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Solution Manual for Developmental Mathematics: College Mathematics and Introductory Algebra, 10th Edition

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    R ESOURCE M ANUAL D EVELOPMENTAL M ATHEMATICS : C OLLEGE M ATHEMATICS AND I NTRODUCTORY A LGEBRA T ENTH E DITION Marvin L. Bittinger Indiana University Purdue University Indianapolis Judith A. Beecher

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    iii C ONTENTS Introduction iv Available Supplements v Collaborative Learning Activities CLA-1 Mini-Lectures ML-1 Mini-Lecture Graph Answers MLGA-1 Course Diagnostic CD-1 Answer Key for Course Diagnostic CDA-1 Chapter Tests CT-1 Answer Keys for Chapter Tests CTA-1 Final Examinations FE-1 Answer Keys for Final Examinations FEA-1

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    CLA-1 C OLLABORATIVE L EARNING A CTIVITIES

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    CLA-2 Name Section Date Activity 1.4 Solving equations in pairs. Focus Solving linear equations Time 10–20 minutes Group size 2 Materials Sets of twenty 3 × 5 cards with linear equations like those in section 1.4 of the text written on one side. Background This activity will give you practice solving linear equations and checking solutions. Instructor notes The equations written on the 3 × 5 cards will be solvable in one step, like problems 5 – 56 in section 1.4. Write one equation per card. Make duplicate sets, so that each group may have a set of twenty equations. 1. The person with the shortest first name will divide the 3 × 5 cards into two equal piles, equation side down. Each person should have the same number of cards in his or her stack. 2. When the instructor gives the signal to begin, each person will pick up the top card on his or her pile and solve the equation, showing the work on the card. 3. Exchange cards with your partner and each of you check the solution your partner found in the original equation on that card. Show your work. 4. If the solution checks, put the card face up in a “finished” pile. 5. If the solution does not check, try to find the error and, working with your partner, correct it. When you are in agreement and the solution checks, put the card in the “finished” pile and pick up the next two cards. 6. Continue steps 2 through 5 until all equations have been solved correctly. Conclusion Participating in this activity gives you practice in solving linear equations. You will be solving equations throughout the remainder of this course and in future mathematics courses. The procedures you learn will prove useful to you whenever you are solving equations.

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    CLA-3 Name Section Date Activity 1.5 Make a budget for a road trip to you favorite destination. Focus Problem solving and estimation Time 20–30 minutes Group size 3–4 Materials State and local highway maps for each group, calculators (optional) Background Planning a road trip involves several mathematical computations. For instance, the total distance to be traveled and the estimated cost for gas can be calculated using the concepts learned in this chapter. 1. Before you begin your calculations, select a destination for a road trip you could take on long weekend. Decide on one destination for your group. Origin: _____________________________ Destination: _____________________________ 2. Using the appropriate state and local highway map(s), highlight the route you would take to get to your destination and back home again. Calculate the total distance you would need to drive. Round this distance to the nearest hundred miles. Total distance:______________________________ Estimated distance: _____________________________ 3. Estimate the gallons of gas you would need for your trip. Use the miles per gallon (mpg) rating on one of your group member’s vehicle. Then calculate the total cost of the gas. Use the price per gallon of gas in your area. Gallons of gas needed:______________________________ Total cost for gas: _____________________________

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    CLA-4 4. Now decide how many days and nights it would take to complete the trip. Then calculate the cost for the accommodations. Days of travel: _____________________________ Number of nights accommodation: _____________________________ Total cost for accommodations: _____________________________ 5. Based on the days of travel, calculate how many meals you would need to eat during the trip. Then calculate the cost of the meals for all the people on this trip. Number of meals per person: _____________________________ Total number of meals: _____________________________ Cost for meals: _____________________________ 6. Summarize your estimated costs below. Include a reasonable figure for the cost of miscellaneous items. These might include the cost of admission tickets, souvenirs, parking, and tolls. Item Estimated Cost Gas: Accommodation: Meals Miscellaneous: Total: Conclusion As you can see, planning a budget for a road trip involves the operations of addition, subtraction, multiplication, and division, as well as estimation. Use the steps given in this activity to plan a road trip for yourself and family or friends.

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    CLA-5 Name Section Date Activity 1.6 Use the order of operations as a group to simplify expressions. Focus Order of operations Time 20–30 minutes Group size 3 Background Simplifying expressions using the rules for order of operations can be quite confusing for complicated expressions. Learning to simplify expressions as a group will help clarify the process. Rules for Order of Operations Do all calculations within parentheses before operations outside. E Evaluate all exponential expressions. MD Do all multiplications and divisions in order from left to right. AS Do all additions and subtractions in order from left to right. 1. Before you begin simplifying expressions, study the rules for order of operations above. Assign each group member to one of the steps listed. Write the name of the group member next to his or her assigned task in the table above. Note that the first step (calculations within parentheses) is not assigned. All group members will do this step together. 2. Now you are ready to simplify expressions as a group. Analyze the expression together and decide on the first step. If there are parentheses, decide whether the expression inside the parentheses needs to be simplified. Following the order of operations, E will perform his or her task before MD, and MD will perform his or her task before AS. Practice with the example on the next page. (This is Example 10, Section 1.6 in your textbook.) The first step has been done for you: Subtract inside the parentheses. AS will do this step, writing “AS” in the left box, and writing the new expression below the original expression. Continue simplifying the expression by passing the problem to the appropriate group member for the next step. When you are done, compare your steps to those in Example 10, Section 1.6 in your textbook. If there are any discrepancies, discuss

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    CLA-6 them within your group. Compare your result with the other groups. Are they the same? Discuss any differences with the other groups. Example 10, Section 1.6 2 3 4 (10 9 1) 3 5 ÷ − + ⋅ − AS 2 3 4 (1 1) 3 5 ÷ + ⋅ − 3. Once you understand the process, choose an expression from Exercise Set 1.6 in your textbook to simplify as a group. Use the table on the next page to organize your work. Make as many copies as you need. Alternatively, you can draw the table on a blank sheet of paper. Do as many problems as you can in the time allotted. Reassign tasks (E, MD, AS) to different members of the group. Do this at least twice so that each member of the group has a turn performing each of the tasks. Make sure you choose at least one of the more complicated expressions from Exercises 59 – 68. Conclusion This activity should help you gain a better understanding of the rules for order of operations. You can also use this group method when you encounter the order of operations in Sections 2.6, 3.4, and 7.8.

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    CLA-7 Original expression ____________________________________________________

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    CLA-8 Name Section Date Activity 1.7 Find all the prime numbers less than 100, using the Sieve of Eratosthenes. Focus Prime and composite numbers Time 10–15 minutes Group size 2 Material Colored pencils (optional) Background One of the methods for finding prime numbers was developed around 200 BC by a mathematician named Eratosthenes. He used the process of elimination to “sift” out the composite numbers, leaving only prime numbers. His method became known as the Sieve of Erastosthenes. 1. In Section 1.7 of your textbook, a prime number is defined as a natural number that has exactly two different factors, itself and 1. For example, the number 7 is prime because it has only the factors 1 and 7. The number 14, on the other hand, is not prime because 7 is a factor of 14. Looking at the definition from another point of view, any number that is a multiple of another number (besides 1) will not be prime. In the example above, 14 is a multiple of 7, so 14 is not prime. In this activity, you will cross off all multiples of prime numbers from a grid of numbers. When you are done, the remaining numbers will be prime. 2. Look at the grid on the next page. The number I has already been crossed off, as it is not a prime number. The smallest number that is not crossed off is 2. Begin by circling the number 2 on the grid. Then, list the first 10 multiples of 2 in the space below: Now, cross off these numbers from the grid. You may want to use a colored pencil to cross off the numbers. Continue crossing off multiples of 2 until you reach the end of the grid. 3. Next, look for the smallest number that is not crossed off and circle it. This is the next prime number. List the first 10 multiples of this number in the space below: Cross off these numbers from the grid. Continue, as before, crossing off multiples of the number until you reach the end of the grid.

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    CLA-9 4. Repeat step 3 until all multiples are crossed off. The circled numbers are the prime numbers less than 100. Write the list of circled numbers in the space below: 5. Compare this list with the table of primes given in section 1.7 of your textbook. Are there any differences between the lists? If there are, check your grid to see if you crossed off all multiples. Check also that you did not accidentally cross off a number that is not a multiple. Conclusion The Sieve of Eratosthenes can be used anytime you need to list the first few prime numbers. For example, if you need all the prime numbers up to 50, make a list of the numbers from 1 to 50, and start crossing out the multiples of 2, 3, 5, etc.

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    CLA-10 Name Section Date Activity 1.8 Use the divisibility rules and properties of numbers to discover an unknown number. Focus Rules for divisibility, place value Time 20–30 minutes Group size 2 Background The rules for divisibility given in Section 1.8 of your textbook provide you with fast ways of determining whether numbers are divisible by 2, 3, 4, 5, 6, 8, 9, and 10. This activity will provide practice with these rules, as well as experience in problem solving. Instructor notes In step 4, show Puzzles A and B, revealing clues one at a time. You can find more puzzles in the book Logic Number Problems, available from Dale Seymour Publications. For your convenience, the divisibility rules from Section 1.8 are repeated here. 2 A number is divisible by 2 (is even) if it has a ones digit of 0, 2, 4, 6, or 8. 3 A number is divisible by 3 if the sum of its digits is divisible by 3. 4 A number is divisible by 4 if the number named by its last two digits is divisible by 4. 5 A number is divisible by 5 if its ones digit is 0 or 5. 6 A number is divisible by 6 if its ones digit is 0, 2, 4, 6, or 8 (is even) and the sum of its digits is divisible by 3. 8 A number is divisible by 8 if the number named by its last three digits is divisible by 8. 9 A number is divisible by 9 if the sum of its digits is divisible by 9. 10 A number is divisible by 10 if its ones digit is 0.

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    CLA-11 1. Each puzzle in this activity gives you clues to the value of an unknown number. The objective is to determine the unknown number by using the fewest number of clues. The clues will be given to you one at a time. 2. First, practice on the following set of clues. Read the clues one at a time, using a sheet of paper to cover up the clues further down. Notice that some clues must be considered together (clues 5, 6, and 7), and that only the first 8 clues are needed to solve this puzzle. . . . . . . . . . . . . . . .

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    CLA-12 3. Here’s another puzzle to practice on. One group member writes down the possible solutions, as was done in the example on the previous page. Use complete sentences when writing the reasons for each possible solution. Read the clues one at a time, using a sheet of paper to cover up the clues further down. Clue Possible solution(s) Reasoning 1 It is a 3-digit number. 2 It is an odd number. 3 One of the digits in 7. 4 It is divisible by 5. 5 It is less than 700. 6 It has no even digits. 7 It is divisible by 3. 8 It is greater than 200. 9 Each digit is different. 10 It is a multiple of 25.

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    CLA-13 When you are done, compare your group’s result with the results of the other groups in your class. How many clues did your group need to solve this puzzle? Could you have determined the unknown number with fewer clues? Did you use the remaining clues (if any) to check your answer? 4. Now, let’s add a little competition to the problem-solving process. Each group will work as a team to solve a puzzle. Take turns, so each group member has a chance to do the writing. Your instructor will reveal the clues one at a time. The goal is to be the first group to correctly deduce the unknown number by using the fewest number of clues. Your instructor will discuss the scoring scheme; alternatively, the class can propose a scheme that is acceptable to all. The scoring scheme should take into account the correctness of the number, the penalty for a wrong number, the number of clues used, and the penalty for using more clues than needed. Conclusion This activity should help you gain experience in applying the divisibility rules. As a side benefit, the problem solving techniques used in solving the puzzles will be useful in solving the applications in your textbook.

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