Uncovering Algebraic Connections in Dice-y Problem Solving Situations Steve Benson Education Development Center Newton MA 02458 [email protected] Electronic

Uncovering Algebraic Connections in Dice-yProblem Solving Situations

Steve BensonEducation Development Center

Newton MA 02458

[email protected]

Electronic versions of handouts not appearing on CD will be availableat http://www2.edc.org/cme/showcase.html

Mathematics Teachers in Appalachia: Future and Present, February 25-26, 2005

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


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.

x x x x

x x x x x x x

x x x x x x x

0 1 2 3 4 5 6

x x x

x x x x

x x x x x x

x x x x x x

0 1 2 3 4 5 6


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?


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?


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.


Theoretical and Experimental frequencies(two-dice differences with 36 rolls)

Theoretical frequency

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

Experimental frequencies

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.



Think about it

Is there a “best” set-up for the Difference Game?


Think about it


Will it always work (i.e., will you always win with it)?


Think about it


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)?


What about sums?

What are the theoretical probabilities of each possible two-dice sum?


What about sums?

What are the theoretical probabilities of each possible two-dice sum?


Theoretical and Experimental frequencies(two-dice sums with 36 rolls)

Theoretical frequency Experimental frequencies





The Law of Large NumbersFrequencies of two-dice sums with 400 rolls






What about three-dice sums?

Can you use the two-dice sum information to determine thethree-dice sum probabilities?










Abstract Clotheslines

Aren’t you itching to factor it?



equals

which equals

A perfect square!



equals

which equals

which equals






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).




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 + x2 + x3 + x4 + x5 + x6)n

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


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Uncovering Algebraic Connections in Dice-y Problem Solving Situations Steve Benson Education Development Center Newton MA 02458 [email protected] Electronic