Pathways to Teacher Leadership in Mathematics Wednesday, July 2, 2014 CCSSM: Operations and Algebraic Thinking (OA) Progression Common Core State Standards for Mathematics Pathways to Teacher Leadership in Mathematics Wednesday, July 2, 2014 Session Description: Take a journey into the “Core” to inspect progressions of mathematical ideas and student learning, to surface shifts from current practice, and to consider implications for instruction, curriculum, and assessment.

Learning Intention & Success Criteria We are learning to: Understand three essential aspects of operations important to arithmetic and algebra. We will be successful: When we can identify the three essential aspects in work with whole numbers, fractions, and variable expressions.

OA Domain

Operations and Algebraic Thinking Dr. Jason Zimba Professor of Physics and Mathematics Bennington College, Vermont Lead Writer, Common Core Standards for Mathematics The Hunt Institute Video Series Common Core State Standards: A New Foundation for Student Success www.youtube.com/user/TheHuntInstitute#p

Meanings of the Operations Properties of the Operations Contextual Situations Which other ones are coming out? Plan: Geometry (K-HS)

Meanings of the Operations Properties of the Operations Contextual Situations Meanings of the Operations

What do students say? Addition Addition means plus. It means to put two things together and then add them like to see what the amount is at the end. Subtraction Subtraction means borrow. Take away. Take the number at the top and the number at the bottom and subtract how many the number is at the bottom. And then put the answer down.

What do students say? Multiplication Times. It means to take the number at the top and take it how many ever times that the bottom number is. Division It’s something like the easy way to subtraction. It’s to see how many numbers are in a number.

‘“Addition, subtraction, multiplication, and division have meanings, mathematical properties, and uses that transcend the particular sorts of objects that one is operating on, whether those be multi-digit numbers or fractions or variables or variables expressions.” --Jason Zimba In what ways do these three purposes in writing the standards, support our work in teaching with the Common Core?

Meanings for the Operations Each group selects one operation. Addition or Multiplication • Discuss and define using language that would be meaningful to your students. • Write your definition on chart paper and post.

Criteria: Definitions that Work Well Visualize actions on or relationships among quantities. Encompass many interpretations, uses, and situations (not limiting to just one view). Accurate in the long run (doesn’t set up misconceptions). Support seeing relationships among the operations.

Comment on aspects of the definitions that seem capable of serving students well across grades. Comment on aspects that might need further revision to avoid leading to misconceptions or limited views of operations and their uses.

Contextual Situations Properties of the Operations Contextual Situations Meanings of the Operations

Put together/Take apart Addition and Subtraction Situations Add to Take from Put together/Take apart Table 1, CCSS Appendix Compare

Multiplication and Division Situations Compare Arrays, Area Equal Groups Table 2, CCSS Appendix

In Grades K-8, how many standards reference “real-world contexts” or “word problems”? Grade K: OA Grade 1: OA Grade 2: OA, MD Grade 3: OA, MD Grade 4: OA, NF, MD Grade 5: NF, MD, G Grade 6: RP, EE, NS, G Grade 7: RP, EE, NS, G Grade 8: EE, G 54 standards Total of 229 standards, grades K-8…. so 25% 24% of K-8 standards

Properties of the Operations Contextual Situations Meanings of the Operations

Mental Math Solve in your head. No pencil or paper! 72 – 29 = ? 24 x 25 = ? Mental Math Solve in your head. No pencil or paper! 72 – 29 =

72 – 29 = ? 24 x 25 = ? Turn and share your reasoning. Discuss how you used: Composing and decomposing Place value in base ten Properties of the operations 72 – 29 =

24 x 25 = ? I would think what 25 x 25 is then subtract 25 or I would think what 20 x 20 is then add it to 5 x 4. I figured that there are 4 twenty-fives in 100, and there are 6 fours in 24, so 100 x 6 = 600. I thought 25 x 25 = 625 and then I subtracted 25. 625 – 25 = 600. I thought 24 x 100 = 2400, and 2400 ÷ 4 = 600.

24 x 25 = ? 25 x 4 = 100, 6 x 100 = 600, 600 + 100 = 700. Well, 10 x 25 = 250, 2(10 x 25) = 500, 500 x 4 = 2000. “I would try to multiply in my head, but I can't do that.”

The properties of operations. Associative property of addition (a + b) + c = a + (b + c) Commutative property of addition a + b = b + a Additive identity property of 0 a + 0 = 0 + a = a Existence of additive inverses For every a there exists –a so that a + (–a) = (–a) + a = 0 Associative property of multiplication (a × b) × c = a × (b × c) Commutative property of multiplication a × b = b × a Multiplicative identity property of 1 a × 1 = 1 × a = a Existence of multiplicative inverses For every a ≠ 0 there exists 1/a so that a × 1/a = 1/a × a = 1 Distributive property of multiplication over addition a × (b + c) = a × b + a × c Not just learning them, but learning to use them. 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, and the complex number system. •Also called “rules of arithmetic” , “number properties Nine properties are the most important preparation for algebra •Just nine: foundation for arithmetic •Exact same properties work for whole numbers, fractions, negative numbers, rational numbers, letters, expressions. •Same properties in 3rd grade and in calculus •Not just learning them, but learning to use them

In Grades K-8, how many standards reference “properties of the operations”? Grade 1: OA, NBT Grade 2: NBT Grade 3: OA, NBT Grade 4: NBT, NF Grade 5: NBT Grade 6: NS, EE Grade 7: NS, EE Grade 8: NS 28 standards 12% of K-8 standards

Standard 3.OA.5 Apply properties of operations as strategies to 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.) Footnote: 2 Students need not use formal terms for these properties.

In Grades K-8, how many standards reference using “strategies”? Grade K: CC Grade 1: OA, NBT Grade 2: OA, NBT Grade 3: OA, NBT Grade 4: NBT, NF Grade 5: NBT Grade 7: NS, EE 26 standards 11% of K-8 standards

CCSSM Glossary Computation strategy Purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at converting one problem into another.

Develop & use strategies Develop and use strategies for single-digit computation facts... before any expectation of knowing facts from memory. Develop and use strategies to add, subtract, multiply, and divide multi-digit whole numbers, fractions, decimals…. before use of standard algorithms. As Hank noted, “CCSSM assumes about three years of development of concepts and strategies before demonstrating fluency.

Homework

Readings Due Monday, July 7, 2014 • Carpenter: Chapters 4 & 6 • Revisit: PtA: Representations p. 24-29. • Revisit OA Progressions, pp. 3-20, 36-38. Appendix. • EE Progressions, p. 6-7. ------------------------- • Thornton (1978). Thinking strategies for basic facts. • PtA: Fluency p. 42-48. • Russell (2000). Computational fluency.

Homework Email by Saturday night: • One key idea related to “fluency” and one question or wondering about developing fluency with your students. • One key message from the Thornton article on basic facts.

Course Assignment: Sequence of Equations Email by Sunday night: Sequence of T/F or Open Number Sentences and Rationale (5% of grade) Equations Rationale z

Disclaimer Pathways to Teacher Leadership in Mathematics Project University of Wisconsin-Milwaukee, 2014-2017   This material was developed for the Pathways to Teacher Leadership in Mathematics project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Any other use of this work—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors—without prior written permission is prohibited. This project was supported through a grant from the Wisconsin ESEA Title II Improving Teacher Quality Program.