Title
AFC Pre-Season Standings (John Long)
Goals
(1) Students will explore the concept of group-ranking as it relates to election theory.
(2) Students will work in small groups to come up with a consensus on how to rank the team. The students will be able to defend their method.
(3) Students will see the benefit of the Borda method of group ranking.
(4) Students will explore insincere voting and recognize its power and potential problems.
Abstract
This activity, which is set in the context of having students vote on the pre-season standings of the AFC Western Division football teams, focuses on group-ranking. Students are asked to rank-order the AFC West teams. The individual preferences are tallied and, in small groups, a group ranking is determined. The students will be asked to explain and defend their ranking. The students will then regroup in order to explore the practice of insincere voting.
Problem Statement
The football season is about to begin and you wanted to get an idea on how the class ranks the potential of the AFC West teams. To do this each student must first rank-order the teams.
Instructor Suggestions
(1)This activity can be used prior to any sport season. To set the stage, distributing or reading some material regarding each teams potential could be beneficial.
(2)Distribute the "AFC West Pre-Season Standings" activity sheet.
(3)Have students rank-order the teams individually and tally their results on the board
(4)Divide student into small groups(3 or 4) to find a ranking that represents the entire class
(5)Have each group present their ranking and explain their method. Encourage discussion.
(6)Discuss the students work as it relates to rank-ordering. Explain how Borda's Method is an excellent method to rank-order.
(7)Re-group the student according to their team preference. Have student experiment with changing their original votes in order to manipulate the result in their favor.
(8)Discuss the students work as it relates to insincere voting.
Materials
Overhead projector and materials, Activity sheet.
Time
Introduction (5 min.),individual work(5 min.),small group work(15 min.), explanations(20 min.), regroup work (20 min.), discussion(15 min.)
Mathematics Concepts
Discrete Mathematics Concepts
Group-Ranking plurality winner, Majority Winner, Borda Method, Runoff Method, Sequential Runoff Method, Condorcet Method(paradox), Arrow's Conditions, Recurrence Relations, Insincere voting
Related Mathematics Concepts
Matrices, Sequences and Series
NCTM Standards Addressed
Problem Solving, Communication, Reasoning, Connections, Algebra, Geometry, Discrete Mathematics
Colorado Model Content Standards Assessment
Data Collection and Analysis (3), Problem Solving techniques (5), Linking Concepts and Procedures (6)
Curriculum Integration
This problem can be used with any class. What I feel is important is the time you introduce it. If a popular sport team is about to begin it season, many students will easily buy into the activity.
Further Investigation
Has insincere voting ever been accused of an individual or individuals in sports?
(For Player of the year, college football national champion etc.)
References/Resources
Crisler, N., Fisher, P., & Froelich, G. (1994). Discrete mathematics through applications. New York: W. G. Greeman and Company.