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- 1. 166 Cycle 2 6. When trying to find the solution to the problem 6 , 3, we can think “? * 3 = 6?” Let’s apply this idea to the following problems. Rewrite each division problem as a multiplication problem and then find the solution. Verify your answer by using your calculator. For the answer that results in an error on the calculator, write “undefined.” a. 0 , 6 6 * ? = 0 ? = 0 b. 6 , 0 0 * ? = 6 undefined c. Try a few more similar problems in which 0 is divided by a number or a number is divided by zero. What conclusions can be drawn? We cannot divide by zero. Zero divided by any number except zero is zero. 7. For each problem, state positive, negative, or it depends. If you state it depends, give two examples to support your answer. a. Positive + Positive Positive b. Positive + Negative depends Possible examples: 3 + (-7) = -4; 7 + (-3) = 4 It c. Negative + Negative Negative d. Positive - Negative Positive e. Positive * Positive Positive f. Positive * Negative Negative g. Negative - Positive Negative 8. Suppose your friend reads the following instructions to you from a book of math tricks: Pick a number. Subtract 1. Multiply by 3. Add 6. Divide by 3. Subtract the original number. She then correctly guesses your final number is 1. You repeat this and arrive at the same result. Try this once with a negative number and once with a fraction. Do you get 1 each time as an answer? Write out your calculation steps. Yes. The result is always 1, even if you start with a negative number or a fraction. 45–50 min Explore 5–10 min Instructor note: Instruct students to work in groups on the Explore. Discuss responses. Instructor note: If you are using them, distribute the counters or chips that students can use in the following problem while groups are working on the Explore. ALMY8454_01_AIE_C02.indd 166 2.5 An Ounce of Prevention 1. A student is taking a math class that has five exams, each worth 100 points. After four exams, her average in the class is 80%. She tells her friend, “If I get a 100 on the last exam, my grade will be an A. 80 and 100 average to be a 90, which is an A.” Her friend replies, “No, it’s not even possible for you to get an A in the class.” Who is right and why? The friend is correct. Currently, the student has 80% of 400 points, or 320 points. To earn an A in the class, she needs 90% of 500 points, or 450 points. She would need to earn 130 points on the last exam, which is not possible. 5/31/13 5:12 AM
- 2. LESSON 2.5 An Ounce of Prevention Discover 5 min Instructor note: Discuss the Look It Up. look it up 167 Most grades are found by calculating an average or mean. Let’s learn more about this measure of a data set. Mean The mean or average of a set of numbers is one measure of the data’s center or middle. To find the mean of a data set, add the data values and divide by the number of values. For example, John buys dinner at the same restaurant three times in one month with bills of $24, $38, and $40. His average meal price is: $24 + $38 + $40 3 = $34meal. This means if he had spent the same total on three meals, $102, but paid the same amount each time, he would have spent $34 on each meal. 45 38 40 35 30 40 34 34 24 25 24 20 38 40 15 10 5 0 Instructor note: Mention that the amount in the bars above the 34 line would fill the gap above the first bar. In the second picture, we can think of the mean as the balancing point (or fulcrum) of a teeter-totter. 20 min Instructor note: Students will complete #2–6 in groups. Walk around to keep groups progressing. 2. Two months into taking a biology course, you have taken three quizzes with these scores (out of 10 points each): 8, 8, 5. Your friend has the same quiz average as you, but she earned the same score on each of her quizzes. What was her score each time? If your friend scored the same on each quiz with the same total, 21, her score was 7 each time. 3. Let’s visualize the average of these quizzes. Draw squares, stacked vertically, to represent counters. Each stack should represent the points on one of the quizzes: 8, 8, 5. Now, earrange the stacks into three even stacks. How many counters are in each r stack? How could you have predicted that number? 7; This is the average, the number of counters in each stack so the stacks have the same height. Start: End: ALMY8454_01_AIE_C02.indd 167 5/31/13 5:12 AM
- 3. 168 Cycle 2 4. Suppose on the fourth quiz, you score 3 points out of 10. Compute your new average. How does this low score affect your grade? Why is that? 6; It lowers the average. Since you scored below your old average of 7 on the last quiz, it brought down your average. Since the total did not increase by 7 and the total has to be d istributed into four columns, the average will be lower than before. Start: End: Remember? 5. How many points lower is your average after quiz #4 compared to after quiz #3? What is the percent change in your average as a result of the last score? To compute the percent change, find the amount of change and divide it by the original value. An average of 7 dropping to 6 is one point; 1 ≈ 14% 7 6. Unhappy with your current average, you decide that you want to raise it to a C (7 out of 10). If the fifth quiz is also worth 10 points, what will you need to score on it to bring your average back to the C range? To find the answer, use the counters to model the first four quiz scores. Explain how to use the counters to answer the question. If you make stacks for 8, 8, 5, and 3, how high would a new stack need to be so you could make them all at least 7 counters high? You would take one off the first stack, one off the second stack, and add them both to the third stack. But you would need four more to add onto the fourth stack and still need 7 for the last stack. So you would need 11 counters. Start: Middle: End: Check your answer by averaging the five scores. The scores 8, 8, 5, 3, and 11 average to 7. Is this possible? If the instructor offered a bonus point on the quiz, would it be p ossible? Is it likely? Not possible on a 10-point quiz. If there was a bonus point, then it could happen. It is u nlikely since the student never scored that high on the earlier material in the course. ALMY8454_01_AIE_C02.indd 168 5/31/13 5:12 AM
- 4. LESSON 2.5 An Ounce of Prevention Instructor note: Set up the balance for demonstration. After all the quiz scores are plotted physically, students should notice that the average is the balancing point. 5 min If you do have not have access to a balancing model (options described in the accompanying instructor page), complete #7 by drawing a teeter-totter similar to the one in the Look It Up. 7 will be the fulcrum. 169 7. Use a number line to represent a balance. Let 7 represent the balancing point. Plot your original three quiz scores: 8, 8, 5. What is the average in this context? The balancing point Now plot the fourth quiz score, 3. What will you need to score on the fifth quiz to bring the average back to 7? 11 Instructor note: Ask students whether the balance model eminds them of anything a teeter-totter or seesaw). Ask what they did to r ( balance it if there was a lot of weight on one side. Go out very far on the other side. Another example involves shutting a door. Try to close a door by pushing next to the hinge with one finger—it’s difficult, if not impossible. Now push with one finger near the handle—it’s very easy to close the door. The reason has to do with the distance compounding with the force to create the desired leverage. Connect 5 min 8. Suppose you have 60% of 100 possible points in a course so far. If you get 90 points out of 100 points on the next test, what is your average in the course? 75% Instructor note: Discuss these problems as a class. 9. Suppose you have 60% of 900 possible points in a course so far. If you get 90 points out of 100 points on the next test, what is your average in the course? 63% 5 min Instructor note: Explain the contents of the Wrap-Up. Have tudents write an answer to s the ycle question, “Why does it c m atter?” A prompt is provided to help students. Discuss homework. Wrap-Up lesson Reflect What’s the point? Means are one measure of a data set’s center. They are often used when calculating grades. Understanding the math behind means can help you succeed in your classes. What did you learn? How to find the mean of a data set How to use means in applied problems Cycle 2 Question: Why does it matter? What does this expression mean? “An ounce of prevention is worth a pound of cure.” It is often easier and simpler to prevent a problem than to deal with the consequences later. ALMY8454_01_AIE_C02.indd 169 5/31/13 5:12 AM
- 5. 170 Cycle 2 2.5 Homework Skills MyMathLab • Find the mean of a data set. • Use means in applied problems. 1. Find the mean of this data set: -2, 7, 8, 4, -1, -10, 1 1 2. A student has a class with four tests, each worth 100 points, and has earned these scores on the first three: 75, 74, 71. What is her average now? What must she earn on the fourth test to get a B (80) in the class? Her current average is 73.3. She must earn a 100 on the last test to have a B average. Concepts and Applications • Use means in applied problems. 3. Suppose you have 3 quizzes and want an average of 7 points. List at least five ways this average can be accomplished. Keep in mind that each score can be no more than 10 points. Any 3 scores, between 0 and 10, that add to 21 will work. 4. One of the take-away points of the lesson is that some things are difficult to undo. Several things fall into the category of being easy to do but hard to undo. List two examples from real life and two examples from math. Easy to gain weight, hard to lose it. Easy to get in debt, hard to get out of it. Addition is usually easier than subtraction. Multiplication is easier than division. 5. Suppose you have dinner with four friends, and the total check (with tax and tip) is $155. How much should you each pay if you split the check evenly? What does this amount represent? You would each pay $31. This amount represents the average of your individual bills. 6. Consider the following two sets of incomes. Each income is in thousands of dollars per year. Group 1: 32, 36, 38, 39, 42, 43, 44, 47, 49, 50 Group 2: 32, 36, 38, 39, 42, 43, 44, 47, 49, 150 ALMY8454_01_AIE_C02.indd 170 5/31/13 5:12 AM
- 6. LESSON 2.5 An Ounce of Prevention 171 a. Find the average or mean salary for each list. Group 1: $42,000/year; Group 2: $52,000/year b. Make a conjecture about what happens to the mean of a data set when the data set includes an extreme value. The mean is affected by an extreme value. c. Do you think the mean of the second set is a good measure of the center of data for the salaries? Explain. No. When there is an extreme value, the mean is not always a good measure of the center of the data. 7. . Create a data set with 5 different values that have a mean of 50. a Answers will vary. b. Add 10 to each of your 5 data values. List the new values. Answers will vary. c. Find the new mean. Answers will vary. d. What do you notice? By adding 10 to each score, the mean also increases by 10. e. If a teacher wanted to increase a class test average of 62 to 70, what could he or she do to each student’s test score to achieve that? Add 8 points to each student’s test score. 8. . A student scores a 50 and 100 on two tests in a class. What is his average? a 75 b. Another student scores a 75 and 75 on two tests in a class. What is her average? 75 c. The mean gives us information about a data set’s center. What does it not describe? It does not tell whether the data is close to the mean or spread out. ALMY8454_01_AIE_C02.indd 171 5/31/13 5:12 AM
- 7. 172 Cycle 2 9. . Find the mean of this data set: 7, 13, 6, 14, 5, 15 a 10 b. If you were to draw a number line and label all the points in the data set as well as the mean, where should the mean fall? Do this to confirm your conjecture. 10 is the balancing point if we think of the number line as a teeter-totter. 10. Looking forward, looking back For each statement, incorrect units have been used. Correct the statement and explain why it was wrong. a. The average height of the males in our class is 69 square inches. It should be 69 inches. Height is a linear measurement, not an area. b. The average square footage of homes in this town is 1,500 feet. It should be 1,500 square feet. Square footage is a measurement of area, which uses square units. 65–70 min Explore 5–10 min 2.6 Measure Up Here are some commonly used geometric formulas. A = lw Parallelogram A = bh Triangle A = 1 2 bh Trapezoid A = 1 2 h (B Circle d = 2r, A = πr 2, C = 2πr = πd Cylinder V = πr 2h, S = 2πr 2 + 2πrh Cone V = 1 2 3 πr h Sphere V = 4 3 3 πr , Rectangular Prism Instructor note: Instruct students to work in groups on the Explore. Rectangle V = lwh + b) S = 4πr 2 1. There are many variables in these formulas. List what each variable represents. A area V volume S surface area B longer base of a trapezoid l length ALMY8454_01_AIE_C02.indd 172 w width b base r radius d diameter h height 5/31/13 5:12 AM
- 8. LESSON 2.6 Measure Up 173 2. Ana is making brownies and debating between an 8-inch round cake pan and an 8-inch square pan. a. If she likes thick brownies, which pan should she use? Explain why and use an a ppropriate calculation to support your answer. She should use the round pan because the bottom has a smaller area, forcing the brownies to be thicker. The area for the round pan is approximately 50.3 square inches, whereas the area for the square pan is 64 square inches. Instructor note: Bring the class together and go over answers. Discuss the Discover introduction. Discover 30 min Instructor note: To help students see a calculation with units, write this volume calculation: V = π(4 cm)2(8 cm). Instructor note: Instruct students to work in groups to complete the third column of #3 without a calculator. Give students 5 minutes. 1 b. If she wants as much crispy edge as possible, which pan should she use? Explain why and use an appropriate calculation to support your answer. She should use the square pan because that pan has a larger perimeter, producing more crust on the brownies. The perimeter for the square pan is 32 inches, whereas the p erimeter for the round pan is approximately 25 inches. When using geometric formulas to find perimeter, area, volume, or surface area, it is common to work only with numbers in the formulas and then list the units with our answer. This method of leaving the units until last can sometimes be problematic since it is easy to forget to include the units at the end. And even if we do remember to list the units, we may forget which type of unit goes with the problem being solved. Including units in the calculation solves both of these problems, but it presents new ones. How do we work with units? 3. Recall that exponents are used to express repeated multiplication. Instead of writing 2 # 2 # 2 # 2, we can write 24. The factor being repeated is 2, and the number of repetitions is 4. In this example, 2 is the base, and 4 is the exponent. Let’s work with this notation to learn more about it. 2 3 4 5 Expression Rewrite expression without exponents Rewrite result from the third column using exponents Conjecture a. 53 # 54 5#5#5#5#5#5#5 57 When multiplying numbers with like bases, keep the base and add the exponents. b. (3 # 5)2 (3 # 5) # (3 # 5) 32 # 52 When taking a product to a power, apply the exponent to each factor. c. 2 2 a b 3 2 2 a b#a b 3 3 d. (73)2 (7 # 7 # 7)2 = (7 # 7 # 7) # (7 # 7 # 7) 76 When taking a power to a power, multiply the exponents. e. 115 11 # 11 # 11 # 11 # 11 11 # 11 # 11 # 11 # 11 = 11 # 11 11 # 11 113 When dividing numbers with like bases, keep the base and subtract the exponents (top – bottom). 112 22 32 When taking a quotient to a power, apply the exponent to the numerator and denominator. Instructor note: As a class, go through each row, checking the 3rd column result and completing the 4th column together. For each row, instruct students to look only at the 2nd and 4th columns and make a conjecture allowing them to skip directly from the 2nd to the 4th column. List conjectures in the last column. Point out that examples do not prove a rule but are helpful in this case. ALMY8454_01_AIE_C02.indd 173 5/31/13 5:12 AM
- 9. 174 Cycle 2 Instructor note: Instruct students to work in groups on #4 and #5. Give them 5 minutes. 4. Simplify the four expressions using exponents. Do not use your calculator. 44 43 42 41 = = = = 256 64 16 4 Look at the problems and results to find a pattern. Write the next problem and result. 40 = 1 Make a conjecture about an exponent of zero. Answers will vary. A number to the zero power is one. 5. Simplify the four expressions using exponents. Do not use your calculator. 04 = 0 03 = 0 02 = 0 Instructor note: Go over #4–5 as a class. Explain that since 00 doesn’t have a unique answer, we say this is an indeterminate form. 01 = 0 Based on the pattern, what do you think the result should be for 00? 0 Based on #4, what do you think the result should be? 1 Use your calculator to find 00. Make a conjecture for the result you see on your calculator. Calculators should give an error message. Zero raised to the zero power is not defined since it seems it could be equal to two different numbers. We have investigated some important properties of whole-number exponents. Let’s s ummarize them. Since we are generalizing these number properties, we will use variables to represent numbers. ALMY8454_01_AIE_C02.indd 174 5/31/13 5:12 AM
- 10. LESSON 2.6 Measure Up 175 To simplify expressions with exponents: a and b are whole numbers; x and y are real numbers. 1. x ax b = x a + b W hen multiplying two numbers with the same base, keep the base and add the exponents. Instructor note: Talk about the How It Works, focusing on the version of the rules in words. 2. note Sticky nt If you forget an expone the probrule, write out . lem without exponents example, should you add For nts for or multiply the expone 4 2? Writing it out, you get x x is x · x · x · x · x · x which 6 We can see the rule is to x . the keep the base and add exponents. xa xb = xa-b W hen dividing two numbers with the same base, keep the base and subtract the exponents, top exponent – bottom exponent. 3. x 0 = 1 A number to the zero power is 1. This is true for all numbers x except zero. 4. (x a)b = x a b When taking a power to a power, multiply the exponents. 5. (x y) a = x a y a When taking a product to a power, apply the exponent to each factor. x a xa 6. a b = a y y W hen taking a quotient to a power, apply the exponent to the numerator and denominator. For example: Work with the coefficients and 1. ( - 3x 2x 5) ( - 6x 2) = ( -3x 7) ( -6x 2) = 18x 9 the exponents separately. Be 2. ( - 2x 4)4 = ( -2)4 (x 4)4 = 16x 16 careful to apply the exponent of 4 to -2, not just to 2. x 15 x 15 Simplify the numerator 3. 2 3 = 6 = x 9 and denominator first. (x ) x Instructor note: Work #6 as a class. Talk through how you choose to start and which rules you are using. Instructor note: Ask students for the base in 6d and 6e. Identifying the base will help when simplifying the expression. Instructor note: Point out that 6f can be simplified in more than one way: Apply the exponent to the numerator and denominator and then simplify or simplify first and then apply the exponent. You might want to simplify the expression both ways and discuss the pros and cons of each. ALMY8454_01_AIE_C02.indd 175 Applying exponent properties can be a challenge. Practice using them to improve your understanding. 6. Simplify each expression. a. 17x2 # 2x5 34x7 b. (4x4)2 16x 8 c. ( -5x 3)2 25x 6 d. ( -8)0 1 e. -80 -1 f. a 3x4 2 x 2 b 4 6x3 5/31/13 5:12 AM
- 11. 176 Cycle 2 We now have the skills to work with geometric formulas and units simultaneously. 10-15 min Instructor note: Complete 7a with the class to model the correct work. Instruct students to work in groups on #7b–8. 7b allows s tudents to see where cc’s occur. Go over answers. tech Tip Finding half a number is the same as dividing by 2. So if you have to calculate 1 2 (5)(20), you can type (1 , 2) # 5 # 20 or 5 # 20 , 2. 7. . ind the area of a triangle with base 5 inches and height 20 inches. a F A = 1 1 bh = (5 in.)(20 in.) = 50 in.2 2 2 b. Find the volume of a cylinder whose radius is 2 centimeters and height is 1 centimeter. Find an approximate answer by using 3.14 for π and round to the nearest tenth. V = πr 2h = π (2 cm)2 (1 cm) = π (4 cm2)(1 cm) ≈ 12.6 cm3 or 12.6 cc c. For a sphere, find the ratio of its volume to its surface area. 4 3 4 r πr 4 1 V 3 3 4 1 r = r , 4 = r# = or r = = S 4 3 3 4 3 3 4πr 2 8. . Find the area of a square whose sides have lengths of 3 centimeters. a Why do you think 32 is sometimes read as “three squared”? 9 cm2; The quantity 32 gives the area of a square with sides of length 3. b. Find the volume of a cube whose sides have lengths of 5 feet. Why do you think 53 is sometimes read as “five cubed”? 125 feet3; The quantity 53 gives the volume of a cube with sides of length 5. Connect 5–10 min Instructor note: Explain the formula in #9 and emphasize the difference between exponents and subscripts. The idea of magnets may help illustrate the formula. Have students complete #9 and go over answers. Instructor note: Part a p reviews the concepts of direct and inverse variation, developed in Cycle 3. Instructor note: Point out that understanding formulas and the units involved will be useful if students will be taking more science classes. ALMY8454_01_AIE_C02.indd 176 9. The following formula calculates the gravitational force between two objects based on their masses (m1 and m2) and the distance between them (r). m1m2 F = G 2 r a. If the masses increase, what happens to the force between them? It increases as well. If the distance between the objects increases, what happens to the force between them? It decreases. b. m1 and m2 have units of kilograms, and r has units of meters. What units must the constant G have in order for the units on F to be newtons (N)? Nm 2 kg 2 5/31/13 5:12 AM
- 12. LESSON 2.6 Measure Up 5 min Instructor note: Explain the contents of the Wrap-Up. Have tudents write an answer to s the ycle question, “Why does it c m atter?” A prompt is provided to help students. Discuss homework. Wrap-Up lesson Reflect 177 What’s the point? Including units when working with formulas illuminates the units for the answer. To do calculations this way, we must understand the properties and rules of exponents. What did you learn? How to apply whole-number exponent rules How to use basic geometric formulas Cycle 2 Question: Why does it matter? Working with calculations and units at the same time can be more work. Why does it matter that we do this? It makes clear the type of units to be included in the answer. Area and surface area will be in square units. Volume will be in cubic units. If we do not use units in calculations, the units of the results must be memorized and might have little meaning. Also, it is easy to forget to put units in the answer if you have not been using them in the calculations. 2.6 Homework Skills MyMathLab • Apply whole-number exponent rules. • Use basic geometric formulas. 1. Simplify. a. (14ab)3(ab) 2,744a 4b 4 b. 150(3x2)3 27x 6 c. ALMY8454_01_AIE_C02.indd 177 4m9 m 6 8m3n 2n 5/31/13 5:12 AM
- 13. 178 Cycle 2 2. a. n interior designer is designing a round table whose radius is 24 inches. She A would like to use a special kind of trim, which is quite expensive, for the edge of the table. Instead of building the table and measuring for the length of trim, she is working on paper in case the trim is too expensive and she needs to make the table smaller. How much trim will she need? Round up to the nearest whole inch to ensure that she has enough. 151 inches b. If she wants to cover the top of the table with a special crackle paint, how many square feet of surface will the paint need to cover? Round up to the nearest whole square foot to ensure she has enough. 13 ft2 Concepts and Applications • Apply whole-number exponent rules. • Use basic geometric formulas. 3. The volume of a substance is related to its mass and density by this formula: V = m. If d the mass is in grams and the density is in grams per cubic centimeter, what will be the units for volume? cubic centimeters 4. The distance a projectile will travel when it is fired at an initial angle of 30 degrees 2 can be found using the formula D = vg # 0.87. If velocity is in meters/second and g is in meters/second2, what will be the units for distance? meters 5. The formula T = 2π L gives the time T for a full cycle of a pendulum’s swing where Ag the pendulum’s length in feet is L, and g is the acceleration due to gravity, 32.2 ft/sec2. When T is calculated from this formula, what will the units be? seconds FP 50–55 min Explore 5–10 min Instructor note: Instruct students to work in groups on the Explore. Go over answers. 2.7 Count Up 1. Find the following sums. List the items being counted in each problem. a. 2 cats + 3 cats 5 cats Items: cats 2 1 b. + 5 5 3 5 Items: fifths 3 2 c. + 7 7 5 7 Items: sevenths ALMY8454_01_AIE_C02.indd 178 5/31/13 5:12 AM