The National Council of Supervisors of Mathematics

1. The National Council of Supervisors of Mathematics The Common Core State Standards Illustrating the Standards for Mathematical Practice: Look for and express regularity in repeated reasoning Construct viable arguments www.mathedleadership.org

2. Module Evaluation Facilitator: At the end of this Powerpoint, you will find a link to an anonymous brief e-survey that will help us understand how the module is being used and how well it worked in your setting. We hope you will help us grow and improve our NCSM resources!

3. Mathematics Standards for Content Standards for Practice Common Core State Standards

4. To explore the mathematical standards for Content and Practice To consider how the Common Core State Standards (CCSS) are likely to impact your mathematics program and plan next steps In particular, participants will Examine opportunities to help students express regularity and repeated reasoning and construct viable arguments (MP3, MP8) Today’s Goals

5. Standards for Mathematical Practice “The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important ‘processes and proficiencies’ with longstanding importance in mathematics education.”(CCSS, 2010)

6. Content Goals These standards of practice will be examined in the context of the following content standard: Understand properties of the operations. Grade 1: p. 14 Grade 2: p. 18 Grade 3: p. 22 Grade 4: p. 28 Grade 5: p. 33 (paragraph #2)

7. What generalization is suggested by these problems?

8. Video 1 – Will it always work? Ms. Kaye’s 3rd Grade CTB/McGraw-Hill; Mathematics Assessment Resource Services, 2003

9. Video 2 – Will it always work?

10. Video 3 – Will it always work?

11. Adding 1 to a Factor Writing prompt: In a multiplication problem, if you add 1 to a factor, I think this will happen to the product…

12. Students’ Articulation of the Claim “The number that is not increased is the number that the answer goes up by.” “The number that is staying and not going up, increases by however many it is.” “I think that the factor you increase, it goes up by the other factor.”

13. Choose which of the original equations you want to work with. Then do one of these… • Draw a picture for the original equation; then change it just enough to match the new equations. • Make an array for the original equation; then change it just enough to match the new equations. • Write a story for the original equation; then change it just enough to match the new equations. Example: Original equation 7 x 5 = 35 New equations 7 x 6 = 42 8 x 5 = 40

14. Frannie’s Story Context There are 7 jewelry boxes and each box has 5 pieces of jewelry. There are 35 pieces of jewelry altogether.

15. Jewelry Boxes 7 x 5 Seven boxes with five pieces of jewelry in each box 35 pieces of jewelry

16. Jewelry Boxes 7 x 5  8 x 5 Eight Seven boxes with five pieces of jewelry in each box 35 pieces of jewelry + 5 pieces of jewelry 40 pieces of jewelry

17. Jewelry Boxes 7 x 5 Seven boxes with five pieces of jewelry in each box 35 pieces of jewelry

18. Jewelry Boxes 7 x 5  7 x6 six Seven boxes with five pieces of jewelry in each box 35 pieces of jewelry + 7 pieces of jewelry 42 pieces of jewelry

19. Other Stories Baskets of bouncy balls Tanks with salmon eggs Baskets of mozzarella sticks Rows of chairs

20. 9 x 4 = 36 9 x 5 = 45 10 x 4 = 40

21. Making Sense of Multiplication Explain how the array changes from 7 x 5 to 8 x 5 and from 7 x 5 to 7 x 6.

22. Video 4

23. Video 5

24. Importance of Contrasting Operations • We were talking about the addends changing by 1 and what happens to the sum. • Now we’re talking about the factors changing by 1 and what happens to the product.

25. Pause and Reflect • How did the mathematical practices play out in these sessions? • What properties of addition and multiplication were under consideration? • How did the practices of noticing regularities and constructing arguments support learning about these properties and about the meaning of the operations?

26. Associative Property of Addition a + (b + c) = (a + b) + c a + (b + 1) = (a + b) + 1 5 + (7 + 1) = (5 + 7) + 1

27. Distributive Property of Multiplication over Addition a (b + c) = ab + ac a (b + 1) = ab + a 5 (7 + 1) = 35 + 5 (a + b) c = ac + bc (a + 1) c = ac + c (5 + 1) 7 = 35 + 7

28. Importance of Contrasting Operations • Commutative property of addition: a + b = b + a • What about subtraction? Is a – b equal to b – a?

29. Video 6

30. 2nd grade • How were the students engaged in the 8thpractice: noticing and expressing regularity? • How were they engaged in the 3rdpractice, creating viable arguments? • How did the practices of noticing regularities and constructing arguments support learning about properties and about the meaning of the operations?

31. Today’s Goals • To explore the mathematical standards for Content and Practice • To consider how the Common Core State Standards (CCSS) are likely to impact your mathematics program and plan next steps In particular, participants will • Examine opportunities to help students express regularity and repeated reasoning and construct viable arguments (MP3, MP8)

32. Reflection on Standards for Mathematical Practice • Individually review the Standards for Mathematical Practice. • Choose a partner at your table and discuss a new insight you had into the Standards for Mathematical Practice. • Then discuss the following question. What implications might the Standards for Mathematical Practice have on your classroom?

33. End of Day Reflections • Are there any aspects of your own thinking and/or practice that our work today has caused you to consider or reconsider? Explain. • 2. Are there any aspects of your students’ mathematical learning that our work today has caused you to consider or reconsider? Explain.

34. This module was developed by Deborah Schifter and Susan Jo Russell, based on work from the projects Teaching to the Big Ideas, Foundations of Algebra in the Elementary and Middle Grades, and Using Routines as an Instructional Tool for Developing Students’ Conceptions of Proof. Video clips are used with permission from these projects. Acknowledgements • This work was funded in part by the National Science Foundation through grants ESI-0242609 to EDC and ESI-0550176 and DRL-1019482 to TERC. Any opinions, findings, conclusions, or recommendations expressed here are those of the authors and do not necessarily reflect the views of the National Science Foundation.

35. Join us in thanking theNoyce Foundationfor their generous grant to NCSM that made this series possible! http://www.noycefdn.org/

36. NCSM Series Contributors • Geraldine Devine, Oakland Schools, Waterford, MI • Aimee L. Evans, Arch Ford ESC, Plumerville, AR • David Foster, Silicon Valley Mathematics Initiative, San José State University, San José, California • Dana L. Gosen, Ph.D., Oakland Schools, Waterford, MI • Linda K. Griffith, Ph.D., University of Central Arkansas • Cynthia A. Miller, Ph.D., Arkansas State University • Valerie L. Mills, Oakland Schools, Waterford, MI • Susan Jo Russell, Ed.D., TERC, Cambridge, MA • Deborah Schifter, Ph.D., Education Development Center, Waltham, MA • Nanette Seago, WestEd, San Francisco, California • Hope Bjerke, Editing Consultant, Redding, CA

37. Help Us Grow! The link below will connect you to a anonymous brief e-survey that will help us understand how the module is being used and how well it worked in your setting. Please help us improve the module by completing a short ten question survey at: http://tinyurl.com/samplesurvey1