Warming the Bench Using Estimation and Benchmark Percents 2 WARM UP Compute each product. 1. 1 10 3 350 2. 1 100 3 350 3. 1 10 3 670 4. 1 100 3 670 LEARNING GOALS Order fractions, decimals, and percents. Estimate the percent of a quantity shaded in a model. Use benchmark percents to calculate common percents of quantities. Estimate percents using benchmarks. KEY TERM benchmark percents You have used reasoning to calculate areas, volumes, decimal and fractional values, and equivalent ratios. How can reasoning be used to solve percent problems? LESSON 2: Warming the Bench M2-123
Getting Started Putting It All in Perspective In your opinion, what does each famous quotation or saying really mean? 1. Genius is one percent inspiration and ninety-nine percent perspiration. -Thomas Edison 2. Success is 99 percent failure. -Soichiro Honda 3. You miss 100 percent of the shots you never take. -Wayne Gretzky 4. "Always give 110%. It's the extra 10% that everyone remembers." -Frank Sonnenberg M2-124 TOPIC 2: Percents
ACTIVITY 2.1 Ordering Fractions, Decimals, and Percents Each student has been given a note card that contains a number expressed as a fraction, decimal, or percent. As a class, order the set of numbers from least to greatest. Think about all of the different ways to express your number. 1. Explain the strategies used by your class to order the numbers. Noah and Dylan were assigned the numbers 0.0 6 and 0.1% but they disagreed on which was larger. Noah says that 0.0 6 is less than 0.1, so 0.0 6 is less than 0.1%. Dylan says that since 0.1% is the same as as 0.001 and 0.001 is less than 0.0 6, 0.1% is less than 0.0 6. 2. Who is correct? Explain your reasoning. 3. Order the numbers from least to greatest. 0.99, 1 9, 17 20, 95%, 25%, 3 8, 70%, 4.3%, 0.81, 0.64 LESSON 2: Warming the Bench M2-125
ACTIVITY 2.2 Estimating Percents from Pictures You know that 100% means one, or the whole, and 50% means half. You can estimate a lot of percents when using a visual model. A laptop computer uses an icon of a battery on the toolbar to show how much power is left in the battery. When you glance at the icon, you can get a good estimate of how much battery life remains before you need to recharge the battery. 1. Estimate how much battery power remains by writing the percent under each battery icon. a. b. c. Are your estimates the same as your partner's? d. e. f. 2. Estimate the shaded part of each circle shown, and write it as a percent. a. b. c. M2-126 TOPIC 2: Percents
d. e. f. 3. Estimate the shaded part of each model, and write it as a fraction, a decimal, and a percent. Write the fraction in lowest terms. a. b. c. d. e. f. Can I determine the percent shown if the shading isn't all together and the parts are not all the same size? 4. Describe the strategies that you used to make your estimations. LESSON 2: Warming the Bench M2-127
ACTIVITY 2.3 Benchmark Percents A benchmark percent is a percent that is commonly used, such as 1%, 5%, 10%, 25%, 50%, and 100%. With fractions and decimals, benchmarks can be used to make estimations. With percents, however, you can use benchmarks to calculate any whole percent of a number. 100% 50% 50% 25% 25% 25% 25% Remember, you worked with the benchmark fractions of 0, 1, and 1. 2 20% 20% 20% 20% 20% 10% 10% 10% 10% 10% 10% 10% 10% 10% 10% 1. Use the tape diagram to state each relationship. a. How is 50% related to 100%? b. How is 25% related to 100%? How is 25% related to 50%? c. How is 10% related to 100%? How is 10% related to 50%? M2-128 TOPIC 2: Percents
2. Continue the pattern from the tape diagram to state each relationship. a. How is 5% related to 10%? Remember that 1% = 0.01. b. How is 1% related to 10%? How is 1% related to 5%? 3. Use the benchmark percents to determine each value if 600 is 100%. a. 50% b. 25% c. 10% d. 5% e. 1% LESSON 2: Warming the Bench M2-129
4. Use your calculator to determine the percent of each number. a. 1% of 28 5 b. 10% of 28 5 c. 1% of 234 5 d. 10% of 234 5 e. 1% of 0.85 5 f. 10% of 0.85 5 g. 1% of 5.86 5 h. 10% of 5.86 5 i. 1% of 98.72 5 j. 10% of 98.72 5 k. 1% of 1085.2 5 l. 10% of 1085.2 5 M2-130 TOPIC 2: Percents
NOTES 5. What patterns do you notice in your answers in Question 4? 6. Write a rule to calculate 1% of any number. 7. Write a rule to calculate 10% of any number. 8. Use the patterns you recognized in Question 4 to calculate each value. a. 10% of 45.21 b. 1% of 45.21 c. 10% of 0.72 d. 1% of 0.72 e. 10% of 2854 f. 1% of 2854 LESSON 2: Warming the Bench M2-131
ACTIVITY 2.4 Determining Percents using Benchmarks Deciding how much tip to leave a server at a restaurant is one way that percents are used in the real world. Akuro eats at the Eat and Talk Restaurant and decides to leave a 15% tip. Akuro says, I can easily calculate 10% of any number, and then calculate half of that, which is equal to 5%. I can then add those two percent values together to get a sum of 15%. 1. Is Akuro s method reasonable? 2. How much should he leave for a tip of 15% on $16.00? 3. What is 15% of each restaurant check total given? Explain how you calculated your answer. Round to the nearest hundredth if necessary. a. $24.00 b. $32.56 c. $47.00 You can determine any whole percent of a number by using 10%, 5%, and 1%. 4. How can you use 10%, 5%, and/or 1% to determine each percent given? Explain your reasoning. a. 18% b. 25% c. 37% M2-132 TOPIC 2: Percents
5. Calculate each value using 1%, 5%, and 10%. a. 27% of 84 b. 43% of 116 c. 98% of 389 d. 77% of 1400 e. 12% of 1248 6. About 12% of the United States population is left-handed. Use this estimate to determine about how many left-handed students there would be for each class of the given size. a. 150 students b. 200 students So, if 12 percent of the U.S. population is left-handed, what percent of the population is right-handed or "both"-handed? c. 375 students LESSON 2: Warming the Bench M2-133
NOTES TALK the TALK Brain Weights A chimpanzee s brain weight can be compared to the brain weight of other mammals. Assume that the weight of an average chimpanzee s brain is 400 grams. The table provides the average brain weight of various mammals as a percent of a chimp s brain weight. Lion Sheep Cat Rabbit Human Bear Average Brain Weight as a Percentage of a Chimp s Brain Weight Average Brain Weight (grams) 60% 35% 7% 2.5% 350% 119% 1. Order from least to greatest the brain weights of the mammals in the table, along with the chimpanzee, based on percents. 2. Use benchmarks to determine the average brain weights for each animal. Show all of your work. 3. Does the order of the percents match the order of the brain weights? Why or why not? M2-134 TOPIC 2: Percents
Assignment Write Explain how to use benchmark percents to order and estimate the value of other percents. Remember Benchmarks percents 1%, 5%, 10%, 25%, 50%, and 100% can be used to perform mental estimation and calculation of percents. Values of benchmark percents can be added and subtracted to calculate the value of other percents. Practice The students at Penncrest Middle School sold various products for a fall fundraiser. The table shows the percent of profit the school earned and the total amount sold for each type of product. Product Percent Profit Amount Sold Candy 65% $6400 Wrapping paper 40% $1200 Stationery 50% $900 Calendars 25% $3120 1. Use benchmark percents to calculate the amount of profit the school earned on the sale of each product. a. Candy b. Wrapping paper c. Stationary d. Calendars 2. Suppose that the students also sold $4500 worth of pens and pencils, which earned a 42% profit. Calculate the profit the school earned on pens and pencils. LESSON 2: Warming the Bench M2-135
Stretch Assume the weight of an average chimpanzee s brain is 400 grams. If the average hedgehog s brain weight is 0.8% of a chimp s brain weight, use benchmark percents to determine the average weight of a hedgehog s brain. Review 1. Complete the table. Write each as a fraction, decimal, and percent. Fraction Decimal Percent 3% 1.5 13 20 2 3 2. Miss Jenn is the teacher of a preschool class at Kids Unlimited Daycare. She must split the children s time between playing and learning. For every 30 minutes, the children will spend 18 minutes playing and 12 minutes learning. Complete the table of equivalent ratios. Total amount of time 30 90 Playing time 18 144 Learning time 12 48 3. Use the standard algorithm to determine each quotient. a. 8302 4 28 b. 39.13 4 4.3 M2-136 LESSON TOPIC 2: 1: Percents A Trip to the Moon