MATHEMATICS OF FAIR GAMES
Grades 4 - 5

The students will learn about mathematicians' notion of fairness in games of chance. They will work in pairs to perform three different experiments using macaroni and paper bags. They will record their results on charts to compare data and make conjectures regarding the role that fairness over time plays.

Square One TV: Challenge Round - Focus on Probability (Children's Television Workshop)

Students will be able to:

• Describe what makes a game of chance fair or unfair.
• Explain why repeated trials are necessary in an investigation.
• Make predictions about whether a given game is fair or unfair.
• Carry out several investigations.
• Display data in a chart.
• Compute class totals using a calculator.
• Draw conclusions based on the data obtained in the investigations.
• List the mathematical possibilities for all possible outcomes.
• Compare the results of the investigations to the true mathematical probabilities.

Each pair of students:

• small paper bag
• three shell shaped macaroni
• two elbow shaped macaroni (or any other small contrasting shapes or colors)
• six sheets of three inch by three inch Post-It notes
• magic marker
• tally sheet and a calculator
• one large piece of newsprint paper should be prepared ahead of time to receive the Post-It tallies for each pair

• fair game
• chance
• probability
• tally
• data
• investigation

Ask the students,"Have you ever heard someone flip a coin to decide something and say 'Heads I win, tails you loose.?' What are the elements that make a game fair?" Discuss what fairness means in games. Students should understand that although they may have heard "That's not fair" during a game, any good game must be fair so that each player has an equal chance at winning.

The focus for viewing is a specific responsibility or task(s) that focuses and engages student's viewing attention. Tell the students to think about what it means to be a fair coin.

START the video at the beginning and PAUSE it after the game show host says, "The focus is probability." Ask the students what experiences they have had with probability and what it means to them. RESUME the video. PAUSE it after the game show host says, "I have a fair coin." Discuss with the students what it means to have a fair coin and how this connects to the concept of probability. RESUME the video. PAUSE it after the game show host says, "Which is more likely on my next flip, heads or tails?" Ask the students how they think the celebrities will respond. RESUME the video. After the celebrities have given their responses, PAUSE the video when the game show host says, "Think about it and write down your answers." Give the students a few minutes to do the same. RESUME and PLAY video to the end.

Remind the students that the reason the coin used in the video was a fair coin was because each side had an equal chance of turning up when the coin was flipped. The probability of getting a head was one-half and the probability of getting a tail was also one-half each time the coin was flipped. Discuss how this concept might apply to fair games and then pass out the prepared materials necessary to play the games. Call the students' attention to the newsprint chart at the front of the room. Tell them that they are going to test the fairness of the three games described on the newsprint. Ask for a volunteer to be your partner to help demonstrate the following:

Place two shells and one elbow in the paper bag. Ask them to think about game # three. (See newsprint.) If partner # one was to score a point each time the two noodles selected from the bag were the same and partner # two scored a point each time the noodles selected were not the same, would this be a fair game or does one of the players have a better chance? Demonstrate this scenario twenty times for the students keeping a tally on the board for when the noodles were the same and when they were different.

Call their attention to the three games for investigation (on the newsprint).
Game #1 - three shells, one elbow
Game #2 - two shells, two elbows
Games #3 - two shells, one elbow

Tell them that their job is to investigate which, if any, of these games is fair. Discuss their predictions. Have them experiment twenty times with each game, keeping a tally for each trial. As they finish each game they record the totals of "same" and "different" on their post-its and place the post-its on the newsprint chart.

Ask students who finish early to calculate the class totals. (There will be opportunities for six pairs of students to do this.) Have each partner double-check the total found with a calculator and record it on the newsprint chart. When all of the statistics have been recorded and calculated, have the class draw conclusions about fairness based on the data obtained in the investigations.

Only one version of the game is mathematically fair - version # one. This seems counter intuitive to many students and the data may or may not support it. List the possibilities for drawing two noodles from the bag to explain this concept. (See attached worksheet.)

• Students may take a field trip to a toy store that has a variety of different types of games and discuss some of the games with a knowledgeable representative. For instance, which games are games of chance and which are games of strategy? They could compile a list of questions to ask the representative.
  
  
• A speaker from a toy company may be invited to the classroom to explain the process involved in determining rules for fair games.
  
  
• Students may wish to write to companies who create board games. Addresses can be found in a library or Web site. Students can inquire about how companies think up new games, the guidelines they use to design them, and the marketing concerns they need to consider.

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Students can bring to class games of their own and explain how they demonstrate this concept. Tell them to consider whether it is a game of chance or of strategy. You may wish to tell them to analyze the rules for fairness and probability.

Students can create their own games describing the rules for play and explaining why the game is fair and then present the game to the class.

Students can investigate games from different cultures such as Dreidel which is traditionally played at Chanukah.

Science: Students can investigate how this concept applies to dominant and recessive genes.

Students can discuss how they think chance affects population growth in a region.

Master Teacher: Mary Ellen Baron
Springfield School Department, Springfield, MA