1. 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.

2. 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.

3. OA Domain

4. 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

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

6. Meanings of the Operations
Properties of the Operations
Contextual
Situations
Meanings of the Operations

7. 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.

8. 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.

9. ‘“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?

10. 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.

11. 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.

12. 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.

13. Contextual Situations
Properties of the Operations
Contextual
Situations
Meanings of the Operations

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

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

16. 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

17. Properties of the Operations
Contextual
Situations
Meanings of the Operations

18. 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 =

19. 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 =

20. 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 = 600.
I thought 24 x 100 = 2400, and 2400 ÷ 4 = 600.

21. 24 x 25 = ?
25 x 4 = 100, 6 x 100 = 600, = 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.”

22. 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

23. 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

24. 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 = (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) = = 56. (Distributive property.)
Footnote: 2 Students need not use formal terms for these properties.

25. 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

26. 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.

27. 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.

28. Homework

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

30. Homework
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.

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

32. Disclaimer Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee,
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.