The probability of rolling dice is another classic explorations used with children in the elementary school. They have vast experience rolling dice to play board games, so there is inherent interest in doing some experiments and analyzing dice games. For the PST, I like to take this analysis a little further. We will spend time in class rolling traditional six sided dice and analyzing the probabilities of rolling certain sums and products. After spending some time examining theoretical and experimental probabilities with these, I like to introduce a dice rolling applet from a project designed by The Shodor Education Foundation "Project Interactive" .
The feature of this applet that makes it useful in my classroom is its option to choose other configurations of six sided die [for example a die with two 1's, two 2's and two 3's], or design your own die [ which you can assign any number 1-6 to any/all of the six sides of the die].
Launching the lesson
I open this discussion of dice by looking at one die and asking some questions regarding the probability of rolling certain numbers [ For example: a three, an even number, and odd number, a number less than five]. I also like to turn the question around and give them a probability like 4/6 and ask them to describe a situation that could have that probability. [I can extend this line of questioning into a review of certainty and impossibility as well.]
I then go into a discussion on fair and unfair games. We talk about what their conception of a fair game is. [They usually make statements involving each person having fair chance to win the game, which I'll accept at this time.] I introduce Dice Game 1, in which two players take turns rolling a pair of traditional dice. Player 1 scores a point if the sum of the dice is even, Player 2 scores a point if the sum is odd. I ask them to think about this situation for a minute or so and then we will "vote" as to whether we think the game is fair or not. I keep track of the votes on the board and ask for some sort of justification from a member of each of the groups as to why they voted this way. I then have them play the game, keeping track of not only whether the sum was odd or even, but also making a line plot of the number of times each sum is rolled. We will collect expriemental probabilities of odd vs. even from each pair and tally the totals on the board, giving us a larger number of trials to examine. During this time we will review what we have learned about the Law of Large Numbers and I'll ask them again if the game seems fair based on this larger pool of data. from here will we do a theoretical analysis of the probability using various methods to examine all of the possible outcomes. I try to leave the methods used up to the students, and I have found that some of them will come up with the idea of using a 6 x 6 matrix to represent the numbers rolled and thier sums. [ I think this method makes the most sense, but that's just me.] From these results we can conclude that the probability of rolling an even sum and an odd sum are equal.
It is here that I introduce the idea of expected value as what determines whether or not a game is fair.
We then go through a similar procedure with Dice Game 2, where we examine products instead of sums. In looking at the expected value for this game, we can se that P(even product) = 3/4 and P(odd product) = 1/4. If each player only recieves 1 point for each outcome, the game is not fair. In order to make it fair, Player 1 would need to get one point for and even sum and Player 2 would need to get 3 points for each odd sum.
Introducing the technology
After some experience with traditional dice, I introduce the Dice Rolling applet. I will show them the preset dice that are included in the applet as well as how to set up their own die. After doing so, I'll ask them to choose some configuaration of dice and as a pair design, play, and analyze a game using their dice. I want them to turn in a summary of the rules of the game, the dice configuration, a list of their experimental probabilities from playing their game, and a theoretical analysis of the game's fairness. In addition, if their game is not fair, I want them to explain how to make it fair.
This example is how I use this applet in my class. there are of course any other applications that could be thought of and used in both a methods class as well as a content course for elementary mathematics.