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

2. 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

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

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

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

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

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

8. 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)

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

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

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

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

13. Critical Path
Assembly line
Baking
Chemistry Lab

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

15. Hamiltonian Path
Traveling Salesman

16. Euler’s path
Seven Bridges

17. 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

18. 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

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

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

21. 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
by the National Council of Teachers of Mathematics
Summary: I used the geometry standards and some of the
problems listed on the website.
Summary: I used ideas for classroom activities from this website