Rolling the Dice: Finding Algebraic Connections by Chance

1. Rolling the Dice:Finding Algebraic Connections by Chance Steve Benson Education Development Center Newton MA 02458 sbenson@edc.org Electronic versions of handouts will be available athttp://www2.edc.org/cme/showcase.html Rolling the Dice NCTM Anaheim, April 7, 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 Rolling the Dice NCTM Anaheim, April 7, 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.  Rolling the Dice NCTM Anaheim, April 7, 2005

4. 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. • 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? Rolling the Dice NCTM Anaheim, April 7, 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? Rolling the Dice NCTM Anaheim, April 7, 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. Rolling the Dice NCTM Anaheim, April 7, 2005

7. Theoretical and Experimental frequencies (two-dice differences with 36 rolls) Theoretical frequency Experimental frequencies Rolling the Dice NCTM Anaheim, April 7, 2005

8. Think about it Is there a “best” set-up for the Difference Game? Rolling the Dice NCTM Anaheim, April 7, 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)? Rolling the Dice NCTM Anaheim, April 7, 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 studentsmight think of (along with their justifications)? Rolling the Dice NCTM Anaheim, April 7, 2005

11. What about sums? What are the theoretical probabilities of each possible two-dice sum? Rolling the Dice NCTM Anaheim, April 7, 2005

12. What about sums? What are the theoretical probabilities of each possible two-dice sum? Rolling the Dice NCTM Anaheim, April 7, 2005

13. Theoretical and Experimental frequencies (two-dice sums with 36 rolls) Theoretical frequency Experimental frequencies Rolling the Dice NCTM Anaheim, April 7, 2005

14. The Law of Large Numbers Frequencies of two-dice sums with 400 rolls Rolling the Dice NCTM Anaheim, April 7, 2005

15. What about three-dice sums? Can you use the two-dice sum information to determine thethree-dice sum probabilities? Rolling the Dice NCTM Anaheim, April 7, 2005

16. What about three-dice sums? Rolling the Dice NCTM Anaheim, April 7, 2005

17. What about three-dice sums? Rolling the Dice NCTM Anaheim, April 7, 2005

18. What about three-dice sums? Rolling the Dice NCTM Anaheim, April 7, 2005

19. What about three-dice sums? Rolling the Dice NCTM Anaheim, April 7, 2005

20. Abstract Clotheslines Aren’t you itching to factor it? Rolling the Dice NCTM Anaheim, April 7, 2005

21. Abstract Clotheslines equals which equals A perfect square! Rolling the Dice NCTM Anaheim, April 7, 2005

22. Abstract Clotheslines equals which equals which equals Rolling the Dice NCTM Anaheim, April 7, 2005

23. Abstract Clotheslines Rolling the Dice NCTM Anaheim, April 7, 2005

24. Abstract Clotheslines Make a conjecture:What is the distribution polynomial for n-dice sums? Rolling the Dice NCTM Anaheim, April 7, 2005

25. Abstract Clotheslines 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 yourprediction works (e.g., for the 3-dice sum polynomial). Rolling the Dice NCTM Anaheim, April 7, 2005

26. Abstract Clotheslines 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 yourprediction works (e.g., for the 3-dice sum polynomial). Prove your conjecture:Prove that the n-dice polynomial must be equal to(x + x2 + x3 + x4 + x5 + x6)n Rolling the Dice NCTM Anaheim, April 7, 2005

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 athttp://www2.edc.org/wttam Rolling the Dice NCTM Anaheim, April 7, 2005