Title
Union Representation: How Can It Be Apportioned Fairly? (Anne Smelker)
Goals
(1) Students will explore the concept of apportioning discrete objects among a number of people fairly.
(2) Students will work in small groups to discuss and derive a method that divides or apportions a number of representative seats. They will explain how they determined their results, methods, and criteria that makes the apportionment fair.
(3) Students will analyze and discuss apportionment methods and how apportionment results may effect the outcomes and number of liaisons from each division.
Abstract
This activity focuses on the concept of fair apportionment. Students will work in small groups to devise methods to apportion fairly the number of seats to represent each division of the Union. This activity can be used to introduce the concept of apportionment and fairness. Discussion of students methods can extend to the exploration and understanding of the apportionment methods of Hamilton, Jefferson, Webster, and Hill, and to debate of not only the paradoxes that arise but of which method is the fairest. Through employing discrete mathematics in devising methods that fairly apportion the number of liaisons, students will explore and compare different methods of apportionment and the paradoxes that arise. Students can then compare, contrast, and analyze their methods, other apportionment methods, and their validity or fairness.
Problem Statement
Everywhere we look, we have people representing us and our beliefs, from the government where we have representatives from each state, to the representatives of student council. In some work places, Unions are formed to represent the employees and divisions of a company. In a Union, there are representatives from each division that serve as liaisons on the Board of Directors. This activity explores how these representatives are apportioned to represent all divisions of a company fairly.
Instructor Suggestions
(1) Begin the activity by reviewing and discussing fairness of division of continuous objects, such dividing 25 cookies between four people, and then introduce the problem statement.
(2) Have students develop their own solutions and methods to the problem statement in small groups after distributing the "Union Representation - How Can It Be Apportioned Fairly?" activity sheet.
(3) After the small groups are finished, have a spokesperson from each group share their method, how they determined their method, how or why they feel their method is the fairest method, and why they feel each of the siblings will be satisfied with their share.
(4) Discuss each of the groups methods as it relates to apportionment, the paradoxes that may arise, and the fairness of each method. At this time, you may introduce the methods of Hamilton, Jefferson, Webster, or Hill.
Materials
"Union Representation - How Can It Be Apportioned Fairly?" activity sheet
Time
Introduction of Problem Statement: 5 minutes
Group Work 20 minutes
Presentations/Class Discussion 20 minutes
Mathematics Concepts
Discrete Mathematics Concepts
Apportionment Algorithms for Discrete Objects (Jefferson Method, Webster Method, Hill Method), Apportionment Paradoxes, Apportionment for Continuous Objects, 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 in a Algebra or Geometry class when matrices, arithmetic mean, geometric mean are being developed.
Further Investigation
Extend the problem to explore other areas where these methods may be applied, such as government, student council, etc.
Variations/Comments
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.
National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: Author.