Overview

Students toss coins and then use mathematical reasoning to determine if a geometric game board is fair. Then they work in partners to find theoretical probabilities of area models.

 

Outcome

Students will be able to calculate theoretical probability of area models by finding the ratio of wanted areas to total area.

 

Standards Addressed

Mathematics — Data Analysis

Grade 7

Probability, Benchmark I

07. Compute probabilities of compound events; e.g., multiple coin tosses or multiple rolls of number cubes, using such methods as organized lists, tree diagrams and area models.

Materials

• Pennies — enough for 20 per pair of students

Procedure

1. Begin the class with a discussion of games the students have played with other children. What does it mean to have a fair game? They should understand that each child would have an equal chance of winning, in other words, a 50-50 chance, or 50 percent probability of winning.

2. Divide students into pairs. Hand out the game board. One player will pick shaded and the other white. They will each toss 20 pennies onto the game board. If the majority of the penny lands on their color, they score a point. After 40 tosses, the player with the highest score wins.

3. As a class, record how many shades won and how many whites won. (There should be more whites.) Ask if the game is fair or rigged. This should promote discussion. By dividing the grid into one-sixteenths, they should see there are nine-sixteenths white areas and seven-sixteenths shaded areas. Write out the probability for the students, showing that white’s probability is more than 50 percent and shade’s is less than 50 percent, definitely unfair.

4. Discuss experimental vs. theoretical probability. In the long run, the player who picked white should always win, even if the experiment didn’t show that.

5. After this initial discussion, hand out the Penny Tossing Fools? handout and have the students work in partners to calculate the probabilities of the area models.

6. Extension: Have the students create their own game boards. Ask for both a triangle and rectangle to be included. They can create both a fair and unfair game.

Answers for Penny Tossing Fools

1. P(shade) = 8/16 or 1⁄2 or .5
   P(white) = 8/16 or 1⁄2 or .5

2. P(red) = 1/25
   P(white) = 8/25
   P(blue) = 16/25

3. a. P(shade) = 17/36
   b. P(shade) = 6/18 or 1/3

4. a. P(white) = 7/16
   b. P(shade) = 9/16

Evaluation

All 9 correct	Clear evidence of understanding probability.
8-7 correct	Adequate evidence of understanding probability.
6-5 correct	Some evidence of understanding probability.
4-0 correct	Little evidence of understanding probability.