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

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    R ESOURCE M ANUAL C OLLEGE M ATHEMATICS FOR T RADES AND T ECHNOLOGIES T ENTH E DITION Cheryl Cleaves Southwest Tennessee Community College Margie Hobbs Southwest Tennessee Community College

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    iii 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 P ractice 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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    iv 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

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    4 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

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    COLLEGE MATHEMATICS 5 Presentation • 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. Deadlines • (Date) Brief description of project, team members, and team assignments • (Date) Draft of written report • (Date) 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 de- emphasized. 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 one- digit 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.

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    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-fraction- decimal 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.

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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 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.

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    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.

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    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 3 x − is 3 x just like the reciprocal of 2 3 is 3 2 . We use this property to change the sign of an exponent when necessary. For example, we can rewrite 1 2 a − as 2 a 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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    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. x 2 – 5 x + 6 can be graphed as the function y = x 2 – 5 x + 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 x - axis. 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.

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    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 1 1 . n 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.

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    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.

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    College Mathematics , Tenth Edition Cleaves and Hobbs Reproducible Activities 13 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 in- depth 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. 17 í 20) 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. 21 í 23) 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. 29 í 32) Outcome: Develop a procedure for comparing fractions to 1 1, 2 or 1 4 by inspection. SIZE OF FRACTIONS (pp. 33 í 35) Outcome: Categorize fractions based upon their relationship to 1 1 3 , , 4 2 4 , and 1. ANALYZING NUTRITION LABELS (pp. 36 í 40) Outcome: Use proportions and percents to analyze nutrition labels. WHAT PERCENT TAX DO I REALLY PAY? (pp. 41 í 43) 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.

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    14 REPRODUCIBLE ACTIVITIES College Mathematics , Tenth Edition Cleaves and Hobbs Reproducible Activities ㌇ 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. 46 í 47) 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. 49 í 50) Calculating Vehicular Speed from Skid Marks and Road Conditions Outcome: Use quadratic equations in career applications. WHAT IS THE NATURAL EXPONENTIAL e ? (p. 51) Outcome: Discover the effect of large values of n in the expression 1 1 n n § · + ¨ ¸ © ¹ . COMPOUNDED AMOUNT AND COMPOUND INTEREST (p. 52) Outcome: Compare the two compound amount formulas for compound interest, 1 and . m m r A p A pe n § · = + = ¨ ¸ © ¹ GRAPHING EQUATIONS PROJECT (pp. 53 í 54) Instructions for Group Projects for Five-Member Groups GRAPHING EQUATIONS (pp. 55 í 63) 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 = ax 2 . Graphing Activity 5 Outcome: Examine quadratic equations in the form y = ax 2 + b . Graphing Activity 6 Outcome: Examine quadratic equations in the form y = ( x + b ) 2 .

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    COLLEGE MATHEMATICS 15 College Mathematics , Tenth Edition Cleaves and Hobbs Reproducible Activities 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. 66 í 67) 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.

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