NCTM Teaching Standards:
Target Audience: Grades 7-12
Tools Needed: Graphing calculator with a random integer generator or an online random number generator (look here for example)
(After demo mode): Students working in groups of 3 (two opponents, the 3rd calls out the numbers and keeps score; roles are rotated)
Sample Classroom Scenario
Who thinks they can beat me at a game of chance? I will demo the game, then I will play against an opponent. If you beat me two out of three, you are the new champ and you can pick your opponent. After 10 minutes, you will be playing in small groups and recording the results.
Using a random integer generator we will generate random digits, one at a time, from 1 through 9, inclusive (no zeros). The object is to build a 5-digit integer which is greater than your opponent's by placing each 'called' digit into one of the five place-value positions. Once you place a digit you cannot change it!
Let's try it... Ok, Marissa, turn on the random integer generator, press Enter and call out the first integer. FOUR!
Ok, I'll place it here: ___ ___ 4 ___ ____
Call out the next integer: SIX!
I'll place it here: ___ 6 4 ___ ___
___ 6 4 ___ 2
___ 6 4 4 2
Last digit! FIVE!
5 6 4 4 2
How did I do? Could I have used a better strategy? Do you think you could have beaten me?
Who wants to play! To win, you have to beat me two out of three. Ok, Dimitri, I will work on my paper and you work on yours. Remember, you cannot change a digit's position once you place it...
Brief Discussion of Strategy Based on Probability Arguments:
Suppose the first two digits called are 3 and 6 in that order. Would it be better to place the 6 in the thousands' place or the ten thousands' (leftmost) position? If you place the digits here:
___ 6 ___ 3 ___, what is the probability that at least one of the next three digits chosen will be 6,7,8, or 9. (Otherwise, your strategy would have backfired). To compute this, we look at the complementary condition, i.e., we determine the probability that the next 3 digits chosen will all be in the range 1 through 5. The probability of this is (5/9)(5/9)(5/9) or approximately 17%, so the probability that our strategy works is about 83%, odds that seem worth playing! Experienced game players often compute these probabilities mentally or have seen these situations so many times they know these probabilities by heart!
(1) Students may not know there is a Random Integer generator built into many graphing calculators. For example on the TI-84, press MATH, then PRB, then 5:randInt(.
From the home screen, Enter randInt(1,9), ENTER. Each time you press ENTER another "random" digit will be displayed. The person calling these out must be instructed to announce only ONE digit at a time!
(2) Why 5-digit numbers? This seems to make the game fairly interesting and moving at a good pace. Expect ties of course!
Perhaps this is a good activity before the holidays. Have fun and let me know how it goes!
Sunday, December 14, 2008
NCTM Teaching Standards: