Lesson Plans: Applications of Graph Theory

1. An E3 Presentation Ralph Cox with Special Thanks to Dr. S. Butenko TAMU and Ronnie Gonzales Lesson Plans: Applications of Graph Theory

2. Overview • Ronnie & I studied Delta Airlines • Converted flight schedules to a matrix • Calculated and plotted • Teach the TEKS through Graph Theory • Lesson 1 • Create graphs • Count edges • Create pie charts and histograms • Lesson 2 • Solve two step equations • Miscellaneous Resources

3. Delta Airline Schedule • Flight schedule to matrix • Undirected graph • Each city a node • Each flight an edge • Calculate edge density • Calculate Degree vs Node Regression • Plot the graph • Convert from matrix to .dot file • Define Edge: Dallas--Houston is an edge. • Use Graphviz to generate plot

4. Delt Flight Map

5. Simplified Version • Lesson 1 • Student learns to draw graph given edge • Student learns to draw graph given matrix • Student learns to count edges to find degrees • Student practices making table of degrees and nodes • Student practices making pie chart out of table • Student practices making histogram out of table • Student practices making scatter plot out of table • Student uses calculator to find linear and exponential regression

6. Simplified Version • Lesson 2 • Student practices solving linear equation using concrete methods • Student makes graph of processes that can be used to solve linear equation • Student sees simple path is standard algorithm • Student learns to solve simple linear equation • Computer Time • Students can play on computer with graphviz and make graphs • Students can go to website for tutorial • Students can play logic games

7. Lesson 1 • Analyze small set flight data • Start with list of edges • Create list of nodes • Create Graph • Larger set of data from airline.xls • Start with matrix • Create list of nodes and edges • Figure out degree vs. nodes • Draw pie chart and histogram • Calculate regression

8. Activity 1 • Draw the graph for this set of edges • Louisville – OrlandoTampa – ColumbusColumbus – OrlandoKey West – OrlandoOrland -- Fort LauderdaleOrlando – Nassau • List the nodes • Draw the several different graphs

9. Activity 1 • Graph from Graphviz

10. Activity 2 • Given the data in Matrix from • List Nodes • List Edges • Draw Graph • Count edges on each node to find degree • Make table • List Degrees • How many nodes of each degree • Color the nodes • No adjacent node same color • Minimum color

11. Activity 2

12. Activity 3 • Start with Table created in prior activity • Identify dependent and independent variable • Identify domain and range • Bin data • Combine a few degrees into one point • For example count every 5 • Make Pie chart • Figure percentage for each interval • How many degrees for each interval • Make Histogram • Look at range to figure out scale

13. Activity 3

14. Activity 3

15. Activity 3

16. Activity 4 • Make a linear scatter plot • What is independent variable and domain • What is the dependent variable and range • Label axis • Make sure to review how to number axis! • Make Log-Log scatter plot • Same except for on Log paper • Exponential becomes a line! • Use a ruler to draw best line

17. Activity 4

18. Activity 4

19. Activity 4 • Enter data into the calculator • Reset calculator 2nd mem 7,1,2 • Press stat and select edit • Enter data in L1 and L2 • Adjust window and turn on plot with 2nd 'stat plot' • Check out graph • Calculate regression • Linear regression: Stat Calc linreg • Exponential: Stat Calc expreg • Enter both equations into y=

20. Review Lesson 1 • Convert list to list of edges and list of nodes • Make a graph • Count edges on each node • Create Table • Not necessary to bin • Create Pie Chart and Histogram • Create Scatter Plot • Calculate regression

21. Lesson 2 • Student discovers optimal solution for solving simple linear equation • Use graphs to show several processes • Start with any equation • Student suggest steps • Continue until reach an answer • Examine graph • Graph will have several paths that lead to the solution • One will be shorter than the rest

22. Activity 1 • Start by solving equations using concrete techniques • Then graph various processes • Start with 2x+3=7 • Accept student suggestions, write in graph form • Continue until have several paths from equation to solution. • Which path is shortest? • Student should move from guess and check to optimal process • At end student should be able to apply standard algorithm.

23. Activity 1

24. Review lesson 2 • Review solving linear equations using concrete methods • Start graphing process to solve linear equation • Choose shortest path for best process • Generalize to standard algorithm • Practice applying algorithm

25. Resources • Graphviz • Uses simple text files and simple commands • Students can make own graphs on computer • See .dot files in lesson. • American Scientist- Practical Graph Theory • http://www.americanscientist.org/template/AssetDetail/assetid/14708 • Graph theory tutorial • http://www.utm.edu/cgi-bin/caldwell/tutor/departments/math/graph/intro

26. Games! • Logic games • http://plastelina.net/ • Wolf sheep and cabbage • Family Crisis • Missionary and Cannibals • Four Random Books • Plait, Philip; Bad Astronomy; 2002 • Levin, Janna; How the Universe Got it's Spots; 2002 • Kranz, Gene; Failure is not an Option; 2001 • Murray, Margaret; Women Becoming Mathematicians; 2000