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

INSTRUCTOR’S 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 © 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 Instructor’s 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 iii Copyright © 2019 Pearson Education, Inc. 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 Copyright © 2019 Pearson Education, Inc. 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. Instructor’s Resources Suggestions for using the Reproducible Activities, teamwork projects, and career applications are given in the Teaching Tips section of this Instructor’s 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. v Copyright © 2019 Pearson Education, Inc. 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 © 2019 Pearson Education, Inc. INSTRUCTOR’S 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 students’ 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 students’ 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 1 Copyright © 2019 Pearson Education, Inc. 2 INSTRUCTOR’S 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 authors’ 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 students’ 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 Copyright © 2019 Pearson Education, Inc. 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 − n a ≠ 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 © 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 Instructor’s 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 • • One report for each team Narrative portion of report will be three to five word-processed, double-spaced pages. • Narrative portion should include: 1. Statement of problem 2. Method for solving problem 3. Explanation of each team member’s responsibility 4. Findings 5. Conclusion 4 Copyright © 2019 Pearson Education, Inc. COLLEGE MATHEMATICS Presentation • • • Deadlines • • • 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 student’s number sense and to increase the student’s 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 student’s 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. Copyright © 2019 Pearson Education, Inc. 6 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 student’s 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 student’s 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 student’s 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 student’s 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 students’ 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. Copyright © 2019 Pearson Education, Inc. 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 student’s 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 students’ 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. Copyright © 2019 Pearson Education, Inc. 8 TEACHING TIPS Making connections between methods for solving equations and inequalities is important in this chapter. Understanding the “and” and “or” 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 “at least” or “at most” can be substituted for “equal.” 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 student’s 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 “look like.” 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. Copyright © 2019 Pearson Education, Inc. COLLEGE MATHEMATICS 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 students’ critical thinking skills and writing skills. An electronic appendix on Matrices and Determinants that includes Cramer’s 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 −3 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 −2 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. Copyright © 2019 Pearson Education, Inc. 10 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 student’s 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 – 5x + 6 can be graphed as the function y = x2 – 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 – 2)(x – 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 – 2)(x – 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 student’s 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 “holes” or excluded values and some will not. Whether the excluded values are shown sometimes depends on the range settings of a particular calculator. Copyright © 2019 Pearson Education, Inc. COLLEGE MATHEMATICS 11 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 student’s 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 1· § ¨1 + ¸ . n © ¹ 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 students’ 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. Copyright © 2019 Pearson Education, Inc. 12 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 30°, 60°, 90° triangle and the 45°, 45°, 90° 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 © 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. 17í20) 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. 21í23) 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. 29í32) Outcome: Develop a procedure for comparing fractions to 1, 1 1 or by inspection. 2 4 SIZE OF FRACTIONS (pp. 33í35) Outcome: Categorize fractions based upon their relationship to 1 1 3 , , , and 1. 4 2 4 ANALYZING NUTRITION LABELS (pp. 36í40) Outcome: Use proportions and percents to analyze nutrition labels. WHAT PERCENT TAX DO I REALLY PAY? (pp. 41í43) 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 Copyright © 2019 Pearson Education, Inc. All rights reserved. 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. 46í47) 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. 49í50) 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 § 1· Outcome: Discover the effect of large values of n in the expression ¨㌇1 + ¸ . © n¹ COMPOUNDED AMOUNT AND COMPOUND INTEREST (p. 52) Outcome: Compare the two compound amount formulas for compound interest, m § r· A = p ¨1 + ¸ and A = pe m . © n¹ GRAPHING EQUATIONS PROJECT (pp. 53í54) Instructions for Group Projects for Five-Member Groups GRAPHING EQUATIONS (pp. 55í63) 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 Copyright © 2019 Pearson Education, Inc. 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. 66í67) Outcome: Estimate linear and circular measure in inches. WHAT IS PI, ʌ ? (p. 68) Outcome: Discover the relationship between the circumference and diameter of a circle. College Mathematics, Tenth Edition Cleaves and Hobbs Reproducible Activities Copyright © 2019 Pearson Education, Inc. 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 Copyright © 2019 Pearson Education, Inc. All rights reserved. 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, “Continue until you reach the given number” 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 Copyright © 2019 Pearson Education, Inc. All rights reserved. 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 Copyright © 2019 Pearson Education, Inc. All rights reserved. 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 Copyright © 2019 Pearson Education, Inc. 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 × 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 × 1, 4 = 2 × 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. Let’s 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 Copyright © 2019 Pearson Education, Inc. All rights reserved. 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 (í2, 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, í1) B = (í2, í3) C = (í1, 5) College Mathematics, Tenth Edition Cleaves and Hobbs Reproducible Activities D = (4, 2) Copyright © 2019 Pearson Education, Inc. All rights reserved. 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 © 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 © 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 everyone’s 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 © 2019 Pearson Education, Inc. All rights reserved.