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Module 2:Content and Curriculum, K-8Mathematics

Agenda• Common Core State Standards Overview• Standards Progression: A Closer Look

– Number Systems:• K-5 Base Ten Operations • 3-5 Fractions• 6-8 Rational Numbers

• Mathematical Practices in Action• Closing

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Math Class Needs a Makeover

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Common Core State Standards“These standards define what students should understand and be able to do in their study of mathematics.”

National Governors Association Center for Best Practicesand Council of Chief State School Officers (2010)Common Core State Standards for Mathematics

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Two Types of Standards• Standards for Mathematical Practice

Eight standards common to all grade levels K-12

• Mathematical Content StandardsStandards that vary by grade level and conceptual category

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Mathematical Habits of Mind“Mathematical habits of mind allow one to bring coherence to many different parts of the curriculum… Building coherence in the development of mathematical ideas across grades is key to improving students’ mathematical learning.”

Cuoco, Goldenberg, & Mark (2010)Organizing a Curriculum around Mathematical Habits of Mind

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Pedagogical Shifts with Common Core State StandardsFocus• Identifies key ideas,

understandings, and skills for each grade or course

• Stresses deep learning, applying concepts and skills within the same grade or course

Coherence• Articulates a progression

of topics across grades• Connects ideas with

other ideas, enabling students to learn with understanding, develop skill proficiency, and solve problems

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Grade-Level Standards• Locate the CCSS for Mathematics handout.• Read the introduction for your grade level.• Highlight critical focus areas and mark as follows:

Expectations similar to CA 1997 Standards! New expectations? Questions I have regarding these expectations

• Be ready to discuss.

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Format of the OverviewDomains:

Overarching ideas that

connect topics across the

grades

Clusters:Illustrate the

progression of increasing

complexity from grade to grade

© Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

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Format of the StandardsDomain:

Cluster:

Standards: Define what

students should

know and be able to

do for each grade level or course.

© Copyright 2010. National Governors Association Center for Best Practicesand Council of Chief State School Officers. All rights reserved.

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Domains and Conceptual Categories

Distribution

Findell & Foughty (2011)College and Career-Readiness Through the

Common Core State Standards for Mathematics

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Example of Domain Flows

Elementary to High School

Calahan (2011)Delving Deeper into the Common Core

Operations & Algebraic Thinking

Expressions & Equations

Algebra

Number: Base Ten

Number & Quantity

Number:Fractions

Geometry Geometry Geometry

Measurement & Data

Statistics &Probability

Statistics &Probability

Functions Functions

The NumberSystem

K - 5 6 - 8 9 - 12

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Base Ten Place Value (K-5)

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Base Ten Place Value (K-5) Chart

On

es

Ten

s

Hu

nd

red

s

Th

ou

san

ds

x 10x 10x 10

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Base Ten Place Value (K-5) Chart

On

es

Ten

s

Hu

nd

red

s

Th

ou

san

ds

x 10x 10x 10 x 10

Ten

ths

x 10

Hu

nd

red

ths

x 10

Th

ou

san

dth

s

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Symmetry with

Respect to the Ones Place

On

es

Ten

s

Hu

nd

red

s

Th

ou

san

ds

Ten

ths

Hu

nd

red

ths

Th

ou

san

dth

s

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Base Ten Place Value (K-5):

On

es

Ten

s

Hu

nd

red

s

Th

ou

san

ds

Ten

ths

Hu

nd

red

ths

Th

ou

san

dth

s

100

30

10

3

Fractional Notation

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Base Ten Place Value (K-5):

On

es

Ten

s

Hu

nd

red

s

Th

ou

san

ds

Ten

ths

Hu

nd

red

ths

Th

ou

san

dth

s

Decimal Notation

1000 100 10 1 0.1 0.01 0.001

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Base Ten Place Value (K-5):

On

es

Ten

s

Hu

nd

red

s

Th

ou

san

ds

Ten

ths

Hu

nd

red

ths

Th

ou

san

dth

s

103 102 101 100

Exponential Notation

1000 = 103 0.001 = 1.0 x 10-3

10-1 10-2 10-3

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Part 1: Number and Operations

Base Ten Progression• Grade K: Work with numbers 11-19 to gain foundation for

place value. (K.NBT.1)• Grade 1: Extend the counting sequence; understand place

value. (1.NBT.2• Grade 2: Understand place value and use this

understanding and properties of operations to add and subtract. (2.NBT.1)

• Grade 3: Use place value understanding and properties of operations to perform multi-digit arithmetic. (

• Grade 4: Generalize place value understanding for multi-digit whole numbers; use this understanding to perform multi-digit arithmetic. (4.NBT.1 – 4.NBT.3)

• Grade 5: Understand the place value system and perform operations with multi-digit whole numbers and with decimals to hundredths. (5.NBT.1 through 5.NBT.4)

What part of the activity depends on the expectations of the K-2 domains?

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Connecting to Current PracticeThink/Pair/Share• How does the place value demonstration with money

highlight the progression of expectations in the K-5 Number and Operations in Base Ten Domain?

• How does the progression of standards within the Base Ten domain correspond to your current practice?

• What are some things you might do now to help students transition to the CCSS?

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Fractions Domain (3-5)Solve this problem on your own.

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Fraction Misconceptions

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Video Debrief

Fractions MisconceptionsThink/Pair/Share• What were this student’s strengths?• What were his misunderstandings regarding

the division of fractions?• How might you address these misunderstandings?

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“Represent a fraction a/b on a number line diagram by marking off the lengths 1/b from 0…”

Represent 2/3 on this number line:

Number and Operations: Fractions Progression (3-5)

Standard 3.NF.2a

0 1

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Number and Operations: Fractions Progression (3-5)

Standard 4.NF.2“Compare two fractions… by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2.”

0 1½

64?

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Number and Operations: Fractions Progression (3-5)

Standard 5.NF.2“Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.”

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Determine whether the number sentence is correct without using common denominators. Explain how you know.

Number Systems Progression (6-8)

Standard 6.NS.6c“Find and position integers and other rational numbers on a horizontalor vertical number line diagram…”

-1 10

-1

1

0

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Number Systems Progression (6-8)

Standards 7.NS.5 and 8.NS.2“Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram…”

-1 4

Where does this belong?

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Number Systems Progression

Area = 1

Area = 424 11

3

Where does this belong?

11

24

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Number Systems Progression (6-8)

Standards 7.NS.5 and 8.NS.2“Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram…”

-1 4

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Common Core State Standards for

Mathematical Practice21st Century Skills• Communication• Collaboration• Creativity• Critical Thinking• Construction of

Knowledge

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Part 1: Standards for

Mathematical Practice in Action • Locate the MARS Tasks and Student

Work handout.• Complete the task on your own.• Once task is completed, discuss the prompts:

– What mathematical skills and understanding are needed to successfully complete the task?

– Which Standards for Mathematical Practice did you use to solve the task?

• Use the Standards of Mathematical Practice Look-For Tool to identify practices used.

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Part 2

Mathematical Practices in Action • Using the MARS Tasks and Student Work

handout, examine the student work provided for the task you completed.

• Determine which Standards for Mathematical Practice are represented in the student work.

• Compare your work with the student work. – How are the practices similar? – How are the practices different?

• More tasks available: – www.insidemathematics.org

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Connecting to Current PracticeThink/Pair/Share• How does the progression of standards within the

Number and Operations in Base Ten, Number and Operations in Fractions, and Number Systems domains correspond to your current practice?

• What are some things you might do now to help students transition to the CCSS?

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Next Steps• Think about what you learned today.• Decide on one thing you will do differently

to start transitioning to the CCSS.• Share your ideas with a partner.

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For Further Investigation• California’s Common Core State Standards

http://www.cde.ca.gov/ci/cc• Common Core State Standards Initiative

http://www.corestandards.org• Tools for the Common Core

(includes the Progressions)http://www.commoncoretools.wordpress.com

• Inside Mathematics (MARS Tasks & Resources) www.insidemathematics.org

• K-5 Math Teaching Resources www.k-5mathteachingresources.com

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