Title
Snowboarding at Eldora (Dan Snook)
Goals
1. Students will explore the concepts of paths, circuits, and shortest routes.
2. Students will apply various algorithms to find a shortest circuit. They will also
attempt to write their own algorithm.
Abstract
This activity, set in the context of finding paths through a ski area, focuses on the concept of finding minimal routes. Students will make graphs based on maps of a familiar location. Students will work individually and in groups to find a shortest route through the ski area. Students will need to explore algorithms previously presented in class.
Problem Statement
Explain to the students that this problem, though it sounds similar, is different than other examples that we have discussed. In the past, the problem has been to reach each vertex in the shortest manner. It was not necessary to use each edge. In this problem, it is necessary to traverse each edge also. Therefore, previously learned algorithms may fail to yield the desired results.
Instructor Suggestions
1. It should be noted that Nederland High and Eldora Ski Area have a good relationship. Many of our students ski and snowboard there regularly. It would be advisable to allow time for the students to look at the naps and discuss their recent "adventures" there. (This will occur anyway, so plan time into the lesson.)
2. Distribute the "Snowboarding at Eldora" worksheets. Explain to the students that it could have as easily been the "Skiing at Eldora" problem. (If you ski, then you know what I'm talking about!) Describe the parameters of the problem to the students.
3. Allow students to work individually on the problem. Then have them compare their results in small groups.
4. Have each group report their findings. Discuss the results, paying close attention to questions 3 through 6 ? Make sure students realize that the paths (with the exception of the Corona Traverse) are directed.
Materials
"Snowboarding at Eldora" worksheets, Eldora trail maps, dry erase board
Time
Distribution and examination of trail maps (10 minutes), Introduction of problem statement (5 minutes), Individual work (10 minutes), Small group work (10 minutes), Presentation of small group work and large group discussion (20 minutes)
Mathematics Concepts
Discrete Mathematics Concepts:
Paths, Circuits, Spanning Trees, Shortest Routes, Euler Paths and Circuits, Hamiltonian
Circuits and Paths, Graphs
Related Mathematics Concepts:
Algorithms
NCTM Standards Addressed
Problem Solving, Communication, Reasoning, Connections, Discrete Mathematics
Colorado and District Standards Addressed
Algebraic Methods (2), Data Collection and Analysis (3), Problem Solving Techniques (5), Linking Concepts and Procedures (6)
Curriculum Integration
This activity could be integrated into a Geometry class as an extension of reasoning and drawing conclusions.
Further Investigation
Once the shortest route is found, an estimate of the time needed to traverse the route could be found. Lift times are listed on the map, as well as the length of the longest run. Students would need to approximate their speed and the total downhill distance to find a solution.
Variations/Comments
Eldora Ski Area provides our school with 4 passes to be given to deserving students over the weekend. A student, nominated by a teacher, picks up the pass Friday after school, and returns the pass Monday morning. As added incentive, I would award the passes to the group presenting the best solution.
References/Resources
Colorado Model Mathematics Standards Task Force. (1995) Colorado model content standards for mathematics.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author: