Grade Level

The Prisoner’s Dilemma is a math problem that incorporates ideas from game theory. It aims to mathematically represent behavior in a situation where an individual's success in making choices depends on the choices of others. Because it does not have a “standard probability”, and instead, it involves recursive choices, it’s recommended that the game targets high school students. Less complex versions of the game, however, can be played and understood by students with a basic grasp of probability and deductive logic skills.

The Problem

Two suspects are arrested by the police. The police have insufficient evidence for a conviction, and, having separated both prisoners, visit each of them to offer the same deal. If one testifies for the prosecution against the other (“betrays”) and the other remains silent (“cooperates”), the betrayer goes free and the silent accomplice receives the full twenty-year sentence. If both remain silent, both prisoners are sentenced to only five years in jail for a minor charge. If each betrays the other, each receives a nine-year sentence. Each prisoner must choose to betray the other or to remain silent. Each one is assured that the other would not know about the betrayal before the end of the investigation. How would you act if you were one of the prisoners?

Materials List

• (100 x # of students in class) = pieces of candy needed (skittles/m&ms recommended)

• Screens made of poster board, cardboard, etc. that can stand between two desks

• Non-transparent containers

• Decks of 54 cards – one deck is needed for every 13 pairs of students

Required Setup

• Arrange student’ desks into pairs, face to face, creating an inner circle and an outer circle.

• Make sure a screen is between each pair of desks so that students cannot communicate with their “partners-in-crime”.

• Each student should be handed 100 pieces of candy, which represents “100 years of life”.

• A non-transparent container should be placed in the middle of each pair of desks, which symbolizes “years in prison”. Every skittle dropped into the container is a year in prison.

• Students in the inner circle should be handed a black and red card of even value (ie: black six of spades and red six of hearts).

• Students in the outer circle should be handed a black and red card of odd value (ie: black jack of clubs and red jack of diamonds).

• Emphasize the students should not communicate with anyone else and should not talk during the progression of the game.

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Probability-based Bingo using legumes and a pair of dice

Overview and NCTM Standard Objectives

This game is intended for 9th grade mathematics students.

This game is intended to provide students with an opportunity to develop a working

knowledge of probability based on comparisons of empirical observations with

theoretical probability distributions. Students must work in teams to formulate and

articulate winning strategies based on conclusions drawn from the relative frequency of

certain events, their intuitive grasp of likelihood and probability, and a theoretical

analysis of the game’s probability distribution.

This game helps teachers meet the following NCTM standards:

Content- Data Analysis and Probability

Process- Reasoning and Proof, Communication, Connections

Specifically regarding Data Analysis and Probability NCTM objectives, this game will

help enable students to:

- Formulate questions that can be addressed with data and collect, organize, and display

relevant data to answer them. This will occur through the collection of data during the

game, the display of data using graphs, and the formulation of strategy based on the

graphical representation of data.

- Develop and evaluate inferences and predictions that are based on data through the

use of simulations to explore the variability of sample statistics from a known population

and to construct sampling distributions. This game should help students understand how

sample statistics reflect the values of population parameters and how to use sampling

distributions as the basis for informal inference;

-Understand and apply basic concepts of probability. This game will help students

understand the concepts of sample space and probability distribution, construct sample

spaces and distributions in simple cases, and use simulations to construct empirical

probability distributions.

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Description: “Get Lucky” is a fast-paced thinking game that requires students to be creative in the ways that they can manipulate basic operators and randomly given integers to reach a “lucky number.” “Algebra Jeopardy” is a team-based activity that tests the knowledge students have acquired in the classroom with review questions categorized by topic. The combination of these games is appropriate for students in 6th through 9th grade (Algebra 1).

How to Play:

1. Separate into teams of 2-4 people.

2. Once teams are made, the game leader picks a “lucky” number (any integer) and each team draws three cards randomly from a deck of cards that has been shuffled and sprawled out on a desk.

3. Once your group has your three cards, quickly determine which combination of cards will put your team closest to the “lucky number” the game leader has chosen. Teams can choose two of the four basic operations (addition, subtraction, multiplication, or division) to achieve a number closest to the one selected by the teacher. However, no card or operator can be used twice. Additionally, Aces are worth 1, Jacks are worth 10, Queens are worth 12, and Kings are worth 13.

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By Alie Doolittle and Kimberly Gonzales

Grade Level: 9th grade Algebra

Materials:

• Worksheet provided
• 6 sided cube dice (one per team)
• 4 sided tetrahedron dice (one per team)
• Mini M&Ms (enough for 20 m&ms per student or student group)

Learning Goals:

• Build intuition about probabilities
• Explore the difference between expected and theoretical probability
• Develop strategies using probabilities

NCTM Standards:

• Understand and apply basic concepts of probability
• Understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases;

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