Title
Division of Real Property (Jo Ann Ellerbrock)
Goals
Students will explore the concept of fair division of continuous, irregular-shaped objects.
Abstract
This activity, which is set in the context of dividing an irregular piece of land between three heirs, asks students to determine a method of dividing the land so that every one is satisfied that they received a fair share. Students should have been introduced to the concept of the fair division of a cake in which one person divides the cake and the others get to choose. In this problem, each heir proposes a division, does not see the other's divisions, and is assigned one of the pieces he proposed.
Problem Statement
When individuals must share things, is desirable that the way in which things are divided by considered fair by those concerned. At times, things are divided by some authority such as a mother, a teacher, or a judge. Another method of fair division allows the people involved to help determine how the division will be made. The goal is to devise a method of dividing the property so that each person feels that the part he has received is a fair share.
Instructor Suggestions
(1) Students need to have been exposed to several methods of dividing continuous quantities.
(2) Establish (or have students determine) a definition of fairness. Some common criteria are:
a. Each individual is capable of dividing the land into portions he feels are equal
b. If each individual receives a portion of the land that he divided into equal shares, he will feel he received his fair share.
(3) If students are having difficulty, suggest they start with two people dividing the land in half.
(4) Have them form groups of three or four where they role play the parts of the heirs and the administrator who assigns the pieces.
(5) Solution: Place all of the lines on one map. Assign the first (left-most) piece to the person who has their first line furthest left. Remove all other lines of that person. Assign the second piece to the person who has their second line the furthest left, etc. This method works regardless of the number of participants.
Materials
Division of Real Property worksheet, pencils, erasers
Time
One class period
Mathematics Concepts
Discrete Mathematics Concepts
Fair division of continuous objects, Algorithms
Related Mathematics Concepts
Fractions, Axioms in a mathematical/logical system (the conditions of fairness), Mathematical applications in the social sciences, Formulating a vaguely stated problem, Communicating results.
NCTM Standards Addressed
Problem Solving, Communication, Reasoning, Discrete Mathematics
Colorado Model Content Standards Addressed
Number Sense (1), Algebraic Techniques (2), Problem Solving Techniques (5), Linking Concepts and Procedures (6)
Curriculum Integration
This activity could be integrated into a traditional Geometry class as an extension of area activities. The activity would also be appropriate to a Calculus class as part of the study of the area under a curve. In both situations, students may want to actually compute the areas received by each heir.
Further Investigation
Show that the method can be extended to any number of heirs.
Variations/Comments
This exercise is based on a problem found in HiMap Module 9, Fair Divisions: Getting Your Fair Share, available from the Consortium for Mathematics and Its Applications (COMAP), 57 Bedford Street, Suite 210, Lexington, MA 02173
References/Resources
COMAP (1987). Fair Divisions: Getting Your Fair Share
Crisler, N., Fisher, P., & Froelich, G. (1994). Discrete mathematics through applications. New York: W.H. Freeman and Company
Kenney, M.J., & Hirsch, C.R. (Eds.). (1991). Discrete mathematics across the curriculum, K-12. Reston, VA: National Council of Teachers of Mathematics.
Steen, Lynn A. (Ed.). (1991). For All Practical Purposes. New York: W.H. Freeman and Company