Objective To broaden studentsвЂ™ understanding of calculating percents to include discounts. 1 materials Teaching the Lesson Key Activities Students estimate and then calculate the percent of a number using fractions and decimals. They identify and calculate percent discounts and explore the percent key on their calculators. Щ— Math Journal 2, pp. 277 and 278 Щ— Student Reference Book, pp. 263вЂ“265 Щ— Study Link 8 8 бњ Key Concepts and Skills вЂў Calculate percents and discounts, and describe strategies used. [Number and Numeration Goal 2] вЂў Convert between fractions, decimals, and percents. [Number and Numeration Goal 5] вЂў Use ratios expressed as percents to solve problems. Щ— slates Щ— calculator Щ— overhead calculator (optional) See Advance Preparation [Operations and Computation Goal 7] Key Vocabulary discount Ongoing Assessment: Recognizing Student Achievement Use Mental Math and Reflexes. [Number and Numeration Goal 5] 2 Ongoing Learning & Practice Students practice and maintain skills through Math Boxes and Study Link activities. 3 Students review finding the percent of a number. ENRICHMENT Students explore a variety of situations involving discounts. Additional Information Advance Preparation Many calculators have a percent key, but not all calculators work the same. For Part 1, familiarize yourself with the use of the percent key on the calculators used by your class. Directions for the TI-15 and Casio fx-55 have been provided. 664 Unit 8 Fractions and Ratios Щ— Math Journal 2, p. 279 Щ— Study Link Master (Math Masters, p. 238) materials Differentiation Options READINESS materials Щ— Teaching Masters (Math Masters, pp. 239 and 240) Щ— calculator (optional) Technology Assessment Management System Mental Math and Reflexes See the iTLG. Getting Started Mental Math and Reflexes а¬™ Math Message Have students write each percent as a decimal and as a fraction in simplest form. Suggestions: 1 20% 0.20, or 0.2; бЋЏ5бЋЏ 1 бЋЏ 29% 0.29; бЋЏ 100 50% 0.50, or 0.5; бЋЏ2бЋЏ 10% 0.10, or 0.1; бЋЏ1бЋЏ0 1 100% 1.00; бЋЏ1бЋЏ 1 9 45% 0.45; бЋЏ2бЋЏ0 It would cost $150,000 to rent a large amusement park for a private party. Would you rather have this price reduced by $35,000 or discounted 25%? 7 29 35% 0.35; бЋЏ2бЋЏ0 3 110% 1.10; бЋЏ1бЋЏ0 11 12% 0.12; бЋЏ2бЋЏ5 Study Link 8 8 Follow-Up бњ Have partners compare answers and resolve differences. Ongoing Assessment: Recognizing Student Achievement Mental Math and Reflexes а¬™ Use the Mental Math and Reflexes problems to assess studentsвЂ™ facility with converting between fractions, decimals, and percents and expressing fractions in simplest form. Students are making adequate progress if they have correctly written both the decimal and the fraction. [Number and Numeration Goal 5] 1 Teaching the Lesson б¤ Math Message Follow-Up WHOLE-CLASS DISCUSSION Have small groups share their solution strategies and then choose one strategy to represent the group. Ask volunteers to present the group strategies. A 25% discount is $37,500; therefore, it is a better deal than a $35,000 discount. б¤ Finding the Percent of WHOLE-CLASS DISCUSSION a Number Pose the following percent-of-a-number problem: The flu hit Roosevelt School hard. 40% of the 480 students were absent at least one day last week. How many students were absent at least one day? Write 40% of 480 П students absent at least one day on the board or a transparency. Ask students to share their solution strategies for this problem. Expect a variety of responses such as the following: бџ The unit-percent approach. 480 is 100%, or the whole. 1% of 480 is 4.8. 40% of 480 is 40 вЂ« ШЎвЂ¬4.8, or 192. Lesson 8 9 бњ 665 Student Page Date Time LESSON Remind students that to find 1%, they could divide the whole by 100 or think: What times 100 equals the whole? Finding a Percent of a Number 8 9 бњ 1. The Madison Middle School boysвЂ™ basketball team has played 5 games. The table at the right shows the number of shots taken by each player and the percent of shots that were baskets. Study the example. Then calculate the number of baskets made by each player. Example: Bill took 15 shots. He made a basket on 40% of these shots. 40 Shots Taken Percent Made Baskets Bill 15 40% 6 Amit 40 30% Josh 25 60% Kevin 8 75% Mike 60 25% Zheng 44 25% Player 4 бЋЏ, or бЋЏбЋЏ 40% П бЋЏ 100 10 4 Вє 15 4 4 15 60 бЋЏбЋЏ of 15 П бЋЏбЋЏ вЂ« ШЎвЂ¬бЋЏбЋЏ П бЋЏбЋЏ П бЋЏбЋЏ П 6 10 Вє 1 10 10 1 10 Bill made 6 baskets. Sample answers: 2. On the basis of shooting ability, AndrГ© 50 10% David 25 20% Bob 18 50% Lars 15 20% Justin 28 25% бџ The 10-percent approach. 480 is 100%, or the whole. 12 15 6 15 11 5 5 9 3 7 10% of 480 is 48. 40% of 480 is 4 вЂ« ШЎвЂ¬48, or 192. Remind students that to find 10%, they could divide the whole by 10 or think: What times 10 equals 480? 2 бџ The equivalent-fraction approach. 40% is equal to бЋЏ5бЋЏ. which five players would you select as the starting lineup for the next basketball game? 2 2 40% of 480 is the same as бЋЏ5бЋЏ of 480, or бЋЏ5бЋЏ вЂ« ШЎвЂ¬480, which is 192. Bill, Amit, Josh, Kevin, and Bob бџ The equivalent-decimal approach. 40% is equal to 0.40. Explain your choices. They have the highest percent of baskets made. 40% of 480 is the same as 0.40 of 480, or 0.40 вЂ« ШЎвЂ¬480, which is 192. Kevin Why? Of the few shots taken, he has made baskets 75% of the time. 3. Which player(s) would you encourage to shoot more often? AndrГ© Of the many shots taken, he has made baskets only 10% of the time. 4. Which player(s) would you encourage to pass more often? Links to the Future Why? 277 This lesson stresses the use of equivalent fractions and equivalent decimals. Lesson 8-10 will address unit percents. Math Journal 2, p. 277 б¤ Using Fractions to Find the PARTNER ACTIVITY Percent of a Number (Math Journal 2, p. 277) Work through the example at the top of the journal page as a class. Emphasize the following points: бџ A percent can be thought of as a fraction with a denominator of 100. 43 Student Page Date Time LESSON 8 9 бњ Calculating a Discount Example: The list price for a toaster is $45. The toaster is sold at a 12% discount 12 бЋЏ (12% off the list price). What are the savings? (Reminder: 12% = бЋЏ 100 П 0.12) Paper and pencil: 12% of $45 540 бЋЏ П 12 вЂ« ШЎвЂ¬45 П бЋЏ 100 100 вЂ« ШЎвЂ¬1 П $5.40 Г† Calculator B: Enter 0.12 45 45 and interpret the answer of 5.4 as $5.40. Then use your calculator or paper and pencil to calculate the discount. (If necessary, round to the nearest cent.) Estimated Discount Item List Price Percent of Discount Clock radio $33.00 20% $6.00 $6.60 Calculator $60.00 7% Sweater $20.00 42% $4.00 $8.00 $4.20 $8.40 Calculated Discount $54.00 Tent $180.00 30% $54.00 Bicycle $200.00 17% $30.00 $34.00 Computer $980.00 25% $250.00 $245.00 Skis $325.00 18% $65.00 $58.50 $29.99 15% $4.50 $4.50 $110.00 55% $55.00 $60.50 278 Math Journal 2, p. 278 666 Unit 8 Fractions and Ratios 1 1 бЋЏ бЋЏбЋЏ бЋЏбЋЏ Example: 50% of 18 is бЋЏ 100 or 2 of 18, or 2 вЂ« ШЎвЂ¬18; and 25% of 28 1 1 is бЋЏ4бЋЏ of 28, or бЋЏ4бЋЏ вЂ« ШЎвЂ¬28. NOTE Some students might write number models in the form 60% вЂ« ШЎвЂ¬25, rather Sample answers: Jacket 100 бџ Many commonly used percents are equivalent to easy fractions: 1 1 25% П бЋЏ4бЋЏ, 50% П бЋЏ2бЋЏ, and so on. 50 and interpret the answer of 5.4 as $5.40. First use your percent sense to estimate the discount for each item in the table below. The discount is the amount by which the list price of an item is reduced. It is the amount the customer saves. Double CD 7 бџ Finding the percent of a number can be thought of as finding a fraction of that number. 12 12 45 бЋЏ вЂ« ШЎвЂ¬45 П бЋЏбЋЏ вЂ« ШЎвЂ¬бЋЏбЋЏ ПбЋЏ 100 100 1 Calculator A: Enter 0.12 99 бЋЏ бЋЏбЋЏ бЋЏбЋЏ бЋЏбЋЏ Example: 43% is бЋЏ 100 ; 99% is 100 ; 7% is 100; and 100% is 100, or 1. 60 6 бЋЏ бЋЏбЋЏ than бЋЏ 100 вЂ« ШЎвЂ¬25, 10 вЂ« ШЎвЂ¬25, or 0.6 вЂ« ШЎвЂ¬25. All of these notations are acceptable. Assign the journal page. Circulate and assist. б¤ Calculating a Percent Discount PARTNER ACTIVITY (Math Journal 2, p. 278) Refer students to journal page 278. As a class, work through the example at the top of the page. Remind students that a discount Student Page is the amount to be subtracted from a given whole. In this case, it is by how much the list price is reduced. Date Time LESSON Math Boxes 8 9 бњ 1. Rename each fraction as a mixed number Ask students to complete the Estimated Discount column of the table on journal page 278. When most students have finished, have volunteers share their solution strategies. For example: бџ 20% is equal to 1 бЋЏбЋЏ. 5 79 a. бЋЏбЋЏ 8 П 45 b. бЋЏбЋЏ 9 П 7 бЋЏбЋЏ 8 9 5 37 21 21бЋЏ35бЋЏ 111 c. бЋЏбЋЏ П 3 126 d. бЋЏбЋЏ П 6 Clock radio: 20% of $33 108 e. бЋЏбЋЏ П 5 One-fifth of $33 is between $6 and $7. 2. Find the area of the rectangle. or a whole number. 6 3 6 вЂ«( ШЎвЂ¬3 П© 5) П A, (6 вЂ« ШЎвЂ¬3) П© (6 вЂ« ШЎвЂ¬5) П A, or 6 вЂ« ШЎвЂ¬8 П A Answer: 62 3. Sam has 8 pounds of oats. A cup of oats is about бЋЏ1бЋЏ a pound. How many cups of 2 бџ $980 is about $1,000. So 25% of $980 is about or $250. Yes. If Amy does the dishes on 6 nights, she will make more than Julie: 6 вЂ« ШЎвЂ¬0.75 П 6 6 3 18 75 1 бЋЏбЋЏ вЂ« ШЎвЂ¬бЋЏбЋЏ П бЋЏбЋЏ вЂ« ШЎвЂ¬бЋЏбЋЏ П бЋЏбЋЏ П 4бЋЏбЋЏ, 1 1 4 4 100 2 of $1,000, or 4.5 П $4.50. 79 80 Jacket: 55% of $110 189 dishes. She pays her sister Amy $0.75 each time Amy does the dishes for her. Is that fair? Explain. 16 c 1 бЋЏбЋЏ 4 48 units2 4. Julie makes $4.00 per week for washing oats does Sam have? Computer: 25% of $980 5 Number model: 42 5. a. Plot the following points on the grid: (4,2); (2,4); (2,7); (6,7) 1 бџ 55% is about бЋЏ2бЋЏ. So 55% of $110 is about $55. 8 7 b. Connect the points with line 6 segments in the order given above. Then connect (6,7) and (4,2). What shape have you drawn? Ask students to calculate and record the actual discount using any method they choose. Encourage them to try calculator and paper-and-pencil calculations. 5 4 Quadrangle, or quadrilateral 3 2 1 0 0 1 2 3 4 5 6 7 8 208 279 Math Journal 2, p. 279 б¤ Exploring the Percent Key PARTNER ACTIVITY on a Calculator (Student Reference Book, pp. 263вЂ“265) Refer students to Student Reference Book, pages 263вЂ“265. As a class, discuss the examples for using the percent key to solve percent-of problems and to display percents as decimals and as fractions in simplest form. If your class calculators differ from the Student Reference Book examples, demonstrate the key sequences using an overhead calculator or by drawing the key sequences on the board. Ask students to use the percent key on their calculators to find 25% of 180. 45 Have volunteers share the key sequences they entered. Partners then make up percent-of problems for each other to solve. Study Link Master Name Date STUDY LINK 89 1. Complete the table so each number is shown as a fraction, decimal, and percent. Fraction 45 бЋЏбЋЏ, 100 б¤ Math Boxes 8 9 INDEPENDENT ACTIVITY (Math Journal 2, p. 279) Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 8-11. The skill in Problem 5 previews Unit 9 content. or бЋЏ9бЋЏ 20 бЋЏ2бЋЏ 10 15 бЋЏбЋЏ, 100 2. бњ Decimal 3 бЋЏбЋЏ 10 2 Ongoing Learning & Practice Time Fractions, Decimals, and Percents бњ or 3 бЋЏбЋЏ 20 26 48 51 83 89 Percent 0.45 45% 0.3 30% 0.2 20% 0.15 15% Use your percent sense to estimate the discount for each item. Then calculate the discount for each item. (If necessary, round to the nearest cent.) Sample answers: Item List Price Percent of Discount Estimated Discount Calculated Discount Saguaro cactus with arms $400.00 25% $100.00 $100.00 Life-size wax figure of yourself $1,500.00 $1,600.00 $10,000.00 16% Manhole cover $78.35 10% $8.00 $7.84 Live scorpion $14.98 5% $0.75 $0.75 10,000 honeybees $29.00 30% $9.00 $8.70 Dinner for one on the Eiffel Tower $88.00 6% $4.50 $5.28 MagicianвЂ™s box for sawing a person in half $4,500.00 18% $900.00 $810.00 Fire hydrant $1,100.00 35% $350.00 $385.00 Source: Everything Has Its Price Math Masters, p. 238 Lesson 8 9 бњ 667 Teaching Master Name Date LESSON 89 бњ Time Finding the Percent of a Number The unit percent is 1% or 0.01. For example, the unit percent of 100 is 1; the unit percent of 200 is 2; the unit percent of 10 is 0.1. 20 10 1% of 100 1% of 200 1 mm 0 cm 1 2 3 4 5 6 7 8 9 10 1% of 10 cm Another way to think of the unit percent of a number is to think: What number times 100 equals the whole? For example, 1 вЂ« ШЎвЂ¬100 П 100; 2 вЂ« ШЎвЂ¬100 П 200; 0.1 вЂ« ШЎвЂ¬100 П 10 1 100 Writing/Reasoning Have students write a response to the following: Explain how to rename an improper fraction as a mixed number. Use Problem 1a as an example. Sample 79 answer: To rename бЋЏ8бЋЏ as a mixed number, first find how many groups of 8 are in 79. There are 9 groups of 8 because 9 вЂ« ШЎвЂ¬8 П 72. Then subtract 72 from 79. The difference is the fraction part of 79 72 7 the mixed number. It tells how many eighths are left: бЋЏ8бЋЏ ПЄ бЋЏ8бЋЏ П бЋЏ8бЋЏ. 79 7 7 So бЋЏ8бЋЏ П 9 П© бЋЏ8бЋЏ, or 9бЋЏ8бЋЏ. To find the unit percent of a whole, multiply by 0.01 or бЋЏбЋЏ. Solve. 1. 1% of 84 П 0.84 2. 0.35 1% of 35 П 3. 6.28 1% of 628 П The unit percent can be used to find other percents of a whole. For example, if you want to find 8% of 200: бњ Calculate the unit percent: 1% of 200 П 200 вЂ« ШЎвЂ¬0.01 П 2 б¤ Study Link 8 9 бњ INDEPENDENT ACTIVITY (Math Masters, p. 238) бњ Check your answer: 2 вЂ« ШЎвЂ¬100 П 200. бњ Multiply your answer by the percent you are finding: 2 вЂ« ШЎвЂ¬8 П 16; 8% of 200 П 16 Solve. 15.96 25.2 19% of 84 П 7. Think about the steps you followed in Problems 4вЂ“6. First you multiplied the unit percent by 0.01, and then you multiplied the product by the number of percents. How can you find the percent of a number by multiplying only once? Provide an example. 5. 72% of 35 П 6. 37% of 628 П 232.36 4. Home Connection Students convert between fractions, decimals, and percents. Then they estimate and calculate discounts for various items. I can change the percent to a fraction and multiply the number by that fraction. 19 19 84 1,596 19% of 84: бЋЏ10бЋЏ0 вЂ« ШЎвЂ¬84 П бЋЏ10бЋЏ0 вЂ« ШЎвЂ¬бЋЏ1бЋЏ П бЋЏ10бЋЏ 0 П 15.96 Math Masters, p. 239 3 Differentiation Options READINESS б¤ Finding the Percent of SMALL-GROUP ACTIVITY 15вЂ“30 Min a Number (Math Masters, p. 239) Teaching Master Name LESSON 89 бњ Date Time Calculating Discounts There are 2 steps to finding a discounted total: бњ Calculate the amount that represents the percent of discount. бњ Subtract the calculated discount from the original total. This is the discounted total. Calculate the discounted total for the following problems. Show your work on the back of this sheet. 1. A computer store has an Internet special for their customers. If Carla spends $50.00 or more, she gets 10% off her order. The shipping and handling charge is 4% of the original total. Carla buys $68.00 in software. What is her total charge? To review finding the percent of a number, have students use unit fractions. Ask students to describe a unit fraction. Sample answer: A unit fraction has a 1 as its numerator and names one of the equal parts that make the whole. Have a volunteer use counters to demonstrate how to use the unit fraction to 3 1 3 find бЋЏ4бЋЏ of 20. бЋЏ4бЋЏ of 20 is 5 because 20 divided by 4 equals 5. So, бЋЏ4бЋЏ of 20 is 5 вЂ« ШЎвЂ¬3, or 15. Explain that the unit percent is similar to a unit fraction. You can find the percent of a whole by finding the unit percent and multiplying. Have students use mental math, paper and pencil, and/or calculators to complete the Math Masters page. $63.92; Discount: 68 вЂ« ШЎвЂ¬0.10 П 6.80; shipping and handling: 68 вЂ« ШЎвЂ¬0.04 П 2.72; total charge: 68.00 П© 2.72 ПЄ 6.80 П 63.92 2. The Hartfield School District wants to get the government discount for telephone service. The discount is based on the percent of students qualifying for the National School Lunch Program. 32% of students in this urban district qualify. The district pays about $3,500 per month for telephone service. Use the table below to find how much the district would save. Percent of Students Urban Discount Rural Discount Less than 1% 20% 25% 1% to 19% 40% 50% 20% to 34% 50% 60% 35% to 49% 60% 70% 50% to 74% 80% 80% 75% to 100% 90% 90% 50% discount. The district will save The Hartfield School District is eligible for a about $1,750.00 per month for its telephone service. With the government discount, the district will pay about $1,750.00 per month. 3. At the Goose Island Family Restaurant, if the original bill is $75.00 or more, the kidsвЂ™ meals are discounted 3%. If the original bill is $95.70, with $23.00 for kidsвЂ™ meals, what is $0.69 What is the discounted total? $95.01 the discounted amount? Math Masters, p. 240 668 Unit 8 Fractions and Ratios ENRICHMENT б¤ Calculating Discounts PARTNER ACTIVITY 15вЂ“30 Min (Math Masters, p. 240) To apply studentsвЂ™ understanding of calculating percents, have them find and calculate the discounts for a variety of situations. Students solve number stories by calculating the percent discount and the discounted total.