# College Mathematics for Trades and Technologies, 10th Edition Solution Manual

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INSTRUCTORโS
RESOURCE MANUAL
C OLLEGE M ATHEMATICS
FOR T RADES AND T ECHNOLOGIES
TENTH EDITION
Cheryl Cleaves
Southwest Tennessee Community College
Margie Hobbs
Southwest Tennessee Community College
The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
development, research, and testing of the theories and programs to determine their effectiveness. The author and publisher
make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this
book. The author and publisher shall not be liable in any event for incidental or consequential damages in connection with,
or arising out of, the furnishing, performance, or use of these programs.
Reproduced by Pearson from electronic files supplied by the author.
Copyright ยฉ 2019, 2014, 2009 Pearson Education, Inc.
Publishing as Pearson, 330 Hudson Street, NY NY 10013
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form
or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the
publisher. Printed in the United States of America.
ISBN-13: 978-0-13-470761-7
ISBN-10: 0-13-470761-3
Instructorโs Resource Manual to accompany
College Mathematics for Trades and Technologies,
Tenth Edition
Cheryl Cleaves and Margie Hobbs
Contents
Preface
v
Introduction
1
Teaching Tips
4
Reproducible Activities
13
Teaching Aids
69
Transparency Masters
94
Selected Solutions
161
Even-Numbered Chapter Review Exercises
All Concept Analysis Exercises
Even-Numbered Practice Test Exercises
Even-Numbered Cumulative Practice Tests
Chapter 1
Review of Basic Concepts
163
Chapter 2
Review of Fractions
173
Chapter 3
Percents
185
Chapter 4
Measurement
198
Chapter 5
Signed Numbers and Powers of 10
205
Chapter 6
Statistics
213
Chapter 7
Linear Equations and Inequalities
224
Chapter 8
Formulas, Proportion, and Variation
239
Chapter 9
Linear Equations, Functions, and Inequalities in Two Variables
245
Chapter 10
Systems of Linear Equations and Inequalities
261
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Chapter 11
Powers and Polynomials
277
Chapter 12
Roots and Radicals
282
Chapter 13
Factoring
288
Chapter 14
Rational Expressions, Equations, and Inequalities
294
Chapter 15
Quadratic and Other Non-Linear Equations and Inequalities
306
Chapter 16
Exponential and Logarithmic Equations
336
Chapter 17
Geometry
347
Chapter 18
Triangles
356
Chapter 19
Right-Triangle Trigonometry
366
Chapter 20
Trigonometry with Any Angle
374
iv
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PREFACE
We have prepared this manual as a resource for instructors who will use this text. Having both
taught a number of years, we are well aware of the workloads that most instructors must manage
and the limited amount of time available for developing classroom activities and implementing
new materials into their programs. In addition, many programs have a large percentage of adjunct
faculty who bring many rich experiences to the classroom from a variety of careers; however, the
time constraints under which they function preclude development of supplementary curricular
materials.
Our aim is to assist you in your role as the facilitator of learning. We recognize that instructors
need a variety of materials to enable them to adapt their personal teaching style to accommodate a
variety of learning styles. We hope these materials will strengthen your program and we welcome
your comments and suggestions as these materials are continually refined and new materials are
developed.
Textbook
College Mathematics for Trades and Technologies, Tenth Edition, is designed for use in
face-to-face classrooms, online classrooms, business and industrial training programs, or learning
laboratories. It is easily adapted to a variety of instructional delivery modes.
Examples show solutions step-by-step with explanatory marginal notes. Rules, formulas,
procedures, and definitions are highlighted. Tip boxes draw attention to special cautions,
procedures, and calculator suggestions.
Section Exercises are placed at the end of each section and are coded by number to the Learning
Outcomes for the section. Selected Section Exercises reference a specific example or examples
from the section to enable students to practice by modeling. Every example in the section is
referenced in at least one exercise. The Chapter Review Exercises are located at the end of each
chapter and are identified by section number. Each chapter has a Chapter Review of Key Concepts
that lists outcomes, rules, and examples. Concepts Analysis exercises and a Practice Test finish
the chapter. The answers to all Section Exercises, the odd-numbered Chapter Review Exercises,
the odd-numbered Practice Test exercises, and the Cumulative Practice Text are located at the end
of the text.
We use a practical, learn-by-doing approach to mathematics with both informal language and
formal mathematical terminology. The active learning emphasis is also promoted through the
Reproducible Activities in this manual.
For a more thorough description and examples of the features of the text, please refer to the
Preface in the text. You will also want to help your students become familiar with the text and the
resources that accompany the text.
Instructorโs Resources
Suggestions for using the Reproducible Activities, teamwork projects, and career applications
are given in the Teaching Tips section of this Instructorโs Resource Manual. These resources can
be used as classroom activities, collaborative activities, or individual or group projects.
The detailed solutions to all even-numbered problems of the Chapter Review Exercises,
Practice Tests, and Cumulative Practice Test and to all Concepts Analysis exercises are included.
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vi
PREFACE
Transparency masters or images for PowerPoint Presentations for selected figures or
illustrations in the text and other reproducible teaching aids are included for the convenience of
the instructor.
MyLab Math, Test Item File and Computerized Test-Generator
MyLab Math and a test item file are available to users of the text. This file is a self-contained
computerized data base which allows instructors to generate work sheets, practice sheets, and
diagnostic or post tests from available test items. Each test item is coded by chapter, section,
learning outcome, and level of difficulty and is available in hardcopy form or as a downloadable
file.
Student Solutions Manual
The Student Solutions Manual shows detailed solutions to odd-numbered exercises of the
Chapter Review Exercises, Practice Tests, and Cumulative Practice Tests. This manual can be used
as an optional or required resource for students.
Multimedia Support
Multimedia features accompany the tenth edition of College Mathematics for Trades and
Technologies, including MyLab Math. This multimedia support will give instructors the resources
needs to offer courses via the Internet as well as resources for traditional classes.
Acknowledgments
We appreciate the suggestions received from students and instructors who have used the
previous editions of the text.
We wish you much success in your use of College Mathematics for Trades and Technologies,
Tenth Edition, and its supporting materials. If you have suggestions for improving these materials,
please give them to your Pearson representative or email Cheryl Cleaves at [email protected].
Cheryl Cleaves
Margie Hobbs
Copyright ยฉ 2019 Pearson Education, Inc.
INSTRUCTORโS RESOURCE MANUAL
INTRODUCTION
You are no doubt aware of the national focus on improving the quality of student learning and
instruction at the college level, especially in mathematics. Employers of our students and
instructors in more advanced mathematics or technical courses continue to emphasize that
students, in general, have difficulty applying their knowledge to different situations and they rely
more on memorizing than on reasoning to solve problems.
Many studies indicate that students can greatly enhance their thinking skills by proper guidance
and active learning experiences. We have tried to make available to you a wide variety of
supplementary materials to assist you in adapting your course to meet the studentsโ needs in several
different delivery modes.
Included in this manual are samples of assignments and activities that are designed to guide
students into a deeper understanding of certain topics. Adopters of College Mathematics for
Trades and Technologies, Tenth Edition are welcome to make copies of these assignments for
classroom use or to modify or adapt them to studentsโ particular needs.
Calculator Usage
We recommend that all students be required to use a scientific or graphing calculator. Students
should become proficient in the use of a calculator and be encouraged to use estimation skills to
determine the reasonableness of an answer. We understand that opinions vary widely on the extent
of calculator usage in a mathematics program and the use of calculators in assessment may vary
with individual instructors; however, calculator proficiency is an important part of preparing
students for success in future mathematical and technical courses and in the work environment.
The text includes general tips for using the calculator. These tips help students learn to use a
calculator by experimenting with a variety of keystroking options rather than memorizing a
sequence of steps. Again, the focus is on understanding how a calculator handles a specific
operation. Many of the reproducible activities in this manual require the use of a calculator.
Assessing Student Progress
In addition to the traditional assessment of student progress through periodic tests, projects,
and assignments, we suggest that students become more aware of their responsibility for learning
by assessing their own progress. These self-assessment strategies help students become more
independent learners. Some examples of student self-assessment forms are provided in this manual
as teaching aids.
Teaching Delivery Modes
Many colleges offer a particular course in a variety of delivery modes. The resources that
accompany the text will assist you in adapting your courses to traditional, independent study,
active learning, laboratory based, and electronic (online) delivery modes. The text provides
students in all delivery modes a comprehensive learning guide and reference.
To support active learning approaches, we have provided numerous suggestions for
collaborative learning techniques, writing-to-learn activities, and experimentation to be used to
supplement classroom discussion. Students should be urged to form study groups to accomplish
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INSTRUCTORโS RESOURCE MANUAL INTRODUCTION
out-of-class assignments and activities and to develop team-building skills that will carry over to
the workplace.
College Mathematics Addresses Curriculum and Pedagogy Standards
College Mathematics for Trades and Technologies, Tenth Edition, addresses curriculum and
pedagogy standards set by both the American Mathematical Association of Two-Year Colleges
(AMATYC) and the National Council of Teachers of Mathematics (NCTM). The goals as
presented by NCTM are for students to learn to value reason mathematically, to communicate
mathematically, to become confident of mathematical abilities, and to become mathematical
problem solvers.
The text helps students to learn to value mathematics by providing real-life situations in both
examples and exercises. Mathematical reasoning is promoted through explanatory comments in
the text narrative and in examples. We use both informal language and mathematically precise
terminology in the text to allow the student to read and understand mathematical concepts as
presented in the text. We also present several projects in the text and in this manual that promote
mathematical communication between students and teachers.
To further promote mathematical communication, the new mathematical terms are defined in
the chapters and are included in the glossary and index. The glossary and index of the text are far
more extensive than in most mathematical texts. The intent is for students to come to value the text
as a reference even after they have completed their formal study of the text.
The Section Exercises with all the answers in the text are designed to help students become
confident of their mathematical abilities before attempting the Chapter Review Exercises where
only the odd answers are provided. The body of the text contains many examples with explanatory
comments provided at each step for the student to study. The Chapter Review of Key Concepts
and Practice Test also provide students with means of gaining confidence in their ability to assess
their own understanding of the concepts presented in the chapter.
Crossroads and Beyond Crossroads
The AMATYC Crossroads and Beyond Crossroads documents identify the following as
abilities to be developed by students in college mathematics programs: number sense, symbol
sense, spatial and geometric sense, probability and statistical sense, and problem solving sense.
College Mathematics for Trades and Technologies, Tenth Edition, develops number sense
through an emphasis on estimation. Whenever calculators are used, students are encouraged to
estimate the answer before they begin the calculations and to check the answer to ensure it is
reasonable. Even though many mathematics professionals still disagree about the use of calculators
in assessing student learning, the authorsโ position is that students will come to view the calculator
as a tool that is invaluable in the workplace and as such will learn to use the calculator in the
classroom setting for appropriate circumstances and not necessarily to make calculations that can
be done mentally.
We encourage a dual approach for problems with reasonable calculations to develop the
studentsโ confidence in both their computational skills and their ability to use the calculator. For
complex calculations, we encourage the use of the calculator so that the student can focus on the
problem-solving aspects of the situation.
Symbol sense is addressed by presenting many rules, formulas, and properties in both symbols
and words. The students should begin to appreciate the value of using mathematical symbolism to
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COLLEGE MATHEMATICS
3
write a complex mathematical relationship in a concise form. For example, in the division law of
exponents, written symbolically as
am
= a m โ n a โ 0,
n
a
students should recognize that (1) this rule applies only to factors with like bases, (2) the quotient
will retain the like base, (3) the exponent of the quotient will be the difference of the exponents of
the numerator and the denominator, and (4) when division is involved, the operation is restricted
to nonzero values in the denominator.
The standards strongly recommend that mathematical concepts be integrated rather than
presented as the traditional arithmetic, algebra, and geometry courses. Spatial and geometric
concepts are integrated throughout the text so that they are presented when the mathematics
required to solve problems involving the concepts is presented. Numerous visual illustrations are
provided with word problems. Sometimes, the entire problem is presented as a visual
representation which must be interpreted by the student. For example, a problem may consist of a
blueprint with missing parts. The students must analyze the given information, decide how it
relates to the missing information, and devise a plan for finding the missing information. In
addition, students are also expected to be able to devise illustrations from written problems.
The text incorporates an entire chapter on data analysis with emphasis on probability and
statistics and the visual representation of data. When problem solving is first introduced, a sixstepped problem-solving plan is developed for investigating and analyzing the situation presented
in the problem to determine what tools to use to solve the problem.
In summary, College Mathematics for Trades and Technologies, Tenth Edition is rich with
mathematical experiences that promote the students’ ability to make connections from one
mathematical concept to another and from mathematical concepts to the workplace and everyday
life situations.
Copyright ยฉ 2019 Pearson Education, Inc.
TEACHING TIPS
College Mathematics for Trades and Technologies, Tenth Edition, and its accompanying instructional
resources have been designed to enable students to experience mathematics. Every instructor brings a different
personality and teaching style to the mathematics classroom. We encourage you to experiment with a wide variety
of activities and projects that are included in the text and this manual. We have found the activities that are most
beneficial will vary with the personality of a class. A variety of approaches will help an instructor provide rich
mathematical experiences for students with a wide range of learning styles.
Impromptu Classroom Collaborative Activities
College students often are reluctant to form study groups with their classmates. It is helpful if the instructor
encourages collaboration by giving students opportunities during class time to realize its benefits and to get to know
some fellow classmates. Mini-sets of exercises can be used during class time. The exercises can be selected from the
even-numbered Chapter Review Exercises, or Practice Test at the end of the chapter, since these answers are not in
the text. The Stop and Check and section exercises or odd-numbered end-of-chapter exercises can be used and
students can check their answers with the answers in the text. A plan for incorporating collaborative learning could
be:
1. Select one to three problems for students to work individually in class. Allow a reasonable amount of time
for students to complete the problems.
2. Have each student compare results with a study partner and each pair should reach a consensus on the
correct answers.
3. Have each pair of students compare results with another pair of students. The four students should reach a
consensus on the correct answers.
4. Record and display the answers of each group of four. If more than one answer is given for any problem,
engage the class in a discussion to reach a class consensus on the correct answers.
Team Projects
Some of the activities suggested in this manual are short activities that can be completed in one class period or
less or in an overnight assignment. However, others will require a longer commitment of time. We suggest that some
activities be accomplished using teamwork. Our distinction between teamwork and collaborative activities is that a
teamwork activity has teammates working on individual tasks that will fit together to complete a product or report.
A collaborative activity involves classmates working collectively in a group to complete a task. The text and this
manual give some guidance on forming teams and working in a team environment.
We are including a plan that has been successful in our classes for building teamwork skills and for helping
students realize the usefulness of mathematics in the team project.
Students can form their own working groups or they can be assigned to a group of three or four students. Each
team selects from activities the instructor provides from this Instructorโs Resource Manual. Even if two teams select
the same project, the teams generally develop an entirely different approach to the project. In the Reproducible
Activities we have included a sample handout to orient students to the team process. Following are sample
instructions that can be used for team projects.
Teams will be three to four students. Each student will have a specific responsibility in the project. Peer
evaluations will be part of the total project grade (Teaching Aid 17 on p. 94).
Written Report
โข
โข
One report for each team
Narrative portion of report will be three to five word-processed, double-spaced pages.
โข
Narrative portion should include:
1. Statement of problem
2. Method for solving problem
3. Explanation of each team memberโs responsibility
4. Findings
5. Conclusion
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COLLEGE MATHEMATICS
Presentation
โข
โข
โข
Deadlines
โข
โข
โข
5
Ten to fifteen minutes
Share the five components of the written report with the class. Do NOT read the written
report. Summarize the written report and use appropriate visual aids for the class.
Presentation does not have to involve all team members; but each team member must be
prepared to answer questions about the project.
(Date)
(Date)
(Date)
Brief description of project, team members, and team assignments
Draft of written report
Final copy of written report Oral presentation
Chapter-by-Chapter Suggestions
For many of us, our workload is so demanding that we may not always have quality time for developing
new ideas or activities. We are providing some chapter-by-chapter suggestions for presenting the concepts of the
chapters.
CHAPTER 1: Review of Basic Concepts
The purpose of this chapter is to review or develop the studentโs number sense and to increase the studentโs
understanding of some very important basic mathematical concepts while focusing on whole numbers and decimals.
Mathematical concepts introduced in this chapter form the foundation for many of the topics found later in the text.
By encountering the concepts of exponents, roots, and powers of 10 while working with whole numbers and
decimals, students can make a smoother transition to the symbolic representations and manipulations involving these
concepts later in the text.
If your course does not allow time to include this chapter, it is advisable to encourage your students to
independently study the Chapter Review of Key Concept and assess themselves using the Practice Test or a
diagnostic test or assignment in MyLab Math.
The authors strongly encourage the early introduction of the scientific or graphing calculator. Students who
develop proficiency with these tools while increasing estimating skills and mental arithmetic skills will have a better
understanding of the mathematical concepts and will have more confidence in their own mathematical abilities.
Understanding of concepts will need to be emphasized while proficiency in manipulation skills will need to be deemphasized.
A structured problem-solving approach is introduced in this chapter so that students can become familiar with
the steps needed to analyze and solve a problem. Even though students may not be required to use a structured
format in writing solutions of a problem, they should be directed to at least mentally consider each step in the
process. This process will help in developing the necessary problem-solving skills for more complex problems.
Activities
The 100-Cell Grid for Multiplication and Fraction Activities (Teaching Aid 4) can be used to introduce onedigit multiplication facts. As an in-class discussion, students can see that basic concepts like the commutative
property of multiplication, zero property of multiplication, multiplication by 1, and multiplication as repeated
addition reduce the amount of rote memorization necessary to learn one-digit multiplication facts. Students should
be encouraged to examine in detail the grid to identify numerous patterns involving the products of two one-digit
numbers. This activity encourages the studentโs development of number sense.
Transposing Digits gives an opportunity for students to investigate a property of numbers that they may or may
not have known before. Students can experience an often used research technique of verifying a suspected pattern or
property.
Factoring Into Pairs of Factors helps students strengthen their recall of multiplication facts and recognize the
relationship between multiplication and division. This activity is sometimes repeated in later chapters to help
students make a connection between previously learned skills and new concepts of reducing fractions, finding
common denominators, factoring polynomials, etc. Estimating Radicals is a powerful activity that helps students
develop some number sense with square roots. For the adult student who is experiencing low self-esteem for
beginning a study of college mathematics, this activity can be a real morale booster. Many advanced students can
improve their number sense with square roots.
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TEACHING TIPS
CHAPTER 2: Review of Fractions
Understanding the concepts of fractions is best developed using fractions with small denominators. Tedious
calculations with fractions having large common denominators or fractions with no common factors can be handled
efficiently using decimal equivalents or a calculator with a fraction key.
Emphasis should be placed on the studentโs ability to identify which of the four basic operations is needed to
solve application problems. For higher-level mathematics, an understanding of equivalent fractions and adding
fractions with unlike denominators is important. However, examples and problems to be solved by hand should be
limited to reasonable numbers. Concepts such as division by zero, multiplying or dividing by 1, and dividing a
number by itself should be emphasized and related to appropriate topics in fractions.
Activities
The fraction activities, Common Fractions, Decimal Fractions, Fractions in Simplest Form, Fraction
Relationships, and Size of Fractions are designed to develop the studentโs number sense about fractions and
decimals and to enable students to discover the relationship between fractions and decimals. These activities can be
separated and all or some of the outcomes can be introduced. However, the series of activities helps students
develop estimating skills with fractions. The activities emphasize the relationship of the numerator to the
denominator of a fraction and the effect this relationship has on the value of the fraction. The calculator is integrated
into the activities to reinforce the concept that fractions and decimals are different notations for expressing the same
value and to increase the studentโs proficiency and confidence in using the calculator.
Factoring Into Pairs of Factors is appropriate preparation for understanding prime factorization, reducing
fractions, and finding common denominators.
CHAPTER 3: Percents
This chapter includes fraction and decimal equivalents. These equivalents help students to develop a number
sense with percents. Focus should be placed on the idea that a specific value can be written in three different
notations. While memorization of some common equivalents will come automatically with practice, the authors
stress understanding the relationships rather than focusing on the memorization of the common percent-fractiondecimal equivalents as isolated facts.
Various versions of percentage formulas are also introduced. These formulas are commonly used in solving
percent problems involving applications such as interest.
Activities
Percents is one of the most useful mathematical topics to introduce problem-solving techniques and to build
critical thinking skills. Real-life exposure to percents creates a natural motivation for students to want to understand
and use percents. Numerous in-class and out-of-class activities can be generated by using local or national daily
newspapers, technical journals, and consumer buying magazines. Projects may also compare strategies for buying or
financing major purchases or planning and developing costs for building or decorating projects. Some students may
be interested in researching various retirement strategies. The scope of these activities could range from having
students develop problems from ads or news articles to extended individual or collaborative projects that incorporate
skills in mathematics, written and oral communication, consumer finance, economics, marketing, and various
technical fields.
Analyzing Nutrition Labels and What Percent Tax Do I Really Pay? apply our knowledge of percents. Many
students from a variety of career choices will be interested in these topics. These activities may be started in class
and completed as out-of-class assignments or projects.
CHAPTER 4: Measurement
Discussion of measurements in this chapter should support the studentโs development of spatial sense. Students
should be able to approximate in the metric system the length, volume, and weight measures of various objects.
Verification of cross-system measurements can be made using the conversion factors for both systems. Being able to
use resources to find conversion factors is more valuable than studentsโ memorizing these factors. The authors see
no value in having students memorize the conversion factors because when employees must convert between
systems, specific levels of precision are required and these factors are readily available to the employee.
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COLLEGE MATHEMATICS
7
Reading measuring instruments is an important skill in the workplace. Detailed instructions and numerous
visual aids are given for a variety of measuring instruments in an electronic appendix. The appendix follows the
same format as the text and is reinforced with MyLab Math support.
Activities
Estimating measures is becoming a lost art. Students should estimate the measure of an object before measuring
it. Hands-on activities can be designed to enable students to verify their estimates by actually measuring various
objects for length, volume, and weight. Students can research the measurements that are most frequently used in
their selected field of study.
CHAPTER 5: Signed Numbers and Powers of 10
The early introduction of signed numbers is important for many technical programs. Visualization of signed
numbers on the number line and on the rectangular coordinate axes is also helpful for many students.
Activities
Activities that simulate workplace situations reinforce the practical need for integers. Arranging Global
Interactive Communications, Preparing a Reference Chart for Global Communications, and Locating Coordinates
on a Sphere are activities included in this manual. Scientific and graphing calculators should be encouraged for use
in this chapter, especially when the student is evaluating an expression with several steps. Emphasis should be
placed on making decisions about the appropriate use of the calculator. Mental calculations should be encouraged!
The mind is still quicker than the hand for many calculations. A thorough understanding of the signed number rules
is very important and the calculator can be used to verify answers obtained by applying the rules.
CHAPTER 6: Statistics
Data interpretation and analysis is often neglected in studies of mathematics. However, it is becoming
increasingly important in the workplace, especially where statistical process control (SPC) is used. Students usually
find the study of data analysis very interesting and motivating because they can see applications more readily than
with other topics.
Activities
The activity Circle, Bar, and Line Graphs may be used here to strengthen the studentโs ability to interpret data
from various types of graphs. It is very beneficial for students to plan, design, and conduct an investigation in which
the findings are reported in writing and graphically. Another interesting point to emphasize is that graphs that appear
in newspapers, magazines, and other publications are not always meaningful or easy to interpret. Critiquing Graphs
helps students judge the usefulness and appropriateness of graphs.
CHAPTER 7: Linear Equations and Inequalities
As students prepare for an in-depth study of solving equations, it is important that studentsโ skills in the basic
operations of signed numbers and the order of operations are fresh. Also, translating from words to symbols is an
important skill in preparing for problem-solving strategies.
Linear equations are introduced through applications. Scales can be used to illustrate the concept of performing
the same operation on both sides of the equation in the simplification process.
Problem solving is emphasized in this chapter and the six-step problem-solving strategy is used for approaching
applied problems. Students should practice their critical thinking skills to interpret problems using algebraic
techniques. Students should not be penalized for devising a sequence of arithmetic calculations to solve the problem;
however, students should be encouraged to express this sequence of arithmetic calculations algebraically. This often
helps to bridge the gap from arithmetic to algebra and to see the usefulness of expressing problems symbolically and
using systematic steps to solve an equation. Many times, easy problems are used to demonstrate the algebraic
process. Once the algebraic process is mastered, it can then be used to solve more complex problems.
The concepts for solving equations are extended to equations containing fractions and decimals. Rate, time, and
work problems are also included.
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TEACHING TIPS
Making connections between methods for solving equations and inequalities is important in this chapter.
Understanding the โandโ and โorโ relationships is also important. However, it is NOT important that students
memorize the symbols for the various sets of numbers. Rather, it is more important that they understand how the
various sets of numbers relate to each other.
Students should realize that they are only extending their knowledge of the linear equations to include the
concept of inequalities.
Activities
Chapter 7 is the first opportunity that many students will have to translate the conditions of a word problem to
an algebraic equation. Many students come to our classes with preconceived notions that they cannot work applied
problems. For the math-anxious student it is very important to experience success with problem solving. We suggest
that students be allowed to work in groups to solve the applied problems in this chapter. There will be opportunities
in later chapters for students to concentrate on their individual problem-solving skills.
To emphasize the parallels between equations and inequalities, have students reword some of the word
problems in the sections for equations to require the use of inequalities. Phrases such as โat leastโ or โat mostโ can
be substituted for โequal.โ
CHAPTER 8: Formulas, Proportion, and Variation
Proportions are presented with a wider variety of applications. This chapter offers many opportunities to help
students make connections with outcomes from earlier chapters. Situations that are directly or inversely proportional
provide a good opportunity for using common sense to predict a reasonable answer. This approach to estimating
allows students to find their own errors in setting up a proportion problem.
Activities
In formula rearrangement, students can increase their proficiency with the calculator by verifying that the new
formula is equivalent to the original formula. To do this they can assign values for each variable in the formula.
These values are used to evaluate both the original and new formulas. When the results are not the same, students
will be able to revise their work in rearranging the formula or in using the calculator.
Analyzing Nutrition Labels can be used to strengthen proportion concepts and to illustrate direct proportions.
A good application of proportions used in the workplace is found in working with blueprints. Projects or
activities can be planned around actual blueprints for various industries. Students could be asked to design a
blueprint for a home project.
The potential for developing activities in this section is extensive and activities can easily be customized to the
studentโs interest. A team or individual project that has students investigate the common formulas of a particular
career will greatly enhance this chapter. Oral presentations of the findings of the students will give the entire class
an overview of the power of mathematics in broadening career choices.
CHAPTER 9: Linear Equations, Functions, and Inequalities in Two Variables
Critical thinking skills can be strengthened by allowing students to make decisions about which graphing
method is best to graph equations with various characteristics. Students should be able to visualize the graph of an
equation by looking at the various components of the equation. Students should be able to write an equation by
examining the visual graph of the equation. Graphing calculators or computer software may be used to electronically
graph equations.
First, we examine the graphs of equations. Then, we examine the equations of graphs.
The definition of slope should be visualized by graphing the two points and drawing the line that passes through
the two points. Students should be able to predict the value of the slope from the graph BEFORE they calculate the
slope. That is, they should be able to predict if the graph rises or falls from left to right or if the graph is more steep
or more flat. Also, students should be encouraged to make the calculation first, then predict what the graph of the
line connecting these points should โlook like.โ Finally, graph the two points to confirm their prediction.
Similarities and differences between equations of parallel lines and perpendicular lines should be presented in a
visual format. Students should be able to see examples of pairs of equations and their parallel graphs. They should
also be able to see examples of pairs of lines that are perpendicular and compare them with their equations.
Comparison of the equation pairs that graph into parallel and perpendicular lines will help students make
connections between the relationship of the equations and their graphs.
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9
The graphing calculator is a very helpful tool in visualization of these concepts.
Activities
Activities 1 through 3 of Graphing Equations can be used to help students expand their understanding of
patterns associated with linear equations. Instructions for using the activities as a group assignment are given to use
with the activity sheets. Commodities Market Investing illustrates the use of graphing in consumer applications.
Students should be encouraged to work together and discuss the visualization activities that are important for a
clear understanding of algebraic concepts in this chapter.
CHAPTER 10: Systems of Linear Equations and Inequalities
The notion of systems of equations can be compared to finding the overlapping regions or intersection of more
than one condition in a problem. The idea is to determine if all the conditions of the problem can be met
simultaneously.
There are numerous applications for systems of equations. Students may be encouraged to write applied
problems that would be appropriate for a given system of equations. This strengthens the studentsโ critical thinking
skills and writing skills.
An electronic appendix on Matrices and Determinants that includes Cramerโs rule for solving systems of
equations is available in MyLab Math.
Activities
While most of the conditions of problems in the text are designed so that all the conditions can be met
simultaneously, students may investigate situations where the conditions cannot be met simultaneously. When this
occurs, they should list the various ranges, if any, under which the conditions may be acceptable.
Making Business Choices can be used to show how systems of equations can be used in decision-making.
Applications problems in this chapter also can be solved using equations with one variable. Students should
investigate a wide range of strategies for solving the problems.
CHAPTER 11: Powers and Polynomials
Students should treat the laws of exponents like the signed number rules. These rules are used with many
concepts in mathematics. Practice should be sufficient so that applications of the laws of exponents become mental
tools that students use instinctively.
Students often confuse the two words exponent and power. In informal language they are sometimes used
interchangeably. However, the formal mathematical definitions distinguish between these two concepts. Using the
mathematical definition, 24 and 16 are both powers. The 24 is a power written in exponential form and the 16 is a
power written in ordinary number form. A power in exponential form includes a base and an exponent.
The definition of exponent and the way we visualize exponents depend on the kind of number used as the
exponent. For example, with positive whole numbers, the exponent is visualized as implying repeated
multiplication. When the exponent is zero, the value of the expression is defined to be one (when the base is other
than zero). When the exponent is a negative integer, the student should visualize the exponent as indicating the
reciprocal of the power with an exponent of the same absolute value. Students should also understand that the
reciprocal of any power can be formed by writing the power with the opposite of the original exponent. For
example, the reciprocal of x โ3 is x3 just like the reciprocal of 2 is 3 . We use this property to change the sign of
3
an exponent when necessary. For example, we can rewrite
1
a โ2
2
as a 2 These two expressions are equal.
Emphasis should be placed on connections between the notations for expressing roots with radicals and with
fractional exponents. Students should understand that the laws of exponents are the same for fractional exponents as
they are for integer exponents.
As with any discussion involving fractions, it should be noted that the exceptions to rules are always made
when denominators are zero. With powers, exceptions to rules are made when the base is zero.
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TEACHING TIPS
CHAPTER 12: Roots and Radicals
Emphasis should be placed on connections between notations for expressing roots with radicals and with
fractional exponents. Students should understand that the laws of exponents covered in Chapter 11 are the same as
those used for fractional exponents. This is an excellent opportunity to show students MANY ways to find powers
and roots using a scientific or graphing calculator.
See teaching tips for Chapter 11 for a discussion of the meaning of exponents depending upon the kind of
number represented by the exponent.
As with any discussion involving fractions, it should be noted that the exceptions to rules are always made
when denominators are zero. With powers, exceptions to rules are made when the base is zero.
The introduction of imaginary and complex numbers is optional. However, this brief treatment helps students to
understand why even roots of negative numbers are undefined when restricted to real numbers. Applications of
complex numbers are included in the discussion on vectors in Chapter 21.
Activities
Estimating Radicals is an activity that is designed to expand the studentโs number sense to include irrational
numbers. It distinguishes between squares and square roots and between rational and irrational numbers. This
activity also enables the students to determine the proper placement of irrational numbers on the number line.
CHAPTER 13: Factoring
The presentation and emphasis for the topic of factoring are in transition due to the various curriculum and
pedagogy standards initiatives and due to the integration of technology in the teaching and learning of mathematics.
The concepts of factoring are still beneficial in recognizing patterns and in understanding the simplifying process for
rational expressions. However, the focus should be on expressions that are relatively easy to factor. The time
required in factoring difficult expressions could be more beneficially spent in having the student use the graphing
calculator to investigate alternative strategies for solving problems that we currently use factoring to solve. This
chapter also presents a number of special products.
Activities
Factoring into Pairs of Factors can be used as an introduction to factoring trinomials. Many manipulatives that
can be used to illustrate or reinforce factoring are commercially available.
Graphing calculators and computer algebra software offer the potential for many activities. Trinomials can be
graphed as functions. By examining where the graph crosses the x axis (when y = 0), the students can determine the
factors of the trinomial. x2 โ 5x + 6 can be graphed as the function y = x2 โ 5x + 6. The graph of this function crosses
the x-axis at 2 and 3. This indicates that the trinomial can be factored into (x โ 2)(x โ 3). Students will naturally
question why the factors have signs opposite the signs of the coordinates of the points where the graph crosses the xaxis. This curiosity will give the instructor an opportunity to lead into the next chapter, which includes the zero
property of multiplication and the solutions for equations like (x โ 2)(x โ 3) = 0.
CHAPTER 14: Rational Expressions, Equations, and Inequalities
This is a topic that will receive decreased emphasis in the transition to the curriculum and pedagogy standards.
This chapter currently serves to integrate several mathematical concepts and to use these concepts in a different set
of circumstances from those in which they have been previously used. For example, students will strengthen their
understanding of the concepts of basic arithmetic such as reducing fractions and finding common denominators.
Emphasis in this chapter should be placed upon the studentโs understanding of the concept of excluded values.
Students often find it difficult to conceptualize this idea and often omit that constraint in giving the solution.
Activities
Patient Charting Efficiency is an activity that uses rational equations in career applications. Students can use
graphing calculators to graph rational expressions. Some graphing calculators will show the โholesโ or excluded
values and some will not. Whether the excluded values are shown sometimes depends on the range settings of a
particular calculator.
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CHAPTER 15: Quadratic and Other Non-Linear Equations and Inequalities
The various strategies for solving quadratic equations give a good opportunity to develop critical thinking skills.
Students should be encouraged to examine the advantages and disadvantages of the various methods and to select
the most appropriate method for the particular problem.
While the text focuses on approximating the roots of solutions for quadratic equations for technical applications,
students should understand that the solution in simplest radical form is the exact number solution and is desired in
some situations.
An electronic appendix on Conics is available in MyLab Math if you need to expand your coverage to parabolas
that are not a function, ellipses, circles, and hyperbolas.
Activities
Activities 4 through 6 of Graphing Equations can be used to enable students to visualize patterns associated
with graphs of quadratic equations. Calculating Vehicular Speed from Skid Marks and Road Conditions shows the
usefulness of quadratic equations in career applications. Graphing calculators can be used to visualize the solutions
of quadratic equations. Students may also be encouraged to research technical literature for applications of quadratic
equations.
CHAPTER 16: Exponential and Logarithmic Equations
This chapter comes after three chapters with several abstract concepts but few motivational applications. Most
students are interested in money and how it grows. The business and investment formulas help students see the
power of mathematics and its usefulness in making life and business decisions.
The section on logarithms is intended to be an introduction to the concept and not a thorough comprehensive
study of the topic. The formula for finding the amount of time necessary for achievement of an investment goal
helps to illustrate the relationship between logarithmic and exponential equations.
Activities
Several activities are available for this chapter. What is the Natural Exponential e? should be used to enable the
student to understand how the constant e is derived. This is also the studentโs first experience with the concept of
limits and this activity enables the student to discover the effect of using increasingly large values in the expression
n
1ยท
ยง
ยจ1 + ยธ .
n
ยฉ
ยน
The activity, Compounded Amount and Compound Interest, enables students to see applications of e and also
illustrates the effect of frequency of compounding on the compounded amount or compound interest.
Many business applications have formulas with variable exponents. Many students get interested in investment
formulas and it can easily lead to a discussion of planned investment programs.
The investment formulas that are given provide the tools for many investigative activities. They also open the
door for some personal finance discussions that are useful for college students.
CHAPTER 17: Geometry
Many areas of work require knowledge and understanding of geometric principles. While many concepts have
been integrated throughout the text, the authors think this chapter is very important to the overall development of the
studentsโ mathematical knowledge base. It is also essential for students who study trigonometry, calculus, and other
mathematical topics.
Emphasis should be placed on understanding and using formulas rather than memorizing them.
Activities
These topics are rich with real-world applications. Students should be encouraged to make connections in a
variety of real-world situations.
Students may estimate the number of degrees in a given radian measure and estimate the number of radians in a
given degree measure. Also, students may sketch angles with given degree or radian measures.
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TEACHING TIPS
CHAPTER 18: Triangles
Triangles play a very important role in the study of mathematics. The Pythagorean theorem, similar triangles,
and special right triangles such as the 30ยฐ, 60ยฐ, 90ยฐ triangle and the 45ยฐ, 45ยฐ, 90ยฐ right triangle are used in solving
many real-world applications. The distance formula is an application of the Pythagorean theorem.
Activities
Individual and team projects can be designed to find missing measures with similar triangles. Students can
investigate careers that use geometry and trigonometry. A good understanding of the properties of triangles is an
important prerequisite for the study of trigonometry.
CHAPTER 19: Right-Triangle Trigonometry
Use the trigonometric ratios to solve applied problems. Classroom discussion should focus on why one function
is used rather than another, different ways a problem can be solved, why one way may be more advantageous than
others, and why answers may vary significantly especially when using rounded calculated values to find other
values. The number of places an answer should be rounded to should also be discussed. Students should research
situations where varying levels of accuracy are desired.
Students should also be encouraged to develop spatial sense by sketching figures to show proper relationships
among the sides and angles. Estimation skills should be developed by encouraging students to predict the result of
calculations and compare the predicted values with the calculated values.
Activities
Students should measure the sides and angles of various right triangles, then confirm measures and the
trigonometric ratios by using the ratios to calculate various angles and sides. The triangles that are measured can be
figures drawn by students or objects in the real world.
CHAPTER 20: Trigonometry with Any Angle
Section 20-1 should be related to graphing ordered pairs from Chapter 9.
Students should be encouraged to critically examine problems to decide whether to use the law of sines or the
law of cosines.
Activities
Students should measure the sides and angles of various oblique triangles, then confirm measures by using the
laws of sines and cosines to calculate various angles and sides. The triangles that are measured can be figures drawn
by students or objects in the real world.
Surveying technology offers many rich experiences for applying the concepts of this chapter.
Copyright ยฉ 2019 Pearson Education, Inc.
REPRODUCIBLE ACTIVITIES
The activities in this section are designed to be self-contained units of work. They can be used in class or out of
class, as assignments to be completed individually by students or in groups, or as long-term projects that can be
completed by students individually or in groups.
The activities are intended to enrich the presentation in the textbook and to guide the students through an
investigative or discovery process. It is hoped that, after having completed an activity, a student will have a more indepth understanding of the underlying mathematical principles and will have developed a richer number sense.
The following activities are available in this supplement. The authors are continually exploring new activities
and as they are refined they will be made available in future editions. More information on when and how to use
each activity is given in the teaching tips.
TRANSPOSING DIGITS (p. 16)
Outcome: Verify that the difference between two numbers that have two digits transposed is divisible by nine.
FACTORING INTO PAIRS OF FACTORS (pp. 17รญ20)
Outcome 1: Find all possible pairs of factors of a given number.
Outcome 2: Distinguish between prime and composite numbers.
Outcome 3: Find numbers that are perfect squares.
Outcome 4: Find the square root of a number using a calculator.
LOCATING COORDINATES ON A SPHERE (pp. 21รญ23)
Outcome: Find locations on a sphere using ordered pairs.
ARRANGING GLOBAL INTERACTIVE COMMUNICATIONS (p. 24)
Outcome: Determine an optimum time for a global interactive communication.
PREPARING A REFERENCE CHART FOR GLOBAL COMMUNICATIONS (p. 25)
Outcome: Gather and organize data to improve efficiency.
COMMON FRACTIONS (p. 26)
Outcome: Distinguish between proper and improper common fractions.
DECIMAL FRACTIONS (p. 27)
Outcome: Examine decimal equivalents of common fractions.
FRACTIONS IN SIMPLEST FORM (p. 28)
Outcome: Examine equivalent fractions in simplest form.
FRACTION RELATIONSHIPS (pp. 29รญ32)
Outcome: Develop a procedure for comparing fractions to 1,
1
1
or by inspection.
2
4
SIZE OF FRACTIONS (pp. 33รญ35)
Outcome: Categorize fractions based upon their relationship to
1 1 3
, , , and 1.
4 2 4
ANALYZING NUTRITION LABELS (pp. 36รญ40)
Outcome: Use proportions and percents to analyze nutrition labels.
WHAT PERCENT TAX DO I REALLY PAY? (pp. 41รญ43)
Outcome 1: Determine the percent of tax withheld for a given taxable income.
Outcome 2: Determine the percent of income tax for a given annual taxable income.
College Mathematics, Tenth Edition
Cleaves and Hobbs
Reproducible Activities
13
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14
REPRODUCIBLE ACTIVITIES
CIRCLE, BAR, AND LINE GRAPHS (p. 44)
Outcome 1: Interpret data from various types of graphs.
Outcome 2: Plan, design, and conduct an investigation in which the findings are reported in writing and
graphically.
CRITIQUING GRAPHS (pp. 45)
Outcome: Critique graphs found in recent publications.
ESTIMATING RADICALS (pp. 46รญ47)
Squares and Square Roots
Outcome 1: Distinguish between squares and square roots and rational and irrational numbers.
Outcome 2: Determine between which two whole numbers a given irrational number will fall.
RATIONAL EQUATIONS (p. 48)
Patient Charting Efficiency
Outcome: Use rational equations in career applications.
QUADRATIC EQUATIONS (pp. 49รญ50)
Calculating Vehicular Speed from Skid Marks and Road Conditions
Outcome: Use quadratic equations in career applications.
WHAT IS THE NATURAL EXPONENTIAL e? (p. 51)
n
ยง 1ยท
Outcome: Discover the effect of large values of n in the expression ยจใ1 + ยธ .
ยฉ nยน
COMPOUNDED AMOUNT AND COMPOUND INTEREST (p. 52)
Outcome: Compare the two compound amount formulas for compound interest,
m
ยง rยท
A = p ยจ1 + ยธ and A = pe m .
ยฉ nยน
GRAPHING EQUATIONS PROJECT (pp. 53รญ54)
Instructions for Group Projects for Five-Member Groups
GRAPHING EQUATIONS (pp. 55รญ63)
Graphing Activity 1
Outcome: Examine equations in the form y = mx.
Graphing Activity 2
Outcome: Examine equations in the form y = mx + b.
Graphing Activity 3
Outcome: Examine equations in the form y = k and x = k.
Graphing Activity 4
Outcome: Examine quadratic equations in the form y = ax2.
Graphing Activity 5
Outcome: Examine quadratic equations in the form y = ax2 + b.
Graphing Activity 6
Outcome: Examine quadratic equations in the form y = (x + b)2.
College Mathematics, Tenth Edition
Cleaves and Hobbs
Reproducible Activities
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All rights reserved.
COLLEGE MATHEMATICS
15
GRAPHICAL REPRESENTATION (p. 64)
Commodities Market Investing
Outcome: Use graphing in consumer applications.
SYSTEMS OF EQUATIONS (p. 65)
Making Business Choices
Outcome: Use systems of equations to make good business choices.
ESTIMATING MEASURES (pp. 66รญ67)
Outcome: Estimate linear and circular measure in inches.
WHAT IS PI, ส ? (p. 68)
Outcome: Discover the relationship between the circumference and diameter of a circle.
College Mathematics, Tenth Edition
Cleaves and Hobbs
Reproducible Activities
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All rights reserved.
16
REPRODUCIBLE ACTIVITIES
TRANSPOSING DIGITS
Outcome: Verify that the difference between two numbers that have two digits transposed is divisible by nine.
We have been told that the difference between two numbers that have two digits transposed is divisible by nine. Is
this always true? Does it matter how many digits are in the number or if more than one pair of digits are transposed?
Find the difference between the following numbers. To avoid using negative numbers, subtract the smaller number
from the larger number. Then divide the difference by nine.
1.
58 and 85
2.
72 and 27
3.
36 and 63
4.
285 and 825
5.
285 and 258
6.
417 and 147
7.
417 and 471
8.
3842 and 8342
9.
3842 and 3482
10. 3842 and 3824
11.
3842 and 8324
12.
13,574 and 31,574
13. 13,574 and 15,374
14.
13,574 and 13,754
15.
13,574 and 13,547
16. 13,574 and 31,754
17.
13,574 and 31,547
18.
13,574 and 15,347
Summarize your conclusions to the following questions:
Is it generally true that the difference between two numbers that have two digits transposed is divisible by nine?
Does it matter how many digits are in the number?
Does it matter if more than one pair of digits are transposed?
Verify your conclusions with additional examples.
Summarize what you learned in this activity:
College Mathematics, Tenth Edition
Cleaves and Hobbs
Reproducible Activities
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COLLEGE MATHEMATICS
17
FACTORING INTO PAIRS OF FACTORS
Outcome 1: Find all possible pairs of factors of a given number.
Numbers that are multiplied together are called factors. The result of the multiplication is called the product. In this
activity we will limit ourselves to looking at pairs of factors, or two numbers that multiply together to give a
particular product. Some numbers may have only one pair of factors while other numbers may have two, three, or
more pairs of factors.
We will list p airs of factors of a given number by following a pattern. Start with the number 1. Can it p air with
another number to result in a product of the given number? Try 2. Try 3. Continue until you reach the given number.
EXAMPLE 1.
List all pairs of factors of 12.
12
1, 12
2,
6
3,
4
5,
3
6,
2
12,
1
Since multiplication is commutative, the pairs 3, 4 and 4, 3 are the same.
Similarly 2, 6 and 6, 2 are the same, and 1, 12 and 12, 1 are the same.
We can modify our procedure for listing all pairs of factors by replacing the statement, โContinue until you reach the
given numberโ with the following statement: Continue until the same pair of numbers appears but in the opposite
order. For example, when listing the pairs of factors of 12, if one pair is 3 and 4 and another pair is 4 and 3, all
remaining pairs of factors will also be duplicates of other pairs of factors that are already listed. Another way to
determine when factors are beginning to repeat is when the first factor is larger than the second. This observation
applies only when you begin with 1 and examine every number.
1.
How many different pairs of factors are there for the number 12? List them.
2.
State the divisibility rule that shows that 5 cannot pair with another whole number to give a product of 12.
Look at Example 2.
EXAMPLE 2.
List all pairs of factors of 210.
210
1, 210
2, 105
3, 70
5, 42
6, 35
7, 30
10, 21
14, 15
College Mathematics, Tenth Edition
Cleaves and Hobbs
Reproducible Activities
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18
REPRODUCIBLE ACTIVITIES
Factoring into Pairs of Factors, page 2
3.
Using individual and combined divisibility rules, state a reason why each of the following given numbers
cannot pair with another whole number to give a product of 210.
(a)
4
(b)
8
(c)
9
(d)
11
(e)
12
(f)
13
(g)
16
4.
How do you know that ALL the pairs of factors of 210 have been listed?
5.
For each number listed below, list all pairs of factors of the number.
15
24
18
36
13
16
20
22
17
28
30
23
25
1
6
8
40
48
Summarize what you learned about factor pairs.
College Mathematics, Tenth Edition
Cleaves and Hobbs
Reproducible Activities
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COLLEGE MATHEMATICS
19
Factoring into Pairs of Factors, page 3
Prime and Composite Numbers
Outcome 2: Distinguish between prime and composite numbers.
A whole number greater than 1 whose only pair of factors is 1 and the number is said to be prime. All other whole
numbers are said to be composite. The smallest prime is 2. Then, all multiples of 2 (4, 6, 8, . . .) are composite. Next,
3 is prime and multiples of 3 (6, 9, 12, . . .) are composite.
6.
Circle each prime number beginning with the smallest prime 2 in the following list of numbers. Then,
remove all multiples of the prime by putting a slash through the number. Continue with the next smallest
prime until all numbers are either circled or marked out.
16
31
46
61
76
91
2
3
4
5
6
7
8
9
10
11
12
13
14
15
17
32
47
62
77
92
18
33
48
63
78
93
19
34
49
64
79
94
20
35
50
65
80
95
21
36
51
66
81
96
22
37
52
67
82
97
23
38
53
68
83
98
24
39
54
69
84
99
25
40
55
70
85
100
26
41
56
71
86
27
42
57
72
87
28
43
58
73
88
29
44
59
74
89
30
45
60
75
90
7.
List the prime numbers that are less than 100.
8.
List all pairs of factors for the following numbers.
100
125
132
230
248
143
144
121
195
350
278
203
Summarize what you have learned about prime and composite numbers.
When a number has several different pairs of factors, it is often desirable to find a particular pair that has a certain
property. Questions 9 and 10 give some practice in this concept.
9.
Looking at the lists of factors for the numbers in Exercise 5, find a pair of factors that meet each stated
condition.
(a) factors of 24 whose sum is 11
(b) factors of 36 whose difference is 5
(c) factors of 18 whose sum is 9
(d) factors of 28 whose difference is 12
(e) factors of 36 that are the same number
10. Looking at the lists of factors for the numbers in Exercise 8, find a pair of factors that meet the stated
conditions.
(a) factors of 132 whose difference is 1
(b) factors of 230 whose sum is 33
(c) factors of 350 whose sum is 39
(d) factors of 195 whose difference is 80
(e) factors of 144 that are the same number
College Mathematics, Tenth Edition
Cleaves and Hobbs
Reproducible Activities
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All rights reserved.
20
REPRODUCIBLE ACTIVITIES
Factoring into Pairs of Factors, page 4
Perfect Squares
Outcome 3: Find numbers that are perfect squares.
When a number has a pair of factors that are the same number, such as 6 ร 6 = 36, this product is called a perfect
square.
11. Circle each perfect square in the following list of numbers.
1
2
3
4
5
6
7
8
9
10
11
12
13
14 15
16
31
46
61
76
91
17
32
47
62
77
92
18
33
48
63
78
93
19
34
49
64
79
94
20
35
50
65
80
95
21
36
51
66
81
96
22
37
52
67
82
97
23
38
53
68
83
98
24
39
54
69
84
99
25
40
55
70
85
100
26
41
56
71
86
27
42
57
72
87
28
43
58
73
88
29
44
59
74
89
30
45
60
75
90
12. For each circled number in Exercise 11, give the pair of like factors. Example: 1 = 1 ร 1, 4 = 2 ร 2, etc.
13. Extend your list of perfect squares to include all perfect squares between 100 and 1000.
Square Roots
Outcome 4: Find the square root of a number using a calculator.
A perfect square has a factor pair of identical factors. The number in the identical factors is the principal square root
of the perfect square. Other numbers also have square roots, but the square roots are not whole numbers. To find the
approximate value of a number that is not a perfect square, use your calculator and the square root key
.
In listing pairs of factors of a given number, how can we be certain we have them all? We have been testing every
whole number until we get a pair with the larger factor first or until a pair of factors is repeated but in the opposite
order. Letโs see if we can find another procedure.
14. Use your calculator to find the square root of each number in Exercise 8.
15. For each number in Exercise 8, compare the factor pair that has the largest first factor with the square root
of the original number.
16. Make a generalization about when to be sure you have found all factor pairs of a number.
Test your generalization from Exercise 16 with the following numbers. Find all pairs of factors of each number.
17.
85
18.
120
19.
136
20.
225
Summarize what you learned in this activity.
College Mathematics, Tenth Edition
Cleaves and Hobbs
Reproducible Activities
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COLLEGE MATHEMATICS
21
LOCATING COORDINATES ON A SPHERE
Outcome: Find locations on a sphere using ordered pairs.
Materials: World globe
Two number lines can be placed together by aligning zero on each line and making the lines meet to form a square
corner. These two number lines are often referred to as the rectangular coordinate system. The zero point where the
two lines meet is called the origin. The horizontal line is called the x-axis and the vertical line is the y-axis.
A point in a rectangular coordinate system is located with two directional numbers. The first shows the amount of
horizontal movement and the second shows the vertical movement. The point indicated by (รญ2, 3) means that you
move two units from the origin to the left. Then, from that point, move three units up.
1.
Locate the following points on a rectangular coordinate system.
A = (3, รญ1)
B = (รญ2, รญ3)
C = (รญ1, 5)
College Mathematics, Tenth Edition
Cleaves and Hobbs
Reproducible Activities
D = (4, 2)
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22
REPRODUCIBLE ACTIVITIES
Locating Coordinates on a Sphere, page 2
On a world globe we have a spherical coordinate system for locating points. Using resources such as an
encyclopedia, a world atlas, or the Internet, develop a strategy for locating points on a world globe.
2.
Locate the following reference points on a world globe.
Equator
Prime Meridian
International Date Line
3.
How are these points similar to the x- and y-axes on a rectangular coordinate system?
Equator:
Prime Meridian:
International Date Line:
4.
What point on a world globe is similar to the origin on a rectangular coordinate system?
5.
Locate the following reference points on a world globe.
North Pole
South Pole
6.
How are the directions North, South, East, and West used to identify locations on a world globe?
North:
South:
East:
West:
7.
How do the four directions compare to the positive and negative directions on the x- and y-axes on a
coordinate plane?
8.
How is locating points on a sphere different from locating points on a coordinate plane?
9.
Locate the following cities using the given coordinates.
Juneau, Alaska: 58N/135W
Rio de Janeiro, Brazil: 23S/43W
London, United Kingdom: 51N/0
Phnom Penh, Cambodia: 12N/105E
College Mathematics, Tenth Edition
Cleaves and Hobbs
Reproducible Activities
Copyright ยฉ 2019 Pearson Education, Inc.
All rights reserved.
COLLEGE MATHEMATICS
23
Locating Coordinates on a Sphere, page 3
10. Write your strategy for locating a point on a globe.
11. Why is this skill useful in the world of business?
Next, find familiar locations on a globe and write these coordinates.
12. Give the coordinates of the following cities.
Latitude
Longitude
Memphis
Seattle
Miami
Tokyo
Nairobi
13. What are some resources available for finding the coordinates of global locations?
14. Look up the coordinates of five cities and locate the cities on a world globe.
City, Country
Coordinates
City 1:
City 2:
City 3:
City 4:
City 5:
Summarize what you learned in this activity:
College Mathematics, Tenth Edition
Cleaves and Hobbs
Reproducible Activities
Copyright ยฉ 2019 Pearson Education, Inc.
All rights reserved.
24
REPRODUCIBLE ACTIVITIES
ARRANGING GLOBAL INTERACTIVE COMMUNICATIONS
Outcome: Determine an optimum time for a global interactive communication.
Materials: World globe, world time zone chart
An interactive teleconference must be arranged for five directors at FedEx. Each director works a normal 8:00 AM
to 5:00 PM day. Select a time for the teleconference that fits within everyoneโs normal working hours, if possible. If
not, select a time that will be most convenient for the largest number of directors.
Location of Directors:
Memphis, TN
Tokyo, Japan
Toronto, Canada
San Juan, Puerto Rico
Frankfurt, Germany
Describe your plan for accomplishing this task.
Time for Teleconference for Each Director:
Memphis, TN:
Toronto, Canada:
Frankfurt, Germany:
Tokyo, Japan:
San Juan, Puerto Rico:
Comment on any inconveniences this teleconference might cause.
Summarize what you learned in this activity:
College Mathematics, Tenth Edition
Cleaves and Hobbs
Reproducible Activities
Copyright ยฉ 2019 Pearson Education, Inc.
All rights reserved.

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