Title
Nature Trails (Greta Lawlor)
Goals
1) The students will explore the concept of graph theory using trees to determine the least number of trails possible to connect nature sites.
2) Students will individually use a strategy using trees to develop a system of trails.
3) Students will work cooperatively in small groups to determine the minimum number of trails and be able to communicate and justify their answer.
Abstract
This activity asks students to determine a minimum number of edges to connect the vertices of a graph in the context of developing nature trails to connect nature sites. Students are asked to work individually and then in small groups to determine the "best" answer.
Problem Statement
Discuss with students the impact of people on natural areas and the overcrowding of state and national parks. Colorado is struggling with accommodating campers, hikers, and tourists in the state parks. This activity will demonstrate how mathematics is used in every day life to help find solutions to problems.
Instructor Suggestions
1) Set the state by discussing the problem statement with your students.
2) Distribute the "Nature Trails" handout and allow the students to work individually on the problem.
3) After the students have had sufficient time to work toward a solution individually, have them come together in small groups and compare their work. They need to come to a consensus about the fewest number of trails and be ready to present and justify their answer.
4) When the small groups have come to consensus, have them present their solutions.
5) Discuss as a whole class the solutions presented and the advantages and disadvantages of each.
Materials
"Nature Trails" handout, overhead transparency, marker(s).
Time
Introduction (5 minutes), Individual work (10 minutes), Group work and presentations (25 minutes), group discussion (15 minutes.)
Mathematics Concepts
Discrete Mathematics Concepts
Graph theory - trees; paths and circuits
Related Mathematics Concepts
Matrices, algorithms
NCTM Standards Addressed
Problem Solving, Communication, Reasoning, Discrete Mathematics
Colorado Model Content Standards Addressed
Data Collection and Analysis (3), Problem Solving Techniques (5), Linking Concepts and Procedures (6)
Curriculum Integration
This activity could be integrated into an Algebra or Geometry class when trees or matrices are introduced. It may also be used when trees are introduced in probability/statistics. The variety of uses for trees should be stressed.
Further Investigation
This activity could be extended by giving distances between sites and asking for a trail that would hit all sites with a minimum distance. The question could be changed by allowing a trail to a site to be doubled back, or each trail used only once. The trails could be one direction only.
Variations/Comments
This activity could be used as an exploratory exercise to introduce trees, or as an assessment of graph theory using any set of restrictions.
References/Resources
Dossey, John A., Otto, Albert D., et. al. Discrete Mathematics. 1972. Glenview, Illinois:
Scott, Foresman and Company.