Session 36-12 Mathematics

Common Core State Standards

Bell Work Activity

• Handout #1

The Common Core

State Standards for

Mathematics from A to Z

• List words that begin with each letter of the alphabet that identify aspects of the Common Core State Standards for Mathematics.

Handout #1

Common Core From A to Z

Handout #1

Common Core From A to Z

2

WELCOME

Introductions

• June Campbell• ORTTTA

• Nancy Kinard• Palm Beach

• Michael Chase• Volusia County

• Amy Coltharp• BFEP

• Kim Wuellner• St. Johns

• Martha Gioielli• DA

Agenda

• The Common Core State Standards for Mathematics from A to Z

• Exploring the Standards for Mathematical Practice

• Exploring the Common Core State Standards for Mathematics Critical Areas

• Exploring the Mathematics Progressions

3

“We live in a time of vast changes that include accelerating globalization, mounting quantities of information, the dominating influence of science and technology, and the clash of civilizations. Those changes call for new ways of learning and thinking in school, business, and the professions.” -Howard Gardner

Five Minds for the Future (2007)

4

Expected Outcomes

• Become familiar with the structure of the Common Core State Standards for Mathematics.2

Enhance knowledge of the Common Core Standards for Mathematical Practice. 3

4Understand how the critical areas bring focus to key mathematical concepts for students to learn at each grade level.

5 Consider how the learning progressions can be used to inform curriculum and guide instruction.

• Enhance knowledge base of the Common Core Standards for Mathematics.1

5

Session 36-12 Mathematics

Digital Checkpoint

Digital Checkpoint

• Poll Everywhere• http://www.pollev

erywhere.com/

7

How To Vote via Texting

1. Standard texting rates only (worst case US $0.20)2. We have no access to your phone number3. Capitalization doesn’t matter, but spaces and spelling do

TIPS

EXAMPLE

How to Vote via Web Response

Rate your level of understanding of the Common Core State Standards

for Mathematics?

• No understanding 125724• Limited understanding 125730• Fair understanding 125803• Good understanding 125804• Excellent understanding 125809

Submit response at http://PollEV.com

Send to: 37607Text Code # to respond

10

Virtual Parking Lot

Submit response at http://PollEV.com

Send to: 37607Text: 220274 plus your message

11

Digital Resources for the Common Core

• Apple

• http://itunes.apple.com/us/app/com

mon-core-standards/id439424555

• Android

• https://play.google.com/store/apps/

details?id=com.edutater.corestand

ards

• Common Core State Standards for

Mathematics

• http://www.corestandards.org/

12

Understanding Teachers

Aligned

McCallum

Guided by Principles

Standards

National Focus NGA

Habits of

Mind

Organized

International Benchmarked

Building Foundation

CCSSO Daro

Progressions

Joint effort

Vision

Quality

Knowledge

Zimba

Robust, Relevant,

Real-world

Whole Child

Approach

Learner-focused Life-long

skills

Ensuring Education Focus

BalancedClarity

Critical Areas

ClustersDomains

Evidenced-based

Illustrative Mathematics

Sense-making

Rigor

Research-based

Procedural fluency

ProficiencyPARCC

OpportunitiesMathematical Practice

XYOU

FluencyApplication

Coherence

Timeline

DevelopmentalLevels

Domains for K-12K 1 2 3 4 5 6 7 8 HS

Counting

and Cardinality

(CC)

Number and

Quantity(N)

Number and Operations in Base Ten (NBT) The Number System (NS)

Number and Operations-Fractions

(NF)

Ratios and Proportional Relationships

(RP)

Operations and Algebraic Thinking (OA)Functions (F)

Expressions and Equations (EE)

Algebra(A)

Geometry (G)

Measurement and Data (MD)

Statistics and Probability (SP)A-Z

Domain

Domain

Domain

Domain

Cluster Headings

Cluster Headings

Cluster Headings

Cluster Headings

16

Grade 7 Overview

Ratios and Proportional Relationships

•Analyze proportional relationships and use them to solve real-world and mathematical problems.

The Number System

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

Expressions and Equations

•Use properties of operations to generate equivalent expressions.

Geometry

•Draw, construct and describe geometrical figures and describe the relationships between them.•Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Statistics and Probability

•Use random sampling to draw inferences about a population.•Draw informal comparative inferences about two populations.•Investigate chance processes and develop, use, and evaluate probability models.

Domain

Cluster Headings

17

High School Conceptual Categories

http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

• Number and Quantity (N)

• Algebra (A)

• Functions (F)

• Modeling (*)

• Geometry (G)

• Statistics and Probability (S)A-Z

Cluster Headings

Cluster Headings

Cluster Headings

Cluster Headings

Domain

Domain

Domain

Domain

A-Z

Cluster Headings

Standards

Domain

Florida’s Numbering ofthe Common Core State Standards

MACC.7.EE.1.1

Subject Grade Domain Cluster Standard

Subject Mathematics Common Core

Grade 7

Domain Expressions and Equations

Cluster Use properties of operations to generate equivalent expressions

Standard Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

20

Florida’s Numbering ofthe Common Core State Standards

MACC.912.N-RN.2.3

Subject Grade Category Domain Cluster StandardSubject Mathematics Common Core

Grade 9-12

Category Number and Quantity

Domain The Real Number System

Cluster Use properties of rational and irrational numbers.

Standard Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is rational; and that the product of a nonzero rational number and an irrational number is irrational. A-Z

Standards for Mathematical Practice

“The Standards for Mathematical Practice are unique in that they describe how teachers need to teach to ensure their students become mathematically proficient. We were purposeful in calling them standards because then they won’t be ignored.”

- Bill McCallum

22

Use appropriate tools strategically5

Attend to precision6

Look for and make sense of structure

7

Look for and express regularity in repeated reasoning

8

8 Standards for Mathematical Practice

Make sense of problems and persevere in solving them

1

Reason abstractly and quantitatively2

Construct viable arguments and critique the reasoning of others

3

Model with mathematics4

23A-Z

Florida’s Common Core State Standards Implementation Timeline

Year/Grade Level K 1 2 3-8 9-12

2011-2012 FL L L L L

2012-2013 F L F L L L L

2013-2014CCSS fully implemented

F L F L F L B L B L

2014-2015CCSS fully implemented and

assessed

F L F L F L F L F L

F - full implementation of CCSS for all content areasL – begin full implementation of content area literacy standards including: (1) use of

informational text, text complexity, quality and range in all grades (K-12), and (2) CCSS Literacy Standards in History/Social Studies, Science, and Technical Subjects (6-12)

B - blended instruction of CCSS with Next Generation Sunshine State Standards (NGSSS); last year of NGSSS assessed on FCAT 2.0

http://www.fldoe.org/bii/pdf/CCSS-ImplementationTimeline.pdf 24A-Z

Session 36-12 Mathematics

Standards for Mathematical Practice

Poll Question

How many Common Core Standards for Mathematical Practice are there?

26

Submit response at http://PollEV.com/

Send to: 37607Text: 129153 plus your message

27

Standards for Mathematical Practice

Reasoning and

Explaining

Seeing Structure and Generalizing

Overarching Habits of Mind of a Productive Mathematical Thinker

2. Reason abstractly and quantitatively

3. Construct viable arguments and

critique the reasoning of others

Modeling and

Using Tools

4. Model with mathematics

5. Use appropriate tools strategically

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoning

1. Make sense of problems and persevere in solving them6. Attend to precision

27

Standards for Mathematical Practice

http://www.youtube.com/watch?v=9pKcO9E4Flw&feature=relmfu http://youtu.be/9pKcO9E4Flw

• Develops dispositions and habits of mind

“Characteristic of an educated person”• Precision in thought• Precision in the use of language and terms• Precision of argument• Sense making happens through conversations

28A-Z

The Standards for Mathematical Practice

• Please locate the Common Core State Standards for Mathematics.

• Take a moment to examine the first three words of the narrative description for each of the 8 mathematical practices.

• What do you notice?

Mathematically Proficient Students…

Page 6

29

30

Digital TaskYour Digital Task is to:

• Read your assigned Mathematical Practice.

• Identify the words (verbs) that illustrate the student actions for this practice.

• Text the words on one continuous line with spaces between each word.

Example: #..... create analysis model describe demonstrate….

Digital Task Text Numbers

Practice #1 – code 130116

Practice #2 – code 50983

Practice #3 – code 71040

Practice #4 – code 71234

Practice #5 – code 71681

Practice #6 – code 71867

Practice #7 – code 72038

Practice #8 – code 6381

Submit responses at http://PollEV.com

Send to: 37607Text a CODE # to respond

31

Mathematical Practice 1 - Make sense of problems and persevere in solving them.

Submit responses at http://PollEV.com

Text 130116 to 37607

Mathematical Practice 2 - Reason Abstractly and Quantitatively

Submit responses at http://PollEV.com

Text 50983 to 37607

Mathematical Practice 3 - Construct viable arguments and critique the

reasoning of others.

Submit responses at http://PollEV.com

Text 71040 to 37607

Mathematical Practice 4 - Model with mathematics.

Submit responses at http://PollEV.com

Text 71234 to 37607

Mathematical Practice 5 - Use appropriate tools strategically.

Submit responses at http://PollEV.com

Text 71681 to 37607

Mathematical Practice 6 - Attend to precision.

Submit responses at http://PollEV.com

Text 71867 to 37607

Mathematical Practice 7 - Look for and make sense of structure.

Submit responses at http://PollEV.com

Text 72038 to 37607

Mathematical Practice 8 - Look for and express regularity in repeated reasoning.

Submit responses at http://PollEV.com

Text 6381 to 37607

http://www.wordle.net/show/wrdl/5360414/Mathematical_Practice_Actions40

Consider the Learners

• Over 240,000 ELLs in Florida

• Almost every district has ELLs

• 300 languages are spoken among ELLs

• 79% of ELLs are in Mainstream/Inclusion model classrooms

• ELLs are learning in the same classrooms as non-ELLs

41

Making the Content Comprehensible

• Use the standards vocabulary as a teaching tool. “Generalize, develop, describe, analyze, apply, measure,” etc. are all words ELLs will hear in the classroom and need to understand.

• ELLs may know how to perform the skill using their language, they just may not yet have the English vocabulary.

• Use pictures, graphs, and charts whenever possible.

• Make use of root words and cognates. 42

Classroom Strategies

• Group ELLs with non-ELLs to work together.

• Allow more wait time for ELLs to respond.– Silence does not necessarily mean the student

does not know the answer, the ELL may be translating the answer and needs more time.

• Remember that ELLs from different countries may display mathematical functions in different ways.

43

Session 36-12 Mathematics

Critical Areas and Mathematics Progressions

K-12 Domains and Critical Areas

45

6th Grade Domains 6th Grade Critical Areas

Ratios and Proportional Relationships The Number System Expressions and Equations Geometry Statistics and Probability

Connecting ratio and rate to whole number multiplication and division using concepts of ratio and rate to solve problems.

Completing understanding of division of fractions and extending the notation of numbers to the system of rational numbers which includes negative numbers.

Writing, interpreting, and using expressions and equations. Developing understanding of statistical thinking.

7th Grade Domains 7th Grade Critical Areas Ratios and Proportional Relationships The Number System Expressions and Equations Geometry Statistics and Probability

Developing understanding of and applying proportional relationships. Developing understanding of operations with rational numbers and

working with expressions and linear equations. Solving problems involving scale drawings and informal geometric

constructions, and working with two-and three-dimensional shapes to solve problems involving area, surface area, and volume.

Drawing inferences about populations based on samples.

8th Grade Domains 8th Grade Critical Areas The Number System Expressions and Equations Functions Geometry Statistics and Probability

Formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations.

Grasping the concept of a function and using functions to describe quantitative relationships.

Analyzing two-and-three dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying Pythagorean Theorem.

Algebra 1 Domains Algebra 1 Critical Areas The Real Number System Quantities Seeing Structure in Expressions Arithmetic with Polynomials and Rational Expressions Creating Equations Reasoning with Equations and Inequalities Interpreting Functions Building Functions Linear, Quadratic, and Exponential Models Interpreting Categorical and Quantitative Data

Relationships between Quantities and Reasoning with Equations; Analyze and explain the process of solving an equation.

Linear and Exponential Relationships; Learn function notation and develop the concept of domain and range.

Descriptive Statistics; Learn formal means of assessing how a model fits (data regression techniques, graphical representations and goodness of fit.)

Expressions and Equations; Create and solve equations, inequalities, and systems of equations involving quadratic expressions.

Quadratic Functions and Modeling; Comparing key characteristics.

Geometry Domains Geometry Critical Areas

Handout #2

Handout #2

Four critical areas in 6th GradeConnecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems

1.

Completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers

2.

Writing, interpreting, and using expressions and equations3.

Developing understanding of statistical thinking4.

46

http://www.corestandards.org/the-standards/mathematics/grade-6/introduction/

Page 39

Identify the 6th Grade Critical AreaCluster Heading Critical Area

Understand ratio concepts and use ratio reasoning to solve problems.Summarize and describe distributions.Compute fluently with multi-digit numbers and find common factors and multiples.Reason about and solve one-variable equations and inequalities.Apply and extend previous understandings of arithmetic to algebraic expressions.Apply and extend previous understandings of numbers to the system of rational numbers.Represent and analyze quantitative relationships between dependent and independent variables.

Develop understanding of statistical variability.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

3-Equations

3-Equations

2-Rational #

1-Ratio

4-Statistics

3-Equations

2-Rational #

47

4-Statistics

2-Rational #

48

Four critical areas in 7th Grade

Page 46

http://www.corestandards.org/the-standards/mathematics/grade-7/introduction/

Developing understanding of and applying proportional relationships1.

Developing understanding of operations with rational numbers and working with expressions and linear equations

2.

Solving problems involving scale drawings and informal geometric constructions, and working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume

3.

Drawing inferences about populations based on samples

4.

Identify the 7th Grade Critical AreasCluster Heading Critical Area

Analyze proportional relationships and use them to solve real-world and mathematical problems.

Draw, construct and describe geometrical figures and describe the relationships between them.

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

Use random sampling to draw inferences about a population.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Draw informal comparative inferences about two populations.

Use properties of operations to generate equivalent expressions.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

1-Proportions, 2-Rational

2-Rational

4-Statistics

3-Geometry

2-Rational, 3-Geometry

3-Geometry

4-Statistics

2-Rational

49

Three critical areas in 8th GradeFormulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations

1.

Grasping the concept of a function and using functions to describe quantitative relationships2.

Analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem

3.

50

http://www.corestandards.org/the-standards/mathematics/grade-8/introduction/Page 52

Mathematics ProgressionsRatio and Proportion in Grades 6-8:

Connections to College and Career Skills

Ratio and Proportion in Grades 6-8

Common Core Progressions Project

http://www.youtube.com/watch?v=5pBOnvzC_Yw&feature=BFa&list=UUF0pa3nE3aZAfBMT8pqM5PA

51

<iframe width="560" height="315" src="http://www.youtube.com/embed/5pBOnvzC_Yw" frameborder="0" allowfullscreen></iframe>

http://youtu.be/5pBOnvzC_Yw

http://commoncoretools.files.wordpress.com/2011/12/ccss_progression_sp_68_2011_12_26_bis.pdf

Five critical areas in Algebra 1Relationships Between Quantities and Reasoning

with Equations 1.

Linear and Exponential Relationships2.

Descriptive Statistics 3.

52

http://www.corestandards.org/the-standards/mathematics/high-school-algebra/introduction/Page 15

Expressions and Equations 4.

Quadratic Functions and Modeling 5.

Six critical areas in GeometryCongruence, Proof, and Constructions1.

Similarity, Proof, and Trigonometry2.

Extending to Three Dimensions3.

53

http://www.corestandards.org/the-standards/mathematics/high-school-geometry/introduction/Page 27

Connecting Algebra and Geometry Through Coordinates4.

Circles With and Without Coordinates5.

Applications of Probability 6.

54

Four critical areas in Algebra 2

Page 36

http://www.corestandards.org/the-standards/mathematics/high-school-algebra/introduction/

Polynomial, Rational, and Radical Relationships

1.

Trigonometric Functions2.

Modeling with Functions3.

Inferences and Conclusions from Data4.

55

Building Fluency

What is meant by fluency?

http://www.youtube.com/watch?v=ZFUAV00bTwA&list=PLD7F4C7DE7CB3D2E6&index=13&feature=plpp_video

http://youtu.be/ZFUAV00bTwA

Mathematics Fluency: A Balanced Approach (1:56)

Grade Standard Key FluencyK MACC.K.OA.1.5 Add/subtract within 5

1 MACC.1.OA.3.6 Add/subtract within 10

2MACC.2.OA.2.2MACC.2.NBT.2.5

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

3MACC.3.NBT.1.2MACC.3.OA.3.7

Add/subtract within 1,000Multiply/divide within 100

4MACC.4.2.4Critical Area #1

Add/subtract within 1,000,000 Develop fluency with efficient procedures for multiplying whole numbers

5MACC.5.NBT.2.5Critical Area #1

Multi-digit multiplicationDeveloping fluency with addition and subtraction of fractions

6MACC.6.NS.2.2MACC.6.NS.2.3

Multi-digit divisionMulti-digit decimal operations

7 MACC.7.EE.2.4a Solve px + q = r, p(x + q) = r8 MACC.8.EE.3.8b Solve simple 22 systems by inspection

Key Fluencies Handout #3

Handout #3

56A-Z

Mathematics Progressions

The Importance of Mathematics Progressions

http://www.youtube.com/watch?v=ONPADo_Nt14&list=UUF0pa3nE3aZAfBMT8pqM5PA&index=6&feature=plcp

Handout #4

Handout #4

57

Gathering Momentum for Algebra

Mathematics Progressions Project• Kindergarten Counting and Cardinality

• K-5 Number and Operations in Base Ten

• 3-5 Number and Operations—Fractions

• K–5 Operations and Algebraic Thinking

• K-5 Measurement and Data (Data part)

• Geometry Progression (coming soon!)

• 6-8 Expressions and Equations

• 6-7 Ratios and Proportional Relationships

• 6-8 Statistics and Probability

http://commoncoretools.me/category/progressions/ 58

• High school Statistics and Probability

Give me a lever long enoughand a fulcrum to place it on,and I can move the world. ‐‐Archimedes

59

Reflective Thoughts

How will the Critical Areas and the Cluster headings help to inform your curriculum and guide your instruction?

2

3 How will you use the Learning Progressions to inform your curriculum and guide your instruction?

How will you use the Common Core Standards for Mathematical Practices to inform your curriculum and guide your instruction?

1

Handout #5

Handout #5

60