Advanced Mathematical
Decision Making:
Unit VII: Networks and Graphs
AMDM is a project of
The Texas Association of Supervisors of Mathematics and
The Charles A. Dana Center at The University of Texas at Austin
With support from the Greater Texas Foundation
2010
Advanced Mathematical Decision Making (2010)
Copyright, acknowledgments, table of contents
Copyright 2010, Charles A. Dana Center at The University of Texas at Austin
This 2010 implementation edition is copyrighted by the Charles A. Dana Center at The
University of Texas at Austin.
Unless otherwise indicated, the materials found in this resource are the copyrighted property
of the Charles A. Dana Center at The University of Texas at Austin (the University). No part of
this resource shall be reproduced, stored in a retrieval system, or transmitted by any means—
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methods yet to be invented—without express written permission from the University, except
under the following conditions:
1) The following guidelines address only the materials owned by the Charles A. Dana Center
at the University of Texas at Austin.
a) The people of Texas are granted by the University a nonexclusive license in
perpetuity—subject to the terms and conditions listed below (item 2)—to use the 2010
edition of the AMDM instructional materials (reflecting improvements based on the
2009–2010 pilot) in Texas classrooms and homes for the education of our children.
b) Teachers and administrators from outside Texas who participate in the Dana Center’s
AMDM professional development in summer 2010 (sign-up information here:
www.utdanacenter.org/pd) may use the 2010 edition of the AMDM instructional
materials in their classrooms and schools, subject to the terms and conditions listed
below (item 2).
c) Organizations or individuals other than those listed above (items a and b) must obtain
prior written permission from the Charles A. Dana Center at The University of Texas at
Austin for the use of the 2010 AMDM instructional materials, the terms of which may
be set forth in a copyright license agreement, and which may include the payment of a
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2) Terms and conditions: Any portion reproduced must retain the original copyright notice,
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Advanced Mathematical Decision Making (2010)
Copyright, acknowledgments, table of contents
About the Charles A. Dana Center at The University of Texas at Austin
The Dana Center works to raise student achievement in K–16 mathematics and science,
especially for historically underserved populations. We do so by providing direct service to
school districts and institutions of higher education; to local, state, and national education
leaders; and to agencies, nonprofits, and professional organizations concerned with
strengthening American mathematics and science education.
The Center was founded in 1991 in the College of Natural Sciences at The University of Texas
at Austin. Our original purpose—which continues in our work today—was to increase the
diversity of students who successfully pursue careers in science, technology, engineering, and
mathematics (STEM) fields.
We carry out our work by supporting high standards and building system capacity; developing
collaborations with key state and national organizations to address emerging issues; creating
and delivering professional supports for educators and education leaders; and writing and
publishing education resources, including student supports.
Our staff of more than 80 researchers and education professionals has worked intensively with
dozens of school systems in nearly 20 states and with 90 percent of Texas’s more than 1,000
school districts. As one of the College’s largest research units, the Dana Center works to
further the university’s mission of achieving excellence in education, research, and public
service. We are committed to ensuring that the accident of where a child attends school does
not limit the academic opportunities he or she can pursue.
For more information about the Dana Center and our programs and resources, see our
homepage at www.utdanacenter.org. To access our resources (many of them free), please
see our products index at www.utdanacenter.org/products. And for updates and background
on the Advanced Mathematical Decision Making project, see www.utdanacenter.org/amdm.
About the Texas Association of Supervisors of Mathematics
The mission of the Texas Association of Supervisors of Mathematics is to assist in promoting
effectiveness in the supervision, coordination, and teaching of mathematics, especially in the
elementary and secondary fields.
TASM accomplishes this by holding meetings for the presentation and discussion of papers; by
conducting public discussion groups, forums, panels, lectures, or other similar programs; by
conducting or promoting investigations for the purpose of improving the teaching of
mathematics; and by the publication of papers, journals, books, and reports, thus vitalizing
and coordinating the work of mathematics supervisors across Texas and bringing the interests
of mathematics to the attention and consideration of the larger education community in
Texas.
For more information about TASM, visit its website at www.tasmonline.net.
Charles A. Dana Center at The University of Texas at Austin
iii
Advanced Mathematical Decision Making (2010)
Copyright, acknowledgments, table of contents
About the development of this resource
The development and production of the AMDM student expectations excerpted in this
resource, as well as the AMDM instructional materials that constitute Units I through VII of
this resource, were supported by the Greater Texas Foundation, based in Bryan, Texas. Any
opinions, findings, conclusions, or recommendations expressed in this material are those of
the author(s) and do not necessarily reflect the views of the Greater Texas Foundation or The
University of Texas at Austin.
•
The AMDM student expectations excerpted in this resource were developed by a writing
group of 20 people, consisting primarily of Texas mathematics educators and
mathematicians at the secondary and postsecondary level, with additional assistance from
mathematicians outside Texas. These AMDM student expectations reflect many months of
development and feedback from across the state—and from expert reviewers outside the
state—as described in updates presented to the Texas State Board of Education in
November 2007, June 2008, and March 2009.
These or any other AMDM student expectations, if adopted by the state of Texas as Texas
Essential Knowledge and Skills, will become part of state law and thus available to all
Texans at no charge. The Dana Center will not claim copyright or charge usage fees for
the use of any AMDM TEKS.
•
The AMDM instructional materials (Units I–VII) were developed by mathematics teachers
and faculty from Texas and beyond, with support from the Texas Association of
Supervisors of Mathematics and others around the state who have donated their time or
served as consultants for the project. See the acknowledgments section for a complete
listing of the authors, reviewers, advisory team, and staff supporting this work. As noted
in the copyright language above, the people of Texas are granted by the University a
nonexclusive license in perpetuity to use the 2010 edition of the AMDM instructional
materials (reflecting improvements based on the 2009–2010 pilot) in Texas classrooms and
homes for the education of our children.
This resource was produced in Microsoft Word 2004 for Mac.
This version constitutes the 2010 implementation edition and reflects enhancements from the
pilot edition (2009). Released June 2010.
Art credits
p. VII-6, Dana Center; p. VII-16, Dana Center; p. VII-21, Dana Center staff; p. VII-22, Dana
Center staff; p. VII-23, Dana Center; p. VII-24, Dana Center; p. VII-26, Dana Center; p. VII-27,
Dana Center; p. VII-29, Dana Center; p. VII-30, Dana Center; p. VII-32, Dana Center; p. VII-33,
Dana Center; p. VII-34, Dana Center; p. VII-36, Dana Center; p. VII-37, Dana Center; p. VII-38,
Dana Center; p. VII-40, Dana Center; p. VII-41, Dana Center; p. VII-42, Dana Center; p. VII-45,
Dana Center; p. VII-51, Dana Center; p. VII-54, Dana Center; p. VII-56, Dana Center; p. VII-58,
Dana Center; p. VII-59, Dana Center; p. VII-60, Dana Center; p. VII-64, Dana Center; p. VII-65,
Dana Center; p. VII-68, Dana Center; p. VII-76, Dana Center; p. VII-78, Dana Center; p. VII-79,
Dana Center; p. VII-80, Dana Center; p. VII-81, Dana Center; p. VII-82, Dana Center; p. VII-83,
Dana Center; p. VII-84, Dana Center; p. VII-85, Dana Center; p. VII-94, Dana Center; p. VII-96,
Dana Center; p. VII-98, Dana Center; p. VII-101, Dana Center; p. VII-105, Dana Center
Charles A. Dana Center at The University of Texas at Austin
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Advanced Mathematical Decision Making (2010)
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Acknowledgments
Here we list alphabetically the authors and reviewers for AMDM Units I through VII. Affiliations
are listed to clarify individuals’ credentials and locations. When affiliations other than the
Charles A. Dana Center at The University of Texas at Austin are listed, it should be understood
that in most cases the individual was acting as an independent consultant and not necessarily
as a representative of his or her organization.
Project director
Cathy Seeley, senior fellow and AMDM project director, Charles A. Dana Center at the
University of Texas at Austin
Director of materials development
Pam Walker, senior program coordinator, Charles A. Dana Center at the University of Texas
at Austin
Pilot coordinator
Mary E. “Molly” Ewing, graduate research assistant, Charles A. Dana Center at the University
of Texas at Austin
Authors
Stuart Boersma, professor of mathematics, Central Washington University, Ellensburg,
Washington
Tom Butts, professor, Department of Science/Mathematics Education, The University of Texas
at Dallas, Dallas, Texas
Paul Gray, director of mathematics and science, Pearland Independent School District,
Pearland, Texas
Susan May, senior program coordinator, Charles A. Dana Center at the University of Texas
at Austin
Michael Ota, graduate research assistant, Charles A. Dana Center at the University of Texas
at Austin
Connie Richardson, instructor of mathematics, Midwestern State University, Wichita Falls,
Texas
Carmen Whitman, independent consultant, Mathematics for All Consulting
Reviewers
Mark L. Daniels, clinical associate professor, UTeach Science Program, The University of Texas
at Austin
Mary E. “Molly” Ewing, graduate research assistant, Charles A. Dana Center at the University
of Texas at Austin
Kelly Flickinger, mathematics teacher, Bowie High School, Austin Independent School District,
Austin, Texas
Stephen Maurer, professor of mathematics and chair, Department of Mathematics and
Statistics, Swarthmore College, Swarthmore, Pennsylvania
Margaret E. “Maggie” Myers, lecturer, Department of Computer Sciences, The University of
Texas at Austin
Cathy Seeley, senior fellow and AMDM project director, Charles A. Dana Center at the
University of Texas at Austin
Rita Tellez, mathematics instructional specialist, Ysleta Independent School District, El Paso,
Texas
Charles A. Dana Center at The University of Texas at Austin
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Advanced Mathematical Decision Making (2010)
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Professional development team
Kelly Flickinger, mathematics teacher, Bowie High School, Austin Independent School District,
Austin, Texas
Susan May, senior program coordinator, Charles A. Dana Center at the University of Texas at
Austin
Bonnie McNemar, independent mathematics consultant, Georgetown, Texas, and former
director of TEXTEAMS
Michael Ota, graduate research assistant, Charles A. Dana Center at the University of Texas
at Austin
Tricia Rothenberg, mathematics curriculum coordinator, Round Rock Independent School
District, Round Rock, Texas
Pam Walker, senior program coordinator, Charles A. Dana Center at the University of Texas
at Austin
Jim Wohlgehagen, mathematics coordinator, secondary curriculum and instruction, Plano
Independent School District, Plano, Texas
Professional development delivery team
Roger Burnham, mathematics teacher, McAllen Memorial High School, McAllen Independent
School District, McAllen, Texas
Joyce Collett, mathematics coordinator 6–12, Clear Creek Independent School District, League
City, Texas
Kelly Flickinger, mathematics teacher, Bowie High School, Austin Independent School District,
Austin, Texas
Jennifer Harper, mathematics teacher, Hightower High School, Fort Bend Independent School
District, Sugar Land, Texas
William “Bill” Jones, mathematics teacher, Clark High School, Northside Independent School
District, San Antonio, Texas
Frakeetta Marshall-Yancey, mathematics teacher, Hightower High School, Fort Bend
Independent School District, Sugar Land, Texas
Bonnie McNemar, independent mathematics consultant, Georgetown, Texas, and former
director of TEXTEAMS
Jonathan Newman, mathematics teacher, Billy Ryan High School, Denton Independent School
District, Denton, Texas
Michael Ota, graduate research assistant, Charles A. Dana Center at the University of Texas
at Austin
David Pulsipher, mathematics teacher, Plano East Senior High School, Plano Independent
School District, Plano, Texas
Connie Richardson, instructor of mathematics, Midwestern State University, Wichita Falls,
Texas
Tricia Rothenberg, mathematics curriculum coordinator, Round Rock Independent School
District, Round Rock, Texas
Miguel Salazar, mathematics teacher and department head, Ralls High School, Ralls
Independent School District, Ralls, Texas
Pam Walker, senior program coordinator, Charles A. Dana Center at the University of Texas
at Austin
Advisory team
Julie Acosta, secondary mathematics coordinator, McAllen Independent School District,
McAllen, Texas
Sandra Browning, assistant professor of mathematics education, University of Houston–Clear
Lake, Clear Lake, Texas
Charles A. Dana Center at The University of Texas at Austin
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Advanced Mathematical Decision Making (2010)
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Tom Butts, professor, Department of Science/Mathematics Education, The University of Texas
at Dallas, Dallas, Texas
Joyce Collett, mathematics coordinator 6–12, Clear Creek Independent School District, League
City, Texas
Wade Ellis, (retired) instructor of mathematics, West Valley Community College, Saratoga,
California
James A. Mendoza Epperson, associate professor of mathematics, The University of Texas at
Arlington, Arlington, Texas
Gregory Foley, Robert L. Morton professor of mathematics education, Ohio University, Athens,
Ohio
Linda Gann, instructional specialist, secondary mathematics, Northside Independent School
District, San Antonio, Texas
Sarah “Sallie” Kay Janes, academic dean, San Jacinto College, North Campus, Houston, Texas
Stephen Maurer, professor of mathematics and chair, Department of Mathematics and
Statistics, Swarthmore College, Swarthmore, Pennsylvania
Susan May, senior program coordinator, Charles A. Dana Center at the University of Texas
at Austin
Bonnie McNemar, independent mathematics consultant, Georgetown, Texas, and former
director of TEXTEAMS
Michael Ota, graduate research assistant, Charles A. Dana Center at the University of Texas
at Austin
Rita Tellez, mathematics instructional specialist, Ysleta Independent School District, El Paso,
Texas
Sherri Waddey, former consultant—teacher preparation and certification, Region 10 Education
Service Center, Richardson, Texas
Pam Walker, senior program coordinator, Charles A. Dana Center at the University of Texas
at Austin
Jim Wohlgehagen, mathematics coordinator, secondary curriculum and instruction, Plano
Independent School District, Plano, Texas
Linda Zientek, assistant professor of mathematics, Department of Mathematics and Statistics,
Sam Houston State University, Huntsville, Texas
Lana Zimmer, high school mathematics teacher, Clear Creek Independent School District,
League City, Texas
Research and evaluation team
Lesley Leach, research scientist associate III, Charles A. Dana Center at the University of
Texas at Austin
Cynthia Schneider, director of research and evaluation, Charles A. Dana Center at the
University of Texas at Austin
Editorial and production team
Amy Dolejs, webmaster and copyeditor, Charles A. Dana Center at the University of Texas
at Austin
Steve Engler, lead editor and production editor, Charles A. Dana Center at the University of
Texas at Austin
Rachel Jenkins, consulting editor, Charles A. Dana Center at the University of Texas at Austin
Sarah Searcy, copyeditor, Charles A. Dana Center at the University of Texas at Austin
Phil Swann, senior designer, Charles A. Dana Center at the University of Texas at Austin
With Special Thanks . . .
To the teachers participating in the 2009–2010 pilot and sharing their ideas for improving this
course.
Charles A. Dana Center at The University of Texas at Austin
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Advanced Mathematical Decision Making (2010)
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Table of Contents
This unit is one of seven that make up the 2010 implementation edition of the Advanced
Mathematical Decision Making instructional materials. The copyright and acknowledgments
for this unit apply to all the AMDM units of this 2010 edition.
Unit
Unit
Unit
Unit
Unit
Unit
I: Analyzing Numerical Data
II: Probability
III: Statistical Studies
IV: Using Recursion in Models and Decision Making
V: Using Functions in Models and Decision Making
VI: Decision Making in Finance
Unit VII: Networks and Graphs
Advanced Mathematical Decision Making
Unit VII: Networks and Graphs
Teacher Materials
Unit VII: Networks and Graphs: Overview .......................................... VII-1
Section VII.A: Circuits, Paths, and Graph Structures: Section Planning................ VII-3
VII.A.1: Euler Circuits and Paths .................................................................. VII-8
VII.A.2: Dominoes ................................................................................ VII-11
VII.A.3: Weighted Graphs ......................................................................... VII-13
VII.A.4: Hamiltonian Circuits and Paths ........................................................ VII-15
VII.A.5: Knight’s Tour .............................................................................. VII-18
VII.A Student
VII.A Student
VII.A Student
VII.A Student
VII.A Student
Activity
Activity
Activity
Activity
Activity
Sheet
Sheet
Sheet
Sheet
Sheet
1 Teacher Version ............................................... VII-21
2 Teacher Version ............................................... VII-29
3 Teacher Version ............................................... VII-32
4 Teacher Version ............................................... VII-36
5 Teacher Version ............................................... VII-40
Section VII.B: Spanning Trees: Section Planning ........................................... VII-43
VII.B.6: High-Speed Internet...................................................................... VII-47
VII.B.7: Minimal Spanning Trees ................................................................. VII-49
VII.B.8: Kruskal’s Algorithm ...................................................................... VII-51
VII.B Student Activity Sheet 6 Teacher Version ............................................... VII-53
VII.B Student Activity Sheet 7 Teacher Version ............................................... VII-58
VII.B Student Activity Sheet 8 Teacher Version ............................................... VII-62
Charles A. Dana Center at The University of Texas at Austin
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Advanced Mathematical Decision Making (2010)
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Section VII.C: Graph Coloring: Section Planning............................................ VII-67
VII.C.9: Map Coloring .............................................................................. VII-71
VII.C.10: Coloring Maps and Scheduling ........................................................ VII-73
VII.C Student Activity Sheet 9 Teacher Version ............................................... VII-76
VII.C Student Activity Sheet 10 Teacher Version ............................................. VII-79
Section VII.D: Program Evaluation and Review Technique (PERT) Charts:
Section Planning ................................................................................ VII-87
VII.D.11: Activity Graphs .......................................................................... VII-90
VII.D.12: Building a Robot......................................................................... VII-92
VII.D Student Activity Sheet 11 Teacher Version ............................................. VII-94
VII.D Student Activity Sheet 12 Teacher Version ........................................... VII-100
Advanced Mathematical Decision Making
Unit VII: Networks and Graphs
Student Materials
VII.A Student Activity Sheet 1: Euler Circuits and Paths ......................................... 1
VII.A Student Activity Sheet 2: Dominoes ........................................................... 9
VII.A Student Activity Sheet 3: Weighted Graphs................................................. 11
VII.A Student Activity Sheet 4: Hamiltonian Circuits and Paths................................ 15
VII.A Student Activity Sheet 5: Knight’s Tour ..................................................... 19
VII.B Student Activity Sheet 6: High-Speed Internet ............................................. 21
VII.B Student Activity Sheet 7: Minimal Spanning Trees ......................................... 25
VII.B Student Activity Sheet 8: Kruskal’s Algorithm .............................................. 29
VII.C Student Activity Sheet 9: Map Coloring ...................................................... 33
VII.C Student Activity Sheet 10: Coloring Maps and Scheduling ................................ 36
VII.D Student Activity Sheet 11: Activity Graphs ................................................. 44
VII.D Student Activity Sheet 12: Building a Robot ................................................ 51
Charles A. Dana Center at The University of Texas at Austin
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