1. Computational Fluency Flexible & Accessible Strategies for Multi-digit Addition and Subtraction Math AllianceMarch 30, 2010 Beth Schefelker and DeAnn Huinker

2. WALT We Are Learning To… Develop flexibility in using computational strategies for addition and subtraction. We will know we are successful when… Solve multi-digit addition and subtraction problems using strategies other than the traditional algorithm.

3. Base-ten Number System: Place Value  Learning about whole number computation must be closely linked to learning about the base-ten number system  The heart of this work is relating the written numeral to the quantity and to how that quantity is composed and can be decomposed. Teacher Note, Computational Fluency and Place Value, Investigations Grade K-5. TERC, 2007

4. The Hundreds Chart A Powerful Tool for Students

7. Computational Fluency  Flexibility  Comfortable with more than one approach.  Chooses strategy appropriate for the numbers.  Efficiency  Easily carries out the strategy, uses intermediate results.  Doesn’t get bogged down in too many steps or lose track of the logic of the strategy.  Accuracy  Can judge the reasonableness of results.  Has a clear way to record and keep track.  Concerned about double-checking results. Source: Russell, S.J. (2000). Developing computational fluency with whole numbers. Teaching Children Mathematics, 7, 154 - 158.

8. Addition Strategies 1. Add each place from left to right 2. Add on the other number in parts 3. Use a nice number and compensate 4. Change to an easier equivalent problem

9. Video Directions  Consider the numbers.  From the list of four strategies, select an appropriate strategy for the numbers.  Solve this problem using the strategy. 581 + 397 = ? Strategies Add each place from left to right Add on the other number in parts Use a nice number and compensate Change to an easier equivalent problem

10. Add Each Place from Left to Right

12. Change to an Easier Equivalent Problem

14. Add On the Other Number in Parts

16. Use a Nice Number & Compensate

18. “Start With” Tasks  Fold a piece of paper into four parts.  Take turns facilitating: Draw a card and present the task to the group.  Everyone solves the problem using the same “start.”  Facilitator leads a discussion comparing solution paths & naming of the strategy.

19. 438 + 295 = ?  Facilitator: Draw a card and present the “start” task to the group.  Everyone uses the same “start” to solve it.  Facilitator: How do our solution paths compare? What is the name of the strategy?

20. 438 + 295 = ?  Start with 400 + 200.  Start with 438 + 200.  Start with 295 + 400.  Start with 295 + 5.  Start with 438 + 300.  Start with 433 + 300.

21. Moving to Subtraction What are some of your thoughts on why subtraction is so much harder for students?

22. No pencils allowed. Solve the problem by reasoning in your mind. 35 – 19 = ? Then turn and share your reasoning.

23. Typical Instructional Sequence Model with Objects Jump to the Standard Algorithm ? What’s missing from this sequence?

24. Developmental Approach to Computation Direct Modeling  Model the situation or action step-by-step using objects or pictures and count, usually by ones, the objects. Counting Approach  Visualize the quantities to count-on, count-up-to, count-down, usually by ones; use fingers to keep track of the counts. Numerical Reasoning Strategy  Use number relationships to strategically work with quantities; break numbers apart and find easier ways to put them back together or to find differences.

25. 35 – 19 = ? Did you...  Use the standard algorithm and “borrow”  Begin with 35 – 20 = 15  Begin at 19, count up by ones or in parts  Begin by subtracting 30 – 10 and 5 – 9  Begin by subtracting 35 – 10  Begin by adding 1 to both numbers

26. Video Solve the problem in two different ways. 674 – 328 = ?

27. Subtraction Strategies 1. Subtract each place 2. Subtract the number in parts 3. Add up from the subtracted number 4. Use a nice number then compensate 5. Change to an easier equivalent problem

28. Add Up from the Subtracted Number

30. Subtract the Number in Parts

32. Use a Nice Number then Compensate

34. Change to Easier Equivalent Problem

35. Change to an Easier Equivalent Problem

36. Subtract Each Place

38. You Choose!  Fold your paper into four parts.  Given a problem, pick a strategy. Then solve it using that strategy.  WAIT.................  Pass your paper to another person.  That person studies the work and identifies which strategy was used.

39. Pick a Strategy. Solve it. 71 – 28 = ? Exchange papers. Study the work. Name the strategy. Strategies Subtract each place Subtract the number in parts Add up from the subtracted number Use a nice number then compensate Change to an easier equivalent problem

40. Pick a strategy. Solve it. 82 – 47 = ? Exchange papers. Study the work. Name the strategy. Strategies Subtract each place Subtract the number in parts Add up from the subtracted number Use a nice number then compensate Change to an easier equivalent problem

41. Pick a Strategy. Solve it. 754 – 273 = ? Exchange papers. Study the work. Name the strategy. Strategies Subtract each place Subtract the number in parts Add up from the subtracted number Use a nice number then compensate Change to an easier equivalent problem

42. Pick a problem. Pick a strategy. 576 – 239 = ? Strategies Subtract each place Subtract the number in parts Add up from the subtracted number Use a nice number then compensate Change to an easier equivalent problem Exchange papers. Study the work. Name the strategy.

43. Pick a Question to Discuss  What are some things students need to know in order to develop computational fluency for addition and subtraction?  In what ways might the use of varied strategies benefit and make computation more accessible for more students?

44. Closing Thought....  The most appropriate computation method can and should change flexibly as the numbers and the context change.  Traditional algorithms are “digit-oriented” and “rigid” and rely on memorizing rules without reasons, and can lead to common errors.  Alternative strategies are “number-oriented” and “flexible” and rely on making sense of working with numbers, and build confidence in students.

45. Homework  Read the article, “Subtraction Strategies from Children’s Thinking.” Reflect on how the information helped you think further about fluency with subtraction.  Practice the five subtraction strategies using either of these problems: 87 – 38 = ? or 574 – 289 = ?  Gather student reasoning on two strategies, either addition or subtraction. Show a work sample (e.g., from class, your own, or the article) to a student and see if he or she can make sense of it. Then give the student a problem to try using the same strategy. Repeat with another work sample. Summarize the student’s reactions and sense-making.