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Illustrative Mathematics and Coherence in the Standards MENA Common Core Conference November 1-2, 2013 Ellen Whitesides Institute for Mathematics and Education, University of Arizona Director of Community Building, Illustrative Mathematics Ellen Whitesides Who do we have in the room? Research suggests that there’s an exceptionally strong relationship between communal learning, collegiality, and collective action (key aspects of professional learning communities) and changes in teacher practice and increases in student learning. Gulamhussein, Allison (2013) Teaching Teachers: Effective Professional Development in an Era of High Stakes Accountability, Center for Public Education Illustrativemathematics.org Illustrative Mathematics is a discerning community of educators dedicated to the coherent learning of mathematics. We collaborate at illustrativemathematics.org, sharing carefully vetted resources for teachers and teacher leaders to give our children an understanding of mathematics and skill in using it. We provide expert guidance to states and districts working to improve mathematics education. Goals of Illustrative Mathematics To illustrate standards with impeccably crafted tasks, videos, lesson plans, and curriculum modules. To be the premier source of freely available online mathematics content for teachers, teacher leaders, assessment developers, curriculum writers, and teacher educators. To be a discerning professional community that creates content and deploys expertise in multiple ways. To engage individuals and small groups, with rare and needed expertise, who are currently isolated in pockets across the country. To provide a space for teachers to share across classrooms and help each other grow in teaching our children mathematics. The Structure of the Common Core The Structure of the Common Core Each domain has standards organized into a small number of clusters The Structure of the Common Core Illustrative Mathematics facilitates viewing standards across grades to see a progression of mathematical ideas The Structure of the Common Core The Structure of the Common Core Organized by conceptual category in high school The Standards for Mathematical Practice The Standards for Mathematical Practice Slab of Soap A slab of soap on one pan of a scale balances ¾ of a slab of soap and a ¾ pound weight on the other pan. How much does the full slab of soap weigh? Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Commenting on Tasks and Rating them +1 Task Talks Web Jams A CCSSM Task Writing Community Jammin’ on the gnarly standards, since 2012! Progressions Documents http://ime.math.arizona.edu/progressions/ Blog http://commoncoretools.me/ The Common Core State Standards in Mathematics K 1 2 3 4 5 Measurement and Data 6 7 Statistics and Probability Ratios and Proportional Relationships CC 8 9 10 11 Statistics and Probability F Functions Number and Operations Fractions Number and Operations in Base Ten The Number System Number and Quantity Algebra Operations and Algebraic Thinking 12 Expressions and Equations Geometry Geometry Modeling Develop understanding of fractions as numbers. 3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. 0 b=3 ⅓ 1 4/3 The Common Core State Standards in Mathematics K 1 2 3 4 5 Measurement and Data 6 7 Statistics and Probability Ratios and Proportional Relationships CC 8 9 10 11 Statistics and Probability F Functions Number and Operations Fractions Number and Operations in Base Ten The Number System Number and Quantity Algebra Operations and Algebraic Thinking 12 Expressions and Equations Geometry Geometry Modeling What fraction does this diagram represent? 2/6 ? 2/3 ? Based on Illustrative Mathematics task: http://www.illustrativemathematics.org/illustrations/833 2? What fraction does this diagram represent? 2/6 Based on Illustrative Mathematics task: http://www.illustrativemathematics.org/illustrations/833 What fraction does this diagram represent? 2/3 Based on Illustrative Mathematics task: http://www.illustrativemathematics.org/illustrations/833 What fraction does this diagram represent? 2/3 Based on Illustrative Mathematics task: http://www.illustrativemathematics.org/illustrations/833 What fraction does this diagram represent? 2 Based on Illustrative Mathematics task: http://www.illustrativemathematics.org/illustrations/833 What fraction does this diagram represent? 2 Based on Illustrative Mathematics task: http://www.illustrativemathematics.org/illustrations/833 Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 4.NF.3Understand a fraction a/b with a > 1 as a sum of fractions 1/b. a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. The Common Core State Standards in Mathematics K 1 2 3 4 5 Measurement and Data 6 7 Statistics and Probability Ratios and Proportional Relationships CC 8 9 10 11 Statistics and Probability F Functions Number and Operations Fractions Number and Operations in Base Ten The Number System Number and Quantity Algebra Operations and Algebraic Thinking 12 Expressions and Equations Geometry Geometry Modeling Add and subtract within 20. 1.OA.5 Relate counting to addition and subtraction (e.g., by counting on 2 to add 2). The Common Core State Standards in Mathematics K 1 2 3 4 5 Measurement and Data 6 7 Statistics and Probability Ratios and Proportional Relationships CC 8 9 10 11 Statistics and Probability F Functions Number and Operations Fractions Number and Operations in Base Ten The Number System Number and Quantity Algebra Operations and Algebraic Thinking 12 Expressions and Equations Geometry Geometry Modeling Relate addition and subtraction to length. 2.MD.6 Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram. The Common Core State Standards in Mathematics K 1 2 3 4 5 Measurement and Data 6 7 Statistics and Probability Ratios and Proportional Relationships CC 8 9 10 11 Statistics and Probability F Functions Number and Operations Fractions Number and Operations in Base Ten The Number System Number and Quantity Algebra Operations and Algebraic Thinking 12 Expressions and Equations Geometry Geometry Modeling What does understanding look like in Algebra? “There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from.” “The student who can explain the rule understands mathematics, and may have a better chance to succeed at a less familiar task such as expanding (a + b + c)(x + y)” From Common Core Standards: www.corestandards.org/the-standards/mathematics Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning MP 7: Look for and make use of structure Mathematically proficient students • look closely to discern a pattern or structure • step back for an overview and shift perspective • see complicated things as single objects, or as composed of several objects © Institute for Mathematics & Education 2011 The Common Core State Standards in Mathematics K 1 2 3 4 5 Measurement and Data 6 7 8 Statistics and Probability Ratios and Proportional Relationships CC 9 10 11 12 Statistics and Probability F Functions Number and Operations Fractions Number and Operations in Base Ten The Number System Number and Quantity Algebra Operations and Algebraic Thinking Expressions and Equations Geometry Geometry Modeling © Copyright 2011 Institute for Mathematics and Education The CCSSM: Pathway to Algebra K 1 2 3 4 5 6 7 8 Number and Operations Fractions 3 7 The Number System 7 9 3 6 7 5 8 4 9 5 3 12 3 2 Arithmetic with Polynomials and Rational Expressions Expressions and Equations 9 Operations and Algebraic Thinking 11 Seeing Structure in Expressions Number and Operations in Base Ten 6 10 Algebra 8 1 9 4 8 Creating Equations Reasoning with Equations and Inequalities © Copyright 2011 Institute for Mathematics and Education Algebra: Seeing Structure in Expressions • Interpret the structure of expressions • Write expressions in equivalent forms to solve problems What does it look like to “see structure in expressions”? Interpret the structure of expressions A-SSE.1 Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P. The Common Core State Standards in Mathematics K 1 2 3 4 5 Measurement and Data 6 7 Statistics and Probability Ratios and Proportional Relationships CC 8 9 10 11 Statistics and Probability F Functions Number and Operations Fractions Number and Operations in Base Ten The Number System Number and Quantity Algebra Operations and Algebraic Thinking 12 Expressions and Equations Geometry Geometry Modeling Delivery Trucks A company uses two different sized trucks to deliver sand. The first truck can transport x cubic yards, and the second y cubic yards. The first truck makes S trips to a job site, while the second makes T trips. What do the following expressions represent in practical terms? a. b. c. d. S+T x+y xS + yT xS+yT S+T If they do this in high school… what needs to come before that high school work? Apply and extend previous understandings of arithmetic to algebraic expressions. 6.EE.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. Use properties of operations to generate equivalent expressions. 7.EE.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.” The Common Core State Standards in Mathematics K 1 2 3 4 5 Measurement and Data 6 7 Statistics and Probability Ratios and Proportional Relationships CC 8 9 10 11 Statistics and Probability F Functions Number and Operations Fractions Number and Operations in Base Ten The Number System Number and Quantity Algebra Operations and Algebraic Thinking 12 Expressions and Equations Geometry Geometry Modeling Writing expressions in different forms for a purpose Students used the following expressions to describe the number of tiles needed to border a square pool with side length s feet. Demonstrate these expressions in the figure: s + s + s + s +4 4(s + 1) 2s + 2(s + 2) Writing expressions in different forms for a purpose 1 s 1 s + s + s + s +4 s+1 s s+1 s s 1 s+1 1 s+1 4(s + 1) 2s + 2(s + 2) Compute fluently with multi-digit numbers and find common factors and multiples. 6.NS.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2). The Common Core State Standards in Mathematics K 1 2 3 4 5 Measurement and Data 6 7 Statistics and Probability Ratios and Proportional Relationships CC 8 9 10 11 Statistics and Probability F Functions Number and Operations Fractions Number and Operations in Base Ten The Number System Number and Quantity Algebra Operations and Algebraic Thinking 12 Expressions and Equations Geometry Geometry Modeling Finding factors for a purpose Nina was finding multiples of 6. She said, 18 and 42 are both multiples of 6, and when I add them, I also get a multiple of 6: 18+42=60. Explain to Nina why adding two multiples of 6 will always result in another multiple of 6. Finding factors for a purpose Source: Primary Sources Report, Early Release, October 2013 Source: Primary Sources Report, Early Release, October 2013 Source: Primary Sources Report, Early Release, October 2013 Thank you! Enjoy the conference. [email protected]