content deepening 7 th grade math september 23, 2013 jeanne simpson amsti math specialist

Content Deepening7th Grade Math

September 23, 2013Jeanne SimpsonAMSTI Math Specialist

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•Name•School•Classes you teach•What are you hoping to learn today?

Welcome

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He who dares to teach

must never cease to

learn.John Cotton Dana

acos2010.wikispaces.com•Electronic version of

handouts•Links to web resources

http://acos2010.wikispaces.com/Home

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Intervention

Assessment

Collaboration

Five Fundamental Areas Required for Successful Implementation of CCSS

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Instruction•Deep

conceptual understanding

•Collaborative lesson design

•Standards for Mathematical Practice

Content• Fewer standards with greater

depth• Understanding, focus, and

coherence• Common and high-demand tasks

Intervention• Common required response to

intervention framework response• Differentiated, targeted, and

intensive response to student needs

• Student equity, access, and support Assess

ment•PLC

teaching-assessing-learning cycle

•In-class formative assessment processes

•Common assessment instruments as formative learning opportunities

Collaboration

How do we teach?

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SMP1 - Make sense of problems and persevere in solving them

SMP2 - Reason abstractly and quantitativelySMP3 - Construct viable arguments and critique the

reasoning of othersSMP4 - Model with mathematicsSMP5 - Use appropriate tools strategicallySMP6 - Attend to precisionSMP7 - Look for and make use of structureSMP8 - Look for and express regularity in repeated

reasoning

Standards for Mathematical Practice

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•Capture the processes and proficiencies that we want our students to possess

•Not just the knowledge and skills but how our students use the knowledge and skills

•Describe habits of mind of the mathematically proficient student

•Carry across all grade levels, K-12

What Are The Practice Standards?

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•√ I already do this.

• ! This sounds exciting!

• ? I have questions.

Standards of Mathematical Practice

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•Problem solving•Demanding tasks•Student understanding•Discussion of alternative

strategies•Extensive mathematics

discussion•Effective questioning•Student conjectures•Multiple representations

High-Leverage Strategies

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Instruction• D

eep conceptual understanding

• Collaborative lesson design

• Standards for Mathematical Practice

Content• F

ewer standards with greater depth

• Understanding, focus, and coherence

• Common and high-demand tasks

Intervention• C

ommon required response to intervention framework response

• Differentiated, targeted, and intensive response to student needs

• Student equity, access, and support

Assessment• P

LC teaching-assessing-learning cycle

• In-class formative assessment processes

• Common assessment instruments as formative learning opportunities

Collaboration

What are we teaching?

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PICS

•Share your section and record what you hear

•Describe one connection you notice•How could you use this with students

Critical Focus Areas

Ratios and Proportional Reasoning

Applying to problemsGraphing and slope

Standards 1-3

Number Systems, Expressions and

Equations

Standards 4-10

Geometry

Scale drawings, constructions, area,

surface area, and volume

Standards 11-16

Statistics

Drawing inferences about populations based on samples

Standards 17-20

Probability – Standards 21-24

Recommend Emphases from PARCC Model Content Framework for Mathematics

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7th Grade Domains

1. Ratios and Proportional Reasoning2. The Number System3. Expressions and Equations4. Geometry5. Statistics and Probability

Analysis Tool

Content Standard

Cluster

Which Standards

in the Cluster Are Familiar?

What’s New or

Challenging in These

Standards?

Which Standards

in the Cluster Need

Unpacking or

Emphasizing?

How Is This Cluster

Connected to the Other 6-8 Domains

and Mathematical Practice?

Draw informal comparative inferences about two populations.

7.SP3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between centers by expressing it as a multiple of a measure of variability.

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Unpacking the Standards

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Unpacking the Standards

Step 1: Target a standardStep 2: Chunk the Main

CategoriesStep 3: Identify all

standard components

Step 4: Identify the Developmental Progression

Step 5: Identify Key Vocabulary

Step 6: Add Clarifying Information

“To increase student achievement by ensuring educators understand specifically what the new standards mean a student must know, understand, and be able to do. (Unpacking) may also be used to facilitate discussion among teachers and curriculum staff and to encourage coherence…(Unpacking), along with on-going professional development is one of many resources used to understand and teach the CCSS.”

-North Carolina Dept of Public Instruction

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Why are we Unpacking Standards?

To understand what the standards are asking students to know, understand, and be able to do

To make time for professional discussion about the standards

To build upon and use common terminology when discussing the implementation of the standards

Unpacking is standards is not a substitute document for the Common Core Standards, it is a

record of the conversation of those who are involved in the process of digging into the

standards.

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Step 1 – Target a Standard

•What standard(s) do you need to explore further?

•Find a group of 2-4 teachers who will explore that topic with you.

222.G.3Partition circles and

rectangles into two, three, or four

equal shares

Partition circles and rectangles into two equal shares, using

the word halves, half of2.G.3

Partition

Builds on 1.G.3 Needed for 3.G.2

partition

Equal shares

rectangle HalvesHalf of

circle

Partition a shape into fourths in different ways

Pattern Blocks

Fraction Bars/Circles

Describe

ThirdsThird of

FourthsFourth of

whole

Identicalwhole

2/2 = one whole

Partition circles and rectangles into three equal shares, using

the word thirds, third of2.G.3

Partition circles and rectangles into four equal shares, using

the word fourths, fourth of

2.G.3

Describe the whole as two halves, three thirds, four fourths

2.G.3

Recognize that equal shares of identical

wholes need not have the same shape

2.G.3

Recognize

The final product….

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Example2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Step 2: Chunk the Main Categories

1.All Standard(s) in the cluster(s)

2.Identify Key Verbs

2.G.3Partition

circles and rectangles into two,

three, or four equal shares

Describe RecognizePartition

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Step 3: Identify all standard components

Components from CCSS:

Analyze nouns and verbsWhat do students need to do?

Include bullets, examples, footnotes, etc.

Take standard apart according to the verbs to separate skills within the standard

What do the students need to know?

lt blue

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Example 2.G.3

Partition circles and rectangles into two, three, or four equal

shares

Partition RecognizeDescribe



2.G.3

Describe the whole as two halves, three thirds, four

fourths2.G.3

Recognize that equal shares of identical wholes need not have

the same shape2.G.3





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Step 4: Identify the Developmental Progression

Questions to consider when looking at the developmental progression of the standards…

•How would you utilize these chunks (blue) for scaffolding toward mastery of the entire standard?

• Where would you start when teaching this standard?

• What is the chunk that demonstrates the highest level of thinking?

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Vertical AlignmentUsing the progression document(s) from Ohio Department of Education and CCSS Writing Team:

Look to the grade level(s) below to see if the standard is introduced.

Look to the grade level(s) above to see if the standard is continued.

Code each standard on the poster with: builds on introduced needed for or mastered

and the grade level to which the standard aligns.

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Introduced?Mastered?

Needed for?Builds on?

2.G.3Partition circles and rectangles into two, three, or four equal

shares

Partition RecognizeDescribe



2.G.3

Describe the whole as two halves, three thirds, four

fourths2.G.3

Recognize that equal shares of identical wholes need not have

the same shape2.G.3





Builds on 1.G.3

Needed for 3.G.2

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Step 5: Identify Key Vocabulary

Identify content vocabulary directly from the standard. Identify additional vocabulary students will need to know to meet the standard.

green



equal shares



Partition


partition

Equal shares


circle

Describe

ThirdsThird of

FourthsFourth of

whole

Identicalwhole





2.G.3


2.G.3



2.G.3

Recognize

Partition circles and rectangles into two equal

shares, using the word halves,

half of2.G.3

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Step 6: Add Clarifying Information

Kid-friendly language to add clarity Clarifying pictures, words, or phrasesDefinitions, examplesSymbols, formulas, pictures, etc.

CAUTION: do not replace important vocabulary that is included in the standard.

yellow



equal shares



Partition


partition

Equal shares


circle


Pattern Blocks


Describe

ThirdsThird of

FourthsFourth of

whole

Identicalwhole

2/2 = one whole





2.G.3


2.G.3



2.G.3

Recognize



half of2.G.3

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Transfer Unwrapping to Chart



equal shares



Partition


partition

Equal shares


circle


Pattern Blocks


Describe

ThirdsThird of

FourthsFourth of

whole

Identicalwhole

2/2 = one whole





2.G.3


2.G.3



2.G.3

Recognize



half of2.G.3

Main Idea of Standard

Key Verbs

Take standard apart according to the verbs to separate skills within the standard.

Use all components of standard. Put in a logical sequence

Vertical alignment

Vocabulary

Clarifying information, student-friendly

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The Number System

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The standards stress not only procedural skill but also conceptual understanding, to make sure students are learning and absorbing the critical information they

need to succeed at higher levels – rather than the current practices by which many

students learn enough to get by on the next test, but forget it shortly thereafter, only to review again the following year.

(CCSSI, 2012)

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Mathematics consists of pieces that make sense; they are not just independent manipulation/skills to be practiced and memorized – as perceived by many students.

These individual pieces progress through different grades (in organized structures we called “flows”) and can/should be unified together into a coherent whole.

Jason Zimba, Bill McCallum

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Mathematical Fluency

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Fluency

•The word fluent is used in the Standards to mean fast and accurate. Fluency in each grade involves a mixture of just knowing some answers from patterns (e.g., “adding 0 yields the same number”), and knowing some answers from the use of strategies.

• Progressions for the Common Core State Standards in Mathematics

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Fluency

•Fluent in the standards means “fast and accurate.” It might also help to think of fluency as meaning more or less the same as when someone is said to be fluent in a foreign language. To be fluent is to flow; fluent isn’t halting, stumbling, or reversing oneself.

Jason Zimba

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Fluency ExpectationsGrade Required Fluency

K Add/subtract within 5

1 Add/subtract within 10

2 Add/subtract within 20Add/subtract within 100 (pencil and paper)

3 Multiply/divide within 100Add/subtract within 1000

4 Add/subtract within 1,000,000

5 Multi-digit multiplication

6 Multi-digit division (6.NS.2)Multi-digit decimal operations (6.NS.3)

7 Solve px + q = r, p(x + q) = r

•Most of the numbers you have worked with in math class this year have been greater than or equal to zero. However, numbers less than zero can provide important information.

•Where have you seen numbers less than zero outside of school?

Integers

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7NS - Understanding• 7.NS.1b – 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.

• 7.NS.1c – Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q).

• 7.NS.2a – 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.

• 7.NS.2b – Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number.

• K–6 Geometry• 6-8 Statistics and Probability• 6–7 Ratios and Proportional Relationships• 6–8 Expressions and Equations• 6-8 Number Systems• 3-5 Number and Operations: Fractions

• These are the documents currently available. They are working on documents for the other domains (Functions, Geometry 7-8).

Progressions Documents

http://ime.math.arizona.edu/progressions/

http://ime.math.arizona.edu/progressions/

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Jigsaw

•Read your assigned section

•Chart paper▫Summarize what you read.▫How can this document help you in your

classroom?

•Be prepared to share

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Adding Integers

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Opposites

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Subtracting Integers

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Subtracting Integers

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Fractions

•Difficulty with learning fractions is pervasive and is an obstacle to further progress in mathematics and other domains dependent on mathematics, including algebra. It has also been linked to difficulties in adulthood, such as failure to understand medication regimens.

National Mathematics Panel Report, 2008

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Fractions

•“Students who are asked to practice the algorithm over and over…stop thinking. They sacrifice the relationships in order to treat the numbers simply as digits.”

Imm, Fosnot, Uittenbogaard (2012)

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Unit Fractions

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Fraction Multiplication in Grade 5

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57


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5th Grade Division

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5th Grade Division

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5th Grade Division Problems

•How much chocolate will each person get if 3 people share ½ pound equally?

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5th Grade Division Problems

•How many 1/3 cup servings are in 2/3 cups of raisins?

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Fraction Division in Grade 6•6.NS.1 – Interpret and compute quotients of

fractions, and solve word problems involving division of fractions, e.g., by using visual fraction models and equations to represent the problem.

•Examples: ▫Create a story context…▫Use a visual fraction model to show the quotient…▫Explain division using its relationship with

multiplication▫Sample problems

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6th Grade Division

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6th Grade Division

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Ratios and Proportional Relationships

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Kim and Bob ran equally fast around a track. Kim started first. When she had run 9 laps, Bob had run 3 laps.

When Bob had run 15 laps, how many laps had Kim run?

Explain your reasoning.

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Solving ProportionsSolve Kanold, p. 94

• The traditional method of creating and solving proportions by using cross-multiplication is de-emphasized (in fact it is not mentioned in the CCSS) because it obscures the proportional relationship between quantities in a given problem situation.

• If two pounds of beans cost $5, how much will 15 pounds of beans cost?

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Ratios and Proportional Relationships Progression, pages 6-7

• Although it is traditional to move students quickly to solving proportions by setting up an equation, the Standards do not require this method in Grade 6. There are a number of strategies for solving problems that involve ratios. As students become familiar with relationships among equivalent ratios, their strategies become increasingly abbreviated and efficient.

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6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.a. Make tables of equivalent ratios relating quantities with

whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

b. Solve unit rate problems including those involving unit pricing and constant speed.

c. Find a percent of a quantity as a rate per 100; 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.

http://threeacts.mrmeyer.com/nana/

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1. The ratio of free throws that Omar made to the ones he missed at practice yesterday was 7:3. If he attempted 90 free throw at practice, how many free throws did Omar make?

made 7

missed 3

Attempted 90

9 x 7 = 6390 ÷ 10 = 9

Omar made 63 free throws.

9

9 9 9

9 9 9 9 9 9

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2. At FDR High School, the ratio of seniors who attend college to those who do not is 5:2. If 98 seniors do not attend college, how many do?

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3. At Mesa Park High School, the ratio of students who have driver’s licenses to those who don’t is 8:3. If 144 students have driver’s licenses, how many students are enrolled at Mesa Park High School?

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4. Of the black and blue pens that Mrs. White has in a drawer in her desk, 18 are black. The ratio of black pens to blue pens is 2:3. When Mrs. White removes 3 blue pens, what is the new ratio of black pens to blue pens?

http://www.risd.k12.nm.us/instruction/singaporemathbook.cfm

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Expressions and Equations

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Understanding

•7.EE.2 – Understand that rewriting an expression in different forms in a problem context can shed light on the problem, and how the quantities in it are related.

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The Mystery Bags Game

Solving Equations

The Mystery Bags Game

There once was a king who loved to

watch his many bags of gold.

• But it can get very boring watching gold all day, so he had the court jester make up games for him to pass the time. The game the king loves best is the Mystery Bags game.

• First, the jester takes one or more empty bags and fills each bag with the same amount of gold. These bags are called the “mystery bags.”

• Next, the jester digs into his collection of lead weights. He takes out his pan balance and places some combination of mystery bags and lead weights on the two pans so that the sides balance.

• The game is to figure out the weight of each mystery bag.

Your Task

• The game sound rather easy, but it can get very difficult for the king. See if you can win the mystery bags game in the various situations described on your worksheet by figuring out how much gold there is in each mystery bag.

• Explain how you know you are correct. You may want to draw diagrams to show what’s going on.

• Write an equation to represent this situation

Use your methods from Problem 3.2 to find the number of gold coins in each pouch

Write down a similar method using the equation that represents this situation

Question 1

• There are 3 mystery bags on one side of the balance and 51 ounces of lead weights on the other side.

51 oz.

2

• There are 1 mystery bag and 42 ounces of weights on one side, and 100 ounces of weights on the other side.

3

• There are 8 mystery bags and 10 ounces of weights on one side, and 90 ounces of weights on the other side.

4

• There are 3 mystery bags and 29 ounces of weights on one side, and 4 mystery bags on the other side.

5

• There are 11 mystery bags and 65 ounces of weights on one side, and 4 mystery bags and 100 ounces of weights on the other side.

6

• There are 6 mystery bags and 13 ounces of weights on one side, and 6 mystery bags and 14 ounces of weights on the other side. (The jester could get in a lot of trouble for this one!)

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• There are 15 mystery bags and 7 ounces of weights on both sides. (At first the king thought this one was easy, but then he found it to be incredibly hard.)

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• The king wants to be able to win easily all of the time, without calling you in. Therefore, your final task in this assignment is to describe in words a procedure by which the king can find out how much is in a mystery bag in any situation.

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•7.EE.4 – Use variables to represent quantities in real-world or mathematical problems, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

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Mathematics Assessment Project• Tools for formative and summative assessment that make

knowledge and reasoning visible, and help teachers to guide students in how to improve, and monitor their progress. These tools comprise:

• Classroom Challenges: lessons for formative assessment, some focused on developing math concepts, others on non-routine problem solving.

• Professional Development Modules: to help teachers with the new pedagogical challenges that formative assessment presents.

• Summative Assessment Task Collection: to illustrate the range of performance goals required by CCSSM.

• Prototype Summative Tests: designed to help teachers and students monitor their progress, these tests provide a model for examinations that may replace or complement current US tests.http://map.mathshell.org/

http://map.mathshell.org/

http://map.mathshell.org/

Steps to Solving EquationsProjector Resources

Writing Algebraic Expressions

P-108

Area of rectangle = _ _ _ _ _ _ _ _ _ _ _ _



P-109

Perimeter of rectangle = _ _ _ _ _ _ _ _ _ _ _ _



P-110

Which two expressions are equivalent?


Which Equations Describe The Story?

P-111

A pencil costs $2 less than a notebook.

A pen costs 3 times as much as a pencil.

The pen costs $9

Which of the four equations opposite describe this story?

Let x represent the cost of notebook.

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Geometry

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The van Hiele Theory of Geometric Thought

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Statistics and ProbabilityResources

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Understanding

•7.SP.1 – Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

Pizza or Broccoli?

At a nearby school, teachers decided to get rid of pizza Fridays. After a survey of all teachers,

counselors, and administrators, it was overwhelmingly decided that pizza would be

replaced with broccoli with ranch dip.

Is this fair???

After surveying 83 students in 3 classes, 70% responded that girls should be allowed to go to lunch two minutes early every day and boys will go at the regular time.

Do you think this is an accurate statistic?

Who do you think the sample population was?

Group Roles Each group will need to assign the following

roles:

Facilitator – keeps group on task and ensures equal participation

Materials Manager – collects and returns materials

Recorder – writes group answer on chart paper

Reporter – presents group answer to the class

Group Work Discuss and complete the handout as a group.

Begin with the multiple choice questions.

Choose one biased survey to present to the class on chart paper. Include the following in your presentation:

Original survey

Why you think it is biased

How you would correct it

Create a Survey

QuestionPopulationSample group

Exit Ticket In a poll of Mrs. Simpson’s math class, 67%

of the students say that math is their favorite academic subject. The editor of the school paper is in the class, and he wants to write an article for the paper saying that math is the most popular subject at the school. Explain why this is not a valid conclusion, and suggest a way to gather better data to determine what subject is most popular.

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Illustrative Mathematics

•Illustrative Mathematics provides guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students experience in a faithful implementation of the Common Core State Standards, and by publishing other tools that support implementation of the standards.

http://www.illustrativemathematics.org/

http://www.illustrativemathematics.org/

What Do You Expect?

Compound Probability

How is probability used in real life?

Event Theoretical probability Experimental probability Outcome

What’s in the Bucket?

What’s in the Bucket?

Without looking in or

emptying the bucket,

how could we

determine the fraction

of blocks that are red,

yellow, or blue?

Problem 1.1

How many blocks drawn by your class were blue?

How many were yellow? How many were red? Which color block do you think there are

the greatest of in the bucket? Which color block do you think there are

the least number of?

Problem 1.1

Based on our experimental data, predict the fraction of blocks in the bucket that are blue, that are yellow, and that are red.

Problem 1.1

How do the fraction of blocks that are blue, yellow, and red compare to the fractions of blue, yellow, and red drawn during the experiment?

Match / No-Match Rules

Spin the spinner twice for each turn. If both spins land on the same color, you

have made a MATCH. Player A scores 1 point.

If the two spins land on different colors, you have made a NO-MATCH. Player B scores 2 points.

Experimental Probability of Match

number of turns that are matches

total number of turns

Experimental Probability of No-Match

number of turns that are no-matches


What are the possible outcomes in this game?

Color on 1st spin – Color on 2nd spin

Are all outcomes equally likely?

Theoretical Probability of No-Match

number of outcomes that are no- matches


Theoretical Probability of Match

number of outcomes that are matches


A. Compare the experimental and theoretical probabilities for match and for no-match.

B. Is Match/No-Match a fair game? If you think the game is fair, explain why. If you think it is not fair, explain how the rules could be changed to make it fair.

Making Purple

BROWN

YELLOW

RED

ORANGE

BLUEGREEN

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Jeanne SimpsonUAHuntsville AMSTI

[email protected]

[email protected]

Contact Information

145

Feedback

Praise

Question

Polish

content deepening 7 th grade math september 23, 2013 jeanne simpson amsti math specialist

Documents

practice standards

equations standards

volume standards

slope standards

repeated reasoning standards

student needs student

coherence common

intensive response