Lesson Plans: Applications of Graph Theory An E 3 Presentation Ralph Cox with Special Thanks to Dr. S. Butenko TAMU and Ronnie Gonzales

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

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

Delt Flight Map

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

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

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

Activity 1 ● Draw the graph for this set of edges – Louisville – Orlando Tampa – Columbus Columbus – Orlando Key West – Orlando Orland -- Fort Lauderdale Orlando – Nassau ● List the nodes ● Draw the several different graphs

Activity 1 ● Graph from Graphviz

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

Activity 2

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

Activity 3

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

Activity 4

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

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

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

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.

Activity 1

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

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 – / /14708 ● Graph theory tutorial – bin/caldwell/tutor/departments/math/graph/intro bin/caldwell/tutor/departments/math/graph/intro –

Games! ● Logic games – – 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