C. right triangle D. rectangle E. rhombus F. square


2015–2016



MAFS.4.MD.3.5:

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

360

1 of a circle is called a “one-degree angle,”

and can be used to measure angles.

b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.

This standard connects angles and circular measurement (360 degrees).

Example: A lawn sprinkler rotates a total of 60°. How many 60° cycles will it go through for the rotation to reach 360°?

Example:

Classify the angles below by putting an X in the each box that applies.

MAFS.4.MD.3.6:

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 (acute, right, and obtuse) to describe angles and rays.

Students should measure angles and sketch angles using a protractor.

120 degrees


2015–2016



MAFS.4.MD.3.7:

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.

Example:

Example: Kyle is adding angles to create other angles. Select the angles Kyle can use to create a 128° angle. Then select the angles that Kyle can use to create a 55° angle.

64° 34° 30° 25°

128°

55°


2015–2016



MAFS.4.G.1.3:

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.

Example:

For each figure, draw all of the lines of symmetry. What pattern do you notice? How many lines of symmetry do you think there would be for regular polygons with 9 and 11 sides. Sketch each figure and check your predictions.

Polygons with an odd number of sides have lines of symmetry that go from a midpoint of a side through a vertex.

MAFS.4.MD.1.3:

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.

Students learn to apply their understanding of area and perimeter to the solution of real-world and mathematical problems.

Example: A rectangular garden has as an area of 80 square feet. It is 5 feet wide. How long is the garden? Here, specifying the area and the width creates an unknown factor problem. (80 square feet = 5 ft. x ℓ). Similarly, students could solve

perimeter problems that give the perimeter and the length of one side and ask the length of the adjacent side.

MAFS.4.MD.1.1:

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)…

Students may use a two-column chart to convert from larger to smaller units and record equivalent measurements. They make statements such as, if one foot is 12 inches, then 3 feet has to be 36 inches because there are 3 groups of 12.

Example:

g ft. in. lb. oz.

1 1000 1 12 1 16

2 2000 2 24 2 32

3 3000 3 36 3 48


2015–2016



MAFS.4.MD.1.2:

Use the four operations to solve word problems1 involving distances, intervals of time, and money, including problems involving simple fractions or decimals2. Represent fractional quantities of distance, and intervals of time using linear models. (1See glossary Table 1 and Table 2.) (2Computational fluency with fractions and decimals is not the goal for students at this grade level.)

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 centimeters, dollars to cents). Students should have ample opportunities to use number line diagrams to solve word problems.

Examples:

Division/fractions: Susan has 2 feet of ribbon. She wants to give her ribbon to her 3 best friends so each friend gets the same amount. How much ribbon will each friend get? Students may draw a diagram of this problem, splitting each foot of the ribbon into

thirds:

Each friend would then get 2 of the pieces, or 3

2of the ribbon (

3

1 of each foot).

Students may also convert the 2 feet to 24 inches: 24 inches ÷ 3 friends = 8 inches.

Some students may understand that 3

1 x 24 inches = 8 inches.

Addition: Mason ran for an hour and 15 minutes on Monday, 25 minutes on Tuesday, and some minutes on Wednesday. He ran a total of 80 minutes. How far did Mason run on Wednesday? (addend unknown)

Subtraction: A pound of apples costs $1.20. Rachel bought a pound and a half of apples. If she gave the clerk a $5.00 bill, how much change will she get back? (result unknown)

Multiplication: Mario and his brother are selling lemonade. Mario brought 3L of lemonade. His brother poured an equal amount of lemonade into 6 pitchers. How much lemonade was in each pitcher? (group size unknown)

*NOTE: See the Operations and Properties Tables in this section for common problem situations.

Number line diagrams that feature a measurement scale can represent measurement quantities. Examples include: ruler, diagram of distance along a road with cities at various points, a timetable showing hours throughout the day, or a volume measure on the side of a container.

1 foot 1 foot


2015–2016







critique the reasoning of

others.



strategically.



structure.

Look for and express

regularity in repeated

reasoning.

Fourth Nine Weeks

FOURTH GRADE – CRITICAL AREAS OF FOCUS

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; and

(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., 9

15=

3

5), 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.

DRAFT

Grade4Mathematics ItemSpecifications

Grade 4 Mathematics Item Specifications Florida Standards Assessments

2 | P a g e M a y 2 0 1 6

The draft Florida Standards Assessments (FSA) Test Item Specifications (Specifications) are based upon the Florida Standards and the Florida Course Descriptions as provided in CPALMs. The Specifications are a resource that defines the content and format of the test and test items for item writers and reviewers. Each grade‐level and course Specifications document indicates the alignment of items with the Florida Standards. It also serves to provide all stakeholders with information about the scope and function of the FSA. Item Specifications Definitions Also assesses refers to standard(s) closely related to the primary standard statement. Clarification statements explain what students are expected to do when responding to the question. Assessment limits define the range of content knowledge and degree of difficulty that should be assessed in the assessment items for the standard. Item types describe the characteristics of the question. Context defines types of stimulus materials that can be used in the assessment items.

Context – Allowable refers to items that may but are not required to have context.

Context – No context refers to items that should not have context.

Context – Required refers to items that must have context.


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Technology‐EnhancedItemDescriptions:TheFloridaStandardsAssessments(FSA)arecomposedoftestitemsthatincludetraditionalmultiple‐choiceitems,itemsthatrequirestudentstotypeorwritearesponse,andtechnology‐enhanceditems(TEI).Technology‐enhanceditemsarecomputer‐delivereditemsthatrequirestudentstointeractwithtestcontenttoselect,construct,and/orsupporttheiranswers.Currently,thereareninetypesofTEIsthatmayappearoncomputer‐basedassessmentsforFSAMathematics.ForstudentswithanIEPor504planthatspecifiesapaper‐basedaccommodation,TEIswillbemodifiedorreplacedwithtestitemsthatcanbescannedandscoredelectronically.

Forsamplesofeachoftheitemtypesdescribedbelow,seetheFSATrainingTests.

Technology‐EnhancedItemTypes–Mathematics

1. EditingTaskChoice–Thestudentclicksahighlightedwordorphrase,whichrevealsadrop‐downmenucontainingoptionsforcorrectinganerroraswellasthehighlightedwordorphraseasitisshowninthesentencetoindicatethatnocorrectionisneeded.Thestudentthenselectsthecorrectwordorphrasefromthedrop‐downmenu.Forpaper‐basedassessments,theitemismodifiedsothatitcanbescannedandscoredelectronically.Thestudentfillsinacircletoindicatethecorrectwordorphrase.

2. EditingTask–Thestudentclicksonahighlightedwordorphrasethatmaybeincorrect,whichrevealsatextbox.Thedirectionsinthetextboxdirectthestudenttoreplacethehighlightedwordorphrasewiththecorrectwordorphrase.Forpaper‐basedassessments,thisitemtypemaybereplacedwithanotheritemtypethatassessesthesamestandardandcanbescannedandscoredelectronically.

3. HotText–a. SelectableHotText–Excerptedsentencesfromthetextarepresented

inthisitemtype.Whenthestudenthoversovercertainwords,phrases,orsentences,theoptionshighlight.Thisindicatesthatthetextisselectable(“hot”).Thestudentcanthenclickonanoptiontoselectit.Forpaper‐basedassessments,a“selectable”hottextitemismodifiedsothatitcanbescannedandscoredelectronically.Inthisversion,thestudentfillsinacircletoindicateaselection.


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b.Drag‐and‐DropHotText–Certainnumbers,words,phrases,or

sentencesmaybedesignated“draggable”inthisitemtype.Whenthestudenthoversovertheseareas,thetexthighlights.Thestudentcanthenclickontheoption,holddownthemousebutton,anddragittoagraphicorotherformat.Forpaper‐basedassessments,drag‐and‐drophottextitemswillbereplacedwithanotheritemtypethatassessesthesamestandardandcanbescannedandscoredelectronically.

4. OpenResponse–Thestudentusesthekeyboardtoenteraresponseintoatextfield.Theseitemscanusuallybeansweredinasentenceortwo.Forpaper‐basedassessments,thisitemtypemaybereplacedwithanotheritemtypethatassessesthesamestandardandcanbescannedandscoredelectronically.

5. Multiselect–Thestudentisdirectedtoselectallofthecorrectanswersfromamonganumberofoptions.Theseitemsaredifferentfrommultiple‐choiceitems,whichallowthestudenttoselectonlyonecorrectanswer.Theseitemsappearintheonlineandpaper‐basedassessments.

6. GraphicResponseItemDisplay(GRID)–Thestudentselectsnumbers,words,phrases,orimagesandusesthedrag‐and‐dropfeaturetoplacethemintoagraphic.Thisitemtypemayalsorequirethestudenttousethepoint,line,orarrowtoolstocreatearesponseonagraph.Forpaper‐basedassessments,thisitemtypemaybereplacedwithanotheritemtypethatassessesthesamestandardandcanbescannedandscoredelectronically.

7. EquationEditor–Thestudentispresentedwithatoolbarthatincludesavarietyofmathematicalsymbolsthatcanbeusedtocreatearesponse.Responsesmaybeintheformofanumber,variable,expression,orequation,asappropriatetothetestitem.Forpaper‐basedassessments,thisitemtypemaybereplacedwithamodifiedversionoftheitemthatcanbescannedandscoredelectronicallyorreplacedwithanotheritemtypethatassessesthesamestandardandcanbescannedandscoredelectronically.

8. MatchingItem–Thestudentchecksaboxtoindicateifinformationfromacolumnheadermatchesinformationfromarow.Forpaper‐basedassessments,thisitemtypemaybereplacedwithanotheritemtypethatassessesthesamestandardandcanbescannedandscoredelectronically.

9.TableItem–Thestudenttypesnumericvaluesintoagiventable.Thestudentmaycompletetheentiretableorportionsofthetabledependingonwhatisbeingasked.Forpaper‐basedassessments,thisitemtypemaybereplacedwithanotheritemtypethatassessesthesamestandardandcanbescannedandscoredelectronically.


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MathematicalPractices:

TheMathematicalPracticesareapartofeachcoursedescriptionforGrades3‐8,Algebra1,Geometry,andAlgebra2.Thesepracticesareanimportantpartofthecurriculum.TheMathematicalPracticeswillbeassessedthroughout.

MAFS.K12.MP.1.1:

Makesenseofproblemsandpersevereinsolvingthem.

Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaningofaproblemandlookingforentrypointstoitssolution.Theyanalyzegivens,constraints,relationships,andgoals.Theymakeconjecturesabouttheformandmeaningofthesolutionandplanasolutionpathwayratherthansimplyjumpingintoasolutionattempt.Theyconsideranalogousproblems,andtryspecialcasesandsimplerformsoftheoriginalprobleminordertogaininsightintoitssolution.Theymonitorandevaluatetheirprogressandchangecourseifnecessary.Olderstudentsmight,dependingonthecontextoftheproblem,transformalgebraicexpressionsorchangetheviewingwindowontheirgraphingcalculatortogettheinformationtheyneed.Mathematicallyproficientstudentscanexplaincorrespondencesbetweenequations,verbaldescriptions,tables,andgraphsordrawdiagramsofimportantfeaturesandrelationships,graphdata,andsearchforregularityortrends.Youngerstudentsmightrelyonusingconcreteobjectsorpicturestohelpconceptualizeandsolveaproblem.Mathematicallyproficientstudentschecktheiranswerstoproblemsusingadifferentmethod,andtheycontinuallyaskthemselves,“Doesthismakesense?”Theycanunderstandtheapproachesofotherstosolvingcomplexproblemsandidentifycorrespondencesbetweendifferentapproaches.

MAFS.K12.MP.2.1:

Reasonabstractlyandquantitatively.

Mathematicallyproficientstudentsmakesenseofquantitiesandtheirrelationshipsinproblemsituations.Theybringtwocomplementaryabilitiestobearonproblemsinvolvingquantitativerelationships:theabilitytodecontextualize—toabstractagivensituationandrepresentitsymbolicallyandmanipulatetherepresentingsymbolsasiftheyhavealifeoftheirown,withoutnecessarilyattendingtotheirreferents—andtheabilitytocontextualize,topauseasneededduringthemanipulationprocessinordertoprobeintothereferentsforthesymbolsinvolved.


6 | P a g e M a y 2 0 1 6

Quantitativereasoningentailshabitsofcreatingacoherentrepresentationoftheproblemathand;consideringtheunitsinvolved;attendingtothemeaningofquantities,notjusthowtocomputethem;andknowingandflexiblyusingdifferentpropertiesofoperationsandobjects.

MAFS.K12.MP.3.1:

Constructviableargumentsandcritiquethereasoningofothers.

Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviouslyestablishedresultsinconstructingarguments.Theymakeconjecturesandbuildalogicalprogressionofstatementstoexplorethetruthoftheirconjectures.Theyareabletoanalyzesituationsbybreakingthemintocases,andcanrecognizeandusecounterexamples.Theyjustifytheirconclusions,communicatethemtoothers,andrespondtotheargumentsofothers.Theyreasoninductivelyaboutdata,makingplausibleargumentsthattakeintoaccountthecontextfromwhichthedataarose.Mathematicallyproficientstudentsarealsoabletocomparetheeffectivenessoftwoplausiblearguments,distinguishcorrectlogicorreasoningfromthatwhichisflawed,and—ifthereisaflawinanargument—explainwhatitis.Elementarystudentscanconstructargumentsusingconcretereferentssuchasobjects,drawings,diagrams,andactions.Suchargumentscanmakesenseandbecorrect,eventhoughtheyarenotgeneralizedormadeformaluntillatergrades.Later,studentslearntodeterminedomainstowhichanargumentapplies.Studentsatallgradescanlistenorreadtheargumentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments.

MAFS.K12.MP.4.1:

Modelwithmathematics.

Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarisingineverydaylife,society,andtheworkplace.Inearlygrades,thismightbeassimpleaswritinganadditionequationtodescribeasituation.Inmiddlegrades,astudentmightapplyproportionalreasoningtoplanaschooleventoranalyzeaprobleminthecommunity.Byhighschool,astudentmightusegeometrytosolveadesignproblemoruseafunctiontodescribehowonequantityofinterestdependsonanother.Mathematicallyproficientstudentswhocanapplywhattheyknowarecomfortablemakingassumptionsandapproximationstosimplifyacomplicatedsituation,realizingthatthesemayneedrevisionlater.Theyareabletoidentifyimportantquantitiesinapracticalsituationandmaptheirrelationshipsusingsuchtoolsasdiagrams,two‐waytables,graphs,flowchartsandformulas.Theycananalyzethoserelationshipsmathematicallytodraw


7 | P a g e M a y 2 0 1 6

conclusions.Theyroutinelyinterprettheirmathematicalresultsinthecontextofthesituationandreflectonwhethertheresultsmakesense,possiblyimprovingthemodelifithasnotserveditspurpose.

MAFS.K12.MP.5.1:

Useappropriatetoolsstrategically.Mathematicallyproficientstudentsconsidertheavailabletoolswhensolvingamathematicalproblem.Thesetoolsmightincludepencilandpaper,concretemodels,aruler,aprotractor,acalculator,aspreadsheet,acomputeralgebrasystem,astatisticalpackage,ordynamicgeometrysoftware.Proficientstudentsaresufficientlyfamiliarwithtoolsappropriatefortheirgradeorcoursetomakesounddecisionsaboutwheneachofthesetoolsmightbehelpful,recognizingboththeinsighttobegainedandtheirlimitations.Forexample,mathematicallyproficienthighschoolstudentsanalyzegraphsoffunctionsandsolutionsgeneratedusingagraphingcalculator.Theydetectpossibleerrorsbystrategicallyusingestimationandothermathematicalknowledge.Whenmakingmathematicalmodels,theyknowthattechnologycanenablethemtovisualizetheresultsofvaryingassumptions,exploreconsequences,andcomparepredictionswithdata.Mathematicallyproficientstudentsatvariousgradelevelsareabletoidentifyrelevantexternalmathematicalresources,suchasdigitalcontentlocatedonawebsite,andusethemtoposeorsolveproblems.Theyareabletousetechnologicaltoolstoexploreanddeepentheirunderstandingofconcepts.

MAFS.K12.MP.6.1:

Attendtoprecision.

Mathematicallyproficientstudentstrytocommunicatepreciselytoothers.Theytrytousecleardefinitionsindiscussionwithothersandintheirownreasoning.Theystatethemeaningofthesymbolstheychoose,includingusingtheequalsignconsistentlyandappropriately.Theyarecarefulaboutspecifyingunitsofmeasure,andlabelingaxestoclarifythecorrespondencewithquantitiesinaproblem.Theycalculateaccuratelyandefficiently,expressnumericalanswerswithadegreeofprecisionappropriatefortheproblemcontext.Intheelementarygrades,studentsgivecarefullyformulatedexplanationstoeachother.Bythetimetheyreachhighschooltheyhavelearnedtoexamineclaimsandmakeexplicituseofdefinitions.


8 | P a g e M a y 2 0 1 6

MAFS.K12.MP.7.1:

Lookforandmakeuseofstructure.

Mathematicallyproficientstudentslookcloselytodiscernapatternorstructure.Youngstudents,forexample,mightnoticethatthreeandsevenmoreisthesameamountassevenandthreemore,ortheymaysortacollectionofshapesaccordingtohowmanysidestheshapeshave.Later,studentswillsee7×8equalsthewellremembered7×5+7×3,inpreparationforlearningaboutthedistributiveproperty.Intheexpressionx²+9x+14,olderstudentscanseethe14as2×7andthe9as2+7.Theyrecognizethesignificanceofanexistinglineinageometricfigureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems.Theyalsocanstepbackforanoverviewandshiftperspective.Theycanseecomplicatedthings,suchassomealgebraicexpressions,assingleobjectsorasbeingcomposedofseveralobjects.Forexample,theycansee5–3(x–y)²as5minusapositivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5foranyrealnumbersxandy.

MAFS.K12.MP.8.1:

Lookforandexpressregularityinrepeatedreasoning.

Mathematicallyproficientstudentsnoticeifcalculationsarerepeated,andlookbothforgeneralmethodsandforshortcuts.Upperelementarystudentsmightnoticewhendividing25by11thattheyarerepeatingthesamecalculationsoverandoveragain,andconcludetheyhavearepeatingdecimal.Bypayingattentiontothecalculationofslopeastheyrepeatedlycheckwhetherpointsareonthelinethrough(1,2)withslope3,middleschoolstudentsmightabstracttheequation(y–2)/(x–1)=3.Noticingtheregularityinthewaytermscancelwhenexpanding(x–1)(x+1),(x–1)(x²+x+1),and(x–1)(x³+x²+x+1)mightleadthemtothegeneralformulaforthesumofageometricseries.Astheyworktosolveaproblem,mathematicallyproficientstudentsmaintainoversightoftheprocess,whileattendingtothedetails.Theycontinuallyevaluatethereasonablenessoftheirintermediateresults.


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ReferenceSheets:•Referencesheetsandz‐tableswillbeavailableasonlinereferences(inapop‐upwindow).Apaperversionwillbeavailableforpaper‐basedtests.•ReferencesheetswithconversionswillbeprovidedforFSAMathematicsassessmentsinGrades4–8andEOCMathematicsassessments.•ThereisnoreferencesheetforGrade3.•ForGrades4,6,and7,Geometry,andAlgebra2,someformulaswillbeprovidedonthereferencesheet.•ForGrade5andAlgebra1,someformulasmaybeincludedwiththetestitemifneededtomeettheintentofthestandardbeingassessed.•ForGrade8,noformulaswillbeprovided;however,conversionswillbeavailableonareferencesheet.•ForAlgebra2,az‐tablewillbeavailable.

Grade Conversions SomeFormulas z‐table3 No No No4 OnReferenceSheet OnReferenceSheet No5 OnReferenceSheet WithItem No6 OnReferenceSheet OnReferenceSheet No7 OnReferenceSheet OnReferenceSheet No8 OnReferenceSheet No No

Algebra1 OnReferenceSheet WithItem NoAlgebra2 OnReferenceSheet OnReferenceSheet YesGeometry OnReferenceSheet OnReferenceSheet No


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Content Standard MAFS.4.OA Operations and Algebraic Thinking MAFS.4.OA.1 Use the four operations with whole numbers to solve problems. MAFS.4.OA.1.1 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.

Assessment Limits Items may not require students to solve for unknown factors that exceed 10 x 10 multiplication facts.

Item must include a verbal description of an equation or a multiplication equation.

Multiplication situations must be a comparison (e.g., times as many).

Calculator No

Item Types Equation Editor GRID Matching Item Multiple Choice Multiselect Open Response

Context Allowable

Sample Item Item Type

Alana has 8 times as many model cars as John. John has 2 model cars. Create a multiplication equation that represents the situation.

GRID


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Reggie has 12 times as many model cars as Jackson. Jackson has 5 model cars. Select all the equations that show how many cars Reggie has. □ 5 x 12 = ? □ 5 + 12 = ? □ 12 + 5 = ? □ 12(5) = ? □ 12(12 + 5) = ?

Multiselect

See Appendix for the practice test item aligned to this standard.


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Content Standard MAFS.4.OA Operations and Algebraic Thinking MAFS.4.OA.1 Use the four operations with whole numbers to solve problems. MAFS.4.OA.1.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

Assessment Limits Multiplication situation must be a comparison (e.g., times as many). Limit multiplication and division to 2‐digit by 1‐digit or a multiple of 10 by a

1‐digit.

Calculator No

Item Types Equation Editor GRID Multiple Choice Multiselect

Context Required


Cassie has 30 marbles. Abdul has m marbles. If Cassie has 10 times as many marbles as Abdul, write an equation that shows how many marbles Abdul has.

Equation Editor



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Content Standard MAFS.4.OA Operations and Algebraic Thinking MAFS.4.OA.1 Use the four operations with whole numbers to solve problems. MAFS.4.OA.1.3 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 reasonableness of answers using mental computation and estimation strategies including rounding.

Assessment Limits Items requiring precise or exact solutions are limited to:

addition and subtraction within 1,000.

multiplication of 2‐digit by 1‐digit or a multiple of 10 by a 1‐digit.

division of 2‐digit by 1‐digit. Items may contain a maximum of 3 steps. Items involving remainders must require the student to interpret and/or use the

remainder with respect to the context. Variables must be represented by a letter, and variables must be defined or

described in the context.

Calculator No

Item Types Equation Editor Multiple Choice Multiselect Open Response

Context Required


Jack bought 2 umbrellas. Each umbrella costs $13. He bought 3 hats, each costing $4. How much did Jack spend in all?

Equation Editor

Jack wants to buy the same number of hats for 3 of his friends. He has $57 dollars, and each hat costs $5. What is the greatest number of hats that Jack buys for each friend?

Equation Editor

Jack bought 2 umbrellas and 3 hats and spent between $30 and $50. Each umbrella costs the same amount. Each hat costs the same amount. The price of a hat is $4. What is the least amount Jack could have spent on an umbrella? What is the most Jack could have spent on an umbrella?

Equation Editor



14 | P a g e M a y 2 0 1 6

Content Standard MAFS.4.OA Operations and Algebraic Thinking MAFS.4.OA.1 Use the four operations with whole numbers to solve problems. MAFS.4.OA.1b Determine the unknown whole number in an equation relating four whole numbers using comparative relational thinking. For example, solve 76 + 9 = n + 5 for n arguing that nine is four more than five, so the unknown number must be four greater than 76. Also Assesses: MAFS.4.OA.1a Determine whether an equation is true or false by using comparative relational thinking. For example, without adding 60 and 24, determine whether the equation 60 + 24 = 57 + 27 is true or false.

Assessment Limits Whole number equations are limited to:

addition and subtraction within 1,000. multiplication of 2‐digit by 1‐digit or a multiple of 10 by a 1‐digit.

division of 2‐digit by 1‐digit. Variables represented by a letter are allowable.

Calculator No

Item Types Editing Task Choice Equation Editor GRID Hot Text Multiple Choice Multiselect Open Response

Context Allowable


Select all the true equations.

□ 72 – 29 = 70 – 31 □ 72 – 29 = 67 – 24 □ 72 – 29 = 70 – 30 □ 72 – 29 = 74 – 31 □ 72 – 29 = 62 – 39

Multiselect

What is the missing number in the equation shown?

102 – 25 = □ – 38

Equation Editor


15 | P a g e M a y 2 0 1 6

Content Standard MAFS.4.OA Operations and Algebraic Thinking MAFS.4.OA.2 Gain familiarity with factors and multiples. MAFS.4.OA.2.4 Investigate factors and multiples.

MAFS.4.OA.2.4a Find all factor pairs for a whole number in the range of 1—100.

MAFS.4.OA.2.4b 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.

MAFS.4.OA.2.4c Determine whether a given whole number in the range 1—100 is prime or composite.

Assessment Limits Items may only contain whole numbers between 1—100. Vocabulary may include prime, composite, factor, or multiple.

Calculator No

Item Types Equation Editor GRID Matching Item Multiple Choice Multiselect Table Item

Context Allowable


What are all the factors of 10? A. 1, 10 B. 2, 5 C. 1, 5, 10 D. 1, 2, 5, 10

Multiple Choice

Which factors do 36 and 42 have in common? □ 1 □ 2 □ 3 □ 4 □ 6 □ 7

Multiselect


16 | P a g e M a y 2 0 1 6


Sarah is arranging the chairs for a recital. She wants to put the 16 chairs into a rectangular array. Complete the table to show three ways that Sarah can arrange the chairs.

Number of Rows

Number of Chairs in Each Row

Arrangement 1

Arrangement 2

Arrangement 3

Table Item

See Appendix for the practice test item aligned to a standard in this group.


17 | P a g e M a y 2 0 1 6

Content Standard MAFS.4.OA Operations and Algebraic Thinking MAFS.4.OA.3 Generate and analyze patterns. MAFS.4.OA.3.5 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.

Assessment Limits Items may only contain whole numbers from 0 to 1,000. Operations in rules are limited to addition, subtraction, multiplication, and

division. Items may not contain rules that exceed two procedural operations. Division rules may not require fractional responses. Rules may not be provided algebraically (e.g., x + 5). Items must provide the rule.

Calculator No

Item Types Editing Task Choice Equation Editor GRID Hot Text Matching Item Multiple Choice Multiselect Open Response Table Item

Context Allowable


The first number in a pattern is 5. The pattern follows the rule “Add 3.” What is the next number in the pattern?

Equation Editor

The first number in a pattern is 80. The pattern follows the rule “Divide by 2.” Complete the table to show the next three numbers in the pattern.

Numbers in the Pattern

80

Table Item



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Content Standard MAFS.4.NBT Number and Operations in Base Ten MAFS.4.NBT.1 Generalize place value understanding for multi‐digit whole numbers. MAFS.4.NBT.1.1 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.

Assessment Limits Items may contain whole numbers within 1,000,000. Items may not compare digits across more than 1 place value.

Calculator No

Item Types Editing Task Choice Equation Editor Hot Text Multiple Choice Multiselect Open Response

Context No context


How many times greater is the value of the 4 in 640,700 than the value of the 4 in 64,070?

Equation Editor



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Content Standard MAFS.4.NBT Number and Operations in Base Ten MAFS.4.NBT.1 Generalize place value understanding for multi‐digit whole numbers. MAFS.4.NBT.1.2 Read and write multi‐digit whole numbers using base‐ten numerals, number names, and expanded form. Compare two multi‐digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

Assessment Limit Given values and item solutions may only be whole numbers between 1 and 1,000,000.

Calculator No

Item Types Equation Editor GRID Matching Item Multiple Choice Multiselect

Context Allowable


Write 6 x 10,000 + 5 x 1,000 + 2 x 100 + 3 x 1 as a number. Equation Editor

Match the name of each number with its numeric form.

600,005 600,050 605,000 650,000

Six hundred five thousand □ □ □ □

Six hundred thousand fifty □ □ □ □

Matching Item

Select all the options with 54,625 written in expanded form. □ 5 ten‐thousands, 46 hundreds, 25 ones □ 5 ten‐thousands, 4 thousands, 62 hundreds, 5 ones □ 50 thousands, 46 hundreds, 20 tens, 5 ones □ 50 thousands, 40 hundreds, 60 tens, 25 ones □ 54 thousands, 6 hundreds, 2 tens, 5 ones

Multiselect



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Content Standard MAFS.4.NBT Number and Operations in Base Ten MAFS.4.NBT.1 Generalize place value understanding for multi‐digit whole numbers. MAFS.4.NBT.1.3 Use place value understanding to round multi‐digit whole numbers to any place.

Assessment Limit Given values and item solutions may only be whole numbers between 1,000 and 1,000,000.

Calculator No

Item Types Equation Editor Matching Item Multiple Choice Multiselect Table Item

Context Allowable


Which numbers round to 4,100 when rounded to the nearest hundred? □ 4,008 □ 4,140 □ 4,060 □ 4,109 □ 4,049

Multiselect

Complete the table to show how each original number was rounded to make the new number.

Original New Nearest 100 Nearest 1,000

3,545 3,500 □ □

14,675 15,000 □ □

16,789 16,800 □ □

Matching Item

A. Round 590,340 to the nearest hundred thousand. B. Round 590,340 to the nearest ten thousand.

Equation Editor



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Content Standard MAFS.4.NBT Number and Operations in Base Ten MAFS 4.NBT.2 Use place value understanding and properties. MAFS.4.NBT.2.4 Fluently add and subtract multi‐digit whole numbers using the standard algorithm.

Assessment Limits Items may only contain whole number factors and solutions greater than 1,000 and within 1,000,000.

Addition expressions may contain up to three addends.

Calculator No


Context No context


An addition problem is shown.

63,829 24,343

+ 1,424

Calculate the sum.

Equation Editor

What is the difference of 31,678 and 28,995? Equation Editor

Enter the missing digit to complete the subtraction statement.

4 0 9, 8 4 5

– 1 □ 6, 6 7 5

2 1 3, 1 7 0

Equation Editor



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Content Standard MAFS.4.NBT Number and Operations in Base Ten MAFS.4.NBT.2 Use place value understanding and properties of operations to perform multi‐digit arithmetic. MAFS.4.NBT.2.5 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.

Assessment Limit Items may require multiplying: four digits by one digit, three digits by one digit, two digits by one digit, or two digits by two digits.

Calculator No

Item Types Equation Editor GRID Multiple Choice Multiselect Open Response

Context No context


Select all the expressions that have a product of 420. □ 35 x 12 □ (3 x 5) x (10 x 2) □ (40 x 10) x (2 x 4) □ 40 x 20 □ 14 x 30

Multiselect



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Content Standard MAFS.4.NBT Number and Operations in Base Ten MAFS.4.NBT.2 Use place value understanding and properties of operations to perform multi‐digit arithmetic. MAFS.4.NBT.2.6 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.

Assessment Limit Items may not require finding a quotient within the factor pairs of 10 x 10.

Calculator No


Context No context


What is 1,356 divided by 3? Equation Editor



24 | P a g e M a y 2 0 1 6

Content Standard MAFS.4.NF Numbers and Operations – Fractions

MAFS.4.NF.1 Extend understanding of fraction equivalence and ordering.

MAFS.4.NF.1.1 Explain why a fraction is equivalent to a fraction 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.

Assessment Limits Denominators of given fractions are limited to: 2, 3, 4, 5, 6, 8, 10, 12, 100. For items with denominators of 10 and 100, focus may not be on equivalence

between these 2 denominators, since this is addressed specifically in standards MAFS.4.NF.5 – 7, but should focus on equivalence between fractions with denominators of 2, 4, and 5, and fractions with denominators

of 10 and 100, e.g., = , = , etc.

Fractions must refer to the same whole, including in models. Fraction models are limited to number lines, rectangles, squares, and circles.

Fractions can be fractions greater than 1 and students may not be guided to

put fractions in lowest terms or to simplify.

Equivalent fractions also include fractions .

Calculator No

Item Types Editing Task Choice Equation Editor GRID Hot Text Matching Item Multiple Choice Multiselect Open Response

Context Allowable


Kari modeled a fraction by shading parts of the circle as shown.

Select sections to model a fraction equivalent to Kari’s fraction.

GRID


25 | P a g e M a y 2 0 1 6


Select all the models that have been shaded to represent fractions equivalent to .

Multiselect

Corey tried to find a fraction equivalent to . His work is shown.

35

35

12

310

Which statement describes Corey’s error?

A. It is impossible to find a fraction equivalent to .

B. He did not multiply by a fraction equal to 1.

C. He incorrectly multiplied and .

D. He should have divided by .

Multiple Choice



26 | P a g e M a y 2 0 1 6

Content Standard MAFS.4.NF Number and Operations – Fractions MAFS.4.NF.1 Extend understanding of fraction equivalence and ordering. MAFS.4.NF.1.2 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 . 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.

Assessment Limits Denominators of given fractions are limited to: 2, 3, 4, 5, 6, 8, 10, 12, 100.

Fractions may be fractions greater than 1 and students may not be guided to

put fractions in lowest terms or to simplify. Two fractions being compared must have both different numerator and different

denominator.

Calculator No

Item Types Editing Task Choice Equation Editor GRID Hot Text Matching Item Multiple Choice Multiselect Open Response

Context Allowable


Kari has two fraction models, each divided into equal‐sized sections. The fraction represented by Model A is greater than the fraction represented by Model B. Model A is divided into 8 sections, and 2 sections are shaded. Model B is divided into 12 sections. What do you know about the number of sections shaded in Model B? Explain your answer.

Open Response



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Content Standard MAFS.4.NF Number and Operations ‐ Fractions

MAFS.4.NF.2 Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

MAFS.4.NF.2.3 Understand a fraction with a > 1 as a sum of fractions .

MAFS.4.NF.2.3a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

MAFS.4.NF.2.3b 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: ; ;2 1 1 .

MAFS.4.NF.2.3c 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.

MAFS.4.NF.2.3d 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.

Assessment Limits Denominators of given fractions are limited to: 2, 3, 4, 5, 6, 8, 10, 12, 100. Mixed numbers and fractions must contain like denominators. Items must reference the same whole. Visual fraction models are limited to circular models, rectangular models, and

number line models.

Calculator No

Item Types Equation Editor GRID Matching Item Multiple Choice Multiselect Open Response

Context Allowable. Required for MAFS.4.NF.2.3d


What is the value of ? Equation Editor


28 | P a g e M a y 2 0 1 6


What is the value of the following expression? 210

910

A.

B.

C.

D.

Multiple Choice

Sue had of a cup of flour. She used of a cup.

How much flour, in cups, does Sue have left?

Equation Editor

What is the sum of 2 and 1 ?

A. Enter your answer as a mixed number. B. Enter your answer as a fraction.

Equation Editor



29 | P a g e M a y 2 0 1 6


MAFS.4.NF.2 Build fractions from unit fractions by applying and extending previous understanding of operations on whole numbers.

MAFS.4.NF.2.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

MAFS.4.NF.2.4a Understand a fraction as a multiple of . For example, use a

visual fraction model to represent as the product 5 , recording the

conclusion by the equation 5 .

MAFS.4.NF.2.4b Understand a multiple of as a multiple of , and use this

understanding to multiply a fraction by a whole number. For example, use a

visual fraction model to express 3 as 6 , recognizing this product as .

(In general, .

MAFS.4.NF.2.4c Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent

the problem. For example, if each person at a party will eat 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?

Assessment Limits Fractions may only be multiplied by a whole number. Denominators of given fractions are limited to: 2, 3, 4, 5, 6, 8, 10, 12, 100.

Calculator None


Context Allowable




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MAFS.4.NF.3 Understand decimal notation for fractions, and compare decimal fractions.

MAFS.4.NF.3.5 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 as , and add

.

Assessment Limits Denominators must be either 10 or 100. Decimal notation may not be assessed at this standard.

Calculator No

Item Types Equation Editor Matching Item Multiple Choice Multiselect

Context Allowable


Create a fraction with a denominator of 100 that is equivalent to . Equation Editor

Which fraction is equivalent to ?

A.

B.

C.

D.

Multiple Choice

An equation is shown.

+ =

What is the missing fraction?

Equation Editor



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MAFS.4.NF.3.6 Use decimal notation for fractions with denominators 10 or 100.

For example, rewrite 0.62 as ; describe a length as 0.62 meters; locate 0.62 on

a number line diagram.

Assessment Limits Denominators are limited to 10 and 100. Decimal notation is limited to tenths and hundredths. Items may contain decimals or fractions greater than 1 and/or mixed numbers.

Calculator No


Context No context


Two values are shown.

0.25 0.83

Use the Add Point tool to correctly plot these values on the number line.

GRID

Select all the fractions that are equivalent to 0.8.

□

□

□

□

□

□

Multiselect



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MAFS.4.NF.3.7 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.

Assessment Limits Decimals may reference the same whole entity. Decimals are limited to tenths and hundredths. Decimals may be greater than 1. Items may not require a comparison of visual models in isolation.

Calculator No

Item Types Editing Task Choice Equation Editor GRID Hot Text Matching Item Multiple Choice Multiselect Open Response Table Item

Context Allowable


Each model shown represents 1 whole.

Click to shade sections in the models to represent 0.2 and 0.3. Then, select the correct comparison symbol.

GRID


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A number line is shown.

A. Drag each number to its correct location on the number line. B. Select the correct comparison symbol.

GRID

Mr. Shelby bought a new plant. The plant grew 2.6 centimeters in the first week and 3.42 centimeters the second week. Select all the true comparisons of the plant growth for the two weeks. □ 2.6 > 3.42 □ 3.42 > 2.6 □ 2.6 < 3.42 □ 3.42 < 2.6 □ 2.6 = 3.42

Multiselect

Zach and Karla each have seeds they will plant in a class garden. Zach’s flower seeds weigh 1.5 grams. Karla’s seeds weigh 1.46 grams. Select the correct symbol for each comparison.

< > =

1.5 □ 1.46 □ □ □

1.46 □ 1.5 □ □ □

Matching Item

The locations of points K and L on the number line represent decimal numbers.

Explain why the value of point L is greater than the value of point K.

Open Response

Allison wrote down a decimal number that is greater than 0.58 but less than 0.62. What is one number Allison could have written down?

Equation Editor



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Content Standard MAFS.4.MD Measurement and Data

MAFS.4.MD.1 Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. MAFS.4.MD.1.1 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), ...

Assessment Limits Measurements may only be whole numbers. For non‐metric conversions, multiplication is limited to 2‐digit numbers by 1‐digit

numbers or a multiple of 10 by a 1‐digit number. Allowable units of measurement include: kilometer, meter, centimeter,

millimeter, liter, milliliter, kilogram, gram, milligram, mile, yard, foot, inch, gallon, quart, pint, cup, ton, pound, and ounce.

Calculator No

Item Types Equation Editor GRID Matching Item Multiple Choice Multiselect Table Item

Context Allowable


Select all the measurements that are about 1 yard long.

□ The length of a student’s desk □ The height of a classroom □ The width of a classroom door □ The length of a movie ticket □ The height of a building

Multiselect

The heights of three boxes are shown. Drag one measurement into each open box to order the heights from shortest to tallest.

GRID



35 | P a g e M a y 2 0 1 6


MAFS.4.MD.1 Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

MAFS.4.MD.1.2 Use the four operations to solve word problems involving distances, intervals of time, and money, including problems involving simple fractions or decimals. Represent fractional quantities of distance and intervals of time using linear models (Computational fluency with fractions and decimals is not the goal for students at this grade level.)

Assessment Limits Measurement conversions are from larger units to smaller units. Calculations are limited to simple fractions or decimals. Operations may include addition, subtraction, multiplication, and division. Item contexts are not limited to distances, intervals of time, and money.

Calculator No


Context Required


Gretchen is baking pies. She needs cup of butter for each pie. One stick of butter

is cup.

How many sticks of butter does Gretchen need to make 4 pies?

Equation Editor



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MAFS.4.MD.1 Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

MAFS.4.MD.1.3 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.

Assessment Limits Figures are limited to rectangles or composite figures composed of rectangles. Fractions are limited to like denominators. Limit multiplication and division to 2‐digit by 1‐digit or a multiple of 10 by 1‐digit.Quotients may only be whole numbers. Limit addition and subtraction to solutions within 1,000. When constructing rectangles, one grid must be labeled with the appropriate

dimension. That dimension must be “1 ____,” as items at this standard may not assess scale.

Calculator No


Context Allowable




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MAFS.4.MD.2 Represent and interpret data.

MAFS.4.MD.2.4 Make a line plot to display a data set of measurements in

fractions of a unit , , . 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.

Assessment Limits Measurement units are limited to halves, quarters, and eighths. Addition and subtraction of fractions is limited to fractions with like

denominators. Limit addition and subtraction to solutions within 1,000.

Calculator No


Context Allowable


Long jump measurements are given.

Long Jump Measurements

(in feet)

4

4

4

4

3

3

Click above the number line to create a correct line plot of the data.

GRID


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Benny recorded the results for his top four long jumps. The total length of all his jumps was 57 feet. Click above the number line to create a possible line plot for these data.

GRID



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Content Standard MAFS.4.MD Measurement and Data MAFS.4.MD.3 Geometric measurement: understand concepts of angle and measure angles. MAFS.4.MD.3.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement. MAFS.4.MD.3.5a 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. MAFS.4.MD.3.5b An angle that turns through n one‐degree angles is said to have an angle measure of n degrees. Also Assesses: MAFS.4.MD.3.6 Measure angles in whole‐number degrees using a protractor. Sketch angles of specified measure.

Assessment Limits Items may contain whole number degree measures within 0° and 360°.

For identification, angles are less than 360. For construction, angles are less than 180. Items may not require estimating the exact measures of angles.

Calculator No


Context Allowable for 4.MD.3.5; no context for 4.MD.3.6.


Select the category of measure for each angle.

Matching Item


40 | P a g e M a y 2 0 1 6


Angle P measures 68. One ray of angle P is shown.

Click on the protractor to show another ray that will create angle P.

GRID

An angle is shown.

What is the measure, in degrees, of the angle?

Equation Editor

See Appendix for the practice test items aligned to these standards.


41 | P a g e M a y 2 0 1 6

Content Standard MAFS.4.MD Measurement and Data MAFS.4.MD.3 Geometric measurement: understand concepts of angle and measure angles. MAFS.4.MD.3.7 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.

Assessment Limit Whole number degree measures, sums, and differences may only be within 0° and 360°.

Calculator No

Item Types Equation Editor Matching Item Multiple Choice Multiselect

Context Allowable


What is the measure of the unknown angle?

A. 40° B. 100° C. 120° D. 180°

Multiple Choice

Kyle is adding angles to create other angles.

Select the angles Kyle can use to create a 128 angle.

Select the angles that Kyle can use to create a 55 angle.

64 34 30 25 128 □ □ □ □

55 □ □ □ □

Matching Item


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A diagram is shown.

What is the sum of all the angles that are labeled?

Equation Editor



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Content Standard MAFS.4.G Geometry

MAFS.4.G.1 Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

MAFS.4.G.1.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two‐dimensional figures.

Assessment Limits Items may not require students to name a given figure. Items may not require knowledge or use of ordered pairs or a defined

coordinate grid system. Items may require students to draw a figure based on multiple attributes (e.g.,

an acute triangle), with the exception of right triangles. Items that include trapezoids must consider both the inclusive and exclusive

definitions. Items may not use the term "kite" but may include the figure.

Calculator No

Item Types GRID Matching Item Multiple Choice Multiselect Open Response

Context Allowable Sample Item Item Type

Which angle is acute?

Multiple Choice


44 | P a g e M a y 2 0 1 6


Select all the attributes that apply to each set of lines.

Matching Item

A. Use the Connect Line tool to draw an acute angle. B. Use the Connect Line tool to draw an obtuse angle.

GRID



45 | P a g e M a y 2 0 1 6


MAFS 4.G.1 Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

MAFS.4.G.1.2 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.

Assessment Limits Triangles: equilateral, equiangular, isosceles, scalene, acute, right, obtuse. Quadrilaterals: parallelograms, rectangles, squares, rhombi, trapezoids. Other polygons may be included where appropriate. Items that include trapezoids must consider both the inclusive and exclusive

definitions. Items may not use the term "kite" but may include the figure.

Calculator No

Item Types Editing Task Choice GRID Hot Text Matching Item Multiple Choice Multiselect Open Response

Context No context


Select all the obtuse triangles.

Multiselect


46 | P a g e M a y 2 0 1 6


Which figure is an acute triangle?

Multiple Choice

Select all the properties that always belong to each shape.

Has a right

angle

Has perpendicular

lines

Has parallel lines

Right Triangle

□ □ □

Rhombus □ □ □

Rectangle □ □ □

Matching Item

The shapes have been sorted into two groups.

Explain what two attributes were used to sort the shapes.

Open Response

The shapes have been sorted into two groups.

Explain what two attributes were used to sort the shapes.

Open Response



47 | P a g e M a y 2 0 1 6


MAFS.4.G.1 Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

MAFS.4.G.1.3 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.

Assessment Limit Items that require constructing lines of symmetry must specify the shape category with regard to the number of sides (quadrilateral, triangle, pentagon, etc.).

Items that include trapezoids must consider both the inclusive and exclusive definitions.

Items may not use the term "kite" but may include the figure.

Calculator No


Context Allowable


Select all the figures that have at least one line of symmetry.

Multiselect

How many lines of symmetry does the following figure have?

Equation Editor

A figure is shown.

How many lines of symmetry does the figure have?

Equation Editor



48 | P a g e M a y 2 0 1 6

Appendix A

The chart below contains information about the standard alignment for the items in the Grade 4

Mathematics FSA Computer‐Based Practice Test at http://fsassessments.org/students‐and‐

families/practice‐tests/.

Content Standard Item Type Computer‐Based Practice Test

Item Number

MAFS.4.OA.1.1 Multiple Choice 9

MAFS.4.OA.1.2 Equation Editor 13

MAFS.4.OA.1.3 Open Response 5

MAFS.4.OA.2.4c Matching Item 3

MAFS.4.OA.3.5 Open Response 18

MAFS.4.NBT.1.1 Multiple Choice 1

MAFS.4.NBT.1.2 Multiselect 11

MAFS.4.NBT.1.3 Table Item 7

MAFS.4.NBT.2.4 Multiple Choice 21

MAFS.4.NBT.2.5 Equation Editor 25

MAFS.4.NBT.2.6 Multiselect 23

MAFS.4.NF.1.1 Multiselect 6

MAFS.4.NF.1.2 Matching Item 26

MAFS.4.NF.2.3b Multiselect 20

MAFS.4.NF.2.4c Equation Editor 12

MAFS.4.NF.3.5 Equation Editor 24

MAFS.4.NF.3.6 Equation Editor 4

MAFS.4.NF.3.7 GRID 16

MAFS.4.MD.1.1 Table Item 17

MAFS.4.MD.1.2 GRID 8

MAFS.4.MD.1.3 Equation Editor 2


MAFS.4.MD.3.5a Multiple Choice 28

MAFS.4.MD.3.6 Multiple Choice 14


MAFS.4.G.1.1 GRID 10

MAFS.4.G.1.2 Multiselect 27

MAFS.4.G.1.3 Multiple Choice 15


49 | P a g e M a y 2 0 1 6

Appendix B: Revisions

Page(s) Revision Date

10‐11 Assessment limits revised. May 2016

12 Assessment limits and item types revised May 2016

14 Combined MAFS.4.OA.1b and MAFS.4.OA.1a standards, added sample items, and item types revised.

May 2016

15‐16 Inserted complete standard language for MAFS.4.OA.2.4b, corrected standard language for MAFS.4.OA.2.4a, and sample items revised.

May 2016

17 Item types revised. May 2016


19 Sample items revised. May 2016

20 Item types and context revised. May 2016



24‐25 Assessment limits, item types, and sample items revised. May 2016

26 Assessment limits and item types revised. May 2016

30 Assessment limits revised. May 2016

32‐33 Item types revised. May 2016



36 Assessment limit and sample items revised. May 2016

37‐38 Assessment limits, item types, and sample items revised. May 2016



45‐46 Assessment limits and item types revised. May 2016


48 Appendix A added to show Practice Test information. May 2016

Common Operation Situations and Properties, page 1 of 3, Brevard Public Schools, 2015 – 2016

Table 1. Common addition and subtraction situations.6

Result Unknown Change Unknown Start Unknown

Add to

Take from

Put together/take apart2

Compare3

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

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

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

Difference Unknown Bigger Unknown Smaller Unknown

(“How many more?” version): Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy?

(“How man 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 Mathematics Learning in Early Childhood, National Research Council (2009, pp. 32, 33).


Table 2. Common multiplication and division situations.7

Unknown Product

3 × 6 = ?

Group Size Unknown (“How many in each

group?” Division)

3 × ? = 18, and 18 ÷ 3 = ?

Number of Groups Unknown

(“How many groups?” Division)

? × 6 = 18, and 18 ÷ 6 = ?

Equal Groups

Arrays4, Area5

Compare3

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?

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?

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 be18 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.


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, a nd the complex number system.

TABLE 4. THE PROPERTIES OF EQUALITY. Here a, b and c stand for arbitrary numbers in the rational, real, or complex number systems.

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 × c > b × c.

If a > b and c < 0, then a × c < b × c.

If a > b and c > 0, then a ÷ c > b ÷ c.

If a > b and c < 0, then a ÷ c < b ÷ c

Associative property of addition

Commutative property of addition

Additive identity property of 0

Existence of additive inverses

Associative property of multiplication

Commutative property of multiplication

Multiplicative identity property of 1

Existence of multiplicative inverses

Distributive property of multiplication over addition

(a + b) + c = a + (b + c)

a + b = b + a

a + 0 = 0 + a = a

For every a there exists –a so that a + (–a) = (–a) + a = 0

(a × b) × c = a × (b × c)

a × b = b × a

a × 1 = 1 × a = a

For every a ≠ 0 there exists 1/a so that a × 1/a = 1/a × a = 1

a × (b + c) = a × b + a × c

Reflexive property of equality

Symmetric property of equality

Transitive property of equality

Addition property of equality

Subtraction property of equality

Multiplication property of equality

Division property of equality

Substitution property of equality

a = a

If a = b, then b = a.

If a = b and b = c, then a = c.

If a = b, then a + c = b + c.

If a = b, then a – c = b – c.

If a = b, then a × c = b × c.

If a = b and c ≠ 0, then a ÷ c = b ÷ c.

If a = b, then b may be substituted for a in any expression containing a.

Standards for

Mathematical Practice

Look for and create

efficient strategies

Use tools and

technology strategically

Do what makes sense and be

persistent

Use number sense when representing

a problem

What do good problem

solvers do?

Be precise with words,

numbers, and symbols

Use math to describe a

real situation or problem

Look for and use

patterns and connections

Make conjectures and prove or

disprove them

What Constitutes a Cognitively Demanding Task?

Lower-level demands (memorization) • Involve either reproducing previously learned facts, rules, formulas, or definitions or committing facts,

rules, formulas or definitions to memory. • Cannot be solved using procedures because a procedure does not exist or because the time frame in

which the task is being completed is too short to use a procedure • Are not ambiguous. Such tasks involve the exact reproduction of previously seen material, and what is to

be reproduced is clearly and directly stated. • Have no connection to the concepts or meaning that underlie the facts, rules, formulas, or definitions

being learned or reproduced.

Lower-level demands (procedures without connections to meaning) • Are algorithmic. Use of the procedure either is specifically called for or is evident from prior instruction,

experience, or placement of the task. • Require limited cognitive demand for successful completion. Little ambiguity exists about what needs to

be done and how to do it. • Have no connection to concepts or meaning that underlie the procedure being used.• Are focused on producing correct answers instead of on developing mathematical understanding.• Require no explanation or explanations that focus solely on describing the procedure that was used.

Higher-level demands (procedures with connections to meaning) • Focus students’ attention on the use of procedures for the purpose of developing deeper levels of

understanding of mathematical concepts and ideas. • Suggest explicitly or implicitly pathways to follow that are broad general procedures that have close

connections to underlying conceptual ideas as opposed to narrow algorithms that are opaque with respect to underlying concepts.

• Usually are represented in multiple ways, such as visual diagrams, manipulatives, symbols, and problemsituations. Making connections among multiple representations helps develop meaning.

• Require some degree of cognitive effort. Although general procedures may be followed, they cannot befollowed mindlessly. Students need to engage with conceptual ideas that underlie the procedures to complete the task successfully and that develop understanding.

Higher-level demands (doing mathematics) • Require complex and non-algorithmic thinking - a predictable, well-rehearsed approach or pathway is

not explicitly suggested by the task, task instructions, or a worked-out example. • Require students to explore and understand the nature of mathematical concepts, processes, or

relationships. • Demand self-monitoring or self-regulation of one’s own cognitive processes.• Require students to access relevant knowledge and experiences and make appropriate use of them in

working through the task.• Require considerable cognitive effort and may involve some level of anxiety for the student because of

the unpredictable nature of the solution process required.

Arbaugh, F., & Brown, C.A. (2005). Analyzing mathematical tasks: a catalyst for change? Journal of Mathematics Teacher Education, 8, p. 530.

Key Ideas in the Mathematics Florida Standards (MAFS)

Focus: Greater focus on fewer topics

Focus deeply on the standards for mastery and the ability to transfer skills.

Focus deeply on the major work of each grade as follows:

In grades K-2: Concepts, skills, problem solving related to addition and subtraction.

In grades 3-5: Concepts, skills, and problem solving related to multiplication and division of

whole numbers and fractions.

In grade 6: Ratios and proportional relationships, and early algebraic expressions and

equations.

This focus will enable students to gain strong foundations, including a solid understanding of

concepts, and the ability to apply the math they know to solve problems inside and outside the

classroom.

Coherence: Linking topics and thinking across grades

Coherence is about making math make sense.

Mathematics is a coherent body of knowledge made up of interconnected concepts.

The standards are designed around coherent progressions from grade to grade.

Learning is carefully connected across grades so that students can build new understanding

onto foundations built in previous years.

Each standard is not a new event, but an extension of previous learning.

It is critical to think across grade levels and examine the progressions to see how major

content is developed across grades.

Rigor: Calls for a balance of tasks that require conceptual understanding,

procedural skills and fluency, and application of mathematics to solve problems

Rigor refers to deep, authentic command of mathematical concepts.

The following three aspects of rigor must be pursued with equal intensity to help students

meet the standards:

Conceptual understanding: The standards call for conceptual understanding of key

concepts. Students must be able to access concepts from a number of perspectives. This

will allow them to see math as more than a set of mnemonics or discrete procedures.

Procedural skills and fluency: The standards call for speed and accuracy in calculation with

a balance of practice and understanding. Students must practice simple calculations such

as single-digit multiplication with meaning, in order to have access to more complex

concepts and procedures.

Application: The standards call for students to have solid conceptual understanding and

procedural fluency. They are expected to apply their understanding and procedural skills in

mathematics to problem solving situations.

-Adapted from www.corestandards.org

Standards for Mathematical Practice The Standards for Mathematical Practice describe behaviors that all students will develop in the Common Core Standards. These practices rest on important “processes and proficiencies” including problem solving, reasoning and proof, communication, representation, and making connections. These practices will allow students to understand and apply mathematics with confidence.

1. Make sense of problems and persevere in solving them.• Find meaning in problems• Analyze, predict, and plan solution pathways• Verify answers• Ask them the question: “Does this make sense?”

2. Reason abstractly and quantitatively.• Make sense of quantities and their relationships in problems• Create coherent representations of problems

3. Construct viable arguments and critique the reasoning of others.• Understand and use information to construct arguments• Make and explore the truth of conjectures• Justify conclusions and respond to arguments of others

4. Model with mathematics.• Apply mathematics to problems in everyday life• Identify quantities in a practical situation• Interpret results in the context of the situation and reflect on

whether the results make sense

When given a problem, I can make a plan to solve it and check my answer.

I can use numbers and words to help me make sense of problems.

I can explain my thinking and consider the mathematical thinking of others.

I can recognize math in everyday life and use math I know to solve problems.

5. Use appropriate tools strategically.• Consider the available tools when solving problems• Be familiar with tools appropriate for their grade or course (pencil and paper, concrete models, ruler,

protractor, calculator, spreadsheet, computer programs, digital content located on a website, and othertechnological tools)

6. Be precise.• Communicate precisely to others• Use clear definitions, state the meaning of symbols and be careful

about specifying units of measure and labeling axes• Calculate accurately and efficiently

7. Look for and make use of structure.• Recognize patterns and structures• Step back for an overview and shift perspective• See complicated things as single objects or as being composed of several objects

8. Look for and identify ways to create shortcuts when doing problems.• When calculations are repeated, look for general methods, patterns and

shortcuts• Be able to evaluate whether an answer makes sense

I can use math tools to help me explore and understand math in my world.

I can be careful when I use math and clear when I share my ideas.

I can see and understand how numbers and shapes are put together as parts and wholes.

I can notice when calculations are repeated.

Standard for Mathematical Practice

Student Friendly Language

1. Make sense of problems andpersevere in solving them.

• I can try many times tounderstand and solve amath problem.

2. Reason abstractlyand quantitatively.

• I can think about the mathproblem in my head, first.

3. Construct viable argumentsand critique the reasoningof others.

• I can make a plan, called astrategy, to solve theproblem and discuss otherstudents’ strategies too.

4. Model with mathematics.• I can use math symbols and

numbers to solve theproblem.

5. Use appropriate toolsstrategically.

• I can use math tools,pictures, drawings, andobjects to solve the problem.

6. Attend to precision.• I can check to see if my

strategy and calculationsare correct.

7. Look for and make useof structure

• I can use what I alreadyknow about math to solvethe problem.

8. Look for and express regularityin repeated reasoning.

• I can use a strategy that Iused to solve another mathproblem.

Carroll County Public Schools, http://www.carrollk12.org/instruction/instruction/elementary/math/curriculum/common/default.asp

Florida State Standards Standards for Mathematical Practice Sample Questions for Teachers to Ask

Make sense of problems and persevere in solving them

Reason abstractly and quantitatively

Construct viable arguments and critique the reasoning of others Model with mathematics

Teachers ask: • What is this problem asking?• How could you start this

problem?• How could you make this

problem easier to solve?• How is ___’s way of solving

the problem like/different fromyours?

• Does your plan make sense?Why or why not?

• What tools/manipulativesmight help you?

• What are you having troublewith?

• How can you check this?

Teachers ask: • What does the number ____

represent in the problem? • How can you represent the

problem with symbols and numbers?

• Create a representation of theproblem.

Teachers ask: • How is your answer different

than _____’s? • How can you prove that your

answer is correct? • What math language will help

you prove your answer? • What examples could prove or

disprove your argument? • What do you think about

_____’s argument • What is wrong with ____’s

thinking? • What questions do you have

for ____? *it is important that the teacherimplements tasks that involve discourse and critiquing of reasoning

Teachers ask: • Write a number sentence to

describe this situation • What do you already know

about solving this problem? • What connections do you see?• Why do the results make

sense?• Is this working or do you need

to change your model?*It is important that the teacherposes tasks that involve real world situations

Use appropriate tools strategically Attend to precision Look for and make use of

structure Look for and express regularity

in repeated reasoning

Teachers ask: • How could you use

manipulatives or a drawing to show your thinking?

• Which tool/manipulative wouldbe best for this problem?

• What other resources couldhelp you solve this problem?

Teachers ask: • What does the word ____

mean? • Explain what you did to solve

the problem. • Compare your answer to

_____’s answer • What labels could you use?• How do you know your answer

is accurate?• Did you use the most efficient

way to solve the problem?

Teachers ask: • Why does this happen?• How is ____ related to ____?• Why is this important to the

problem?• What do you know about ____

that you can apply to thissituation?

• How can you use what youknow to explain why thisworks?

• What patterns do you see?*deductive reasoning (movingfrom general to specific)

Teachers ask: • What generalizations can you

make? • Can you find a shortcut to

solve the problem? How would your shortcut make the problem easier?

• How could this problem helpyou solve another problem?

*inductive reasoning (moving fromspecific to general)

Standards for Mathematical Practice in Action

Practice Sample Student Evidence Sample Teacher Actions

1. Make senseof problemsandpersevere insolving them

Display sense-making behaviors Show patience and listen to others Turn and talk for first steps and/or generate solution plan Analyze information in problems Use and recall multiple strategies Self-evaluate and redirect Assess reasonableness of process and answer

Provide open-ended problems Ask probing questions Probe student responses Promote and value discourse Promote collaboration Model and accept multiple approaches

2. Reasonabstractlyandquantitatively

Represent abstract and contextual situations symbolically Interpret problems logically in context Estimate for reasonableness Make connections including real life situations Create and use multiple representations Visualize problems Put symbolic problems into context

Model context to symbol and symbol to context Create problems such as “what word problem

will this equation solve?” Give real world situations Offer authentic performance tasks Place less emphasis on the answer Value invented strategies Think Aloud

3. Constructviableargumentsand critiquethereasoning ofothers

Questions others Use examples and non-examples Support beliefs and challenges with mathematical evidence Forms logical arguments with conjectures and counterexamples Use multiple representations for evidence Listen and respond to others well Uses precise mathematical vocabulary

Create a safe and collaborative environment Model respectful discourse behaviors “Find the error” problems Promote student to student discourse (do not

mediate discussion) Plan effective questions or Socratic formats Provide time and value discourse

4. Model withmathematics

Connect math (numbers and symbols) to real-life situations Symbolize real-world problems with math Make sense of mathematics Apply prior knowledge to solve problems Choose and apply representations, manipulatives and other

models to solve problems Use strategies to make problems simpler Use estimation and logic to check reasonableness of an answer

Model reasoning skills Provide meaningful, real world, authentic

performance-based tasks Make appropriate tools available Model various modeling techniques Accept and value multiple approaches and

representations

5. Useappropriatetoolsstrategically

Choose appropriate tool(s) for a given problem Use technology to deepen understanding Identify and locate resources Defend mathematically choice of tool

Provide a “toolbox” at all times with all available tools – students then choose as needed

Model tool use, especially technology for understanding

6. Attend toprecision

Communicate (oral and written) with precise vocabulary Carefully formulate questions and explanations (not retelling

steps) Decode and interpret meaning of symbols Pay attention to units, labeling, scale, etc. Calculate accurately and effectively Express answers within context when appropriate

Model problem solving strategies Give explicit and precise instruction Ask probing questions Use ELA strategies of decoding,

comprehending, and text-to-self connections for interpretation of symbolic and contextual math problems

Guided inquiry

7. Look for andmake use ofstructure

Look for, identify, and interpret patterns and structures Make connections to skills and strategies previously learned to

solve new problems and tasks Breakdown complex problems into simpler and more

manageable chunks Use multiple representations for quantities View complicated quantities as both a single object or a

composition of objects

Let students explore and explain patterns Use open-ended questioning Prompt students to make connections and

choose problems that foster connections Ask for multiple interpretations of quantities

8. Look for andexpressregularity inrepeatedreasoning

Design and state “shortcuts” Generate “rules” from repeated reasoning or practice (e.g.

integer operations) Evaluate the reasonableness of intermediate steps Make generalizations

Provide tasks that allow students to generalize Don’t teach steps or rules, but allow students to

explore and generalize in order to discover and formalize

Ask deliberate questions Create strategic and purposeful check-in points

Standards for Mathematical Practice, (from North Carolina Department of Education, http://www.ncpublicschools.org/), Fourth Grade, page 1 of 2, 2013 - 2014

STANDARDS FOR MATHEMATICAL PRACTICE IN FOURTH GRADE The Standards for Mathematical Practice are expected to be integrated into every mathematics lesson for all students Grades K-12. Below are a few examples of how these Practices may be integrated into tasks that students complete.

Practice Explanation and Example

1. Make sense of problemsand persevere in solvingthem.

Mathematically proficient students in fourth grade know that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Fourth grade students may use concrete objects or pictures to help themselves conceptualize and solve problems. They may check their thinking by asking themselves, “Does this make sense?”. They listen to the strategies of others and will try different approaches. They often will use another method to check their answers.

2. Reason abstractly andquantitatively.

Mathematically proficient students in fourth grade should recognize that a number represents a specific quantity. They connect the quantity to written symbols and create a logical representation of the problem at hand, considering both the appropriate units involved and the meaning of quantities. They extend this understanding from whole numbers to their work with fractions and decimals. Students write simple expressions, record calculations with numbers, and represent or round numbers using place value concepts.

3. Construct viablearguments and critiquethe reasoning of others.

Mathematically proficient students in fourth grade may construct arguments using concrete referents, such as objects, pictures, and drawings. They explain their thinking and make connections between models and equations. They refine their mathematical communication skills as they participate in mathematical discussions involving questions like “How did you get that?” and “Why is that true?”. They explain their thinking to others and respond to others’ thinking.

4. Model with mathematics.

Mathematically proficient students in fourth grade experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, making a chart, list, or graph, creating equations, etc. Students need opportunities to connect these different representations and explain the connections. They should be able to use all of these representations as needed. Fourth grade students should evaluate their results in the context of the situation and reflect on whether the results make sense.

http://www.ncpublicschools.org/

Standards for Mathematical Practice, (from North Carolina Department of Education, http://www.ncpublicschools.org/), Fourth Grade, page 2 of 2, 2013 - 2014

Practice Explanation and Example

5. Use appropriate toolsstrategically.

Mathematically proficient students in fourth grade consider the available tools (including estimation) when solving a mathematical problem and decide when certain tools might be helpful. For instance, they may use graph paper or a number line to represent and compare decimals and protractors to measure angles. They use other measurement tools to understand the relative size of units within a system and express measurements given in larger units in terms of smaller units.

6. Attend to precision.Mathematically proficient students in fourth grade develop their mathematical communication skills as they try to use clear and precise language in their discussions with others and in their own reasoning. They are careful about specifying units of measure and state the meaning of the symbols they choose. For instance, they use appropriate labels when creating a line plot.

7. Look for and make useof structure.

Mathematically proficient students in fourth grade look closely to discover a pattern or structure. For instance, students use properties of operations to explain calculations (partial products model). They relate representations of counting problems such as tree diagrams and arrays to the multiplication principal of counting. They generate number or shape patterns that follow a given rule.

8. Look for and expressregularity in repeatedreasoning.

Mathematically proficient students in fourth grade should notice repetitive actions in computation to make generalizations. Students use models to explain calculations and understand how algorithms work. They also use models to examine patterns and generate their own algorithms. For example, students use visual fraction models to write equivalent fractions.

http://www.ncpublicschools.org/

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Getting

to Know

MAFS

Breaking the Code Mathematics Florida Standards

MAFS.5.OA.1.1

MAFS = Mathematics Florida Standards 5 = Fifth Grade OA = Operations and Algebraic Thinking

1 = Cluster – Write and interpret numerical expressions.

1 = Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

K – 5 Domains

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

MD = Measurement and Data G = Geometry

Subject/Standards Domain

Grade Level

Cluster

Standard

Fourth Grade Mathematics Florida Standards 2015 – 2016

Fourth Grade Mathematics Florida Standards, page 1 of 5, Brevard Public Schools, 2015 – 2016

Domain: OPERATIONS AND ALGEBRAIC THINKING

Cluster 1: Use the four operations with whole numbers to solve problems.

MAFS.4.OA.1.1:

Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 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.

MAFS.4.OA.1.2:

Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

MAFS.4.OA.1.3:

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 reasonableness of answers using mental computation and estimation strategies including rounding.

MAFS.4.OA.1.a:

Determine whether an equation is true or false by using comparative relational thinking. For example, without adding 60 and 24,

determine whether the equation 60 + 24 = 57 + 27 is true or false.

MAFS.4.OA.1.b:

Determine the unknown whole number in an equation relating four whole numbers using comparative relational thinking. For example, solve 76 + 9 = n + 5 for n by arguing that nine is four more than five, so the unknown number must be four greater than 76.

Cluster 2: Gain familiarity with factors and multiples.

MAFS.4.OA.2.4:

Investigate factors and multiples.

a. Find all factor pairs for a whole number in the range 1-100.

b. 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 amultiple of a given one-digit number.

c. Determine whether a given whole number in the range 1-100 is prime or composite.

Cluster 3: Generate and analyze patterns.

MAFS.4.OA.3.5:

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.

Domain: NUMBER AND OPERATIONS IN BASE TEN

Cluster 1: Generalize place value understanding for multi-digit whole numbers.

MAFS.4.NBT.1.1:

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.



MAFS.4.NBT.1.2:

Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

MAFS.4.NBT.1.3:

Use place value understanding to round multi-digit whole numbers to any place.

Cluster 2: Use place value understanding and properties of operations to perform multi-digit arithmetic.

MAFS.4.NBT.2.4:

Fluently add and subtract multi-digit whole numbers using the standard algorithm.

MAFS.4.NBT.2.5:

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.

MAFS.4.NBT.2.6:

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.

Domain: NUMBER AND OPERATIONS - FRACTIONS

Cluster 1: Extend understanding of fraction equivalence and ordering.

MAFS.4.NF.1.1:

Explain why a fraction b

a is equivalent to a fraction

bn

an

×

× 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.

MAFS.4.NF.1.2:

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 2

1. 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.



Cluster 2: Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

MAFS.4.NF.2.3:


a with a > 1 as a sum of fractions

b

1.

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:8

3=

8

1+

8

1 +

8

1;

8

3 =

8

1 +

8

2;

28

1 = 1 + 1 +

8

1 =

8

8 +

8

8 +

8

1

c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or byusing 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., byusing visual fraction models and equations to represent the problem.

MAFS.4.NF.2.4:

Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

a. Understand a fractionb

a as a multiple of

b

1. For example, use a visual fraction model to represent

4

5 as the product 5

4

1,

recording the conclusion by the equation 4

5= 5

4

1.

b. Understand a multiple ofb

a as a multiple of

b

1, and use this understanding to multiply a fraction by a whole number. For example,

use a visual fraction model to express 3 5

2 as 6

5

1, recognizing this product as

5

6. (In general, n

b

a =

b

an ×.)

c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to

represent the problem. For example, if each person at a party will eat8

3 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?

Cluster 3: Understand decimal notation for fractions, and compare decimal fractions.

MAFS.4.NF.3.5:

Express a fraction with a 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 10

3 as

100

30, and add

10

3 +

100

4 =

100

34.

MAFS.4.NF.3.6:

Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 100

62; describe a length as 0.62 meters;

locate 0.62 on a number line diagram.

MAFS.4.NF.3.7:

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.



Domain: MEASUREMENT AND DATA

Cluster 1: Solve problems involving measurement and conversion of measurements

MAFS.4.MD.1.1:

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)…

MAFS.4.MD.1.2:

Use the four operations to solve word problems1 involving distances, intervals of time, and money, including problems involving simple fractions or decimals2. Represent fractional quantities of distance, and intervals of time using linear models. (1See glossary Table 1 and Table 2.) (2Computational fluency with fractions and decimals is not the goal for students at this grade level.)

MAFS.4.MD.1.3:

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.

Cluster 2: Represent and interpret data.

MAFS.4.MD.2.4:

Make a line plot to display a data set of measurements in fractions of a unit (2

1,

4

1 ,

8

1). 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.

Cluster 3: Geometric measurement: understand concepts of angle and measure angles.

MAFS.4.MD.3.5:

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 through360

1 a circle is called a “one-

degree angle,” and can be used to measure angles.

b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.

MAFS.4.MD.3.6:

Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

MAFS.4.MD.3.7:

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.



Domain: GEOMETRY

Cluster 1: Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

MAFS.4.G.1.1:

Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

MAFS.4.G.1.2:

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.

MAFS.4.G.1.3:

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.

3rd Grade


STANDARD CODE

REVISED/ DELETED/NEW

STANDARD

MACC.3.MD.1.2 PREVIOUS

Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). 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.

MAFS.3.MD.1.2 REVISED Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units.

4th Grade


STANDARD CODE


STANDARD


Use the four operations to solve word problems1 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.

MAFS.4.MD.1.2 REVISED

Use the four operations to solve word problems1 involving distances, intervals of time, and money, including problems involving simple fractions or decimals.2 Represent fractional quantities of distance and intervals of time using linear models. (1See Table 2 Common Multiplication and Division Situations) (2Computational fluency with fractions and decimals is not the goal for students at this grade level.)

MAFS.4.OA.1.a NEW Determine whether an equation is true or false by using comparative relational thinking. For example, without adding 60 and 24, determine whether the equation 60 + 24 = 57 + 27 is true or false.

MAFS.4.OA.1.b NEW Determine the unknown whole number in an equation relating four whole numbers using comparative relational thinking. For example, solve 76 + 9 = n + 5 for n by arguing that nine is four more than five, so the unknown number must be four greater than 76.

MACC.4.OA.2.4 PREVIOUS

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.

MAFS.4.OA.2.4 REVISED

Investigate factors and multiples. A. Find all factor pairs for a whole number in the range 1–100. B. 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. C. Determine whether a given whole number in the range 1–100 is prime or composite.

5th Grade


STANDARD CODE


STANDARD

MACC.5.G.2.4 PREVIOUS Classify two-dimensional figures in a hierarchy based on properties.

MAFS.5.G.2.4 REVISED Classify and organize two-dimensional figures into Venn diagrams based on the attributes of the figures.

MACC.5.MD.1.1 PREVIOUS 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.

MAFS.5.MD.1.1 REVISED Convert among different-sized standard measurement units (i.e., km, m, cm; kg, g; lb., oz.; l, ml; hr., min, sec) 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.


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 theedge lengths, equivalently by multiplying the height by the area of the base. Representthreefold whole-number products as volumes, e.g., to represent the associative property ofmultiplication.

b. Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of rightrectangular prisms with whole- number edge lengths in the context of solving real world andmathematical 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.

MAFS.5.MD.3.5 REVISED

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 theedge lengths, equivalently by multiplying the height by the area of the base. Representthreefold whole-number products as volumes, e.g., to represent the associative property ofmultiplication.

b. Apply the formulas V = l × w × h and V = B × h for rectangular prisms to find volumes ofright rectangular prisms with whole- number edge lengths in the context of solving real worldand 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.

6th Grade


STANDARD CODE


STANDARD

MACC.6.RP.1.3 PREVIOUS

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, findmissing values in the tables, and plot the pairs of values on the coordinate plane. Use tablesto compare ratios.

b. Solve unit rate problems including those involving unit pricing and constant speed. Forexample, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could bemowed in 35 hours? At what rate were lawns being mowed?

c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 timesthe quantity); solve problems involving finding the whole, given a part and the percent.

d. Use ratio reasoning to convert measurement units; manipulate and transform unitsappropriately when multiplying or dividing quantities.

MAFS.6.RP.1.3 REVISED

Use ratio and rate reasoning to solve real-world and mathematical problems1, 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, findmissing values in the tables, and plot the pairs of values on the coordinate plane. Use tablesto compare ratios.

b. Solve unit rate problems including those involving unit pricing and constant speed. Forexample, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could bemowed in 35 hours? At what rate were lawns being mowed?

c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 timesthe quantity); solve problems involving finding the whole, given a part and the percent.

d. Use ratio reasoning to convert measurement units; manipulate and transform unitsappropriately when multiplying or dividing quantities.

e. Understand the concept of Pi as the ratio of the circumference of a circle to its diameter.

(1See Table 2 Common Multiplication and Division Situations)

Domain Progression, Brevard Public Schools, 2013-2014 Page 1 of 22

DOMAIN PROGRESSION OPERATIONS AND ALGEBRAIC THINKING

Third Grade Fourth Grade Fifth Grade

Represent and solve problems involving multiplication and division. 3.OA.1.1: 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.

3.OA.1.2: Interpret whole number quotients of wholenumbers (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.

3.OA.1.3: Use multiplication and division within 100 tosolve word problems in situations involving equal groups, arrays, and measurement quantities (e.g., by using drawings and equations with a symbol for the unknown number to represent the problem).

Use the four operations with whole numbers to solve problems.

4.OA.1.1: Interpret a multiplication equation as acomparison (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.

4.OA.1.2: Multiply or divide to solve word problemsinvolving multiplicative comparison (e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison).

4.OA.1.3: Solve multi-step word problems posed withwhole 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 reasonableness of answers using mental computation and estimation strategies including rounding.

Write and interpret numerical expressions.

5.OA.1.1: Use parenthesis, brackets, or braces innumerical expressions, and evaluate expressions with these symbols.

5.OA.1.2: Write simple expressions that recordcalculations 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 × (18,932 + 921) is three times as large as 18,932 + 921, without having to calculate the indicated sum or product.

Analyze patterns and relationships. 5.OA.2.3: 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.



3.OA.1.4: Determine the unknown whole number in amultiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = ☐ ÷ 3, 6 × 6 = ?.

Understand properties of multiplication and the relationship between multiplication and division.

3.OA.2.5: Apply properties of operations as strategiesto multiply and divide.

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)

3.OA.2.6: Understand division as an unknown-factorproblem.

For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Gain familiarity with factors and multiples.

4.OA.2.4: Find all factor pairs for a whole number inthe 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.

Generate and analyze problems.

4.OA.2.5: Generate a number or shape pattern thatfollows 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 this way.



Multiply and divide within 100.

3.OA.3.7: Fluently multiply and divide within 100, usingstrategies 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.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

3.OA.4.8: Solve two-step word problems using the fouroperations. 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.

3.OA.4.9: Identify arithmetic patterns (includingpatterns 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.


NUMBERS AND OPERATIONS BASE IN TEN

Third Grade Fourth Grade Fifth Grade Use place value understanding and properties of operations to perform multi-digit arithmetic.

3.NBT.1.1: Use place value understanding to roundwhole numbers to the nearest 10 or 100.

3.NBT.1.2: Fluently add and subtract within 1,000using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

3.NBT.1.3: Multiply one-digit whole numbers bymultiples of 10 in the range 10-90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

Generalize place value understanding for multi-digit whole numbers. 4.NBT.1.1: 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 ÷ 7 = 10 by applying concepts of place value and division.

4.NBT.1.2: Read and write multi-digit whole numbersusing base ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

4.NBT.1.3: Use place value understanding to roundmulti-digit whole numbers to any place.

Use place value understanding and properties of operations to perform multi-digit arithmetic.

4.NBT.2.4: Fluently add and subtract multi-digit wholenumbers using the standard algorithm.

4.NBT.2.5: Multiply a whole number of up to four digitsby 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.

Understand the place value system.

5.NBT.1.1: Recognize that in a multi-digit number, adigit in one place represents 10 times as much as it represents in the place to its right and

101 of what it represents in the place to

its left.

5.NBT.1.2: Explain patterns in the number of zeros ofthe product when multiplying a number of 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.

5.NBT.1.3: Read, write, and compare decimals tothousandths.

a. Read and write decimals to thousandthsusing base-ten numerals, number names, and expanded form (e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 ×

101 + 9 ×

1001 + 2 ×

000,11 ).

b. Compare two decimals to thousandthsbased on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

5.NBT.1.4: Use place value understanding to rounddecimals to any place.



4.NBT.2.6: Find whole number quotients andremainders 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.

Perform operations with multi-digit whole numbers and with decimals to hundredths.

5.NBT.2.5: Fluently multiply multi-digit whole numbersusing the standard algorithm.

5.NBT.2.6: Find whole number quotients of wholenumbers 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.

5.NBT.2.7: Add, subtract, multiply, and divide decimalsto hundredths, 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.


NUMBER AND OPERATIONS - FRACTIONS


Develop understanding of fractions as numbers.

3.NF.1.1: Understand a fraction b1 as the quantity

formed by 1 part when a whole is partitioned into b equal parts; understand a fraction

ba

as the quantity formed by a parts of size b1 .

3.NF.1.2: Understand a fraction as a number on thenumber line; represent fractions on a number line diagram.

a. Represent a fraction b1 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

b1 and the endpoint of the part

based at 0 locates the number b1 on the

number line. b. Represent a fraction

ba on a number line

diagram by marking off a lengths b1 from

0. Recognize that the resulting intervalhas size

ba on the number line.

Extend understanding of fractional equivalence and ordering.

4.NF.1.1: Explain why a fraction ba is equivalent to a

fraction bnan

×× 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.

4.NF.1.2: Compare two fractions with differentnumerators and different denominators (e.g., by creating common numerators and denominators, or by comparing to a benchmark fraction such as

21 ). 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 the visual fraction model).

Extend understanding of fractional equivalence and ordering.

4.NF.2.3: Understand a fraction ba with 𝑎 > 1 as a sum

of fractions b1 .

Use equivalent fractions as a strategy to add and subtract fractions.

5.NF.1.1: Add and subtract fractions with unlikedenominators (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: 32 +

45 =

128 +

1215 =

1223 . (In general,

ba +

dc =

bdbcad )( + ).

5.NF.1.2: Solve word problems involving addition andsubtraction 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

52 +

21 =

73 , by observing that

73 <

21 .



3.NF.1.3: Explain equivalence of fractions in specialcases, 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 equivalentfractions (e.g.,

21 =

42 ,

64 =

32 ). Explain

why the fractions are equivalent (e.g., by using a visual fraction model).

c. Express whole numbers as fractions, andrecognize fractions that are equivalent to whole numbers.

For example: Express 3 in the form 3 = 13 ; recognize

that 16 = 6; locate

44 and 1 at the same point on a

number line diagram.

d. Compare two fractions with the samenumerator 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).

a. Understand addition and subtraction offractions as joining and separating parts referring to the same whole.

b. Decompose a fraction into a sum of fractionswith the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions (e.g., by using a visual fraction model).

For example: 83 =

81 +

81 +

81 ;

83 =

81 +

82 ;

2 81 = 1 + 1 +

81 =

88 +

88 +

81

c. Add and subtract mixed numbers with likedenominators (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 andsubtraction of fractions referring to the same whole and having like denominators (e.g., by using visual fraction models and equations to represent the problem).

4.NF.2.4: Apply and extend previous understandings ofmultiplication to multiply a fraction by a whole number.

a. Understand a fraction ba as a multiple of

b1 .

Use equivalent fractions as a strategy to add and subtract fractions.

5.NF.2.3: Interpret a fraction as division of the numerator by

the denominator (ba = 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 visual fraction models or equations to represent the problem).

For example, interpret 43 as the result of dividing

3 by 4, noticing that 43 multiplied by 4 equals 3, and that

when 3 wholes are shared equally among 4 people, each

person has a share of size 43 . 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?

5.NF.2.4: Apply and extend previous understandings ofmultiplication to multiply a fraction or whole number by a fraction.

a. Interpret the productba × 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

32 × 4 =

38 , and create a story context for this equation. Do

the same with 32 ×

54 =

158 . (In general,

ba ×

dc =

dbac .)



For example: use a visual fraction model to represent

45 as the product 5 ×

41 , recording the conclusion by

the equation 45 = 5 ×

41 .

b. Understand a multiple of ba as a multiple of

b1 ,

and use this understanding to multiply a fraction by a whole number.

For example: use a visual fraction model to express 3 ×

52 as 6 ×

51 , recognizing this product as

56 . (In

general, n × ba =

ban × ).

c. Solve word problems involving multiplication ofa fraction by a whole number (e.g., by using visual fraction models and equations to represent the problem).

For example: if each person at a party will eat 83 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?

b. Find the area of a rectangle with fractional sidelengths 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.

5.NF.2.5: Interpret multiplication as scaling (resizing),by:

a. Comparing the size of a product to the size ofone factor on the basis of the size of the other factor, without performing the indicated multiplication.

b. Explaining why multiplying a given number by afraction 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

ba =

bnan

×× to the effect of

multiplying ba by 1.



Understand decimal notation for fractions, and compare decimal fractions.

4.NF.3.5: Express a fraction with denominator 10 as anequivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example: express

103 as

10030 , and add

103 +

1004 =

10034 .

4.NF.3.6: Use decimal notation for fractions withdenominators 10 or 100.

For example: rewrite 0.62 as 10062 ; describe a length

as 0.62 meters; locate 0.62 on a number line diagram.

4.NF.3.7: Compare two decimals to hundredths byreasoning 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).

5.NF.2.6: Solve real world problems involvingmultiplication of fractions and mixed numbers (e.g., by using visual fraction models or equations to represent the problem).

5.NF.2.7: Apply and extend previous understandings ofdivision to divide unit fractions by whole numbers and whole numbers by unit fractions.

a. Interpret division of a unit fraction by a non-zerowhole number, and compute such quotients.

For example, create a story context for 31 ÷ 4, and use

a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that

31 ÷ 4 =

121 because

121 × 4 =

31 .

b. Solve real-world problems involving division ofunit fractions by non-zero whole numbers and division of whole numbers by unit fractions (e.g., by using visual fraction models and equations to represent the problem).

For example, how much chocolate will each person get if 3 people share

21 lb. of chocolate equally? How

many 31 cup servings are in 2 cups of raisins?


MEASUREMENT AND DATA


Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

3.MD.1.1: Tell and write time to the nearest minute andmeasure 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).

3.MD.1.2: Measure and estimate liquid volumes andmasses of objects using standard units of grams (g), kilograms (kg), and liters (L). 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).

Represent and interpret data.

3.MD.2.3: Draw a scaled picture graph and a scaledbar 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.

Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

4.MD.1.1: Know relative sizes of measurement unitswithin one system of units including km, m, cm; kg, g; lb., oz.; L, mL; hr., min., and sec. Within a single system of measurement, express measurements in a larger 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),…

4.MD.1.2: Use the four operations to solve wordproblems 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.

Convert like measurement units within a given measurement system.

5.MD.1.1: Convert among different sized standardmeasurement 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.

Represent and interpret data.

5.MD.2.2: Make a line plot to display a data set ofmeasurements in fractions of a unit (

21 ,

41 ,

81 ). 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.



3.MD.2.4: Generate measurement data by measuringlengths 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.

Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

3.MD.3.5: Recognize area as an attribute of planefigures and understand concepts of area management.

a. A square with side length 1 unit, called “a unitsquare,” is said to have “one square unit” of area,and can be used to measure area.

b. A plane figure that can be covered without gapsor overlaps by n unit squares is said to have anarea of n square units.

3.MD.3.6: Measure areas by counting unit squares(square cm, square m, square in., square ft., and improvised units).

3.MD.3.7: Relate area to the operations ofmultiplication and addition.

a. Find the area of a rectangle with whole numberside lengths by tiling it, and show that the area isthe same as would be found by multiplying theside lengths.

4.MD.1.3: Apply the area and perimeter formulas forrectangles 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.

Represent and interpret data. 4.MD.2.4: Make a line plot to display a data set of

measurements in fractions of a unit (

21 ,

41 ,

81 ). 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.

Geometric measurement: understand concepts of angles and measure angles.

4.MD.3.5: Recognize angles as geometric shapes thatare 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

3601 of a

circle is called a “one-degree angle,” and can be used to measure angles.

Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

5.MD.3.3: Recognize volume as an attribute of solidfigures and understand concepts of volume measurement.

a. A cube with side length 1 unit, called a “unitcube,” is said to have “one cubic unit” of volume,and can be used to measure volume.

b. A solid figure which can be packed without gapsor overlaps using n unit cubes is said to have avolume of n cubic units.

5.MD.3.4: Measure volumes by counting unit cubes,using cubic cm, cubic in., cubic ft., and improvised units.

5.MD.3.5: Relate volume to the operations ofmultiplication and addition and solve real- world and mathematical problems involving volume.

a. Find the volume of a right-rectangular prism withwhole 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. Multiply side lengths to find areas of rectangleswith whole number side lengths in the context ofsolving real world and mathematical problems,and represent whole number products asrectangular areas in mathematical reasoning.

c. Use tiling to show in a concrete case that thearea of a rectangle with whole number sidelengths a and b + c is the sum of a × b and a × c.Use area models to represent the distributiveproperty in mathematical reasoning.

d. Recognize area as additive. Find areas ofrectilinear figures by decomposing them into non-overlapping rectangles and adding the areas ofthe non-overlapping parts, applying thistechnique to solve real world problems.

Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

3.MD.4.8: Solve real world and mathematical problemsinvolving 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.

b. An angle that turns through n one-degree anglesis said to have an angle measurement of ndegrees.

4.MD.3.6: Measure angles in whole number degreesusing a protractor. Sketch angles of a specified measure.

4.MD.3.7: Recognize angle measure as additive. Whenan 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).

b. Apply the formulas V = l × w × h andV = B × h for rectangular prisms to findvolumes of right-rectangular prisms withwhole number edge lengths in the context ofsolving real world and mathematicalproblems.

c. Recognize volume as additive. Find volumesof 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.


GEOMETRY

Third Grade Fourth Grade Fifth Grade Sixth Grade

Reason with shapes and their attributes.

3.G.1.1: Understand that shapes indifferent 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.

3.G.1.2: Partition shapes into parts withequal 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

41 of the area of

the shape.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

4.G.1.1: Draw points, lines, linesegments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

4.G.1.2: Classify two-dimensionalfigures based on the presence or absence of parallel or perpendicular lines, or presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

4.G.1.3: Recognize a line of symmetryfor 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.

Graph points on the coordinate plane to solve real-world and mathematical problems.

5.G.1.1: Use a pair of perpendicularnumber 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.1.2: Represent real-world andmathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

Solve real-world and mathematical problems involving area, surface area, and volume.

6.G.1.1: 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.

6.G.1.2: 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.


Third Grade Fourth Grade Fifth Grade Sixth Grade

Classify two-dimensional figures into categories based on their properties.

5.G.2.3: Understand that attributesbelonging 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.

5.G.2.4: Classify two-dimensionalfigures in a hierarchy based on properties.

6.G.1.3: Draw polygons in thecoordinate 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 mathematical problems.

6.G.1.4: Represent three-dimensionalfigures 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.


RATIOS AND PROPORTIONAL RELATIONSHIPS

Sixth Grade Seventh Grade Understand ratio concepts and use ratio reasoning to solve problems. 6.RP.1.1: 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.

6.RP.1.2: Understand the concept of a unit rate ba 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

43 cup of flour for each cup of sugar. We paid $75 for 15 hamburgers, which is a

rate of $5 per hamburger.

6.RP.1.3: Use ratio and rate reasoning to solve real-world and mathematicalproblems (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 numbermeasurements, finding missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

b. Solve unit rate problems including those involving unit pricing and constantspeed.

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?

c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantitymeans

10030 times the quantity); solve problems involving finding the

whole, given a part and the percent. d. Use ratio reasoning to convert measurement units; manipulate and

transform units appropriately when multiplying or dividing quantities.

Analyze proportional relationships and use them to solve real-world and mathematical problems. 7.RP.1.1: 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

21 mile in each

41 hour, compute the unit rate as

the complex fraction 1214 miles per hour, equivalently 2 miles per hour.

7.RP.1.2: 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 constraint 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 t = pn.

d. Explain what a point (x, y) on the graph of a proportional relationshipmeans in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate

7.RP.1.3: Use proportional relationships to solve multi-step ratio and percentproblems.

Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error


THE NUMBER SYSTEM

Sixth Grade Seventh Grade Apply and extend previous understandings of multiplication and division to divide fractions by fractions. 6.NS.1.1: Interpret and compute quotients of fractions, and solve world 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 32 ÷

43 and use a visual fraction model to

show the quotient; use the relationship between multiplication and division to explain that

32 ÷

43 =

98 because

43 of

98 is

32 (In general

ba ÷

dc =

bcad .) How

much chocolate will each person get if 3 people share 21 lb. of chocolate equally?

How many 43 cup servings are in

32 cup of yogurt? How wide is a rectangular strip

of land with a length of 43 mile and an area of

21 square mile?

Compute fluently with multi-digit numbers and find common factors and multiples. 6.NS.2.2: Fluently divide multi-digit numbers using the standard algorithm. 6. NS.2.3: Fluently add, subtract, multiply, and divide multi-digit decimals using the

standard algorithm for each operation. 6.NS.2.4: 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).

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

7.NS.1.1: 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.

7.NS.1.2. Apply and extend previous understandings of 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. Interpret products of rational numbers by describing real-world contexts.


Sixth Grade Seventh Grade

Apply and extend previous understandings of numbers to the system of rational numbers.

6.NS.3.5: Understand that positive and negative numbers are used together todescribe 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.

6.NS.3.6: Understand a rational number as a point on the number line. Extendnumber 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 oppositesides 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 inquadrants 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 orvertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

6.NS.3.7: Understand ordering and absolute value of rational numbers.a. Interpret statements of inequality as statements about the relative position

of two numbers on a number line diagram.

b. Understand that integers can be divided, provided that the divisor is notzero, and every quotient of integers is a rational number. If p and q are integers, then –

qp =

qp− =

qp−

. 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 to 0s or eventually repeats.

7.NS.1.3. Solve real-world and mathematical problems involving the four operationswith rational numbers.


Sixth Grade Seventh Grade

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.

b. Write, interpret, and explain statements of order for rational numbers in real-world contexts.

For example, write -3°C > -7°C to express the fact that -3°C is warmer than -7°C. c. Understand the absolute value of a rational number as its distance from 0

on the number line; interpret absolute value as magnitude for a positive or negative quantity in real-world situation.

For example, for an account balance of -30 dollars, write |-30| = 30 to describe the size of debt in dollars.

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. 6.NS.3.8: 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.


EXPRESSIONS AND EQUATIONS

Sixth Grade Seventh Grade Apply and extend previous understandings of arithmetic to algebraic expressions. 6.EE.1.1: Write and evaluate numerical expressions involving whole number

exponents. 6.EE.1.2: 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.

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.

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 = 6s2 to find the volume and surface area of a cube with sides of length s =

21 .

6.EE.1.3: 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.

Use properties of operations to generate equivalent expressions. 7.EE.1.1: Apply properties of operations as strategies to add, subtract, factor, and

expand linear expressions with rational coefficients. 7.EE.1.2: 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”. Solve real-life and mathematical problems using numerical and algebraic expressions and equations. 7.EE.2.3: 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 reasonableness of answers using mental computation and estimation strategies.

For example: If a woman making $25 an hour gets a 10% raise, she will make an additional

101 of her salary an hour, or $2.50, for a new salary of $27.50. If you want

to place a towel bar 9 43 inches long in the center of a door that is 27

21 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. 7.EE.2.4: 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.


Sixth Grade Seventh Grade 6.EE.1.4: Identify when two expressions are equivalent (e.g., 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 because they name the same number regardless of which number y stands for.

Reason about and solve one-variable equations and inequalities. 6.EE.2.5: 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.

6.EE.2.6: Use variables to represent numbers and write expressions when solving areal-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.

6.EE.2.7: Solve real-world and mathematical problems by writing and solvingequations of the form x + p = q and px = q for cases in which p, q and x are all non-negative rational numbers.

6.EE.2.8: Write an inequality of the form x > c or x < c to represent a constraint orcondition in a real-world or 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.

Represent and analyze quantitative relationships between dependent and independent variables. 6.EE.3.9: 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 the 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.

a. Solve word problems leading to equations of the formpx + 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?

b. Solve word problems leading to inequalities of the formpx + x > 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 a least $100. Write an inequality for the number of sales you need to make and describe the solutions.


STATISTICS AND PROBABILITY

Sixth Grade Seventh Grade Develop understanding of statistical variability.

6.SP.1.1: Recognize a statistical question as one that anticipates variability in thedata 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.

6.SP.1.2: Understand that a set of data collected to answer a statistical questionhas a distribution which can be described by its center, spread, and overall shape.

6.SP.1.3: Recognize that a measure of center for a numerical data set summarizesall of its values with a single number, while a measure of variation describes how its values vary with a single number.

Summarize and describe distributions.

6.SP.2.4: Display numerical data in plots on a number line, including dot plots,histograms, and box plots.

6.SP.2.5: 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 ofthe data distribution and the context in which the data were gathered.

Use random sampling to draw inferences about a population.

7.SP.1.1: Understand that statistics can be used to gain information about apopulation 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.

7.SP.1.2: Use data from a random sample to draw inferences about a populationwith 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.

Draw informal comparative inferences about two populations.

7.SP.2.3: Informally assess the degree of visual overlap of two numerical datadistributions 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.

7.SP.2.4: Use measures of center and measures of variability for numerical datafrom 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 words in a chapter of a fourth grade science book.


Sixth Grade Seventh Grade Investigate chance processes and develop, use, and evaluate probability models. 7.SP.3.5: 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 21 indicates an event that is neither unlikely nor likely, and a probability

near 1 indicates a likely event. 7.SP.3.6: Approximate the probability of a chance event a probability around 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. 7.SP.3.7: 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 indata 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 passed on the observed frequencies?

7.SP.3.8: Find probabilities of compound events using organized lists, tables, treediagrams, and simulation.

a. Understand that, just as with simple events, the probability of a compound event is thefraction of outcomes in the sample space for which the compound event occurs.

b. Represent sample spaces for compound events using methods such as organized lists,tables, and tree diagrams. For an event described in everyday language (e.g., “rollingdouble 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?

FOURTH GRADE DOMAIN/CLUSTER DESCRIPTORS AND CLARIFICATION

Fourth Grade Domain and Cluster Descriptors and Clarifications (from Ohio Department of Education, www.ode.state.oh.us/), page 1 of 15, 2013 - 2014

Domain: Operations and Algebraic Thinking Cluster: Use the four operations with whole numbers to solve problems.

Students need experiences that allow them to connect mathematical statements and number sentences or equations. This allows for an effective transition to formal algebraic concepts. They represent an unknown number in a word problem with a symbol. Word problems which require multiplication or division are solved by using drawings and equations. Students need to solve word problems involving multiplicative comparison (product unknown, partition unknown) using multiplication or division as shown in Table 2 of the Common Core State Standards for Mathematics, page 89. They should use drawings or equations with a symbol for the unknown number to represent the problem. Students need to be able to distinguish whether a word problem involves multiplicative comparison or additive comparison. Present multi-step word problems with whole numbers and whole number answers using the four operations. Students should know which operations are needed to solve the problem. Drawing pictures or using models will help students understand what the problem is asking. They should check the reasonableness of their answer using mental computation and estimation strategies.

http://www.ode.state.oh.us/



Domain: Operations and Algebraic Thinking Cluster: Gain familiarity with factors and multiples.

Students need to develop an understanding of the concepts of number theory such as prime numbers and composite numbers. This includes the relationship of factors and multiples. Multiplication and division are used to develop concepts of factors and multiples. Division problems resulting in remainders are used as counter-examples of factors.

Review vocabulary so that students have an understanding of terms such as factor, product, multiples, and odd and even numbers.

Students need to develop strategies for determining if a number is prime or composite. Starting with a number chart of 1 to 20, use multiples of prime numbers to eliminate later numbers in the chart. Encourage the development of rules that can be used to aid in the determination of composite numbers. Definitions of prime and composite numbers should not be provided, but determined after many strategies have been used in finding all possible factors of a number.

Using area models will also enable students to analyze numbers and arrive at an understanding of whether a number is prime or composite. Have students construct rectangles with an area equal to a given number. They should see an association between the number of rectangles and the given number for the area as to whether this number is a prime or composite number.

Provide students with counters to find the factors of numbers. Have them find ways to separate the counters into equal subsets. For example, have them find several factors of 10, 14, 25 or 32, and write multiplication expressions for the numbers.

Another way to find the factor of a number is to use arrays from square tiles or drawn on grid papers. Have students build rectangles that have the given number of squares. For example, if you have 16 squares:




The idea that a product of any two whole numbers is a common multiple of those two numbers is a difficult concept to understand. For example, 5 x 8 is 40; the table below shows the multiples of each factor.

5 10 15 20 25 30 35 40 45

8 16 24 32 40 48 56 64 72

Knowing how to determine factors and multiples is the foundation for finding common multiples and factors in Grade 6. Writing multiplication expressions for numbers with several factors and for numbers with a few factors will help students in making conjectures about the numbers. Students need to look for commonalities among the numbers.

Common Misconceptions When listing multiples of numbers, students may not list the number itself. Emphasize that the smallest multiple is the number itself. Some students may think that larger numbers have more factors. Having students share all factor pairs and how they found them will clear up this misconception.

(continued)




Domain: Operations and Algebraic Thinking Cluster: Generate and analyze patterns.

In order for students to be successful later in the formal study of algebra, their algebraic thinking needs to be developed. Understanding patterns is fundamental to algebraic thinking. Students have experience in identifying arithmetic patterns, especially those included in addition and multiplication tables. Contexts familiar to students are helpful in developing students’ algebraic thinking. Students should generate numerical or geometric patterns that follow a given rule. They should look for relationships in the patterns and be able to describe and make generalizations. As students generate numeric patterns for rules, they should be able to “undo” the pattern to determine if the rule works with all of the numbers generated. For example, given the rule, “Add 4” starting with the number 1, the pattern 1, 5, 9, 13, 17, … is generated. In analyzing the pattern, students need to determine how to get from one term to the next term. Teachers can ask students, “How is a number in the sequence related to the one that came before it?”, and “If they started at the end of the pattern, will this relationship be the same?”. Students can use this type of questioning in analyzing number patterns to determine the rule. Students should also determine if there are other relationships in the patterns. In the numeric pattern generated above, students should observe that the numbers are all odd numbers. Provide patterns that involve shapes so that students can determine the rule for the pattern. For example, Students may state that the rule is to multiply the previous number of squares by 3.




Domain: Number and Operations base ten Cluster: Generalize place value understanding for multi-digit whole

numbers.

Provide multiple opportunities in the classroom setting and use real-world context for students to read and write multi-digit whole numbers. Students also need to have opportunities to compare numbers with the same number of digits (e.g., compare 453, 698 and 215), numbers that have the same number in the leading digit position (e.g., compare 45, 495 and 41,223), and numbers that have different numbers of digits and different leading digits (e.g., compare 312, 95, 5245 and 10,002). Students also need to create numbers that meet specific criteria. For example, provide students with cards numbered 0 through 9. Ask students to select 4 to 6 cards; then, using all the cards make the largest number possible with the cards, the smallest number possible, and the closest number to 5,000 that is greater than 5,000 or less than 5,000. In Grade 4, rounding is not new, and students need to build on the Grade 3 skill of rounding to the nearest 10 or 100 to include larger numbers and place value. What is new for Grade 4 is rounding to digits other than the leading digit (e.g., round 23,960 to the nearest hundred). This requires greater sophistication than rounding to the nearest ten thousand because the digit in the hundreds place represents 900 and when rounded it becomes 1,000, not just zero. Students should also begin to develop some rules for rounding, building off the basic strategy of; “Is 48 closer to 40 or 50?”. Since 48 is only 2 away from 50 and 8 away from 40, 48 would round to 50. Now students need to generalize the rule for much larger numbers and rounding to values that are not the leading digit.

Common Misconceptions There are several misconceptions students may have about writing numerals from verbal descriptions. Numbers like one thousand do not cause a problem; however a number like one thousand two causes problems for students. Many students will understand the 1,000 and the 2, but then instead of placing the 2 in the ones place, students will write the numbers as they hear them, 10,002 (ten thousand two). There are multiple strategies that can be used to assist with this concept, including place-value boxes and vertical-addition method. Students often assume that the first digit of a multi-digit number indicates the "greatness" of a number. The assumption is made that 954 is greater than 1,002 because students are focusing on the first digit instead of the number as a whole.




Domain: Number and Operations in Base Ten Cluster: Use place value understanding and properties of operations to

perform multi-digit arithmetic.

It is important for students to have seen and used a variety of strategies and materials to broaden and deepen their understanding of place value before they are required to use standard algorithms. The goal is for them to understand all the steps in the algorithm, and they should be able to explain the meaning of each digit. For example, a 1 can represent one, ten, one hundred, and so on. For multi-digit addition and subtraction in Grade 4, the goal is also fluency, which means students must be able to carry out the calculations efficiently and accurately. Start with a student’s understanding of a certain strategy, and then make intentional, clear-cut connections for the student to the standard algorithm. This allows the student to gain understanding of the algorithm rather than just memorize certain steps to follow. It is very important for some students to talk through their understanding of connections between different strategies and standard addition and subtractions algorithms. Give students many opportunities to talk with classmates about how they could explain standard algorithms. Think-Pair-Share is a good protocol for all students. When asking students to gain understanding about multiplying larger number provide frequent opportunities to engage in mental math exercises. When doing mental math, it is difficult to even attempt to use a strategy that one does not fully understand. Also, it is a natural tendency to use numbers that are 'friendly' (multiples of 10) when doing mental math, and this promotes its understanding. Use a variation of an area model. For example, to multiply 23 × 36, arrange the partial products as follows. Then add the four partial products to get 828.

20 3

30 600 90

6 120 18




As students developed an understanding of multiplying a whole number up to four digits by a one-digit whole number, and multiplying two two-digit numbers through various strategies, they should do the same when finding whole-number quotients and remainders. By relating division to multiplication and repeated subtraction, students can find partial quotients. An explanation of partial quotients can be viewed at http://www.teachertube.com, search for Outline of partial quotients. This strategy will help them understand the division algorithm. Students will have a better understanding of multiplication or division when problems are presented in context. Students should be able to illustrate and explain multiplication and division calculations by using equations, rectangular arrays, and the properties of operations. These strategies were used in Grade 3 as students developed an understanding of multiplication. Vocabulary is important. Students should have an understanding of terms such as sum, difference, fewer, more, less, ones, tens, hundreds, thousands, digit, whole numbers, product, factors, and multiples. Common Misconceptions Students often do not notice the need of regrouping and just take the smaller digit from the larger one. Emphasize place value and the meaning of each of the digits.

(continued)




Domain: Number and Operations – Fractions Cluster: Extend understanding of fractions equivalence and ordering.

Students’ initial experience with fractions began in Grade 3. They used models such as number lines to locate unit fractions, fraction bars or strips, area or length models, and Venn diagrams to recognize and generate equivalent fractions and make comparisons of fractions. Students extend their understanding of unit fractions to compare two fractions with different numerators and different denominators. Students should use models to compare two fractions with different denominators by creating common denominators or numerators. The models should be the same (both fractions shown using fraction bars or both fractions using circular models) so that the models represent the

same whole. Students should also use benchmark fractions such as 21 to compare two

fractions. The result of the comparisons should be recorded using ˃, ˂, and = symbols.

43

21

31

Misconceptions Students think that when generating equivalent fractions they need to multiply or divide either

the numerator or denominator, such as, changing 21 to sixths. They would multiply the

denominator by 3 to get 61 , instead of multiplying the numerator by 3 also. Their focus is only

on the multiple of the denominator, not the whole fraction.

Students need to use a fraction in the form of one, such as 33 , so that the numerator and

denominator do not contain the original numerator or denominator.




Domain: Number and Operations – Fractions Cluster: Build fractions from unit fractions by applying and extending

previous understandings of operations on whole numbers.

In Grade 3, students added unit fractions with the same denominator. Now, they begin to represent a fraction by decomposing the fraction as the sum of unit fractions and justify with a

fraction model. For example, 43 =

41 +

41 +

41 .

Students also represented whole numbers as fractions. They use this knowledge to add and subtract mixed numbers with like denominators using properties of number and appropriate fraction models. It is important to stress that whichever model is used, it should be the same for the same whole. Understanding of multiplication of whole numbers is extended to multiplying a fraction by a whole number. Allow students to use fraction models and drawings to show their understanding. Present word problems involving multiplication of a fraction by a whole number. Have students solve the problems using visual models and write equations to represent the problems. Common Misconceptions Students think that it does not matter which model to use when finding the sum or difference of fractions. They may represent one fraction with a rectangle and the other fraction with a circle. They need to know that the models need to represent the same whole.

= + +




Domain: Number and Operations – Fractions Cluster: Understand decimal notations for fractions, and compare decimal

fractions. The place value system developed for whole numbers extends to fractional parts represented as decimals. This is a connection to the metric system. Decimals are another way to write fractions. The place-value system developed for whole numbers extends to decimals. The concept of one whole used in fractions is extended to models of decimals. Students need to make connections between fractions and decimals. They should be able to write decimals for fractions with denominators of 10 or 100. Have students say the fraction

with denominators of 10 and 100 aloud. For example, 104 would be “four tenths” or

10027 would

be “twenty-seven hundredths.” Also, have students represent decimals in word form with

digits and the decimal place value, such as 104 written as 4 tenths.

Students should be able to express decimals to the hundredths as the sum of two decimals or fractions. This is based on understanding of decimal place value. For example 0.32 would be the sum of 3 tenths and 2 hundredths. Using this, students that understand can write 0.32

as the sum of two fractions (103 +

1002 ). Students’ understanding of decimals to hundredths is

important in preparation for performing operations with decimals to hundredths in Grade 5. In decimal numbers, the value of each place is 10 times the value of the place to its immediate right. Students need an understanding of decimal notations before they try to do conversions in the metric system. Understanding of the decimal place value system is important prior to the generalization of moving the decimal point when performing operations involving decimals. Students extend fraction equivalence from Grade 3 with denominators of 2, 3, 4, 6, and 8 to fractions with a denominator of 10. Provide fraction models of tenths and hundredths so that students can express a fraction with a denominator of 10 as an equivalent fraction with a denominator of 100.




When comparing two decimals, remind students that as in comparing two fractions, the decimals need to refer to the same whole. Allow students to use visual models to compare two decimals. They can shade in a representation of each decimal on a 10 x 10 grid. The 10 x 10 grid is defined as one whole. The decimal must relate to the whole.

Flexibility with converting fractions to decimals and decimals to fractions provides efficiency in solving problems involving all four operations in later grades. Common Misconceptions Students treat decimals as whole numbers when making comparisons of two decimals. They think the longer the number, the greater the value. For example, they think that .03 is greater than 0.3.

(continued)




Domain: Measurement and Data Cluster: Solve problems involving measurement and conversion of

measurements from a larger unit to a smaller unit.

In order for students to have a better understanding of the relationships between units, they need to use measuring devices in class. The number of units needs to relate to the size of the unit. They need to discover that there are 12 inches in 1 foot and 3 feet in 1 yard. Allow students to use rulers and yardsticks to discover these relationships among these units of measurements. Using 12-inch rulers and a yardstick, students can see that three of the 12-inch rulers, which is the same as 3 feet since each ruler is 1 foot in length, are equivalent to one yardstick. Have students record the relationships in a two column table or t-chart. A similar strategy can be used with rulers marked with centimeters and a meter stick to discover the relationships between centimeters and meters.

Present word problems as a source of students’ understanding of the relationships among inches, feet, and yards.

Students are 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.

Present problems that involve multiplication of a fraction by a whole number (denominators are 2, 3, 4, 5, 6, 8, 10, 12, and 100). Problems involving addition and subtraction of fractions should have the same denominators. Allow students to use strategies learned with these concepts.

Students used models to find area and perimeter in Grade 3. They need to relate discoveries from the use of models to develop an understanding of the area and perimeter formulas to solve real-world and mathematical problems.

Common Misconceptions Students believe that larger units will give the larger measure. Students should be given multiple opportunities to measure the same object with different measuring units. For example, have the students measure the length of a room with one-inch tiles, one-foot rulers, and yardsticks. Students should notice that it takes fewer yardsticks to measure the room.




Domain: Measurement and Data Cluster: Represent and interpret data.

Data have been measured and represented on line plots in units of whole numbers, halves, or quarters. Students have also represented fractions on number lines. Now students are using line plots to display measurement data in fraction units and using the data to solve problems involving addition or subtraction of fractions.

Have students create line plots with fractions of a unit (21 ,

41 ,

81 ) and plot data showing multiple

data points for each fraction.

Common Misconceptions Students use whole number names when counting fractional parts on a number line. The fraction name should be used instead. For example, if two-fourths is represented on the line plot three times, then there would be six-fourths.




Domain: Measurement and Data Cluster: Geometric measurement: understand concepts of angles and

measure angles.

Angles are geometric shapes composed of two rays that are infinite in length. Introduce angles as acute (less than the measure of a right angle) and obtuse (greater than the measure of a right angle). Have students draw representations of each type of angle. They also need to be able to identify angles in two-dimensional figures. Students can also create an angle explorer (two strips of cardboard attached with a brass fastener) to learn about angles. They can use the angle explorer to get a feel of the relative size of angles as they rotate the cardboard strips around.

Students can compare angles to determine whether an angle is acute or obtuse. This will allow them to have a benchmark reference for what an angle measure should be when using a tool such as a protractor or an angle ruler.

Provide students with four pieces of straw, two pieces of the same length to make one angle and another two pieces of the same length to make an angle with longer rays.

Students are ready to use a tool to measure angles once they understand the difference between an acute angle and an obtuse angle. Angles are measured in degrees. There is a relationship between the number of degrees in an angle and circle which has a measure of 360 degrees. Students are to use a protractor to measure angles in whole number degrees. They can determine if the measure of the angle is reasonable based on the relationship of the angle to a right angle. They also make sketches of angles of specified measure.

Common Misconceptions Students are confused as to which number to use when determining the measure of an angle using a protractor because most protractors have a double set of numbers. Students should decide first if the angle appears to be an angle that is less than the measure of a right angle (90°) or greater than the measure of a right angle (90°). If the angle appears to be less than 90°, it is an acute angle and its measure ranges from 0° to 89°. If the angle appears to be an angle that is greater than 90°, it is an obtuse angle and its measures range from 91° to 179°. Ask questions about the appearance of the angle to help students in deciding which number to use.




Domain: Geometry Cluster: Draw and identify lines and angles, and classify shapes by

properties of their lines and angles.

Students can and should make geometric distinctions about angles without measuring or mentioning degrees. Angles should be classified in comparison to right angles, such as larger than, smaller than, or the same size as a right angle.

Students can use the corner of a sheet of paper as a benchmark for a right angle. They can use a right angle to determine relationships of other angles.

When introducing line of symmetry, provide examples of geometric shapes with and without lines of symmetry. Shapes can be classified by the existence of lines of symmetry in sorting activities. This can be done informally by folding paper, tracing, creating designs with tiles, or investigating reflections in mirrors.

Two-dimensional shapes are classified based on relationships by the angles and sides. Students can determine if the sides are parallel or perpendicular, and classify accordingly. Characteristics of rectangles (including squares) are used to develop the concept of parallel and perpendicular lines. The characteristics and understanding of parallel and perpendicular lines are used to draw rectangles. Repeated experiences in comparing and contrasting shapes enable students to gain a deeper understanding about shapes and their properties.

Informal understanding of the characteristics of triangles is developed through angle measures and side length relationships. Triangles are named according to their angle measures (right, acute, or obtuse) and side lengths (scalene, isosceles, or equilateral). These characteristics are used to draw triangles.

Common Misconceptions Students believe a wide angle with short sides may seem smaller than a narrow angle with long sides. Students can compare two angles by tracing one and placing it over the other. Students will then realize that the length of the sides does not determine whether one angle is larger or smaller than another angle. The measure of the angle does not change.


Additional Resources

Addition and Subtraction Strategies, page 1, 2015 - 2016

ADDITION AND SUBTRACTION STRATEGIES

The development of strategies for addition and subtraction is a critical area in the Common Core State

Standards. By using and comparing a variety of solution strategies students build their understanding

of the relationship between addition and subtraction.

*The following information regarding addition and subtraction strategies has been adapted from:

Van de Walle, J.A., & Lovin, L.H. (2006). Teaching Student-Centered Mathematics, Volume I. Boston:

Pearson. See chapters three and four of this book for further clarification of addition and subtraction

strategies.

Addition Strategies Subtraction Strategies

Zero Think-Addition

One More/Two More Build Up Through Ten

Doubles Back Down Through Ten

Near Doubles Invented

Sums of Ten

Make Ten

Ten Plus

Invented

Commutative Property

Associative Property

SPECIAL NOTES:

Basic facts for addition are combinations of numbers where both addends are less than 10.

Subtraction facts correspond to the addition facts.

Fluency involves accuracy (correct answer), efficiency (a reasonable amount of steps that does

not include counting), and flexibility (using strategies that demonstrate number sense).

Every child, including ESE children, can master the basic facts with efficient mental tools.

Steps to Mastery:

1) Children must develop an understanding of number relationships and the operations.

2) Children need to develop efficient strategies for fact retrieval.

3) Teachers need to provide practice of selection of strategies once they have been developed.

Children who do not learn mental strategies will continue to count on their fingers since they

have no other strategies to solve basic addition and subtraction problems.

AVOID PREMATURE DRILL: if a child does not know a fact and is given a timed test; the

child will revert to counting.

Downplay counting on as a strategy because children often get confused as to why they can

count for some problems but not others. It is used as a crutch where other strategies would be

more efficient.

Many of the strategies apply to more than one fact. Therefore, students need to choose the one

that works best for them through discussion and justification.

Encourage discussion so students can justify and defend their method. This allows the students

to hear other methods that might lead to the development of a more effective strategy.


Zero One addend is always zero

The sum of any addend and zero is the original addend.

There are 19 facts where zero is one of the addends

Be sure to show 0 + 6 and 6 + 0

Children assume that addition sentences result in a larger number

Note: This may seem easy; however, students over generalize that an addition sentence always equals a larger

sum.

One/Two More One addend is 1 or 2

36 facts

Students are ready for these activities when they can identify 1 or 2 more without counting

Doubles The two addends are the same 0 + 0, 1 + 1, 2 + 2, etc.

There are 10 doubles facts

These facts will be anchors for other facts (such as 4 + 4 = 8 so 4 + 5 = 9 , see Near-Doubles)

Near-Doubles All combinations where one addend is more than the other

Note: Some children will double the smaller fact and add up 6 + 6 = 12 so 6 + 7 = 13. Others will double the

greater fact and subtract one 7 + 7 = 14 so 7 + 6 = 13

*Be sure students are exposed to both so they can decide which is better for them.

Sums of Ten The two addends equal the sum of ten

These facts will be anchors for other facts (such as 9 + 1=10, so 9 + 4 becomes 10 + 3)

Ten Plus One addend is 10, 10 + 4, 4 + 10

Children need to recognize that a set of ten and a set of 4 total 14 without counting.

* This is not an appropriate place for the term 1 ten as regrouping for first graders. The term 1 set of ten not a 1 in

the tens place should be used to meet the needs of the early first grade student.

Make-Ten These facts all have 8 or 9 as one of the addends

Children use 10 as a way to “bridge” to get the sum 6 + 8. Start with 8; decompose the 6 into 4 + 2 add the

2 to 8 and get a sum of 10. 10 and the remaining 4 equals 14 so 6 + 8 = 14.

Commutative Property The order of the addends does not change the sum

2 + 5 = 5 + 2

Associative Property The sum is the same regardless of the grouping of the addends.

2 + (6 + 4) = 2 + 10 = 12

NOTE: Counting on is not a sophisticated strategy. Children coming from Kindergarten are expected to

recognize small sets of numbers but may count. Children in first and second grade are expected to take the

next step by creating and using more sophisticated strategies such as the ones listed below.

Addition Strategies (Continued)


Addition Strategies

Remaining 4 Facts 3 + 5 3 + 6 4 + 7 5 + 7

The children have learned or discovered strategies to solve the 4 strategies above. Now encourage the students to

apply and choose a strategy that will work for them.

7 + 4 decompose the 4 into 3 + 1 to make ten, add 1 more

7 + 5 decompose the 5 to make 3 + 2, therefore making a ten creating a fact they know (7 + 3 = 10), then

add 2 more

Invented Students create and/or apply any of the above strategies to other equations.

Students will create ways to solve problems that are not noted above.

Encourage students to create other ways to solve problems other than counting.

Invented strategies are number-oriented, flexible, and constructed by students.

7+5

Make Ten 7 + 3 = 10 +2 12

Invented (using what I

know)

7 + 7 = 14 7 + 5 = 12

Invented (applying a

near double)

7 + 6 = 13 7 + 5 = 12

Circle Map


Subtraction Strategies

Think-Addition

The student understands subtraction as an unknown addend problem.

This strategy works best for sums less than 10 because 64 % of the 100 subtraction facts

fall into this category, for example: 9 – 4 (think 4 + 5 = 9)

Such facts as 7 – 2 would go along well with 2 more, now think 2 less along with

2 + 5 = 7, so 7 – 2 = 5

Build Up Through Ten

This group includes all the facts where the part is either 8 or 9

Start with the 8 or 9 and ask how much to ten and then build up

Back Down Through 10

It is most useful for facts where one digit is close to the number it is being subtracted

from

14 - 6, remove six from a ten frame and then two more to get the eight

Known as decomposing a number leading to a ten in Common Core

Invented

Students will create ways to solve problems that are not noted above

Encourage students to create other ways to solve problems other than counting

Circle Map

14-8

8+6

14 - 8 (4+4)

14 - 4=10-4=6

8+7=15 8+6=14

8+8=16 -2 14 8-2=6

8+2=10 +4 14

Think addition

Back Down Through 10

Invented (using a fact I know)

Invented

Build Up Through 10


MACC.2.NBT.2

Use place value understanding and properties of operations to add and subtract.

The standard algorithm is introduced and taught in fourth grade: 4.NBT.4.

Second and third grade students are encouraged to invent strategies when solving multi digit addition and subtraction problems for the following reasons. Place value concepts are enhanced.

Students make fewer errors as they are focused on the number and number relationships.

Less reteaching is necessary as they are inventing for themselves what makes sense.

Mental computation and estimation are enhanced.

Flexible thinking of number leads to strategies and this thinking is often faster than standard algorithms.

Strategies serve students just as well as traditional algorithms on tests (including FCAT 2.0).

Students who look at the meaning of numbers and use what they know to solve problems know and use more mathematics than those that follow a procedure.

Samples of Invented Strategies for Addition

Place Value: 352 + 675

300 + 50 + 2

600 + 70 + 5

900 + 120 + 7

1,027 = 1,000 + 20 + 7

Friendly Tens: 352 + 675

327 + 700 = 1,027

Compensate: 352 + 675

350 is easier to add to

650

1,000

Now I pick up my 25 + 2 = 27 1,000 + 27 = 1,027

Adding Hundreds or Ones first: 352 + 675

300 + 600 = 900

50 + 70 = 120

2 + 5 = 7

1,027


Samples of Invented Strategies for Subtraction

Place Value: 675 – 352 600 + 70 + 5

300 + 50 + 2 323 = 300 + 20 + 3

Add up: 675 – 352 352 + 8 = 360 360 + 40 = 400 400 + 275 = 675 323

Students may extend 275 + 5 (5+ 3 = 8) to get 280 + 20 (20 + 20 = 40) to get 300 + 23 = 323

See Chapter 6 strategies for whole-number computation in Van de Walle, J.A., & Lovin, L.H. (2006). Teaching Student-Centered Mathematics, Volume I, Boston: Pearson .

Basic Multiplication and Division Fact Strategies, page 1 of 3, adapted from Van de Walle, J. A., Lovin, L. H, Karp, K. S, & Bay-Williams, J. M. (2014). Teaching Student-Centered Mathematics, Volume II. Boston; Pearson.

Strategies for BASIC Multiplication and Division Facts

The development of strategies for multiplication and division is a critical area in the Mathematics Florida Standards (MAFS). By using and comparing a variety of solution strategies students build their understanding of the relationship between multiplication and division.

The following information regarding multiplication and division strategies has been adapted from: Van de Walle, J.A., Lovin, L.H, Karp, K.S, & Bay-Williams, J.M. (2014). Teaching Student-Centered Mathematics, Volume II. Boston; Pearson. See chapters eight, nine, and eleven of this book for further clarification of multiplication and division strategies and.

Multiplication Strategies Division Strategies

Doubles Think Multiplication and then Apply a Known Multiplication Fact Fives

Zeros and Ones

Nifty Nines

Using Known Facts to Derive Other Facts

SPECIAL NOTES:

The use of a problem-based approach and a focus on reasoning strategies are critical to developing mastery of the multiplication and related division facts. Thus, story problems should be used to develop reasoning strategies for basic fact mastery.

BASIC Multiplication Fact Strategies

Doubles

These are facts with 2 as a factor and are equivalent to the addition doubles, so students should already know these.

Students need experiences to help them realize that 2 x 8 is the same as double 8 (8 + 8).

Fives

These are facts that have 5 as a first or second factor.

Mastery development ideas:

Skip count by fives: 0, 5, 10, 15, 20 . . .

Connect counting by fives with arrays that have 5 dots For example three rows is 3 x 5

Connect to counting minutes on the clock.


Zeros and Ones

These are facts that have at least one factor that is either 0 or 1.

While these facts seem easy, they can be confusing to students because of rules for addition. For example, when zero is added to a number, it does not change the number (8 + 0 = 8). However, 8 x 0 = 0. Adding 1 to a number results in the next number, or one more (8 + 1 = 9), but a number multiplied by one does not change the number (8 x 1 = 8).

The use of rules that are strictly procedural, such as “anything times zero is zero” should be avoided.

Nifty Nines Facts with factors of 9 may be among the easiest to learn because of reasoning strategies and

patterns. 9 x 8 is the same as 10 x 8 less one set of 8, or 80 – 8 = 72 The tens digit is always one less than the other factor (the factor other than 9) and the sum of the

digits in the product is always 9. Therefore, for the fact 9 x 8, the tens digit is 7 and since the two digits in the product must add to 9 the ones digit is 2 and the product is 72.

Patterns are not rules without reasons. Students should be challenged to understand why they work.

Using Known Facts to Derive Other Facts Reasoning Strategies:

Double and Double Again This applies to all facts with a factor of 4. For example, 4 x 6 is the same as 2 x 6 doubled. Note that for some facts such as 4 x 8, doubling

the product may result in a difficult addition problem. For 4 x 8, a student knows 2 x 8 is 16, and then doubles 16. Doubling 16 is a difficult addition and simply adding 16 + 16 defeats the purpose of efficient reasoning. Students should use effective and efficient addition strategies such as, “I know 15 + 15 is 30 and 16 + 16 is 2 more, or 32.”

Double and One More This works with facts that have 3 as one factor. For example, 3 x 6 is 2 x 6 and 6 more

(12 + 6 = 18). Note that 3 x 8 and 3 x 9 result in challenging mental additions.

Half then Double This applies to all facts with one even factor. For example, 6 x 8; half of 6 eights is 3 eights,

3 times 8 is 24, double 24 is 48.

Close Fact strategy This involves adding one more set to a known fact. For example, think of 6 x 8 as 6 eights. Five

eights is close and results in 40. Six eights is one more eight, or 48. Using 5 x 8 to figure out 6 x 8, the language “6 groups of eight” or “6 eights” can help students

remember to add 8 more not 6 more. The Close Fact strategy can be used with any multiplication fact. It reinforces students’ number

sense and relationships between numbers.


BASIC Division Fact Strategies

Reasoning

Mastery of basic division facts is dependent on the inverse relationship of multiplication and division. For example, to solve 48 ÷ 6, we might naturally ask ourselves, “Six times what is 48?” The reasoning strategy is to (1) think multiplication, and then (2) apply a known multiplication fact.

Near facts: 60 ÷ 8; mentally review a short sequence of multiplication facts comparing each product to 60: 6 x 8 = 48 (too low), 7 x 8 = 56 (close), 8 x 8 = 64 (too high). It must be 7, so that is 56 with 4 left over.

NOTE: Division with remainders if much more prevalent in the real world than basic division facts

that have no remainders. Students should be able to solve these near fact problems with reasonable speed.

Four-Corners-and-a-Rhombus Math Graphic Organizer

What do you already know? Brainstorm ways to solve this problem. Try two ways to solve the problem here. List words and phrases you need to include in

your communication write up.

What do you need to

find out?

Cognitive Complexity of Mathematics Items

Low Complexity This category relies heavily on the recall and recognition of previously learned concepts and principles. Items typically specify what the student is to do, which is often to carry out come procedure that can be performed mechanically. It is not left to the student to come up with an original method or solution. The list below illustrates some, but not all, of the demands that items in the low complexity category might make:

• Recall or recognize a fact, term, or property. • Identify appropriate units or tools for common measurements. • Compute a sum, difference, product, or quotient. • Recognize or construct an equivalent representation. • Perform a specified operation or procedure. • Evaluate a variable expression, given specific values for the variables. • Solve a one-step problem. • Retrieve information from a graph, table, or figure. • Perform a single-unit conversion.

Moderate Complexity Items in the moderate complexity category involve more flexibility of thinking and choice among alternatives than do those in the low complexity category. They require a response that goes beyond the habitual, is not specified, and ordinarily has more than a single step. The student is expected to decide what to do, using informal methods of reasoning and problem solving strategies, and to bring together skill and knowledge from various domains. The list below illustrates some, but not all, of the demands that items of moderate complexity might make.

• Solve a problem requiring multiple operations. • Solve a problem involving spatial visualization and/or reasoning. • Retrieve information from a graph, table, or figure and use it to solve a problem. • Compare figures or statements. • Determine a reasonable estimate. • Extend an algebraic or geometric pattern. • Provide a justification for steps in a solution process. • Formulate a routine problem, given data and conditions. • Represent a situation mathematically in more than one way. • Select and/or use different representations, depending on situation and purpose.

High Complexity High complexity items make heavy demands on student thinking. Students must engage in more abstract reasoning, planning, analysis, judgment, and creative thought. The item requires that the student think in an abstract and sophisticated way. The list below illustrates some, but not all, of the demands that items in the high complexity category might make:

• Perform a procedure having multiple steps and multiple decision points. • Describe how different representations can be used for different purposes. • Solve a non-routine problem (as determined by grade-level appropriateness). • Analyze similarities and differences between procedures and concepts. • Generalize an algebraic or geometric pattern. • Formulate an original problem, given a situation. • Solve a problem in more than one way. • Explain and justify a solution to a problem. • Describe, compare, and contrast solution methods. • Formulate a mathematical model for a complex situation. • Analyze or produce a deductive argument. • Provide a mathematical justification.

NOTE: The complexity of an item is generally NOT dependent on the multiple-choice distractors. The options may affect the difficulty of the item, not the complexity of the item.

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