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MIDDLE-GRADES STUDENTS SHOULD HAVE opportunities to experiment actively with situations that model probability, including "making hypotheses, testing conjectures, and refining their theories on the basis of new information" (NCTM 1989, 111). These experiences should include a discussion of theoretical probabilities, where appropriate, and the use of correct mathematical terminology and expressions.
Exploring probabilistic thinking not only exercises students' general reasoning abilities but also helps build a foundation for more informed decision making in everyday life. Students, like adults, regularly use their knowledge of probabilities to make choices about important issues, such as health and safety, and to determine strategies in some recreational pastimes, such as sports and games. In this article, we describe a dice game that can be used by students as a basis for exploring mathematical probabilities and making decisions while they also exercise skills in multiplication, pattern identification, proportional thinking, and communication. As we observed and listened to the sixth graders participating in the activity, we gained some interesting insights into student thinking at this age level.
Game Overview
STUDENTS PLAY THE GAME IN PAIRS, ALTHOUGH groups of three or four would also work. On her or his own game board (see fig. 1), each student divides a given number of chips between the two categories of "even" and "odd." Students take turns rolling two dice, finding the product of the values on the faces, and removing-only for their own rollan "even" chip for an even product and an "odd" chip for an odd product. The first to remove all chips for both categories wins. Rolls that come up for a category for which all chips have already been cleared result in a lost turn. Students get an equal number of turns, so whoever rolls the dice second at the start of a two-player game may roll last.
Playing with Tetrahedral Dice
TO BEGIN PLAYING THE GAME, STUDENTS WERE given a pair of tetrahedral, or four-faced, dice; a game board; and sixteen chips. Dice with a greater number of faces could be used. Students were divided into four-member discussion groups. Each student had a game board, and each pair of students had two four-faced dice. Students were instructed how to read these dice. After playing instructions were given, students were asked...