1. Math Games to Build Skills and Thinking Claran Einfeldt, claran@cmath2.com Cathy Carter, cathy@cmath2.com http://www.cmath2.com

2. What is “Computational Fluency”? “connection between conceptual understanding and computational proficiency” (NCTM 2000, p. 35)

3. Conceptual Computational Understanding Proficiency Place value Operational properties Number relationships Accurate, efficient, flexible use of computation for multiple purposes

4. Computation Algorithms: Seeing the Math

5. Computation Algorithms in Instead of learning a prescribed (and limited) set of algorithms, we should encourage students to be flexible in their thinking about numbers and arithmetic. Students begin to realize that problems can be solved in more than one way. They also improve their understanding of place value and sharpen their estimation and mental- computation skills.

6. Before selecting an algorithm, consider how you would solve the following problem. 48 + 799 We are trying to develop flexible thinkers who recognize that this problem can be readily computed in their heads! One way to approach it is to notice that 48 can be renamed as 1 + 47 and then What was your thinking? 48 + 799 = 47 + 1 + 799 = 47 + 800 = 847

7. Important Qualities of Algorithms Accuracy  Does it always lead to a right answer if you do it right? Generality  For what kinds of numbers does this work? (The larger the set of numbers the better.) Efficiency  How quick is it? Do students persist? Ease of correct use  Does it minimize errors? Transparency (versus opacity)  Can you SEE the mathematical ideas behind the algorithm? Hyman Bass. “Computational Fluency, Algorithms, and Mathematical Proficiency: One Mathematician’s Perspective.” Teaching Children Mathematics. February, 2003.

8. Table of Contents Partial Sums Partial Products Partial Differences Partial Quotients Lattice Multiplication Click on the algorithm you’d like to see! Trade First

9. 735 + 246 900 Add the hundreds (700 + 200) Add the tens (30 + 40) 70 Add the ones (5 + 6) Add the partial sums (900 + 70 + 11) +11 981 Click to proceed at your own speed!

10. 356 + 247 500 Add the hundreds (300 + 200) 90 Add the tens (50 + 40) Add the ones (6 + 7) Add the partial sums (500 + 90 + 13) +13 603

11. 429 + 989 1300 100 + 18 1418 Click here to go back to the menu.

12. 56 × 82 4,000 480 100 12 + 4,592 80 X 50 80 X 6 2 X 50 2 X 6 Add the partial products Click to proceed at your own speed!

13. 52 ×76 3,500 140 300 12 + 70 X 50 70 X 2 6 X 50 6 X 2 3,952 Add the partial products

14. 502 40 6 200080 12 300 52 × 46 2,000 300 80 12 2,392 Click here to go back to the menu. A Geometrical Representation of Partial Products (Area Model)

15. 12 7 2 37 2 3 4 5 94 5 9 6 11 2 13 6 4 Students complete all regrouping before doing the subtraction. This can be done from left to right. In this case, we need to regroup a 100 into 10 tens. The 7 hundreds is now 6 hundreds and the 2 tens is now 12 tens. Next, we need to regroup a 10 into 10 ones. The 12 tens is now 11 tens and the 3 ones is now 13 ones. Now, we complete the subtraction. We have 6 hundreds minus 4 hundreds, 11 tens minus 5 tens, and 13 ones minus 9 ones.

16. Click here to go back to the menu. 10 8 0 28 0 2 2 7 42 7 4 7 9 5 12 2 8 14 9 4 69 4 6 5 6 85 6 8 8 13 3 16 7 8

17. Subtract the hundreds (700 – 200) Subtract the tens (30 – 40) Subtract the ones (6 – 5) Add the partial differences (500 + (-10) + 1) 5 0 05 0 0 – 2 4 5– 2 4 5 1 4 9 14 9 1 1 0 7 3 6

18. Subtract the hundreds (400 – 300) Subtract the tens (10 – 30) Subtract the ones (2 – 5) Add the partial differences (100 + (-20) + (-3)) 1 0 01 0 0 – 3 3 5– 3 3 5 7 7 2 0 4 1 2 3 Click here to go back to the menu.

19. 4 1 1 1 1 0 5 1 9 R3 1 2 01 2 0 6 0 2 3 12 3 1 1 2 Click to proceed at your own speed! 5 1 4 8 3 1 9 Students begin by choosing partial quotients that they recognize! Add the partial quotients, and record the quotient along with the remainder. I know 10 x 12 will work…

20. Click here to go back to the menu. 1 01 0 1 1 2 61 1 2 6 5 05 0 2 52 5 8 5 R6 8 0 08 0 0 2 7 2 62 7 2 6 3 2 3 2 63 2 6 3 2 03 2 0 68 58 5 Compare the partial quotients used here to the ones that you chose! 1 6 0 0 1 6 0 0

21. 53 7 2 2 1 1 0 0 6 8 1 6 53 × 72 3500 100 210 6 3816 + 3 5 3 Compare to partial products! 3 × 7 3 × 2 5 × 7 5 × 2 Add the numbers on the diagonals. Click to proceed at your own speed!

22. 16 2 3 1 2 0 3 1 8 3 6 8 16 × 23 200 30 120 18 368 + 0 2 Click here to go back to the menu.

23. Algorithms “If children understand the mathematics behind the problem, they may very well be able to come up with a unique working algorithm that proves they “get it.” Helping children become comfortable with algorithmic and procedural thinking is essential to their growth and development in mathematics and as everyday problem solvers... Extensive research shows the main problem with teaching standard algorithms too early is that children then use the algorithms as substitutes for thinking and common sense.”

24. Importance of Games

25. Provides......regular experience with meaningful procedures so students develop and draw on mathematical understanding even as they cultivate computational proficiency. Balance and connection of understanding and proficiency are essential, particularly for computation to be useful in “comprehending” problem-solving situations.

26. Benefits Should be central part of mathematics curriculum Engaging opportunities for practice Encourages strategic mathematical thinking Encourages efficiency in computation Develops familiarity with number system and compatible numbers (landmark) Provides home school connection

27. Where’s the Math? What mathematical ideas or understanding does this game promote? What mathematics is involved in effective strategies for playing this game? What numerical understanding is involved in scoring this game? How much of the game is luck or mathematical skill?

28. Games Require Reflection Games need to be seen as a learning experience

29. Where’s the Math? What is the goal of the game? Post this for students. Ask mathematical questions and have students write responses. Model the game first, along with mathematical thinking Encourage cooperation, not competition Share the game and mathematical goals with parents

30. Extensions Have students create rules or different versions of the games Require students to test out the games, explain and justify revisions based on fairness, mathematical reasoning

31. Games websites www.mathwire.com http://childparenting.about.com/od/makeathome mathgames/ http://www.netrover.com/~kingskid/Math/math.ht m http://www.multiplication.com/classroom_games. htm http://www.awesomelibrary.org/Classroom/Math ematics/Mathematics.html http://www.primarygames.co.uk/ http://www.pbs.org/teachers/math/