Index

Discrete Mathematics Project
Fair Division Activity

Title

Cakes, Estates, and Seats: Can You Divide Them Fairly? (Dominic D. Peressini)

Goals

  1. Students will explore the concept of fair division as it relates to dividing a continuous object (e.g., a cake) and apportioning discrete objects (e.g., an estate or legislative seats) among a number of people or groups.
  2. Students will work in small groups in order to arrive at a consensual fair division and explain how they arrived at this division and what criteria makes the division fair.

Abstract

This activity, set in the context of (a) dividing a cake among a group of people, (b) dividing an estate among heirs, and (c) apportioning "seats" among legislative/political groups, focuses on the concept of fair division. Students are asked to form groups and devise methods that fairly divide the objects involved in the three scenarios. Each group is then asked to present and explain their methods to the entire class. The activity could be used to introduce the three most common circumstances in which the fair division of an object is required.

Problem Statement

Suggest to your students that almost everyone has felt the frustration of feeling that someone else has gotten the biggest piece of cake (or pizza, dessert, . . .) when a group of people are dining together. This feeling is similar to the feeling that heirs may feel when an estate is divided, or states may feel when seats in congress are divided and assigned. This activity will allow students to explore these division situations and employ discrete mathematics in devising methods that fairly distribute these objects.

Instructor Suggestions

  1. Set the stage by discussing the "Problem Statement" (see above) with your students.
  2. Have the students form small groups.
  3. Distribute the "Cakes, Estates, and Seats: Can You Divide Them Fairly?" activity sheet (see attachment) and ask each group to work on two of the three problems (be sure that you have an equal distribution of problems being examined).
  4. When the small groups are finished, have a spokesperson for each group share there methods for the two problems they chose to work examine.
  5. Discuss the students work as it relates to fair division.

Materials

"Cakes, Estates, and Seats: Can You Divide Them Fairly?" activity sheet

Time

Introduction of Problem Statement (5 min.), small group work (20 min.), presentation of small group work and large group discussion (20 min.)

Mathematics Concepts

Discrete Mathematics Concepts

Estate Division, Apportionment Algorithms for Discrete Objects (Jefferson Method, Webster Method, Hill Method), Apportionment Paradoxes, Apportionment Algorithms for Continuous Objects (Cut-And-Choose Method, Inspection Method), Fairness

Related Mathematics Concepts

Matrices, Quotas, Ratios, Arithmetic Mean, Geometric Mean

NCTM Standards Addressed

Problem Solving, Communication, Reasoning, Connections (within mathematics and across disciplines), Algebra, Geometry, Discrete Mathematics

Colorado Model Content Standards Addressed

Algebraic Techniques (2), Problem Solving Techniques (5), Linking Concepts and Procedures (6)

Curriculum Integration

This activity could be integrated (1) into a traditional Algebra or Geometry class as the topic of Matrix operations are examined--the matrix connection could be made through the use of matrices in the algorithm for dividing an estate. The activity could also be integrated (2) into a traditional Algebra class when ratio and quotas are being discussed--ratio and quotas are essential aspects of several of the apportionment algorithms (Jefferson and Hamilton Methods). Another natural point of integration would be (3) into an Algebra, Probability, or Statistics class when arithmetic and geometric means are examined--both arithmetic and geometric means are essential components of the Webster and Hill methods for apportionment.

Further Investigation

Extend each of the problems by having more people to eat cake, more assets in the estate, more heirs, different proportions of students, and more or less legislative seats. You may ask students to formally define what it means to be "fair."

Variations/Comments

Provide students with actual cakes, money (e.g., Monopoly money), and seats so that they can use these to refine and test their fair division methods. In order to reduce time, you may want to have each group examine only one of the three problem situations.
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References/Resources

Crisler, N., Fisher, P., & Froelich, G. (1994). Discrete Mathematics Through Applications. New York: W. H. Freeman and Company.

Eisner, M. P. (1982). Methods of congressional apportionment. Lexington,MA: COMAP.

Kenney, M. J., & Hirsch, C. R. (Eds.). (1991). Discrete Mathematics Across the Curriculum, K-12. Reston. VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: Author.

Last updated January 16, 1997