Title
High Tech Education (Kim Kendrick)
Goals
(1) Students will explore the concept of group-ranking, approval voting, and weighted voting as it relates to election theory.
(2) Students will be able to individually devise a justify a method for determining a group ranking and apply Arrow's Conditions to their method.
(3) Students will work in small groups in order to arrive at a consensual weighted voting system and be able to explain, an justify, how they arrived at this group-ranking to the rest of the class.
Abstract
This activity, which is set in the context of selecting a site for a new, high tech facility to replace one of the four existing buildings in our District. Students are asked to determine a group-ranking based on a set of already collected preferences. Students are also asked to devise their own "fair" weighted voting system for the selection of the new facility. Students should be familiar with group-ranking and should have experience applying various group-ranking methods and algorithms.
Problem Statement
Inform your students that spending and budgeting within any large organization is always controversial especially when the money that is to be spent comes from public funds. Deciding when and where to remodel or rebuild a school is a difficult task. Today we will be examining how decisions like this are made. We will be looking at a set of already collected preferences. The idea of fairness will be a central issue as we determine the site of the new school.
Instructor Suggestions
(1) Set the stage by discussing the "Problem Statement" (see above) with the class.
(2) Distribute the "High Tech Education" activity sheet (see attachment) and allow the students to individually read and complete the first part of the activity.
(3) When all of the students are finished, have each of the students share their group-ranking for the student surveys that were conducted. Be sure to have the students identify the method they used and why.
(4) Have the students form small-groups and devise a weighted voting system that they believe is fair and be able to explain, and justify this system to the rest of the class.
(5) When the small-groups are finished, have a spokesperson for each group share their preference and present their system and explain its fairness.
(6) Discuss the students' work as it relates to rank-ordering, weighted voting, and approval voting.
Materials
"High Tech Education" activity sheet, chalk board
Time
Introduction of problem statement (5 minutes), Individual work (10 minutes), Individual presentation and discussion (10 minutes), Small group work (15-20 minutes), Presentation of small-group work (15 minutes), Large group discussion (15 minutes)
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.
Related Mathematics Concepts:
Matrices, Permutations and combinations.
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), Data Collection and Analysis (3), Problem Solving techniques (5), Linking Concepts and Procedures (6)
Curriculum Integration
This activity will be used in the first year of the Integrated Math Program in a separate module. It could also be used in a traditional Algebra I class when discussing and applying Borda's method or in an Algebra II class when discussing permutations and combinations.
Further Investigation
This activity can be extended in a variety of ways by including descriptive data about student enrollment, current facilities, and possibly socio-economic information about the students enrolled at each of the schools.
Variations/Comments
This lesson could be modified by re-doing the activity as a site for nuclear waste facilities or new sports arenas for various cities in the Denver area. The methods for selecting sites should remain constant but may change when considering the idea of fairness and representation of the people in each city as the consequences of each site are on opposite ends of the spectrum.
References/Resources
Crisler, N., Fisher, P., & Froelich, G. (1994). Discrete Mathematics Through Applications. New York: W.H. Freeman and Company.