Title
Are We Being Fair? (Dorothy Gieck)
Goals
(1) Students will begin to explore the concept of division as it relates to a continuous object and a discrete object.
(2) Students will work in small groups in order to arrive at a fair division solution and be able to explain, and justify, how they arrived at this division and what makes it fair to the rest of the class.
Abstract
This activity, which is set in the context of having students divide a pizza and an estate, focuses on the concept of fair division. Students are asked to come to a consensus within a small group as to a method for dividing the two situations in a fair manner. These methods and solutions are then shared with the rest of the class. This activity could be used to introduce Fair Division.
Problem Statement
We learn form an early age what our idea of a fair portion is. Discuss with students how a fair division must seem fair to everyone that is getting a portion and that the methods must be devised for a division that will be fair regardless of the number sharing and the amount to be shared. In this activity we will be coming up with methods (algorithms) for dividing things fairly.
Instructor Suggestions
(1) Set the stage by discussing the above problem statement with your students.
(2) Have students form small groups.
(3) Distribute "Are We Being Fair?" activity sheets and allow students to work in their small groups to come up with appropriate solutions.
(4) Circulate between the groups and facilitate the discussions.
(5) When the small groups are finished, have a spokesperson for each group share their solution and explain their method and reasoning involved in arriving at their divisions.
(6) Discuss the students' work as it relates to fair division concepts.
Materials
"Are We Being Fair?" activity sheet, overhead projector, chalkboard
Time
Introduction of problem (5 min), small group work (20 min), small group presentations and whole class discussion (20 min), extension questions (5 min)
Mathematics Concepts
Discrete Mathematics Concepts
estate division, apportionment, algorithms, fairness criteria
Related Mathematics Concepts
matrices, quotas, ratios
NCTM Standards Addressed
Problem Solving, Communication, Reasoning, Connections
Colorado Model Content Standards Addressed
Number Sense (1), Geometric Techniques (4), Problem Solving Techniques (5), Linking Concepts and Procedures (6)
Curriculum Integration
This activity could be integrated into most any level. Discussion could include all discrete mathematics topics related to fair division. With a lower level class you might use this activity to discuss how division is not just an arbitrary conclusion arrived at by the powers that be. This activity could also be used with a traditional Advanced Algebra or Pre-calculus class when the topics of ratio and proportion or matrix operations are examined.
Further Investigation
This activity can be extended by changing the values to be divided and testing the various methods to see if they hold true.
Variations/Comments
References/Resources
Crisler, N., Fisher, P., & Froelich, G. (1994). Discrete mathematics through applications. New York: W. H. Freeman and Company.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
Colorado Model Content Standards for Mathematics (1995)