Shana Norman Dec 11, 2003 Final Project Vertex-Edge Graphs Shana Norman Dec 11, 2003 Final Project

Introduction Geometry curriculum is constantly changing NC trying to evolve to NCTM standards Vertex-edge graphs are in the NCTM curriculum for high school geometry

Why study vertex-edge graphs? Think visually Think spatially Has real world applications

Road Map Vertex-edge graphs Using vertex-edge graphs Famous algorithms NCTM standards

What is a vertex-edge graph? 5 vertices 5 edges

Another vertex-edge graph Edges contain distance Vertices represent a location

Useful vertex-edge graphs Optimal Routes Traveling salesman, Paved roads Efficient Scheduling Classes, Jobs, Tasks Planning Lights, Bridges

Paved Road Problem 7 towns, pave roads so people can get from every town to every other town on a paved road. (minimize the distance)

More Paved Road Minimal spanning tree Two types: Minimizes the sum of the distances Two types: Kruskal’s algorithm Prim’s algorithm

Kruskal’s Algorithm Choose shortest distance AB, BD, EF, BF, FG, CD Total length = 88

Prim’s Algorithm Choose connected shortest path Retrace steps if needed AB, BD, DC, BF, EF, FG Total length = 88

More vertex-edge graphs Critical Path 7 Tasks Hours at each vertex

Critical Path Assembly line Baking Chemistry Lab

Assembly Line Critical Path = 1, 4, 5 or 6, 7

Hamiltonian Path Traveling Salesman

Euler’s path Seven Bridges

Quick Review Vertex-edge graph Example of vertex-edge graphs Paved road problem Krusal’s algorithm Prim’s algorithm Critical path Hamiltonian path Euler’s path

National Council of Teachers of Mathematics Want students to think and reason spatially Vertex-edge problems allow students to think beyond what information is given

Answers to problems Fully specify algorithm Compare algorithms Steps taken to achieve goal Compare algorithms Consider which algorithms are more efficient.

Conclusion Vertex-edge graphs should be incorporated into the high school geometry curriculum Think visually Think spatially Has real world applications

Resources Discrete Mathematics with Applications By Susanna S. Epp Published 1990 Summary: I used vertex-edge problems, Kruskal’s algorithm, Prim’s algorithm, Euler’s path, and the Hamiltonian path from this book http://standards.nctm.org/document/chapter3/geom.htm by the National Council of Teachers of Mathematics Summary: I used the geometry standards and some of the problems listed on the website. http://dmoz.org/Science/Math/Education/Personal_and_Class_Pages Summary: I used ideas for classroom activities from this website