1. Uncovering Algebraic Connections in Dice-y Problem Solving Situations Steve Benson Education Development Center Newton MA 02458 sbenson@edc.org Electronic versions of handouts not appearing on CD will be available at http://www2.edc.org/cme/showcase.html Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

2. The Difference Game* Each player begins with 18 chips and a game board Players start by placing their chips in the numbered columns on their game boards (in any arrangement). Players take turns rolling the dice. The result of each roll is the difference between the number of dots on top of the two dice. Each player who has a chip in the column corresponding to the result of the roll removes one chip from that column. The first player to remove all of the chips from his or her game board is the winner. * Adapted from an activity in MathThematics book 2 Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

3. 1. Play the Difference Game in groups of two or more (a)Before starting, find a way to record the initial set-up of your game board have someone record the result of each roll (you'll refer to both of these in part (b)). (b)In the space provided, record the initial set-up of each of the players at your table and who won. xxxx xxxxxxx xxxxxxx 0123456 xxx xxxx xxxxxx xxxxxx 0123456  Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

4. 2.Did the results of the game give you any new ideas about how you should place the game pieces to start the game? Play the Difference Game, again to see if you're right. Again, record the starting positions of you and your tablemates, as well as who wins. 3.What have you noticed about the frequency of different results (dice differences)? Talk it over with those at your table. Which difference seems to occur most? Which occurs least? Are there any possible differences that haven't come up, yet? Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

5. 4. Roll your two dice twenty times, recording the differences in the table provided. When you're done, combine the data from everyone at your table and create a bar graph to share with the other groups. 5. According to the data from the class, what is the experimental probability of rolling each of the possible differences? Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

6. 5. Now, work with your tablemates to determine the theoretical probability of rolling each of the possible differences. Be ready to share your results, and the methods you used, with the whole group. Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

7. Theoretical and Experimental frequencies (two-dice differences with 36 rolls) Theoretical frequencyExperimental frequencies Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

8. Think about it Is there a “best” set-up for the Difference Game? Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

9. Think about it Is there a “best” set-up for the Difference Game? Will it always work (i.e., will you always win with it)? Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

10. Think about it Is there a “best” set-up for the Difference Game? Will it always work (i.e., will you always win with it)? Is mirroring the theoretical probabilities the only “best” strategy? Can you think of different strategies that students might think of (along with their justifications)? Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

11. What about sums? What are the theoretical probabilities of each possible two-dice sum? Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

12. What about sums? What are the theoretical probabilities of each possible two-dice sum? Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

13. Theoretical and Experimental frequencies (two-dice sums with 36 rolls) Theoretical frequencyExperimental frequencies Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

14. The Law of Large Numbers Frequencies of two-dice sums with 400 rolls Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

15. What about three-dice sums? Can you use the two-dice sum information to determine the three-dice sum probabilities? Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

16. What about three-dice sums? Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

17. What about three-dice sums? Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

18. What about three-dice sums? Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

19. What about three-dice sums? Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

20. Abstract Clotheslines Aren’t you itching to factor it? Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

21. Abstract Clotheslines equals which equals A perfect square! Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

22. Abstract Clotheslines equals which equals Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

23. Abstract Clotheslines Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

24. Abstract Clotheslines Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005 Make a conjecture: What is the distribution polynomial for n-dice sums?

25. Abstract Clotheslines Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005 Make a conjecture: What is the distribution polynomial for n-dice sums? Check your conjecture: Using the data you’ve collected, check to see if your prediction works (e.g., for the 3-dice sum polynomial).

26. Abstract Clotheslines Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005 Make a conjecture: What is the distribution polynomial for n-dice sums? Check your conjecture: Using the data you’ve collected, check to see if your prediction works (e.g., for the 3-dice sum polynomial). Prove your conjecture: Prove that the n-dice polynomial must be equal to (x + x 2 + x 3 + x 4 + x 5 + x 6 ) n

27. Many of the materials used in today’s activity were adapted from Ways to Think About Mathematics: Activities and Investigations for Grade 6-12 Teachers, available from Corwin Press. A Facilitator’s Guide and Supplementary CD (including solutions and additional activities) are also available. More information at http://www2.edc.org/wttam Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005