1. 2010-2011 Workshop Series for YISD Teachers of 6 th Grade Math 2010-2011 Workshop Series for YISD Teachers of 6 th Grade Math Conceptual Understanding & Mathematical Thinking Workshop 3 (Jan 31) Kien Lim Kien Lim Dept. of Mathematical Sciences, UTEP

2. Goals: Strengthen our mathematical knowledge for teaching signed numbers and operations involving them Question for Thought How can we help our students understand deeply, so much so that they can explain clearly, why the product of two negative numbers is positive?

3. Goals: Revisit the idea about the necessity principle Explore how to make learning interesting and fun for students Strengthen our mathematical knowledge for teaching signed numbers and operations involving them Raise your hands if you use games in your classrooms at least 3 times a year.

4. Activities for Today’s Workshop The Zero-Sum Game Discuss the underlying math concepts Discuss pedagogical issues Design a sequence of activities for use in your classrooms

5. The Zero-Sum Game  Number of Groups: 4  Goal: Increase assets and decrease debts  Materials: o 15 debt cards and 15 asset cards (ranging from $1 to $15) o Recording sheets (one per group) o A small board to write and show your group’s answer o One pot (for collecting cards) o An elmo and projector to reveal the cards in the pot  Number of Rounds: 5 o Round 0: Each group randomly picks 3 debt and 3 asset cards. o Rounds 1-4: Put 1 debt and 1 asset into the pot; and then the groups will take turn to pick 1 debt and 1 asset.

6. The Zero-Sum Game  What happens in each round? o A question will be posed. The question in each round is different. o Each group will write their answer on a board and all group members stand-up to signify completion. o The first group that has the correct or closest answer will get to pick cards first, followed by the second group, and so forth. o Speed and accuracy counts! o At the end of each round, record the transactions, update the net worth of each group, and get ready for the next round.

7. The Zero-Sum Game  Instructions for Recording Transactions o Each group will receive a sheet to record what cards are received and what cards are removed o Each group is advised to record their own transactions as well as other groups’ transactions

8. The Zero-Sum Game  Instructions for Recording Transactions o Each group will receive a sheet to record what cards are received and what cards are removed o Each group is advised to record their own transactions as well as other groups’ transactions o It is advantageous to update your records of every group’s net worth after each round Are You Ready?

9.  Randomly pick your 3 asset and 3 debt cards (no cheating)  What cards each group received are public information (each group must honestly report what cards your group received) Round O

10.  Each group puts a debt card and an asset card into the pot (face down; you don’t need to tell others what you put into the pot)  Get ready to write your answer Round 1 Information: Suppose the referee receives all the remaining cards. Question: How much do you predict the referee’s net worth is? Are You Ready?

11.  Let’s see which group is the first to give the correct (or closest answer).  The first group with the correct/closest answer will get to pick a debt card and an asset card from the pot. Announce the cards you pick.  The second, third, and fourth group will pick their cards and announce the cards they pick.  Each group will have 5 minutes to update their recording sheet. Results for Round 1

12. What Can We Learn from Round 1 Strategy-sharing: How did you make your prediction? Why? Note the name of the game!

13.  Did any group write the word “Debt” or “Asset” on the recording sheet?  What are some efficient ways to symbolize Debt and Asset? Let’s be more efficient in recording Are You Ready for Round 2?

14.  Each group puts a debt card and an asset card into the pot, and get ready to for the question Round 2 Are You Ready? Action: The referee turns over all the asset cards in the pot to reveal their value. Question: What is the value of all the debt cards in the pot?

15.  Each group will take turn to pick a debt card and an asset card.  Update your recording sheet. Results for Round 2 What strategy did you use? Did it work? Why, or why not?

16.  Each group puts a debt card and an asset card into the pot, and get ready to for the question Round 3 Are You Ready? Action: The referee turns all cards in the pot. Question: Excluding the debt card with the greatest number, what is the total value in the pot?

17.  Each group will take turn to pick a debt card and an asset card.  Update your recording sheet. Results for Round 3 What strategy did you use? Did it work? Why, or why not?

18.  Each group puts a debt card and an asset card into the pot, and get ready to for the question Round 4 Action: The referee will pose a math-like question using 3 of debt-asset cards and 3 operator cards. (Here is an example)

19. Debt $ 2.00 Asset $ 1.00 Subtract Asset $ 3.00 Add Twice Question: What is the net result? Answer: A debt of $2 Add

20. Action: The referee will pose a math-like question using 3 of debt-asset cards and 3 operator cards. (Here is an example) Question: What is the net result? Debt $ 1.50 Asset $ 2.00 Debt $ 2.50 Add Subtract Add Twice Are You Ready?

21.  Each group will take turn to pick a debt card and an asset card.  Update your recording sheet. Results for Round 4 What strategy did you use? Did it work? Why, or why not?

22. How can we represent these transactions mathematically? Debt $ 1.25 Add Asset $ 0.25 Subtract Asset $ 0.75 Add Twice +– + + (-1.25)(+0.25) (+0.75) + (-1.25) – (+0.25) + 2 (0.75)

23. 1.What mathematical concepts can students learn from playing this game? 2.What challenges do you foresee if you were to use Game 1 in your classroom? 3.How will you address those challenges? (i.e., how will you change the same to suit your classroom) Follow-up Group Discussion

24. Somebody spilled coffee on Fantasia’s net worth statement. She is trying to figure out what transaction took place to give her a net worth of $12,000. List two possible transactions. Fantasia’s Net Worth Statement Net Worth: $10,000 Transaction: Net Worth: $12,000 A Follow Question

25. Possible Activities from an Article Stephan, M. L. (2009). What are you worth? Mathematics Teaching in the Middle School, 15(1), NCTM.

26. Who is worth more, Brad or Angelina? Justify your answer.

27. Which bachelor is financially most favorable?

28. Determine whether each transaction increases or decreases one’s net worth.

29. The Necessity Principle “For students to learn what we intend to teach them, they must have a need for it, where by ‘need’ is meant intellectual need, not social or economic need.” (Harel, 2007) Does this game provoke an intellectual need for students to learn a certain concept? If so, how?

30. The Key Idea Debt $ 2 Subtract  Does adding an asset a good thing?  Does subtracting an asset a good thing?  Does adding a debt a good thing?  Does subtracting a debt a good thing? Asset $ 2 Add = – (-2) = + (+2)

31. The Product of Two Negative Numbers is Positive. Why? For example, why (-3) x (-4) = 12 Think in terms of multiplier and multiplicand. 3 x 4 means adding 3 times of 4 i.e., 3 x 4 = + 4 + 4 + 4 = 12 to 0 3 x -4 means adding 3 times of -4 i.e., 3 x -4 = + (-4) + (-4) + (-4) = -12 0 to 0 0 -3 x 4means subtracting 3 times of 4 i.e., -3 x 4 = – 4 – 4 – 4 = -12 from 0 0 -3 x -4 means subtracting 3 times of -4 i.e., -3 x -4 = – (-4) – (-4) – (-4) from 0 0 = 0 + 4 + 4 + 4

32. The context of debts and assets help students understand Recapitulation the difference between multiplier and multiplicand the difference between a negative operation (subtraction) and a negative value (negative integer) why the value of subtracting a negative number is essentially adding its absolute value (i.e., removing a debt of $2 has the same effect on net worth as adding an asset of $2) why -3 times -4 can be interpreted as repeatedly- subtracting negative 4 three times