Title
Sharing Fairly (Anne Smelker)
Goals
(1) Students will explore the concept of fair division as it relates to dividing a continuous object and to dividing discrete objects
(2) Students will work in small groups to discuss and derive a fair division.
(3) Students will explain and justify how they determined a fair division and what criteria makes the divisions fair.
Abstract
This activity focuses on fair division associated with i) dividing continuous objects such as cookies and ii) discrete objects such as an estate. Students will develop an understanding and definition for fairness. This activity allows students to explore, discover, and develop methods to divide continuous and discrete objects while applying their definition or understanding of fairness. This activity could be used to introduce fair division of objects under different circumstances.
Problem Statement
Sharing occurs throughout our lives. From sharing toys when we are little to sharing food, cars, living space, time, feelings, clothes, and the list could go on as we mature. One aspect of sharing involves fairness. You are with three friends. You are starving. All there is left is only one piece of pizza. How do you share it fairly? Students will form groups and devise methods to divide each situation fairly. The key words are "divide fairly". How would you define fairness? Students are to explore each division situation and devise a method and/or process to divide it fairly.
Instructor Suggestions
(1) As a class discuss only the Problem Statement and what is expected of each group but not how to begin solving the problems.
(2) Have the students work in groups to develop a definition of fairness and a method and/or process to determine how to divide objects fairly.
(3) When the groups are finished, have each group have a spokesperson share their methods and the processes they used to divide the scenarios fairly.
(4) Discuss each of their methods as it relates to fair division.
Materials
"Sharing Fairly" activity sheet
Time
Introduction of Problem Statement (5 minutes), Group Work (20 minutes), Presentations/Class Discussion (20 minutes)
Mathematics Concepts
Discrete Mathematics Concepts
Fair Division, Fairness (Arrow's Conditions), Estate Division, Apportionment Algorithms for Continuous Objects (Cut-And-Choose Method, Inspection Method)
Related Mathematics Concepts
Matrices
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 in a Algebra or Geometry class as the topic of Matrix Operations.
Further Investigation
Extend each of the problems by having more people to share the items, more assets in the estate, or more heirs.
Variations/Comments
Provide students with cookies, fake money (Monopoly money), or fake lotto tickets so they can use them to develop, refine, and test their fair division methods.
References/Resources
Crisler, N., Fisher, P., and Froelich, G. (1994). Discrete Mathematics Through Applications. New York: W.H. Freeman and Company.
Kenney, M. J., and Hirsch, C. R. (Eds.). (1991). Discrete Mathematics Across the Curriculum, K - 12. Reston, VA: National Council of Teachers of Mathematics.
Lynch, D., Martin, W., and Porter, B. (1995). Higher Expectations - Better Results Assessment Workbook. Colorado State Board of Education.
National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: Author.