Rolling the Dice: Finding Algebraic Connections by Chance Steve Benson Education Development Center Newton MA 02458 [email protected]Electronic versions of handouts will be available a http://www2.edc.org/cme/showcase.html Rolling the Dice NCTM Anaheim, April 7, 2005
27
Embed
Rolling the Dice: Finding Algebraic Connections by Chance Steve Benson Education Development Center Newton MA 02458 [email protected] Electronic versions.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Rolling the Dice: Finding Algebraic Connections by Chance
Electronic versions of handouts will be available athttp://www2.edc.org/cme/showcase.html
Rolling the Dice NCTM Anaheim, April 7, 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
Rolling the Dice NCTM Anaheim, April 7, 2005
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
Rolling the Dice NCTM Anaheim, April 7, 2005
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?
Rolling the Dice NCTM Anaheim, April 7, 2005
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
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
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.
QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
Rolling the Dice NCTM Anaheim, April 7, 2005
Think about it
Is there a “best” set-up for the Difference Game?
Rolling the Dice NCTM Anaheim, April 7, 2005
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
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)?
Rolling the Dice NCTM Anaheim, April 7, 2005
What about sums?
What are the theoretical probabilities of each possible two-dice sum?
Rolling the Dice NCTM Anaheim, April 7, 2005
What about sums?
What are the theoretical probabilities of each possible two-dice sum?
Rolling the Dice NCTM Anaheim, April 7, 2005
Theoretical and Experimental frequencies(two-dice sums with 36 rolls)
Theoretical frequency Experimental frequencies
QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
Rolling the Dice NCTM Anaheim, April 7, 2005
The Law of Large NumbersFrequencies of two-dice sums with 400 rolls
QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
Rolling the Dice NCTM Anaheim, April 7, 2005
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
What about three-dice sums?
Rolling the Dice NCTM Anaheim, April 7, 2005
What about three-dice sums?
Rolling the Dice NCTM Anaheim, April 7, 2005
What about three-dice sums?
Rolling the Dice NCTM Anaheim, April 7, 2005
What about three-dice sums?
Rolling the Dice NCTM Anaheim, April 7, 2005
Abstract Clotheslines
Aren’t you itching to factor it?
Rolling the Dice NCTM Anaheim, April 7, 2005
Abstract Clotheslines
equals
which equals
A perfect square!
Rolling the Dice NCTM Anaheim, April 7, 2005
Abstract Clotheslines
equals
which equals
which equals
Rolling the Dice NCTM Anaheim, April 7, 2005
Abstract Clotheslines
Rolling the Dice NCTM Anaheim, April 7, 2005
Abstract Clotheslines
Make a conjecture: What is the distribution polynomial for n-dice sums?
Rolling the Dice NCTM Anaheim, April 7, 2005
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 your
prediction works (e.g., for the 3-dice sum polynomial).
Rolling the Dice NCTM Anaheim, April 7, 2005
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 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
Rolling the Dice NCTM Anaheim, April 7, 2005
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.