Instructor’s Resource Manual
Discrete Mathematics for
Teachers
PRELIMINARY EDITION
Ed Wheeler
Gordon College
Jim Brawner
Armstrong Atlantic State University
John Bakken
Ed Wheeler
Gordon College
Jim Brawner
Armstrong Atlantic State University
John Bakken
Instructor’s Resource Manual
to accompany
Discrete Mathematics for
Teachers
PRELIMINARY EDITION
Ed Wheeler and Jim Brawner
CONTENTS
Preface v
Course Outlines
vi
Solutions to Exercises 1
Chapter 1
An Introduction to Logic 1
Chapter 2
Sets, Functions, and Sequences 23
Chapter 3
An Introduction to Mathematical Proof 35
Chapter 4
Graph Theory 57
Chapter 5
Trees 89
Chapter 6
Combinatorics 111
Chapter 7
Probability 131
Chapter 8
Discrete Applications in Political Theory 143
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PREFACE
In writing Discrete Mathematics for Teachers we have attempted to provide a text that will place
design of the course squarely in the hands of the teacher. Our hopes are that many different kinds of
courses will be taught using this text; indeed, we have already taught many different kinds of courses
using draft versions. To this end we have attempted to preserve flexibility in several different directions.
In the first place, we have attempted to preserve flexibility in terms of the topics chosen and the order
presented. We have indicated this flexibility with a prerequisite chart in the preface of the text and in
suggested course outlines that will follow later in this Instructor’s Resource Manual. However, even in
this chart and in these course outlines we have not indicated the full extent of the flexibility. In teaching
from this text to broad audiences, we often start with Sections 8.1 and 8.2 from the last chapter of the
book to startle students into realizing that this course will not be like many others they have experienced
in mathematics. In a similar vein, we have taught Chapter 6 on combinatorics before teaching Chapter 4
on graph theory, suggesting that instructors do have control of the order in which they present material.
In the second place, we have preserved flexibility in terms of pedagogy. This text can be used in a
laboratory/workshop setting. The instructor can use the Exploratory Exercises and selected regular
exercises for guided cooperative explorations in class and then assign readings from the text, writing
projects, and a few regular exercises for work outside of class. On the other hand, the text is well-suited
for traditional presentation in which the instructor lectures and answers questions in class and sends the
students off to work on exercises and selected exploratory exercises outside of class. We have most
often used the text in classrooms that feature a hybrid of traditional and laboratory settings. We often
involve the students in groups with exploratory exercises either at the beginning of class or after a short
presentation, ask other students to present on exploratory projects completed at home either individually
or in groups, discuss relevant homework, and then send students off with a number of exercises to be
done on their own.
In the third place, we have preserved flexibility in terms of attention to careful mathematical
reasoning. Although we firmly believe that those who specialize in the teaching of mathematics, whether
at the K-5 level or at the middle-grades or secondary level, need to grapple with what constitutes a
mathematical proof, the classes we teach are not always populated by these persons alone. To this end,
we have written the text in such a way that it is very possible to pass through the whole text, skipping
Chapter 3, without attention to these issues. On the other hand, by review of Chapter 1, careful study of
Chapter 3, and careful attention to exercises labeled Proof Exercise in subsequent chapters, students are
provided with an introduction to what it means to move from pattern to conjecture to carefully reasoned
proof in mathematical thought. Our efforts are unusual in the mathematical literature because we discuss
and model the need to prepare to prove theorems, giving careful attention to reflection on examples,
definitions, and previous theorems as we develop a plan to produce a proof.
A Final Note: For optimal use of these materials, it is important to carefully choose the pace at
which the material will be covered. Although the book can certainly be covered with the traditional
section-a-day pace, instructors may find that to occasionally linger on a section, spending some class time
on exploratory exercises in groups and then soliciting discussion from groups after the work is done, will
provide a deeper understanding than we usually expect from our mathematics students. We have enjoyed
the process of writing this text and teaching from it. We hope that you will enjoy it also. We are always
eager to hear suggestions for improvement.
Ed Wheeler
edw@gdn.edu
iv
Jim Brawner
James.Brawner@armstrong.edu
COURSE OUTLINES
Because this text has been written to serve the needs of a variety of different pedagogical approaches,
audiences, and content objectives, there are many different course outlines that work well with the text.
Rather than try to describe all of them, we will describe three outlines in hopes that they may serve as
models in terms of building others. Because some of the pedagogical approaches that might be used with
the text will involve multiple days spent on some sections, we identify 20 - 22 key sections for each
outline along with additional sections that might be added should time permit.
Course Outline 1:
Audience: Broad, pre-service and in-service students, K-5 through Middle School
Preparation: Mixed
Content Objectives: A broad overview of the topics of discrete mathematics
Chapter 1: Sections 1.1, 1.2, 1.3
Chapter 2: Sections 2.1, 2.2, 2.3
Chapter 3: Section 3.1
Chapter 4: Sections 4.1, 4.2, 4.3, 4.4
Chapter 5: Sections 5.1
Chapter 6: Sections 6.1, 6,2, 6.3, 6.4
Chapter 7: Sections 7.1, 7.2
Chapter 8: Sections 8.1, 8.2, 8.3, 8.4
Additional Topics: In Sections 2.4, and 2.5 you will find some very interesting material
linking sequences, web plots, graphing calculators, and spreadsheets that is not widely
available in textbooks. Sections 4.5 and 4.6 on planar graphs and graph coloring and/or
6.5 and 6.6 on the Pigeonhole and Inclusion/Exclusion Principles give the students a
broader look at either graph theory or combinatorics, the core areas in discrete
mathematics. Completing Sections 7.3 and 7.4 will allow students to develop more
understanding of the critical discipline of probability.
Course Outline 2:
Audience: Broad, pre-service and in-service students, K-5 through Middle School
Preparation: Mixed
Content Objectives: Thorough coverage of graph theory and combinatorics
Chapter 2: Sections 2.1, 2.2, 2.3
Chapter 4: Sections 4.1, 4.2, 4.3, 4.4, 4.5, 4.6
Chapter 5: Sections 5.1, 5.2, 5.3, 5.4
Chapter 6: Sections 6.1, 6,2, 6.3, 6.4, 6.5, 6.6
Chapter 7: Sections 7.1, 7.2, 7.3
Additional Topics: In Sections 2.4, and 2.5 you will find some very interesting material
linking sequences, web plots, graphing calculators, and spreadsheets that is not widely
available in textbooks. Completing Section 7.4 will allow students to develop more
understanding of the critical discipline of probability. Chapter 8 on Discrete Applications
in Political Theory will give students a view of the utility of mathematics that they may
have never experienced before.
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Course Outline 3:
Audience: Pre-service and in-service mathematics specialists, K-5, middle grades, and
secondary
Preparation: Stronger
Content Objectives: Thorough coverage of discrete mathematics special attention to
careful mathematical reasoning
Chapter 1: Sections 1.1, 1.2, 1.3, 1.4 as needed
Chapter 2: Sections 2.1, 2.2, 2.3 as needed
Chapter 3: Section 3.1, 3.2, 3.3
Chapter 4: Sections 4.1, 4.2, 4.3, 4.4, 4.5, 4.6 with attention to exercises labeled Proof
Exercise.
Chapter 5: Sections 5.1, 5.2 with attention to exercises labeled Proof Exercise.
Chapter 6: Sections 6.1, 6.2, 6.3, 6.4, 6.5, 6.6 with attention to exercises labeled Proof
Exercise.
Chapter 7: Sections 7.1, 7.2, 7.3 with attention to exercises labeled Proof Exercise.
Additional Topics: Immediately after discussing techniques of proof in Sections 3.1 –
3.3, Sections 3.4 and 3.5 give opportunity to practice those modes of thinking while
investigating elementary number theory. Sections 5.3 and 5.4 enable one to extend
knowledge of trees, one of the most applicable areas of graph theory, while adding
Section 7.4 gives additional depth in probability, depth that is often missing in the
preparation of prospective teachers.
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