Using the Relationship Between Addition and Subtraction to Build Fluency

1. Using the Relationship Between Addition and Subtraction to Build Fluency Common Core Leadership in Mathematics (CCLM)Thursday June 28, 2012 This material was developed for use by participants in the Common Core Leadership in Mathematics (CCLM^2) project through the University of Wisconsin-Milwaukee. Use by school district personnel to support learning of its teachers and staff is permitted provided appropriate acknowledgement of its source. Use by others is prohibited except by prior written permission.

2. Our Motto “Learning is a messy business, and constructing understanding is hard work.” -Fosnot & Dolk, Young Mathematicians At Work p.38.

3. Learning Intention and Success Criteria We are learning to… • Recognize the role unknown addend situations and equations play in developing fluency with single-digit problems. • Understand “decompose a number leading to a ten” and “think addition to subtract” and explore how to build that understanding in students. We will be successful when we... • Understand how to use addition and subtraction word problems to support students’ ability to reason fluently.

4. Fluency

5. “Fluency” in the CCSSM Revisit the discussion of fluencypp.18-19 OA Progression document (Starts with the last paragraph.) Table groups: Discuss and chart a group response to your designated prompt.

6. Fluency • Summarize what it means to be fluent and what kinds of thinking fluency involves for single digit addition and subtraction. • What are the CCSSM expectations for fluency in your assigned grade and what are some examples of what students will be expected to demonstrate? • How does this work on fluency link to the Standards for Mathematical Practice?

7. Decompose a number leading to a ten

8. Thinking about instruction… By the end of the K-2 grade span, students have sufficient experience with addition and subtraction to know single-digit sums from memory . . . this is not a matter of instilling facts divorced from their meanings, but rather as an outcome of a multi-year process that heavily involves the interplay of practice and reasoning. p. 19 OA Progressions document How might this statement impact classroom instruction around “single-digit addition and subtraction” at all levels?

9. Progression toward Fluency – Subtraction Methods for solving single-digit subtraction Level 1: Direct Modeling or Taking Away Count, Count, Count Level 2: Counting on Think addition or find unknown addend Level 3: Convert to an Easier Problem Decompose a number leading to a 10

10. Building Understanding Through Context How would a student at each level approach this problem? Note: For Level 3, be sure to examine both “Build up through ten” & “Back down through ten” There were 13 cookies on the plate. Sam ate 8 of the cookies on the plate. How many cookies are on the plate now?

11. Decompose to ten: 15 – 6 • Place 15 counters on the double ten frame. • Completely fill one frame, place 5 on the other frame.

12. Decompose to ten: 15 – 6 How can you remove 6 counters in parts by decomposing it in a way that gets you to or “leads to a ten”? • Tell a story for 15 – 6.

13. Decompose to ten: 15 – 6 • Place 15 counters on the double ten frame. • Completely fill one frame, place 5 on the other frame.

14. Decompose to ten: 15 – 6 • Decompose 6 to remove it in parts. • Remove 5 counters to get to ten.

15. Decompose to ten: 15 – 6 • 6 = 5 + 1 • Remove 5 counters to get to ten. • Remove 1 more.

16. Decompose to ten: 15 – 6 • Decomposed 6 to remove 5 and then remove 1. • Write an equation(s). • 15 – 5 – 1 = 9 or • 15 – 5 = 1010 – 1 = 9

17. Try it: 13 – 5 16 – 7 • Use ten frames and counters to reason through the “Decompose to Ten” strategy. • Write an equation(s) to show the reasoning. • Share and discuss in your small group. • Brainstorm: What other subtraction facts would lend themselves well to this strategy? Make a list of facts and try them out.

18. Unknown Addend Problems

19. Unknown Addend Problems Read K.OA.2, K.OA.4, 1.OA.1, 1.OA.4, 2.OA.1 • Highlight essential ideas in each. • Discuss the progression from Kindergarten to grade 2 in developing the idea of an unknown addend.

20. Food for thought…. Learning to think of and solve subtractions as unknown addend problems makes subtraction as easy as addition (or even easier), and it emphasizes the relationship between addition and subtraction. • K-5 OA Progressions document, p. 15

21. Problem Situations Scan Table 1 revised“Addition & Subtraction Problem Situations.” • Which are the easier problem situations to solve? Why? • Which problem situations prompt Level 2 and Level 3 reasoning?

22. Addition and Subtraction Problem Situations

23. Pathways… Show how these standards are connected and build across grades by drawing lines and arrows. K.OA: 2, 3 & 4 1.OA: 1, 3, 4 & 6 2.OA: 1 & 2 Graphically demonstrate these pathways on chart paper.

24. Revisit the Practices • Divide your slate in half. • On the left side, choose a SMP that links to the work on Unknown Addend Problems. • On the right side, cite a specific example that illustrates the chosen practice.

25. Learning Intention and Success Criteria We are learning to… • Recognize the role unknown addend situations and equations play in developing fluency with single-digit problems. • Understand “decompose a number leading to a ten” and “think addition to subtract” and explore how to build that understanding in students. We will be successful when we... • Understand how to use addition and subtraction word problems to support students’ ability to reason fluently.