# Math Lit Lessons 2.5 & 2.6

13 de Feb de 2014
1 de 13

### Math Lit Lessons 2.5 & 2.6

• 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