1. Welcome Academic Math Coaches! Let’s Mix It Up! Find a seat at a table. Use the dot on your nametag to make sure each color is represented. Green 1-2 years Blue 3-4 years Yellow 5-6 years Red 7 +

2. Building a Community of Coaches Share one experience you’ve had in September that has impacted your work as a coach. As a table group finish this statement… Being a math coach is like Autumn because…

3. Making Connections across the Common Core Domains Beth Schefelker Bridget Schock Connie Laughlin Hank Kepner Kevin McLeod October 5, 2012

4. Content Goals 2012-2013 Explore how the number system expands across the grades and how operations are extended in such a way that the properties of operations remain unchanged. Analyze and solve mathematical modeling situations and/or problems using single and multi-step word problem structures. Extend understanding of whole number operations to fractions, ratios and proportions. Focus on Math Practice Standards  7 Look for and make use of structure.  8 Look for and express regularity in repeated reasoning.

5. Academic Coach Math Meeting Structure In order to reach the K-12 spectrum:  First meeting will develop foundational understandings of a mathematical idea.  Second meeting will extend those mathematical understandings across grades and domains.

7. Put your pencils down, it’s time for some mental math! 48 + 23 93 – 38 Turn and talk. Share your strategy with a partner! Mental Math

8. Learning Intention & Success Criteria Learning Intention We are learning to …  Understand properties of addition and subtraction to simplify and solve problems.  Understand subtraction as a missing addend. Success Criteria We will be successful when we can…  Apply strategies to reason through addition and subtraction problems.  Justify how properties support the strategies for addition and subtraction.

9. The Importance of Properties

10. Addition and Subtraction “The mathematical foundations for understanding computational procedures for addition and subtraction of whole numbers are the properties of addition and place value.” Developing Essential Understanding of Addition & Subtraction Pre-K – Grade 2, p. 28

11. The properties of operations. Associative property of addition(a + b) + c = a + (b + c) Commutative property of additiona + b = b + a Additive identity property of 0a + 0 = 0 + a = a Existence of additive inversesFor 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 1a × 1 = 1 × a = a Existence of multiplicative inversesFor 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.

12. And in the domain of Operations and Algebraic Thinking, it is those meanings, properties, and uses which are the focus… and it is those meanings, properties, and uses that will remain when students begin doing algebra in middle grades [and beyond]. --Jason Zimba And in the domain of Operations and Algebraic Thinking, it is those meanings, properties, and uses which are the focus… and it is those meanings, properties, and uses that will remain when students begin doing algebra in middle grades [and beyond]. --Jason Zimba

13. In Grades K-8, how many standards reference “properties of the operations”? 28 standards 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 12% of K-8 standards

14. Using properties of operations 1OA3. Apply properties of operations as strategies to add and subtract. 3OA5. Apply properties of operations as strategies to multiply and divide. 4NBT5. Multiply two two-digit numbers using strategies based on place value and the properties of operations. 5NBT6. Find whole-number quotients and remainders with … using strategies based on place value, properties of operations …. 5NBT7. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations….

15. 6EE3. Apply the properties of operations to generate equivalent expressions. 7NS2c: Apply properties of operations as strategies to multiply and divide rational numbers. 7EE1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. and into high school……

16. Develop and use strategies based on properties of the operations

17. Addition Strategies and Properties Whole Numbers

18. CCSSM Connection Understand and apply properties of operations and the relationship between addition and subtraction. 1.OA.3. Apply properties of operations as strategies to add and subtract. 2 Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.) 1.OA.4. Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8. Add and subtract within 20. Math Practice Standards 7 Look for and make use of structure 8 Look for and express regularity and repeated reasoning

19. What’s the relationship between these two number sentences? 4 + 4 = 8 4 + 5 = 9

20. Case 7: Adding 1 to an Addend Combine tables to form groups of 10. A. Role players  Person 1: Teacher  Person 2: Ester, Connie, Tia  Person 3: Joe  Person 4: Terry, Giovanni, Sema  Use cubes to model out the actions of the students in the case study. B. Observers  Attend to the conversations of student thinking

21. Case 7: Adding 1 to an Addend Re-read the case. As you read, consider the following… What mathematical ideas are the students developing? How are students making sense of the mathematics?

22. So what’s happening? We know: 21 + 23 = 44 What about 21 + 24? 21 + 24 = 21 + (23 + 1) = (21 + 23) + 1 = 44 + 1 = 45

23. Articulating a Generalization for Addition If 21 + 23 is 44, then 21 + 24 is 45. If you add 1 to one of the numbers (addends), then you add 1 to the sum. If you add any number to one of the addends, then you add that same number to the sum. Does it work for other numbers?

24. Moving student thinking… How would a student solve19+7? How would you help a student use properties to make sense of 19 + 7?  Where would you start?  What questions would you ask of your students?

25. Let’s Talk About Questions Opportunity for assessment to focus on questioning….

26. Subtraction Strategies and Properties Whole Numbers

27. Number Talks Case 3.3 As you watch the DVD… Look for growth and clarification of students’ understanding Teacher moves

28. 70-59

29. What strategies did you see students using? Why do you plus “it” on again? Students were unsure whether 1 should be added or subtracted. How could you help students understand this thinking? Where are the properties?

30. Student Generalizations What properties underlie computational strategies for addition and subtraction? How do we draw on models for representing addition and subtraction, such as visual images, story contexts, and number lines, to express and justify generalizations?

31. 1.OA.4 Understand subtraction as an unknown- addend problem. How does the conversation in the DVD support the use of thinking about subtraction as a missing addend and using an open number line to justify thinking strategies?

32. Math Practice Standards As you think about what we did today, where do you see these Math Practice Standards?  MP7 Look for and make use of structure  MP8 Look for and express regularity in repeated reasoning

33. Learning Intention & Success Criteria Learning Intention We are learning to …  Understand properties of addition and subtraction to simplify and solve problems.  Understand subtraction as a missing addend. Success Criteria We will be successful when we can…  Apply strategies to reason through addition and subtraction problems.  Justify how properties support the strategies for addition and subtraction.

34. Walk Away What has to happen in the classroom for students to develop an understanding of properties in order to apply them to strategies? What conversations will you have with teachers as they develop student reasoning with number strategies?

35. Professional Practice Find at least two ways properties can help you find solutions to each of the following:  347 + 454  135 – 97 Practice with students  During the next 2 weeks, work with a small group of students on one of the two problems  Chart/Scribe using these representations to capture student thinking, similar to the teacher in the video: Number line Associating decomposed and recomposed numbers

36. Math Practice Standards Read Think Math article explaining  MP7 Look for and make use of structure  MP8 Look for and express regularity in repeated reasoning Highlight important ideas in each explanation of the Practice. Create a poster for the two standards