Understanding elementary algebra with geometry : a course for college students

707 Pages·2006·8.17 MB·English

by  Lewis Hirsch

Study
1.1 Studying Algebra—How Often? 8
Skills
1.2 Previewing Material 15
1.3 What to Do First: Reviewing Material 21
1.4 Doing Exercises 27
1.5 Reading Directions 35
1.6 Forming a StudyGroup 40
2.1 Comparing and Contrasting Examples 54
2.2 Coping with Getting Stuck 62
2.3 Seeking Help 72
2.4 Reviewing Old Material 78
2.5 Reflecting 87
2.6 Checking Your Work and Estimation 94
3.1 Preparing for Exams: Is Doing Homework Enough? 99
3.2 Preparing for Exams: When to Study 108
3.3 Preparing for Exams: Study Activities 122
3.4 Preparing for Exams: Making Study Cards 131
3.5 Preparing for Exams: Using Study Cards 139
3.6 Preparing for Exams: Reviewing Your Notes and Text; Reflecting 143
3.7 Preparing for Exams: Using Quiz Cards to Create a Practice Test 145
4.1 Taking an Algebra Exam: Just Before the Exam 153
4.2 Taking an Algebra Exam: Beginning the Exam 161
4.3 Taking an Algebra Exam: What to Do First 170
4.4 Taking an Algebra Exam: Dealing with Panic 179
4.5 Taking an Algebra Exam: A Few Other Comments About Exams 186
5.1 Reviewing Your Exam: Diagnosing Your Strengths and Weaknesses 213
5.2 Reviewing Your Exam: Checking Your Understanding 227
5.3 Preparing for a Comprehensive Final Exam 238
Number
Rational numbers e.g.,–2, 1.62, 7
3
Systems Irrational numbers
Integers e.g.,–2, 7, –3 e.g., 3, π, 35
Whole numbers e.g., 0, 1, 3
Natural numbers
e.g., 1, 5, 8
Real numbers
Some Fundamental Principle of Fractions
Important (cid:1)a(cid:1) (cid:2) a(cid:1)(cid:1)k b, k (cid:3) 0
Properties b b(cid:1)k
and Formulas
Zero-Product Rule
If a(cid:1)b (cid:2) 0, then either a (cid:2) 0 or b (cid:2) 0 (or both).
Properties of Radicals for Nonnegative a and b
1. (cid:1)a(cid:2)b (cid:2) (cid:1)a(cid:2)(cid:1)b(cid:2)
(cid:3)(cid:4)
a (cid:1)(cid:2)
2. (cid:1) = (cid:1)a b (cid:3) 0
b
(cid:1)(cid:2)
b
Equations of a Line
Point-slope form: y (cid:4) y (cid:2) m(x (cid:4) x )
1 1
Slope-intercept form: y (cid:2) mx (cid:5) b
The Quadratic Formula
The solutions to the equation ax2 (cid:5) bx (cid:5) c (cid:2) 0 are
(cid:4)b (cid:6) (cid:1)b(cid:2)2 (cid:4) 4(cid:2)ac
x (cid:2) (cid:1)(cid:1)(cid:1) (where a =/ 0)
2a
Understanding
Elementary Algebra
with Geometry
A Course for College Students Sixth Edition
Lewis Hirsch
Rutgers University
Arthur Goodman
Queens College,
City University of New York
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Understanding Elementary Algebra with Geometry: A Course for College Students, 6e
Lewis Hirsch, Arthur Goodman
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Preface to the Instructor
Purpose
Understanding Elementary Algebra with Geometry, 6th edition, is an attempt on our
part to offer a textbook that reflects our philosophy—that students can understand
what they are doing in algebra and why.
We offer a view of algebra that takes every opportunity to explain why things
are done in a certain way, and to show how supposedly “new” topics are actually just
new applications of concepts already learned.
This book assumes only a basic knowledge of arithmetic. Appendix A includes
a brief review of the arithmetic of decimals and percents.
Pedagogy
We believe that a student can successfully learn elementary algebra by mastering a
few basic concepts and being able to apply them to a wide variety of situations.
Thus, each section begins by relating the topic to be discussed to material previously
learned. In this way the students can see algebra as a whole rather than as a series of
isolated ideas.
Basic concepts, rules, and definitions are motivated and explained via numeri-
cal and algebraic examples. Formal proofs have been avoided except for those occa-
sions when they illuminate the discussion.
Concepts are developed in a series of carefully constructed illustrative exam-
ples. Through the course of these examples we compare and contrast related ideas,
helping the student to understand the sometimes subtle distinctions among various
situations. In addition, these examples strengthen a student’s understanding of how
this “new” idea fits into the overall picture.
Every opportunity has been taken to point out common errors often made by
students and to explain the misconception that often leads to a particular error.
Basic rules and/or procedures are highlighted so that students can find impor-
tant ideas quickly and easily.
A spiral approach has been used for the presentation of some more difficult top-
ics. That is, a topic is first presented at an elementary level and then returned to at
increasing levels of complexity.
For example,
• Simple rational expressions are covered in Chapter 4, while more complex
rational expressions are dealt with in Chapter 9.
• Factoring is covered in Section 2.3, and in Chapters 8 and 9.
• Applications are covered repeatedly throughout the text.
A number of topics have been motivated by describing an application in which
a particular type of equation is used to model a situation. We then return to this very
same application after the necessary algebraic techniques have been developed. For
example,
• See the introductions to Chapter 4, Chapter 5, Chapter 8, Chapter 9, Section
10.5, and Chapter 11.
iv Preface to the Instructor
Additionally, special attention has been paid to using pattern recognition and
numerical examples to illustrate the mathematical model being used in solving various
applications.
Features
Examples The various steps in the solutions to examples are explained in detail.
Many steps appear with annotations (highlighted in color) that involve the student in
the solution. These comments explain how and why a solution is proceeding in a cer-
tain way.
Exercises There are over 3,000 homework exercises. Not only have the exercises
been matched odd/even, but they have also been designed so that, in many situations,
successive odd-numbered exercises compare and contrast subtle differences in apply-
ing the concepts covered in the section. Additionally, variety has been added to the
exercise sets so that the student must be alert as to what the problem is asking.
For example, the exercise sets in Sections 4.3 and 9.3, which deal primarily with
adding rational expressions, also contain some exercises on multiplying and dividing
rational expressions. The exercise set in Section 4.4 on solving fractional equations also
asks the student to combine rational expressions. The exercise sets in Sections 8.4 and
11.1, which deal primarily with quadratic equations, contain some linear equations
aswell.
Study Skill One of the main sources of students’ difficulties is that they do not
know how to study algebra. In this regard we offer a unique feature. Each section in the
first four chapters concludes with a Study Skill. This is a brief paragraph discussing
some aspect of studying algebra, doing homework, or preparing for or taking exams.
Our students who have used the earlier editions of this book indicated that they found
the Study Skills very helpful. The Algebra Study Skills sections in this text are based
on ideas in the book Studying Mathematicsby Mary Catherine Hudspeth and Lewis R.
Hirsch (1982, Kendall/Hunt Publishing Company, Dubuque, Iowa). For more informa-
tion and ideas on improving mathematics learning, we direct you to that book.
Questions for Thought Almost every exercise set contains Questions for
Thought, which offer the student an opportunity to thinkabout various algebraic ideas.
They may be asked to compare and contrast related ideas, or to examine an incorrect
solution and explain why the solution is wrong. The Questions for Thought are intended
to be answered in complete sentences and in grammatically correct English. The Ques-
tions for Thought can also be used by instructors as a vehicle for having students write
across the curriculum.
Margin Comments Margin Comments have been added where appropriate in
order to involve the students more actively in the learning process as they read the text.
A margin comment will usually seek to emphasize a point made in the text presentation
or ask a question requiring the student to focus attention on a particularly crucial aspect
of the discussion.
Different Perspectives Different Perspectives boxes appear wherever there is an
opportunity to highlight the connection between algebraic and geometric aspects of the
same concept. In this way the student is encouraged to think about mathematical ideas
from more than one point of view.
Thinking Out Loud Thinking Out Loud is a feature in which the solution to
certain examples is presented in a question-and-answer format so that students can
see examples of the thought processes involved in approaching and solving new or
unfamiliar problems. This helps students develop more appropriate problem-solving
strategies.
Preface to the Instructor v
Mini-Review Most sections contain a Mini-Review, which consists of exercises
that allow students to periodically review important topics as well as help them prepare
for the material to come. These Mini-Reviews afford the student additional opportunity
to see new topics within the framework of what they have already learned.
Chapter Summary Each chapter contains a chapter summary describing the
basic concepts in the chapter. Each point listed in the summary is accompanied by an
example illustrating the concept or procedure.
Review Exercises There are over 750 review exercises. Each chapter contains a
set of chapter review exercises and a chapter practice test. Additionally, there are four
cumulative review exercise sets and four cumulative practice tests following Chap-
ters3, 6, 9, and 11. These offer the student more opportunities to practice choosing the
appropriate procedure in a variety of situations.
Answer Section The answer section contains answers to all the odd-numbered
exercises, as well as to allthe mini-reviews, chapter and cumulative review exercises,
and practice test problems. The answer to each verbal problem contains a description
of what the variable(s) represent and the equation (or system of equations) used to solve
it. In addition, the answers to the cumulative review exercises and cumulative practice
tests contain a reference to the section in which the relevant material is covered.
Using a Calculator As in previous editions, exercises that require the use of a
calculator have not been specially designated. We assume that the calculator is at the
student’s side for all problems, and part of the learning process is determining when a
calculator is appropriate and/or necessary. Appendix B discusses the basic use of a sci-
entific calculator. In addition, Appendix C discusses some graphing calculator issues
referenced to examples in the text.
Functions Appendix D, An Introduction to Functions, provides a brief discussion
of the concept of a function and explains and illustrates the use of function notation.
Icons point students to material contained on the Brooks/Cole Website and
ontheInteractiveVideoSkillbuilderCD-ROMthataccompaniesthetext.
New to the Sixth Edition
• The THINKING OUT LOUD feature has been expanded to more examples in
order to more fully support stronger active learning for the students. This feature
presents solutions to certain examples in a question-and-answer format, so that stu-
dents can see examples of the thought processes involved in approaching and solv-
ing new or unfamiliar problems.
• AnewSTUDYSKILLhasbeenaddedtotheSTUDYSKILLSfeature.Thelatest
studyskillencouragesandinstructsstudentsonhowtoformstudygroups.Author
and reviewer experience indicates that students who participate in study groups
enjoymoresuccessandhaveamorepositiveexperienceintheirmathcourses.
• More than 200 ADDITIONAL EXERCISES have been integrated into the text to
allow for amore gradual progression of difficulty as the student moves through the
exercisesets.
• Many of the exercises have been rewritten and numerous applications have been
updated.
• 40 HOURS PER WEEK OF ONLINE TUTORING are available to your students
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vi Preface to the Instructor
Teaching Tools for the Instructor
Complete Solutions Manual (0534-999743) The Complete Solutions
Manual provides worked-out solutions to all of the problems in the text.
Text-Specific Videotapes (0534-999778) These text-specific videotape
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Test Bank (0534-999751) The Test Bank includes multiple tests per chapter as
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iLrn Instructor Version™ (0534-99976X) Providing instructors and stu-
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fully integrated teaching and learning system. iLrn ties together five fundamental learn-
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TLE Labs—The ideal upgrade for any iLrn™ Tutorial! Each of the 15
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ronment. Ask your Thomson representative about TLE Labs! They’re a great value for
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College Success Factors Index Developed by Edmond Hallberg, Kaylene
Hallberg, and Loren Sauer
Help your students assess their college success needs! This unique online assess-
ment system allows students to identify the behaviors and attitudes that will help them
succeed in college. Students can take the 80-statement survey, based on eight important
areas that have been proven to correlate with college success, in a password-protected
area right from their own computer. A special section for instructors offers information
and advice on how to administer, interpret, and report the data. Contact your Thomson
Brooks/Cole representative for packaging information.
JoinIn™ on TurningPoint® Thomson Brooks/Cole is pleased to offer you
book-specific JoinIn™ content for electronic response systems tailored to Under-
standing Elementary Algebra with Geometry. You can transform your classroom
and assess your students’ progress with instant in-class quizzes and polls. Our exclu-
sive agreement to offer TurningPoint®software lets you pose book-specific questions
and display students’ answers seamlessly within the Microsoft®PowerPoint®slides of
your own lecture, in conjunction with the "clicker" hardware of your choice. Enhance
Preface to the Instructor vii
how your students interact with you, your lecture, and each other. Contact your local
Thomson representative to learn more.
For the Student
Student Solutions Manual (0534-999786) The Student Solutions Manual
provides worked-out solutions to the odd-numbered problems in the text.
Thomson Brooks/Cole Mathematics Web site http://mathematics.brooks
cole.com Visit us on the web for access to a wealth of free learning resources, includ-
ing graphing calculator tutorials, tools to address math anxiety, Historical Notes, an
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iLrn™ Tutorial Student Version (0534-999735) Featuring a variety of
approaches that connect with all types of learners, iLrn™ Tutorial offers text-specific
tutorials that require no set up by instructors. Students can begin exploring active exam-
ples from the text by using the access code packaged free with a new book. iLrn Tuto-
rial supports students with explanations from the text, examples, step-by-step problem
solving help, unlimited practice, and chapter-by-chapter video lessons. With this self-
paced system, students can even check their comprehension along the way by taking
quizzes and receiving feedback. If they still are having trouble, students can easily
access vMentor™ for online help from a live math instructor. Students can ask any
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computer microphones to speak with the instructor. While designed for self-study,
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vMentor™…live, online tutoring Packaged FREE with every text! Accessed
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Interactive Video Skillbuilder CD-ROM (0534-999794) Think of it as
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lesson are shown first so that students can try working them before watching the solu-
tion. To help students evaluate their progress, each section contains a 10-question Web
quiz (the results of which can be e-mailed to the instructor) and each chapter contains
a chapter test, with the answer to each problem on each test. A new learning tool on this
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• MathCue Skill Builder—Presents problems to solve, evaluates answers and tutors
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