Proportions and Percent

Transcription

1 Proportions and Percent MODULE 5? ESSENTIAL QUESTION How can you use proportions and percent to solve real-world problems? You can use percent and proportions to find the amount by which real-world quantities have increased or decreased. LESSON 5.1 Percent Increase and Decrease 7.RP.3 LESSON 5.2 Rewriting Percent Expressions 7.RP.3, 7.EE.2, 7.EE.3 LESSON 5.3 Applications of Percent 7.RP.3, 7.EE.3 Real-World Video Sam Dudgeon/Houghton Mifflin Harcourt A store may have a sale with deep discounts on some items. They can still make a profit because they first markup the wholesale price by as much as 400%, then markdown the retail price. Math On the Spot Animated Math 137 Module 5 Go digital with your write-in student edition, accessible on any device. Scan with your smart phone to jump directly to the online edition, video tutor, and more. Interactively explore key concepts to see how math works. Get immediate feedback and help as you work through practice sets. 137

2 Are You Ready? Are YOU Ready? Assess Readiness Complete these exercises to review skills you will need for this module. Use the assessment on this page to determine if students need intensive or strategic intervention for the module s prerequisite skills. Percents and Decimals 2 1 EXAMPLE Enrichment Online Assessment and Intervention Online Practice and Help = Write the percent as the sum of 1 whole and a percent remainder. Write the percents as fractions. = Write the fractions as decimals. = 1.47 Simplify. Write each percent as a decimal. Access Are You Ready? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment. 147% = 100% + 47% Response to Intervention Intervention 1. 22% % % 98% % 1.89 Write each decimal as a percent. Online and Print Resources Skills Intervention worksheets Differentiated Instruction Skill 30 Percents and Decimals Challenge worksheets Skill 46 Find the Percent of a Number 59% % % Find the Percent of a Number PRE-AP EXAMPLE PRE-AP Extend the Math Lesson Activities in TE Real-World Video Viewing Guide 30% of 45 =? 30% = _ 13.5 Write the percent as a decimal. Multiply. 3 Find the percent of each number. After students have watched the video, discuss the following: What is a discount? How is the amount of the discount calculated? Multiply the original price by the discount expressed as a fraction or a decimal % of % of % of % of % of % of 1, Unit 2 PROFESSIONAL DEVELOPMENT VIDEO Author Juli Dixon models successful teaching practices as she explores percent problems in an actual seventh-grade classroom. Online Teacher Edition Access a full suite of teaching resources online plan, present, and manage classes and assignments. Professional Development eplanner Easily plan your classes and access all your resources online. Interactive Answers and Solutions Customize answer keys to print or display in the classroom. Choose to include answers only or full solutions to all lesson exercises. Interactive Whiteboards Engage students with interactive whiteboard-ready lessons and activities. : Online Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated practice tests aligned with Common Core. Proportions and Percent 138

3 Reading Start-Up EL Have students complete the activities on this page by working alone or with others. Strategies for English Learners Each lesson in the TE contains specific strategies to help English Learners of all levels succeed. Emerging: Students at this level typically progress very quickly, learning to use English for immediate needs as well as beginning to understand and use academic vocabulary and other features of academic language. Expanding: Students at this level are challenged to increase their English skills in more contexts, and learn a greater variety of vocabulary and linguistic structures, applying their growing language skills in more sophisticated ways appropriate to their age and grade level. Bridging: Students at this level continue to learn and apply a range of high-level English language skills in a wide variety of contexts, including comprehension and production of highly technical texts. Active Reading Integrating Language Arts Students can use these reading and note-taking strategies to help them organize and understand new concepts and vocabulary. Additional Resources Differentiated Instruction Reading Strategies EL Reading Start-Up Visualize Vocabulary Use the words to complete the triangle. Write the review word that fits the description in each section of the triangle. Understand Vocabulary Complete the sentences using the preview words. 1. A fixed percent of the principal is simple interest. 2. The original amount of money deposited or borrowed is the principal. 3. A percent increase is the amount of increase divided by the original amount. Active Reading proportion compares a number to 100 percent a comparison of two numbers by division ratio Tri-Fold Before beginning the module, create a tri-fold to help you learn the concepts and vocabulary in this module. Fold the paper into three sections. Label the columns What I Know, What I Need to Know, and What I Learned. Complete the first two columns before you read. After studying the module, complete the third. a statement that two ratios are equivalent Vocabulary Review Words proportion (proporción) percent (porcentaje) rate (tasa) ratio (razón) unit rate (tasa unitaria) Preview Words percent decrease (porcentaje de disminución) percent increase (porcentaje de aumento) principal (capital) simple interest (interés simple) Module Focus Coherence Rigor Tracking Your Learning Progression Before Students understand proportional relationships: convert units within a measurement system solve real-world problems involving percent In this module Students represent and solve problems involving proportional relationships: solve problems involving percent increase, percent decrease, and percent of change solve markup and markdown problems use percents to find sales tax, tips, total cost, simple interest After Students will: solve real-world problems using percent 139 Module 5

4 GETTING READY FOR Proportions and Percent GETTING READY FOR Proportions and Percent Understanding the Standards and the vocabulary terms in the Standards will help you know exactly what you are expected to learn in this module. Use the examples on the page to help students know exactly what they are expected to learn in this module. Content Areas CA Common Core Standards Ratios and Proportional Relationships 7.RP Analyze proportional relationships and use them to solve real-world and mathematical problems. 7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. Key Vocabulary proportion (proporción) An equation that states that two ratios are equivalent. ratio (razón) A comparison of two quantities by division. percent (porcentaje) A ratio that compares a part to the whole using 100. What It Means to You You will use proportions to solve problems involving ratio and percent. EXAMPLE 7.RP.3 Find the amount of sales tax if the sales tax rate is 5% and the cost of the item is $ % = 100 = 1 20 Multiply 1 times the cost to find the sales tax = 2 The sales tax is $2. Expressions and Equations 7.EE Cluster Use properties of operations to generate equivalent expressions. Go online to see a complete unpacking of the CA Common Core Standards. 7.EE.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. Key Vocabulary expression (expresión) A mathematical phrase containing variables, constants and operation symbols. What It Means to You You will find helpful ways to rewrite an expression in an equivalent form. EXAMPLE 7.EE.2 A store advertises that all bicycle helmets will be sold at 10% off the regular price. Find two expressions that represent the value of the sale price p for the helmets that are on sale. Sale price = original price minus 10% of the price = p p Equivalently, p p = p(1-0.10) = 0.90p Image Credits: Hemera Technologies/Alamy Images Visit to see all CA Common Core Standards explained. 140 Unit 2 California Common Core Standards Lesson 5.1 Lesson 5.2 Lesson RP.3 Recognize and represent proportional relationships between quantities. 7.EE.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. 7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. Proportions and Percent 140

5 LESSON 5.1 Percent Increase and Decrease Lesson Support Content Objective Language Objective Students will learn to use percents to describe change. Students will show how to use percents to describe change. California Common Core Standards 7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. MP.2 Reason abstractly and quantitatively. Focus Coherence Rigor Building Background Visualize Math Have students work with a partner to shade in two decimal squares each representing a different percent. Then have them compare the percents. 80% 50% The first decimal square shows 30% more than the second decimal square. Learning Progressions In this lesson, students continue to build their understanding of percents. They will use percent to describe change as percent increase and percent decrease. Some key understandings for students are the following: Percent change (increase or decrease) is always the amount of change divided by the original amount. Percent increase describes how much a quantity increases in comparison to the original amount. Percent decrease describes how much a quantity decreases in comparison to the original amount. The original amount and the percent of change can be used to determine the new amount. The concepts of percent increase and percent decrease will be used to solve a variety of real-world problems, such as problems involving price markups and markdowns. Cluster Connections This lesson provides an excellent opportunity to connect ideas in this cluster: Analyze proportional relationships and use them to solve real-world and mathematical problems. Give students the following prompt: One week a store decreased the price of potatoes by 25%. The next week the price was increased by 25%. The original price of 5 pounds of potatoes was $4.00. How does the final price compare with the original price? Have students justify their answer. The final price was less = 1.00, 4-1 = 3. The first week the price was $3. Then = 0.75, = The final price was $3.75, which is less than the original price, $ A

6 Language Support EL PROFESSIONAL DEVELOPMENT California ELD Standards Emerging 2.I.6c. Reading/viewing closely Use knowledge of morphology, context, reference materials, and visual cues to determine the meaning of unknown and multiple-meaning words on familiar topics. Expanding 2.I.6c. Reading/viewing closely Use knowledge of morphology, context, reference materials, and visual cues to determine the meaning of unknown and multiple-meaning words on familiar and new topics. Bridging 2.I.6c. Reading/viewing closely Use knowledge of morphology, context, reference materials, and visual cues to determine the meaning, including figurative and connotative meanings, of unknown and multiple-meaning words on a variety of new topics. Linguistic Support EL Academic/Content Vocabulary percent The word percent may be familiar to speakers of Spanish and other Latin-based languages. The root word percent means one part out of each hundred. Point out that one cent is another word for one penny and that there are 100 pennies in a US dollar. In Spanish the meaning of the word for percent porciento is much more evident because the word ciento is also the word for hundred. Multiple Meaning Words change Explain to students that the word change can have different meanings in mathematics. When working with money, change can indicate an amount of money returned from a transaction, or it can indicate the coins. In this lesson, percent change describes the percent of increase or decrease in an amount compared to the original amount. Review with students the words increase and decrease. Leveled Strategies for English Learners EL Emerging Have students select a problem from this lesson and demonstrate how to find the percentage increase or the percentage decrease. Expanding Pair students and have them select a problem from the lesson and explain it to each other. Provide a sentence frame: This problem is an example of percentage increase/decrease because. Bridging Have students write in their journal when they think showing a percent increase, rather than an actual amount of increase, can be more useful. Math Talk Model for English learners how to begin their responses with a sentence frame. Finding percent increase and finding percent decrease are alike because. Finding percent increase and finding percent decrease are not alike because. Percent Increase and Decrease 141B

7 LESSON 5.1 CA Common Core Standards The student is expected to: Ratio and Proportional Relationships 7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. Mathematical Practices MP.2 Reasoning Percent Increase and Decrease Engage ESSENTIAL QUESTION How do you use percents to describe change? Sample answer: Dividing the amount of the change by the original amount results in a percent increase or decrease. Motivate the Lesson Ask: What does it mean to have a 100% increase in something? Begin the lesson to find out. Explore Motivate the Lesson A pen costs $1, and a jacket costs $199; both prices increase by $1. Have students discuss how they preceive both increases, and use this opportunity to introduce relative increases. Explain ADDITIONAL EXAMPLE 1 The number of people signed up for a bus trip increased from 32 to 45. What is the percent increase? Round to the nearest percent. 41% Interactive Whiteboard Interactive example available online ADDITIONAL EXAMPLE 2 The regular price of a scooter is $ It is on sale for $ What is the percent decrease from the regular price to the sale price of the scooter? 20% Interactive Whiteboard Interactive example available online EXAMPLE 1 Focus on Communication Make sure students can express in their own words that a percent increase is always the amount of change divided by the original amount. Questioning Strategies Mathematical Practices How do you find the amount of change? Subtract the lesser value from the greater value. YOUR TURN Focus on Technology Mathematical Practices If students are using a calculator, make sure parentheses are inputted to find (64-52) 52. Talk About It Check for Understanding Ask: If you were not told a situation is a percent increase, how could you recognize that it is? The new amount is greater than the original amount. EXAMPLE 2 Questioning Strategies Mathematical Practices How could you find 50% of 89? 50% of a number is half the number is Avoid Common Errors Remind students that when changing a fraction to a decimal, the top number is divided by the bottom number. For example, means or YOUR TURN Focus on Modeling Mathematical Practices Have students show symbolically the connection between the original amount, 18, the new amount, 12, the amount of change, 6, and the percent decrease, 33%. 141 Lesson 5.1

8 ? LESSON 5.1 How do you use percents to describe change? Finding Percent Increase Percents can be used to describe how an amount changes. Amount of Change Percent Change = Original Amount The change may be an increase or a decrease. Percent increase describes how much a quantity increases in comparison to the original amount. EXAMPLE 1 Percent Increase and Decrease ESSENTIAL QUESTION Amber got a raise, and her hourly wage increased from $8 to $9.50. What is the percent increase? STEP 1 Find the amount of change. Amount of Change = Greater Value - Lesser Value = Substitute values. = 1.50 Subtract. STEP 2 Find the percent increase. Round to the nearest percent. Amount of Change Percent Change = Original Amount = = % Reflect 1. What does a 100% increase mean? The amount of change is equal to the original amount; the value doubles. YOUR TURN Substitute values. Divide. Write as a percent and round. 2. The price of a pair of shoes increases from $52 to $64. What is the percent increase to the nearest percent? 23% 7.RP.3 7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. Math On the Spot Online Practice and Help Math On the Spot My Notes Its the same because you subtract the lesser value from the greater value to find the amount of change and then divide the amount of change by the original amount to find percent change. Its different because the original amount is greater than the new quantity. Math Talk Mathematical Practices How is finding percent decrease the same as finding percent increase? How is it different? Online Practice and Help Finding Percent Decrease When the change in the amount decreases, you can use a similar approach to find percent decrease. Percent decrease describes how much a quantity decreases in comparison to the original amount. EXAMPLE 2 David moved from a house that is 89 miles away from his workplace to a house that is 51 miles away from his workplace. What is the percent decrease in the distance from his home to his workplace? STEP 1 STEP 2 Find the amount of change. Amount of Change = Greater Value - Lesser Value = Substitute values. = 38 Subtract. Find the percent decrease. Round to the nearest percent. Amount of Change Percent Change = Original Amount = Substitute values Divide. = 43% Write as a percent and round. Reflect 3. Critique Reasoning David considered moving even closer to his workplace. He claims that if he had done so, the percent of decrease would have been more than 100%. Is David correct? Explain your reasoning. No; The least distance David could live from his YOUR TURN 7.RP.3 workplace is 0 miles, which corresponds to a 100% decrease. A decrease greater than this is impossible. 4. The number of students in a chess club decreased from 18 to 12. What is the percent decrease? Round to the nearest percent. 33% 5. Officer Brimberry wrote 16 tickets for traffic violations last week, but only 10 tickets this week. What is the percent decrease? 37.5% Lesson Unit 2 PROFESSIONAL DEVELOPMENT Integrate Mathematical Practices MP.2 This lesson provides an opportunity to address this Mathematical Practice standard. It calls for students to create and use representations to organize, record, and communicate mathematical ideas. Students use verbal equations to model a relationship among the percent increase or decrease, the amount of change, and the original amount. Students use these equations to then write numerical equations to find the percent of change. Math Background The percent of change compares the amount of change to the original amount. When there is a series of percent changes, the original amount changes with each additional percent increase or decrease. For example, the percent increase from 10 to 15, where 10 is the original amount, is a 50% increase ( = 5 = 50% ), but the percent decrease of 15 back to 10 is a 33.3% decrease ( = = 33. _ 3 % ) because 15 is now the original amount. Percent Increase and Decrease 142

9 ADDITIONAL EXAMPLE 3 A shoe sales associate earned $300 in August. In September she earned 8% more than she did in August. How much did she earn in September? $324 Interactive Whiteboard Interactive example available online EXAMPLE 3 Engage with the Whiteboard Cover up the solution and have students read the Example a couple of times. Then invite a student to circle all the information needed to solve the problem and to draw a line through any extraneous information. Questioning Strategies Mathematical Practices Why is 1.15 used as a factor for 115%? 1.15 is the decimal equivalent for 115%. To multiply by a percent, the percent must be represented by either a decimal or a fraction. How do you know whether to add or subtract the amount of change? Since the population increased, the amount of change is added to the original amount. Focus on Critical Thinking Mathematical Practices Be sure students understand how to change any percent to a decimal. Remind them that percents less than 100% will equal decimals less than 1. YOUR TURN Avoid Common Errors When solving Exercises 8 and 9, students may skip Step 2 as shown in Example 3. Remind them that the new amount for a percent increase is found by adding the original amount to the amount of change. The new amount for a percent decrease is found by subtracting the amount of change from the original amount. Elaborate Talk About It Summarize the Lesson Ask: How would you explain percent increase and percent decrease? Percent increase is a ratio of the amount of the increase to the original amount expressed as a percent. Percent decrease is a ratio of the amount of the decrease to the original amount expressed as a percent. GUIDED PRACTICE Engage with the Whiteboard In the space under each of Exercises 1 6, invite volunteers to write the original amount (OM), the amount of change (AC), and the ratio they will simplify to find the percent increase or decrease (PI or PD). So, for Exercise 1 students would write OM = 5; AC = 8-5 or 3; PI = 3 5. Avoid Common Errors Exercise 6 Remind students to use number sense to check their answers for reasonableness. 16 is more than 3 5, so the percent will be more than 200%. Exercise 14 Students might question whether the 3 hours be changed to minutes or the half hour be treated as a fraction or a decimal. Either approach will yield the correct answer. However, the math is much easier if the calculation is performed using hours as the unit of measure. 143 Lesson 5.1

10 DO NOT EDIT--Changes must be made through File info CorrectionKey=B DO NOT EDIT--Changes must be made through File info CorrectionKey=A Using Percent of Change Given an original amount and a percent increase or decrease, you can use the percent of change to find the new amount. EXAMPLE 3 7.RP.3 Math On the Spot Guided Practice Find each percent increase. Round to the nearest percent. (Example 1) 1. From $5 to $8 60% 2. From 20 students to 30 students 3. From 86 books to 150 books 74% 4. From $3.49 to $ % 50% Image Credits: Corbis The grizzly bear population in Yellowstone National Park in 1970 was about 270. Over the next 35 years, it increased by about 115%. What was the population in 2005? STEP 1 STEP 2 Find the amount of change = Find 115% of 270. Write 115% as a decimal. 311 Round to the nearest whole number. Find the new amount. New Amount = Original Amount + Amount of Change = Substitute values. = 581 Add. The population in 2005 was about 581 grizzly bears. Reflect 6. Why will the percent of change always be represented by a positive number? Sample answer: The amount of change is equal to the greater value minus the lesser value, which is always positive. 7. Draw Conclusions If an amount of $100 in a savings account increases by 10%, then increases by 10% again, is that the same as increasing by 20%? Explain. No. An increase of 10% gives a balance of $110. Another 10% increase would give a balance of $121. One increase of 20% would give a balance of $120. YOUR TURN A TV has an original price of $499. Find the new price after the given percent of change % increase $ % decrease $ Add the amount of change because the population increased. Online Practice and Help 5. From 13 friends to 14 friends 8% 6. From 5 miles to 16 miles 7. Nathan usually drinks 36 ounces of water per day. He read that he should drink 64 ounces of water per day. If he starts drinking 64 ounces, what is the percent increase? Round to the nearest percent. (Example 1) Find each percent decrease. Round to the nearest percent. (Example 2) 8. From $80 to $64 20% 9. From 95 F to 68 F 10. From 90 points to 45 points 50% 11. From 145 pounds to 132 pounds 12. From 64 photos to 21 photos 67% 13. From 16 bagels to 0 bagels 14. Over the summer, Jackie played video games 3 hours per day. When school began in the fall, she was only allowed to play video games for half an hour per day. What is the percent decrease? Round to the nearest percent. (Example 2) Find the new amount given the original amount and the percent of change. (Example 3) 15. $9; 10% increase $ cookies; 25% decrease pages; 20% decrease 272 pages members; 50% increase 42 members 21. Adam currently runs about 20 miles per week, and he wants to increase his weekly mileage by 30%. How many miles will Adam run per week? (Example 3)? ESSENTIAL QUESTION CHECK-IN 22. What process do you use to find the percent change of a quantity? Divide the amount of change in the quantity by the original amount, then express the answer as a percent. 220% 78% 28% 83% 36 cookies 19. $29,000; 4% decrease $27, songs; 130% increase 1,863 songs 26 miles 9% 100% Lesson Unit 2 7_MCABESE202610_U2M05L1.indd /10/13 2:42 AM 7_MCAAESE202610_U2M05L1.indd 144 4/23/13 12:39 PM DIFFERENTIATE INSTRUCTION Cooperative Learning Have students work in pairs to solve percent increase and decrease problems. Start with a problem, and have each person complete one step in the process. Have students exchange roles so each person has a chance to complete each step at least once. This helps emphasize that finding percent increase or decrease is a multi-step process. Critical Thinking Ask students to think about percent increase and decrease in the context of integers. For example, if a bank account increases from -$100 to $100, can you use the formula to calculate percent increase? Does the answer make sense? The formula gives a percent increase of -200% in this context. This percent doesn t make much intuitive sense, so percent increase may not be a useful tool for understanding increases from negative to positive. Additional Resources Differentiated Instruction includes: Reading Strategies Success for English Learners EL Reteach Challenge PRE-AP Percent Increase and Decrease 144

11 5.1 LESSON QUIZ Online Assessment and Intervention Online homework assignment available 7.RP.3 Lesson Quiz available online Find each percent increase or decrease to the nearest percent. 1. from 14 books to 40 books 2. from 72 points to 50 points Find the new amount given the original amount and the percent of change. 3. $12; 20% increase grams; 45% decrease 5. If 48 eggs are used in the cafeteria today but the number expected to be used tomorrow is 30% less than that, how many eggs are expected to be used tomorrow? 6. Priscilla currently reads 10 pages in her book each night. She wants to increase the number of pages by 30%. How many pages will Priscilla read each night after the increase? Evaluate GUIDED AND INDEPENDENT PRACTICE 7.RP.3 Concepts & Skills Example 1 Finding Percent Increase Example 2 Finding Percent Decrease Example 3 Using Percent of Change Additional Resources Differentiated Instruction includes: Leveled Practice Worksheets Practice Exercises 1 7, 23, 25 Exercises 8 14, 24, 25 Exercises 15 21, 26 Focus Coherence Rigor Exercise Depth of Knowledge (D.O.K.) Mathematical Practices 23 2 Skills/Concepts MP.2 Reasoning 24 2 Skills/Concepts MP.2 Reasoning 25 2 Skills/Concepts MP.4 Modeling 26 4 Extended Thinking MP.7 Using Structure 27 3 Strategic Thinking MP.7 Using Structure 28 3 Strategic Thinking MP.3 Logic 29 3 Strategic Thinking MP.4 Modeling Answers % increase 2. 31% decrease 3. $ g Lesson 5.1

12 DO NOT EDIT--Changes must be made through File info CorrectionKey=A DO NOT EDIT--Changes must be made through File info CorrectionKey=B Name Class Date 5.1 Independent Practice 23. Complete the table. Item 7.RP.3 Original Price New Price Bike $110 $96 Percent Change Increase or Decrease Scooter $45 $56 Tennis Racket $79 $ % Increase Skis $580 25% Decrease $435 13% 24% 24. Multiple Representations The bar graph shows the number of hurricanes in the Atlantic Basin from a. Find the amount of change and the percent of decrease in the number of hurricanes from 2008 to 2009 and from 2010 to Compare the amounts of change and percents of decrease. 5; 5; 62.5%; 41.7%; the amount of change is the same, but the percent of change is less from 2010 to b. Between which two years was the percent of change the greatest? What was the percent of change during that period? 2009 and 2010; 300% increase Decrease Increase Hurricanes Online Practice and Help Atlantic Basin Hurricanes Year 26. Percent error calculations are used to determine how close to the true values, or how accurate, experimental values really are. The formula is similar to finding percent of change. Experimental Value - Actual Value Percent Error = 100 Actual Value In chemistry class, Charlie records the volume of a liquid as 13.3 milliliters. The actual volume is 13.6 milliliters. What is his percent error? Round to the nearest percent. 2% FOCUS ON HIGHER ORDER THINKING 27. Look for a Pattern Leroi and Sylvia both put $100 in a savings account. Leroi decides he will put in an additional $10 each week. Sylvia decides to put in an additional 10% of the amount in the account each week. a. Who has more money after the first additional deposit? Explain. They have the same. $100 + $10 = $110 and $ %($100) = $110. b. Who has more money after the second additional deposit? Explain. Sylvia has more. Leroi has $110 + $10 = $120 and Sylvia has $ %($110) = $121. c. How do you think the amounts in the two accounts will compare after a month? A year? Because Sylvia will have more after the second additional deposit and she will be depositing increasing amounts, she will always have more in her account. Work Area 25. Represent Real-World Problems Cheese sticks that were previously priced at 5 for $1 are now 4 for $1. Find each percent of change and show your work. a. Find the percent decrease in the number of cheese sticks you can buy for $1. Amount of change = 1; percent decrease = 1_ 5 = 20% b. Find the percent increase in the price per cheese stick. $1.00 = $0.20 each; $1.00 = $0.25 each. Amount of 5 4 change = $0.05; percent increase = = 25% 28. Critical Thinking Suppose an amount increases by 100%, then decreases by 100%. Find the final amount. Would the situation change if the original increase was 150%? Explain your reasoning. The final amount is always 0. A 100% decrease of any amount would leave Look for a Pattern Ariel deposited $100 into a bank account. Each Friday she will withdraw 10% of the money in the account to spend. Ariel thinks her account will be empty after 10 withdrawals. Do you agree? Explain. No. Only the first withdrawal is $10. Each withdrawal after that is less than $10 because it is 10% of the remaining balance. There will be money left after 10 withdrawals. Lesson Unit 2 7_MCAAESE202610_U2M05L1.indd 145 4/23/13 12:39 PM 7_MCABESE202610_U2M05L1.indd /11/13 4:45 PM EXTEND THE MATH PRE-AP Activity available online Activity On a grid draw a 4 4 square. Use the square and what you have learned about percent increase and percent decrease to determine what happens to the area of the square when the sides are increased by 50%. State by what percent the area increases. Then make a third square by decreasing the sides of the second square by 50%. State by what percent the area decreases. By what percent would you have had to change the sides of the 4 4 square to get the third square? The area of the 4 4 square is 16 units 2. Increasing the sides by 50% makes a 6 6 square with an area of 36 units 2. The area of the first square is increased by 125%. Decreasing the second square s sides by 50% makes a 3 3 square with an area of 9 units 2. The area of the second square is decreased by 75%. The sides of the original square could have been decreased by 25% to get the third square. Percent Increase and Decrease 146

13 LESSON 5.2 Rewriting Percent Expressions Lesson Support Content Objective Language Objective Students will learn to rewrite percent expressions to solve markup and markdown problems. Students will demonstrate and explain how to rewrite expressions to help you solve markup and markdown problems. California Common Core Standards 7.EE.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. 7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. MP.5 Use appropriate tools strategically. Focus Coherence Rigor Building Background Visualize Math Knowledge Draw the bar model on the board. Discuss with students what the model shows and how they could use the model to find the missing information. For example, the model shows that the amount 300 is equal to 100%. Four equal parts of the second bar are equal to 100%. So, each part is = 75. The fifth 25% is another 75. So, the total length of the top bar is 375, which is 125% of % % 25% 25% 25% 25% Learning Progressions In this lesson, students extend their skill at using percents to solve problems by rewriting expressions for easier computation. Some key understandings for students are the following: A percent can be written as a decimal or as a fraction. Depending on the problem, one of the two forms may provide a more efficient solution than the other. A markup is an example of a percent increase. The term markup sometimes refers to a percent increase and sometimes to the amount of the increase. A markup of 20% on $150 is a markup of $30. A markdown is an example of a percent decrease. Concepts related to percent and the use of equivalent expressions will continue to be applied in everyday life and in the study of algebra. Cluster Connections This lesson provides an excellent opportunity to connect ideas in this cluster: Analyze proportional relationships and use them to solve real-world and mathematical problems. Give students the following prompt: Elisa and Dan each are calculating the sale prices of a tablet device at two different stores. What is the sale price at each store? Have students complete the table and ask them show a one-step calculation to find each sale price. Store Computer Deals Today s Computers Original Price Percent Discount Sale Price $280 15% $238 $315 25% $ = 238; = A

14 Language Support EL PROFESSIONAL DEVELOPMENT California ELD Standards Emerging 2.I.6c. Reading/viewing closely Use knowledge of morphology, context, reference materials, and visual cues to determine the meaning of unknown and multiple-meaning words on familiar topics. Expanding 2.I.6c. Reading/viewing closely Use knowledge of morphology, context, reference materials, and visual cues to determine the meaning of unknown and multiple-meaning words on familiar and new topics. Bridging 2.I.6c. Reading/viewing closely Use knowledge of morphology, context, reference materials, and visual cues to determine the meaning, including figurative and connotative meanings, of unknown and multiple-meaning words on a variety of new topics. Linguistic Support EL Academic/Content Vocabulary increase or decrease In this lesson, percents are used to solve markup and markdown problems. Point out to English learners the words that cue them to whether the change is increase or decrease. Words that cue an increase: markup (noun), mark up (verb), profit Words that cue a decrease: markdown (noun), mark down (verb), loss, discount Word that cues neither an increase nor a decrease: break even Rules and Patterns co- Point out to English learners any prefixes, suffixes, etc. to help them figure out the meanings of words in word problems. The prefix co-, meaning with, appears in the word coefficient in this lesson. Common words with the prefix co- include coworker, co-author, co-star, co-exist, coed. Notice that some words have a hyphen after the prefix co- and while others do not. Leveled Strategies for English Learners EL Emerging Visual cues, like bar models, can help students at this level of English proficiency understand an abstract idea or concept. Have students draw and label a bar model to demonstrate how to solve a markup problem. Expanding Have pairs of students review and discuss the steps in Example 1 of the lesson before solving one of the word problems in Independent Practice. Bridging Pair students at this level of English proficiency to discuss and review the steps in Example 1 of the lesson. Then have them explain the difference between how to solve a markup vs. markdown problem. Math Talk To help English learners answer the question posed in Example 1 Math Talk, give them a model to begin their answer with: It makes sense to write the retail price as the sum of because. A good reason for writing the retail price as the sum of is. Rewriting Percent Expressions 147B

15 LESSON 5.2 CA Common Core Standards The student is expected to: Expressions and Equations 7.EE.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. Ratio and Proportional Relationships 7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. Also 7.EE.3 Mathematical Practices MP.5 Using Tools Rewriting Percent Expressions Engage ESSENTIAL QUESTION How can you rewrite expressions to help you solve markup and markdown problems? Sample answer: Markups are 1 plus a percent of the cost, and markdowns are 1 minus a percent of a price. Either can be rewritten as a single term. Motivate the Lesson Ask: Did you ever want to figure out the sale price of an item before you got to the check-out counter? Begin the lesson to find out how to do this. Explore Multiple Representations Mathematical Practices Explain that a certain pack of gum costs $1 and that you have 100% of what it costs to buy that gum. Show students four quarters. Explain that you plan to sell the pack of gum to make a profit. You plan a markup of 50%. Ask how much 50% of $1 is. Show the original cost, four quarters, in one hand and the markup, two quarters, in your other hand. Explain that the retail price is now $1.50. Explain ADDITIONAL EXAMPLE 1 A shoe store buys a pair of boots from a supplier for b dollars. The store s manager decides on a markup of 35%. Write an expression for the retail price of a pair of boots. 1.35b Interactive Whiteboard Interactive example available online EXAMPLE 1 Focus on Modeling Mathematical Practices Point out to students that the part of the model in Step 1 labeled as s is equivalent to 1s and represents 100% of the original cost. The part labeled 0.42s shows 42% of the original cost, the amount being added to the original cost, while the entire model represents 142% of the original cost. Questioning Strategies Mathematical Practices How could you use a mathematical property to add 1s s? Use the Distributive Property to write 1s s as ( )s. How could you use the expression to help you determine the retail price of a skateboard that cost the store $50? Substitute $50 for s in 1.42s $50 = $71 YOUR TURN Avoid Common Errors Students may think that tripling a cost means a 300% markup. A cost x tripled is 3x. Breaking apart 3x into the cost plus the markup yields x + 2x. So, you must add 200% of x or 2x to x to get triple the cost, 3x. 147 Lesson 5.2 Talk About It Check for Understanding Ask: Why will 1.1c work as an expression for a 10% markup, no matter what is being sold? c is a variable that can stand for any original cost. 1.1 is a constant that represents 100% plus 10%.

16 DO NOT EDIT--Changes must be made through File info CorrectionKey=A DO NOT EDIT--Changes must be made through File info CorrectionKey=B? L E S S O N 5.2 Rewriting Percent Expressions ESSENTIAL QUESTION Calculating Markups A markup is one kind of percent increase. You can use a bar model to represent the retail price of an item, that is, the total price including the markup. EXAMPLE 1 How can you rewrite expressions to help you solve markup and markdown problems? 7.EE.2, 7.RP.3, 7.EE.3 To make a profit, stores mark up the prices on the items they sell. A sports store buys skateboards from a supplier for s dollars. What is the retail price for skateboards that the manager buys for $35 and $56 after a 42% markup? 7.EE.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. Also 7.RP.3, 7.EE.3 Math On the Spot Online Practice and Help YOUR TURN 2. Rick buys remote control cars to resell. He applies a markup of 10%. a. Write two expressions that represent the retail price of the cars. 1c + 0.1c, 1.1c b. If Rick buys a remote control car for $28.00, what is his selling price? $ An exclusive clothing boutique triples the price of the items it purchases for resale. a. What is the boutique s markup percent? 200% b. Write two expressions that represent the retail price of the clothes. 1c + 2c; 3c STEP 1 STEP 2 STEP 3 Use a bar model. Draw a bar for the cost of the skateboard s. Then draw a bar that shows the markup: 42% of s, or 0.42s. s 0.42s s s These bars together represent the cost plus the markup, s s. Retail price = Original cost + Markup = s s = 1s s = 1.42s Use the expression to find the retail price of each skateboard. s = $35 Retail price = 1.42($35) = $49.70 s = $56 Retail price = 1.42($56) = $79.52 Sample answer: Two terms shows the original cost and the markup. One term allows for quicker calculation. Math Talk Mathematical Practices Why write the retail price as the sum of two terms? as one term? Math On the Spot Animated Math Calculating Markdowns An example of a percent decrease is a discount, or markdown. A price after a markdown may be called a sale price. You can also use a bar model to represent the price of an item including the markdown. EXAMPLE 2 A discount store marks down all of its holiday merchandise by 20% off the regular selling price. Find the discounted price of decorations that regularly sell for $16 and $23. STEP 1 Use a bar model. Draw a bar for the regular price p. Then draw a bar that shows the discount: 20% of p, or 0.2p. 0.2p p p - 0.2p 7.EE.2, 7.RP.3, 7.EE.3 Reflect 1. What If? The markup is changed to 34%; how does the expression for the retail price change? The expression would change to 1s s or 1.34s. Lesson Unit 2 The difference between these two bars represents the price minus the discount, p - 0.2p. 7_MCAAESE202610_U2M05L2.indd /04/13 6:44 PM 7_MCABESE202610_U2M05L2.indd /11/13 8:33 PM PROFESSIONAL DEVELOPMENT Integrate Mathematical Practices MP.5 This lesson provides an opportunity to address this Mathematical Practice standard. It calls for students to use bar models to model the relationship between a mathematical expression and a real-world context regarding either a markup or a markdown. This gives students the opportunity to read a real-world situation and use that information to write an algebraic expression to represent retail and sale prices. Finally, the students use the expression they write to solve problems regarding markups and markdowns. Math Background Not explored in this lesson is a not-so-subtle distinction between the amount used as the basis for a markup and the amount used as the basis for a markdown. As presented in the lesson, a markup is generally based on the cost of an item to the retailer. However, not covered in this lesson, a markdown is generally based on the retail price of an item after a markup has been applied. This means that if an item costing $100 is marked up 20%, it will retail for $120. If this item is later placed on sale at a 20% markdown, the sale price is not $120 - $20, but $120 - (20% of $120) or $120 - $24, which is $96. Rewriting Percent Expressions 148

17 ADDITIONAL EXAMPLE 2 A pet store marks down all of its grooming products by 15% off the regular selling price of p. Write an expression for the sale price. 0.85p Interactive Whiteboard Interactive example available online Animated Math Explore Markups and Markdowns Students discover how markups and markdowns relate to the original cost using virtual manipulatives. EXAMPLE 2 Questioning Strategies Mathematical Practices In Step 1, how do you know how much of the 1p bar to shade to show 0.2p? The amount shaded does not need to be a specific amount, just a portion of the bar to represent 0.2p. How does the bar model for a markdown differ from the bar model for a markup? For a markup, the bar model for the expression is longer than the original cost. For a markdown, the bar model for the expression is shorter than the retail price. Connect Vocabulary EL Remind students that both percents in Example 2 are rational numbers. 20% is equivalent to 0.2 or 2 8, and 80% is equivalent to 0.8 or 10 YOUR TURN Engage with the Whiteboard Have a student volunteer draw the model in part a. Discuss whether the model needs to be drawn to scale for it to be helpful in solving the problem. Focus on Math Connections Mathematical Practices Point out that the Distributive Property also works for subtraction. So, 1b b = (1-0.24)b = 0.76b. 10. Elaborate Talk About It Summarize the Lesson Ask: How does a bar model showing the expression for a sale price compare to one showing the expression for a retail price? Both show the original price and the percent markup or markdown. The bar model for the retail price shows the percent markup added to the bar model, while the sale price shows the percent markdown subtracted from the bar model. GUIDED PRACTICE Engage with the Whiteboard To the right of each row, have students volunteer to write the expression that could be used to find the retail price in Exercises 2 7 and write the expression that could be used to find the sale price for Exercises 8 11 on the write-on lines. Avoid Common Errors Exercise 1c Remind students that once $32 has been substituted for s in the expression 1.35s, they do not need to add $32 to the value of the expression again. Exercises 2 7 Remind students that the markup is an amount found by multiplying the cost by the percent markup. The retail price is the cost plus the markup. Integrating Language Arts EL Encourage English learners to ask for clarification on any terms or phrases that they don t understand. 149 Lesson 5.2

18 STEP 2 Sale price = Original price - Markdown Guided Practice STEP 3 = p - 0.2p = 1p - 0.2p = 0.8p Use the expression to find the sale price of each decoration. p = $16 Sale price = 0.8($16) = $12.80 p = $23 Sale price = 0.8($23) = $18.40 Reflect 4. Conjecture Compare the single term expression for retail price after a markup from Example 1 and the single term expression for sale price after a markdown from Example 2. What do you notice about the coefficients in the two expressions? A markup includes a coefficient greater than 1 and a markdown includes a coefficient less than 1. YOUR TURN 5. A bicycle shop marks down each bicycle s selling price b by 24% for a holiday sale. The amount of a 20% markup and a 20% discount are the same, but one is added and the other is subtracted. Math Talk Mathematical Practices Is a 20% markup equal to a 20% markdown? Explain. 1. Dana buys dress shirts from a clothing manufacturer for s dollars each, and then sells the dress shirts in her retail clothing store at a 35% markup. (Example 1) a. Write the markup as a decimal. 0.35s b. Write two expressions for the retail price of the dress shirt. 1s s, 1.35s c. What is the retail price of a dress shirt that Dana purchased for $32.00? $43.20 d. How much was added to the original price of the dress shirt? $11.20 List the markup and retail price of each item. Round to two decimal places when necessary. (Example 1) Item Price Markup % Markup Retail Price 2. Hat $18 15% 3. Book $ % 4. Shirt $ % 5. Shoes $ % 6. Clock $ % 7. Painting $ % $2.70 $9.45 $25.31 $24.75 $48.60 $ $20.70 $31.95 $59.06 $99.74 $97.20 $ a. Draw a bar model to represent the problem. 0.24b 1b Find the sale price of each item. Round to two decimal places when necessary. (Example 2) 8. Original price: $45.00; Markdown: 22% 9. Original price: $89.00; Markdown: 33% $35.10 $ b b b. What is a single term expression for the sale price? 0.76b 6. Jane sells pillows. For a sale, she marks them down 5%. a. Write two expressions that represent the sale price of the pillows. 1p 0.05p, 0.95p b. If the original price of a pillow is $15.00, what is the sale price? $14.25 Online Practice and Help 10. Original price: $23.99; Markdown: 44% 11. Original price: $279.99, Markdown: 75%? $13.43 ESSENTIAL QUESTION CHECK-IN $ How can you determine the sale price if you are given the regular price and the percent of markdown? Write the percent of markdown as a decimal, subtract the product of this decimal and the regular price from the regular price. Lesson Unit 2 DIFFERENTIATE INSTRUCTION Kinesthetic Experience Have students write expressions that could be used to calculate a 10% increase and a 10% decrease in a distance. Then have each student stand on a start line, toss his/her uniquely decorated cotton ball, measure the distance to the nearest centimeter, and record the data in a table like the one shown below. Then have students use their expressions to calculate a distance that would be 10% more and 10% less than their original distance. Next, have each Original Distance 10% more 10% less Second Distance % more or less student throw his/her cotton ball a second time, trying to throw exactly 10% more or less than their original distance. Finally, have students calculate the percent more or less the cotton ball actually went than their original distance. Cooperative Learning Have each student secretly think of a percent markup or markdown for a hat. Then on an index card, have each student write a one-term expression that could be used to find the retail price or sale price for the hat. Have students exchange cards and decide if the expression they were just given is a markup or a markdown. Then on the reverse side of the card students should draw a bar model verifying that their decision is correct. Students then trade the completed cards with a third person who will determine if the bar model and conclusion about the expression are correct. Additional Resources Differentiated Instruction includes: Reading Strategies Success for English Learners EL Reteach Challenge PRE-AP Rewriting Percent Expressions 150

19 5.2 LESSON QUIZ Online Assessment and Intervention Online homework assignment available Lesson Quiz available online Fred buys flags from a manufacturer for f dollars each and then sells the flags in his store for a 26% markup. 1. Write the markup as a decimal. 2. Write an expression for the retail price of a flag. 3. What is the retail price of a flag for which Fred paid $40? 4. How much was added to the cost of the flag? List the sale price of each item. Round to two decimal places when necessary. 5. Original price: $25; Markdown: 12% 6. Original price: $16.45; Markdown: 33% Answers f f or 1.26f 3. $ $ $22 6. $ RP.3, 7.EE.2, 7.EE.3 Evaluate GUIDED AND INDEPENDENT PRACTICE 7.RP.3, 7.EE.2, 7.EE.3 Concepts & Skills Example 1 Calculating Markups Example 2 Calculating Markdowns Additional Resources Differentiated Instruction includes: Leveled Practice Worksheets Practice Exercises 1 7, 15, 16 Exercises 8 11, Focus Coherence Rigor Exercise Depth of Knowledge (D.O.K.) Mathematical Practices 13 2 Skills/Concepts MP.4 Modeling 14 2 Skills/Concepts MP.2 Reasoning 15 3 Strategic Thinking MP.7 Using Structure 16 2 Skills/Concepts MP.4 Modeling 17 3 Strategic Thinking MP.6 Precision 18 3 Strategic Thinking MP.8 Patterns 19 2 Skills/Concepts MP.7 Using Structure 20 3 Strategic Thinking MP.4 Modeling Exercise 18 combines concepts from the California Common Core cluster Analyze proportional relationships and use them to solve real-world and mathematical problems. 151 Lesson 5.2

20 Name Class Date 5.2 Independent Practice 7.RP.3, 7.EE.2, 7.EE A bookstore manager marks down the price of older hardcover books, which originally sell for b dollars, by 46%. a. Write the markdown as a decimal. 0.46b b. Write two expressions for the sale price of the hardcover book. 1b 0.46b, 0.54b c. What is the sale price of a hardcover book for which the original retail price was $29.00? $15.66 d. If you buy the book in part c, how much do you save by paying the sale price? $ Raquela s coworker made price tags for several items that are to be marked down by 35%. Match each Regular Price to the correct Sale Price, if possible. Not all sales tags match an item. Regular Price $3.29 Regular Price $4.19 Regular Price $2.79 Regular Price $3.09 Regular Price $3.77 Online Practice and Help 16. Represent Real-World Problems Harold works at a men s clothing store, which marks up its retail clothing by 27%. The store purchases pants for $74.00, suit jackets for $325.00, and dress shirts for $ How much will Harold charge a customer for two pairs of pants, three dress shirts, and a suit jacket? $ Analyze Relationships Your family needs a set of 4 tires. Which of the following deals would you prefer? Explain. (I) Buy 3, get one free (II) 20% off (III) 1 4 off Either buy 3, get one free or 1_ off. Either case would 4 result in a discount of 25%, which is better than 20%. FOCUS ON HIGHER ORDER THINKING 18. Critique Reasoning Margo purchases bulk teas from a warehouse and marks up those prices by 20% for retail sale. When teas go unsold for more than two months, Margo marks down the retail price by 20%. She says that she is breaking even, that is, she is getting the same price for the tea that she paid for it. Is she correct? Explain. No; she is taking a loss. Her cost for the tea is t, so the retail price is 1.2t. The discounted price is t, or 0.96t, which is less than t. Work Area Sale Price $2.01 Sale Price $2.45 Sale Price $1.15 Sale Price $ Communicate Mathematical Ideas For each situation, give an example that includes the original price and final price after markup or markdown. a. A markdown that is greater than 99% but less than 100% Sample answer: original price: $100; final price: $0.50 b. A markdown that is less than 1% Sample answer: original price: $100; final price: $99.50 c. A markup that is more than 200% Sample answer: original price: $100; final price: $350 Sale Price $ Problem Solving Grady marks down some $2.49 pens to $1.99 for a week and then marks them back up to $2.49. Find the percent of increase and the percent of decrease to the nearest tenth. Are the percents of change the same for both price changes? If not, which is a greater change? No; first change: 20.1% decrease; second change: 25.1% increase. The second percent change is greater. 20. Persevere in Problem Solving At Danielle s clothing boutique, if an item does not sell for eight weeks, she marks it down by 15%. If it remains unsold after that, she marks it down an additional 5% each week until she can no longer make a profit. Then she donates it to charity. Rafael wants to buy a coat originally priced $150, but he can t afford more than $110. If Danielle paid $100 for the coat, during which week(s) could Rafael buy the coat within his budget? Justify your answer. 11 or 12 weeks; after 11 weeks, the price is $109.32, after 12 weeks, the price is $103.85, and after that Danielle donates the coat. Lesson Unit 2 EXTEND THE MATH PRE-AP Activity available online Activity A shirt is on sale now for $20. Starting today, a morning sales clerk decreases the price by 30%, and then an afternoon sales clerk increases the price by 20%. This pattern continues for several days. Provided the shirt is never purchased, on which day is the shirt marked down to about 75% off the price it is now? 75% off the current price would make the shirt s sale price $5. On the morning of Day 6, the shirt will be priced at $4.90. Rewriting Percent Expressions 152