Transcript essap.tamu.edu

Lesson Plans:
Applications of Graph Theory
An E3 Presentation
Ralph Cox
with Special Thanks to
Dr. S. Butenko
TAMU
and
Ronnie Gonzales
Overview
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Ronnie & I studied Delta Airlines
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Teach the TEKS through Graph Theory
Lesson 1
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Create graphs
Count edges
Create pie charts and histograms
Lesson 2
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Converted flight schedules to a matrix
Calculated and plotted
Solve two step equations
Miscellaneous Resources
Delta Airline Schedule
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Flight schedule to matrix
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Undirected graph
Each city a node
Each flight an edge
Calculate edge density
Calculate Degree vs Node Regression
Plot the graph
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Convert from matrix to .dot file
Define Edge: Dallas--Houston is an edge.
Use Graphviz to generate plot
Delt Flight Map
Simplified Version
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Lesson 1
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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
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Lesson 2
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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
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Students can play on computer with graphviz
and make graphs
Students can go to website for tutorial
Students can play logic games
Lesson 1
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Analyze small set flight data
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Start with list of edges
Create list of nodes
Create Graph
Larger set of data from airline.xls
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Start with matrix
Create list of nodes and edges
Figure out degree vs. nodes
Draw pie chart and histogram
Calculate regression
Activity 1
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Draw the graph for this set of edges
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Louisville – Orlando
Tampa – Columbus
Columbus – Orlando
Key West – Orlando
Orland -- Fort Lauderdale
Orlando – Nassau
List the nodes
Draw the several different graphs
Activity 1
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Graph from Graphviz
Activity 2
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Given the data in Matrix from
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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
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Start with Table created in prior activity
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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
Degrees
Nodes
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5
6
10
11
15
16
20
21
25
26
30
31
35
36
40
41
45
46
50
51
55
56
60
61
65
66
70
71
75
76
80
81
85
86
90
91
95
96
100
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22
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5
5
2
0
2
1
2
1
0
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1
1
0
0
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Mean Degree Nodes
Fit
3
16
42.59
8
16
13.9
13
22
7.99
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19
5.51
23
5
4.17
28
5
3.33
33
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2.76
38
1
2.35
43
2
2.04
48
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1.8
53
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1.61
58
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1.45
63
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1.32
68
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1.21
73
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1.11
78
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1.03
83
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0.96
88
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0.9
93
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0.85
98
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0.8
Activity 3
Pie Chart %nodes at each Degree
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13
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23
28
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38
43
48
53
58
63
68
73
78
83
88
93
98
Activity 3
Sam ple Airline Hist ograph
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1 7.5
Nod es
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1 2.5
10
7.5
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2.5
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28
33
38
43
48
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Deg ree
58
63
68
73
78
83
88
93
98
Activity 4
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Make a linear scatter plot
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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
Lin ear Scatter Plot
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Nodes
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12.5
10
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2.5
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Mean Degree
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80
90
100
Activity 4
Log Log Scat t er Plot
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Nodes
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Degree
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Activity 4
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Enter data into the calculator
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Calculate regression
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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
Linear regression: Stat Calc linreg
Exponential: Stat Calc expreg
Enter both equations into y=
Review Lesson 1
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Convert list to list of edges and list of nodes
Make a graph
Count edges on each node
Create Table
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Not necessary to bin
Create Pie Chart and Histogram
Create Scatter Plot
Calculate regression
Lesson 2
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Student discovers optimal solution for solving
simple linear equation
Use graphs to show several processes
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Start with any equation
Student suggest steps
Continue until reach an answer
Examine graph
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Graph will have several paths that lead to the
solution
One will be shorter than the rest
Activity 1
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Start by solving equations using concrete
techniques
Then graph various processes
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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
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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
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Graphviz
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Uses simple text files and simple commands
Students can make own graphs on computer
See .dot files in lesson.
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http://www.americanscientist.org/template/AssetDetail/assetid
/14708
American Scientist- Practical Graph Theory
Graph theory tutorial
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http://www.utm.edu/cgibin/caldwell/tutor/departments/math/graph/intro
Games!
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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