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MathematicsMathematicsMathematicsEnglish Language Artscom mon core7Did you know that some in-linePerformanceCoachspeed skaters compete in races of up to86 miles? You may not be able to skatethat far, but Performance Coach will helpyou reach your goals on tests this year.7common corePerformanceCoachcom mon core7PerformanceCoachStudent Editionwww.triumphlearning.comSample LessonPhone: (800) 338-6519 Fax: (866) 805-5723 E-mail: customerservice@triumphlearning.comTo order, call 800-338-6519 orvisit www.triumphlearning.com.ISBN-13: 978-1-62362-809-3900009 781623 628093314NASE G7 MATH SE cvr.indd 1314NASEPerformance Coach2/14/14 10:40 AM
CONTENTSStandardsLetter to the Student . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Domain 1: RATIOS AND PROPORTIONAL RELATIONSHIPS . . . 5Lesson 1Computing Unit Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67.RP.1Lesson 2Identifying Proportional Relationships . . . . . . . . . . . . . 137.RP.2.a, 7.RP.2.bLesson 3Representing Proportional Relationships . . . . . . . . . . . 217.RP.2.a, 7.RP.2.c, 7.RP.2.dLesson 4Solving Problems with Ratio and Percent . . . . . . . . . . . . 317.RP.3Domain 1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Domain 2: THE NUMBER SYSTEM . . . . . . . . . . . . . . . . . . . . 47Lesson 5Adding and Subtracting Rational Numbers . . . . . . . . . 48Lesson 67.NS.1.a, 7.NS.1.b, 7.NS.1.cAdding and Subtracting Rational Numbers UsingProperties of Operations . . . . . . . . . . . . . . . . . . . . . . . . 567.NS.1.dLesson 7Multiplying Rational Numbers . . . . . . . . . . . . . . . . . . . . 647.NS.2.a, 7.NS.2.c, 7.NS.3Lesson 8Dividing Rational Numbers . . . . . . . . . . . . . . . . . . . . . . 737.NS.2.b, 7.NS.2.c, 7.NS.3Lesson 9Converting Rational Numbers to Decimals . . . . . . . . . 827.NS.2.dLesson 10Solving Problems with Rational Numbers . . . . . . . . . . 907.NS.3Domain 2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Domain 3: EXPRESSIONS AND EQUATIONS . . . . . . . . . . 105Lesson 11Writing Equivalent Expressions . . . . . . . . . . . . . . . . . . 1067.EE.1, 7.EE.2Lesson 12Factoring and Expanding Linear Expressions . . . . . . 1137.EE.1Lesson 13Adding and Subtracting Algebraic Expressions . . . . 1207.EE.1Lesson 14Solving Problems Using Expressionsand Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.EE.3Lesson 15Solving Word Problems Using Equations . . . . . . . . . . 1367.EE.4.aLesson 16Solving Word Problems Using Inequalities . . . . . . . . 1457.EE.4.bDomain 3 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Duplicating any part of this book is prohibited by law. 2015 Triumph Learning, LLCPerformance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104ii314NASE Mth G7 D1 PDF.indd 221/02/14 2:10 AM
StandardsDomain 4: GEOMETRY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161Lesson 17 Understanding Scale Drawings . . . . . . . . . . . . . . . . . . 1627.G.1Lesson 18Drawing Geometric Shapes . . . . . . . . . . . . . . . . . . . . . 1727.G.2Lesson 19Understanding Cross Sections ofThree-Dimensional Figures . . . . . . . . . . . . . . . . . . . . . 1827.G.3Lesson 20Area and Circumference of Circles . . . . . . . . . . . . . . . 1907.G.4Lesson 21Angle Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1987.G.5Lesson 22Area of Two-Dimensional Figures . . . . . . . . . . . . . . . . 2077.G.6Lesson 23Surface Area of Composite Figures . . . . . . . . . . . . . . . 2167.G.6Lesson 24Volume of Three-Dimensional Figures . . . . . . . . . . . . 2267.G.6Domain 4 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240Domain 5: STATISTICS AND PROBABILITY . . . . . . . . . . . . 241Lesson 25 Understanding Sampling . . . . . . . . . . . . . . . . . . . . . . . 242Lesson 26Understanding Mean and MeanAbsolute Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2517.SP.3, 7.SP.4Making Comparative Inferences aboutTwo Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2627.SP.3, 7.SP.4Lesson 28Understanding Probability of Simple Events . . . . . . . 2717.SP.5, 7.SP.6, 7.SP.7.a, 7.SP.7.bLesson 29Understanding Probability ofCompound Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2817.SP.8.a, 7.SP.8.b, 7.SP.8.cDuplicating any part of this book is prohibited by law. 2015 Triumph Learning, LLCLesson 27Duplicating any part of this book is prohibited by law. 2015 Triumph Learning, LLC7.SP.1, 7.SP.2Domain 5 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298Math Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307iii314NASE Mth G7 D1 PDF.indd 321/02/14 2:10 AM
LESSON4Solving Problems with Ratio and PercentStudent Edition pages 31–40LESSON OVERVIEWStandardObjectives7.RP.3Students will: Use proportions to solve multi-step percent problems about tax,tips, and discounts Solve problems using the simple interest formula Solve problems involving percent increase, percent decrease, andpercent errorDiscussion QuestionsMP1How can you be sure you have answered the question thatwas asked in a multi-step problem?MP7What is the difference between percent increase and percentdecrease?MP3When is a percent of a number greater than the originalquantity? Give an example to illustrate your answer.Key Termscommissioninterestpercentpercent decreasepercent errorpercent increasepercent ofchangeprincipalsimple interestMaterial Math Tool: Percent Rules, p. C5(Student Edition, p. 311)DifferentiationLesson Support Have students use the Math Tool:Percent Rules to make visual models. These modelscan be used to determine if an answer is reasonable.For instance, in Example 2, students must find 15%of 53.74. They can mark one of the percent rules toshow 50 as the total. Then each small box represents 5. Fifteen percent is halfway between 10% and 20%,so the answer should be close to halfway between 5and 10. This technique shows that the computed tip, 8.06, is reasonable.Lesson Extension Many problems in this lesson aresolved by finding a percent using multiplication andthen adding it to the original amount. Show studentsthat an alternate strategy for Example 1 is to addfirst; that is, add 100% 1 8% to get 108%, and thenmultiply 1.08 by 45 to find 108% of 45. Challengestudents to find other problems in the lesson forwhich this strategy is appropriate.1 GETTING THE IDEALesson OpenerA familiar everyday application of percents forstudents is items on sale. Ask: What are somedifferent ways that stores advertise items on sale?(percent off; fractional part off; buy one, get one free)8Domain 1: Ratios and Proportional RelationshipsHave students share stories of great buys they havefound on sale items. When possible, have studentsdescribe their discounts in terms of percents. Forexample, ask: What is the percent off the total regularprice when you buy one and get another at the sameDuplicating this page is prohibited by law. 2015 Triumph Learning, LLC
u Example 1This example shows how a percent of a numbercan be found and then used to solve a multi-stepproblem. Relate the strategy in this example backto what students already know about ratios andproportions from previous lessons, and emphasizethat percents are ratios of numbers to 100. It may helpto relate this idea to students’ real-world experiencesas well. Ask: Imagine that a video game has a pricetag that reads 45? Is that what you will actually payfor the game? If you have exactly 45 in cash, canyou buy this game? (No, if students live in a state withsales tax on luxuries such as games) Common Errors When solving multi-step problems,students may begin solving the problem but stop beforethey are finished. Emphasize that students should alwaysreread the original problem and make sure they haveanswered the question that was asked.u Example 2Tell students that 15% tips are often computedusing mental math, first by finding 10% and then byadding half of that amount. For example, a 15% tipon 40 equals 4 1 2, or 6. You may wish to havestudents practice using mental math to compute15% tips. Journal PromptMP1 Another way to find 15% of 53.74 is to multiply the dollar amount by the decimalDOMAIN 1price free? (50%) Then guide students throughthe Getting the Idea section.0.15. How is this equivalent to solving the proportion15x100 5 53.74 ?u Example 3This example differs from previous examples becausethe amounts are given and students must calculatethe percent.u Examples 4 and 5These examples require students to compute simpleinterest and a percent error. Example 4 also showsthat another way to find the percent of a number is tomultiply the decimal value of the percent—the interestrate—by the amount. If students prefer working withdecimals, suggest that they use decimals to writeratios for Example 5 instead of fractions. ELL Support The words principal and principle arehomophones. Different meanings include:principal: the head of a school; “The principal is in theoffice.”principal: the amount invested or loaned; “Theprincipal invested was 2,000.”principle: a basic truth or law; “I learned an importantphysics principle.”2 COACHED EXAMPLEMonitor students at they work through the CoachedExample. As needed, assist students in crossmultiplying to find the value of each percent.Encourage any students who are struggling to drawloops around the numbers to be cross-multiplied;this method will help them keep track.For answers, see page A3.3 LESSON PRACTICEBefore students solve problem 3, make sure studentsunderstand the term commission. Explain that acommission is a percentage of total sales and is partof a salesperson’s earnings. As students are working,pay special attention to problem 6 since it involvesDuplicating this page is prohibited by law. 2015 Triumph Learning, LLCworking with non-integer percents. Be sure studentscan represent decimal percents such as 4.2% and0.75% as ratios or decimals.For answers, see page A3.Lesson 4: Solving Problems with Ratio and Percent9
LESSON47.RP.3Solving Problems withRatio and Percent1 GETTING THE IDEARatios and proportions can be used to solve many real-world problems, including percent problems.20A percent is a ratio that compares a number to 100. For example, 20% can be represented as100 .There are many everyday examples of percent problems. Percents are used to calculate the total costof a restaurant meal, including sales tax. Sales tax is a percentage of the total cost of the meal. In asit-down restaurant, a customer also pays a tip, or gratuity, to the server. This is also a percentageof the total amount of the bill.A salesperson’s earnings may be calculated by adding the person’s base pay or salary to acommission. The commission is a percentage of the total amount of the sales made.Since percents are ratios, proportions can be used to solve real-world percent problems.Example 1The sales tax on a coat that is priced at 45 is 8%. Find the total cost of the coat.Duplicating any part of this book is prohibited by law. 2015 Triumph Learning, LLCStrategyStep 1Set up and solve a proportion. Then add the amount of tax to the cost of the item.Write the percent as a ratio.88% 5100Step 2Write a ratio comparing the amount of the tax to the price of the coat.The amount of the tax is unknown. The price of the coat is 45.x45Step 3Set the two ratios equal and solve for x.8x100 5 458 ? 45 5 100 ? x360 5 100x3.60 5 xSo, the amount of the tax is 3.60.Lesson 4: Solving Problems with Ratio and Percent 31
Step 4Add the amount of the tax to the cost of the coat. 45.00 1 3.60 5 48.60SolutionThe total cost of the coat is 48.60.Example 2Nancy and two friends paid a 15% tip for a restaurant bill of 53.74. Then they split the total evenly.How much did each person pay?StrategyStep 1Set up and solve a proportion. Then add the amount of the tip to the bill anddivide.Set up a proportion to find the amount of the tip.15Write a ratio for the percent: 15% 5 100 xWrite a ratio comparing the unknown amount of the tip to the cost of the bill: 53.74 15xSet the ratios equal: 100 5 53.74 Step 2Solve the proportion to find the amount of the tip.15x 100 5 53.74 15 ? 53.74 5 100 ? x806.1 5 100x8.061 5 xRound the amount. The tip was about 8.06.Step 3Find the amount each person will pay.53.74 1 8.06 5 61.80There are 3 people—Nancy plus 2 friends. So, divide the total by 3.61.80 4 3 5 20.60Solution32Each person will pay 20.60.Domain 1: Ratios and Proportional RelationshipsDuplicating any part of this book is prohibited by law. 2015 Triumph Learning, LLCAdd to find the total of the bill plus the tip.
A percent of change is the ratio of the change in a quantity to the original quantity, expressed as apercent. If the quantity is increasing, this is called a percent increase. If the quantity is decreasing,it is a percent decrease. A discount or markdown is a type of percent decrease. A markup is a typeof percent increase.Example 3A lamp that originally cost 180 was marked down to 150. Find the percent of the discount.StrategyStep 1Subtract to find the amount of the discount. Set up and solve a proportion to findthe percent.Subtract to find the amount of the discount.180 2 150 5 30Step 2Set up a proportion.Write a ratio to show the unknown percent. Then write a second ratio comparingthe amount of the discount to the original cost.30x 100 5 180 Step 3Solve the proportion.x30 100 5 180 x ? 180 5 100 ? 30180x 5 3,0002x 5 16 3 16.7%The percent of the discount was about 16.7%.Duplicating any part of this book is prohibited by law. 2015 Triumph Learning, LLCSolutionLesson 4: Solving Problems with Ratio and Percent 33
Interest is the amount earned when money is deposited in a savings account or other type ofinvestment. Interest can also be the extra amount paid when money is borrowed.The original amount invested or borrowed is called the principal. To compute simple interest,solve a proportion to find the amount of interest earned in 1 year. Then, multiply by the number ofyears. Alternatively, you can use the simple interest formula.Simple Interest FormulaI prtwhere I is the amount of the interest, p is the principal, r is the interest rateexpressed as a decimal, and t is the time in yearsExample 4Dante invests 3,500 and earns 2% in simple interest each year. Assuming he makes no additionaldeposits or withdrawals, how much will his investment be worth in 5 years?StrategyStep 1Find the interest earned in 1 year. Multiply by the number of years. Then, add thisto the principal. Check your answer using the simple interest formula.Write and solve a proportion to find the interest for 1 year.2x100 5 35002 ? 3,500 5 100 ? x7,000 5 100xIn 1 year, the principal earns 70 in interest.Step 2Multiply by the number of years.5 ? 70 5 350Step 3Add the total interest to the principal.350 1 3,500 5 3,850Step 4Check your answer using the simple interest formula.22% 5100 5 0.02, so:I 5 prt 5 3,500 ? 0.02 ? 5 5 70 ? 5 5 350This is the same amount of interest found using proportions, so the sum will bethe same, 3,850.Solution34In 5 years, the investment will be worth 3,850.Domain 1: Ratios and Proportional RelationshipsDuplicating any part of this book is prohibited by law. 2015 Triumph Learning, LLC70 5 x
The percent error shows how close an estimate or measurement is to the actual value. Percent error isoften used when describing a scientific experiment.Example 5John needs to measure his windows so he can buy new curtains. He measured the height of his3windows to be 44 feet. The actual height is 5 feet. Find the percent error in John’s measurement.StrategyFind the absolute value of the difference between the estimated and actual heights.Use the actual height to write a proportion. Solve to find the percent.Step 1Find the positive difference between John’s measurement and the actual measurement. John’s measurement 2 actual measurement 5 4 34 2 5 5 2 41 5 411The difference is4 foot.Step 2Set up a proportion.You are looking for the percent error, so the first ratio should represent thepercent.The second ratio will compare the difference between the measurements tothe actual measurement.1x4100 5 5Step 3Solve the proportion.1x4100 5 51x ? 5 5 100 ?4Duplicating any part of this book is prohibited by law. 2015 Triumph Learning, LLC5x 5 25x55SolutionThe percent error in John’s measurement was 5%.Lesson 4: Solving Problems with Ratio and Percent 35
2 COACHED EXAMPLEThe chart shows population changes in two towns over a period of 10 years. Which town hadthe greater percent increase in its population?Population Changes in Ten 016,500Determine the amount of the increase forRiverton.Determine the amount of the increase forMountainside.41,200 2 38,500 516,500 2Write a ratio for the percent increase:Write a ratio for the percent increase:amount of increaseoriginal population 5 38,500amount of increaseoriginal population 5rSet that ratio equal to100 . Then solve for r, thepercent increase for Riverton.r538,500100m5100?r? 100 55 r5So, the population of Riverton increased by%.approximatelyThe town of36mSet that ratio equal to100 . Then solve for m,the percent increase for Mountainside. mhad a greater percent increase in its population.Domain 1: Ratios and Proportional Relationships?mDuplicating any part of this book is prohibited by law. 2015 Triumph Learning, LLC? 100 55
LESSON PRACTICEPRACTICE3 LESSON1The number of students enrolled at a community college increased by 12%from last year to this year. The enrollment last year was 15,500.1861,860Use numbers from the box to complete each statement.The amount of the increase in enrollment wasThe number of students enrolled this year is.13,64015,68617,3602Select the items for which the discount is greater than 15%. Circle all that apply.A. 29.95 box set of CDs on sale for 19.95B. 64.49 pair of shoes on sale for 55C. 85 radio on sale for 74.49D. 145 suitcase on sale for 114.993Henry works in a store where he earns a 3% commission on everything he sells.Duplicating any part of this book is prohibited by law. 2015 Triumph Learning, LLCPart AWhat will be his total commission on a day when he sells three items, priced at 86, 245,and 295?Part BHow much must he sell to earn 60 in commission? Show your work.Lesson 4: Solving Problems with Ratio and Percent 37
These charts shows state sales tax as of 2011. Use this chart for problems 4–5.4Alabama4%Louisiana4%Rhode Island7%Florida6%Maryland6%South Carolina6%Georgia4%Michigan6%South Dakota4%Hawaii4%Mississippi7%Tennessee7%Idaho6%New York4%Vermont6%Indiana7%North Dakota5%West ylvania6%Wyoming4%Students in various states are attending a marching band conference. Each uniform costs 135before state sales tax is added. How much will the uniforms cost, with tax, for each group ofstudents?10 students from Idaho5 students from Mississippi15 students from Hawaii538A customer paid 682.50, including sales tax, for a table priced at 650. In which of thesestates could the customer live? Explain your answer.Domain 1: Ratios and Proportional RelationshipsDuplicating any part of this book is prohibited by law. 2015 Triumph Learning, LLC8 students from Wisconsin
67Five people each invested 1,500 in an account that earns simple interest. Assuming noother deposits or withdrawals were made, did each investment earn more than 100?Select Yes or No.A. 2 years at 4.2% Yes NoB. 1.3% for 6 years Yes NoC. 8 years at 0.75% Yes NoD. 5.2% for 1 year Yes NoE. 4 years at 2.8% Yes NoJulie estimated her town’s population at 77,500. The actual population of her town is 75,000.Find the percent error.Duplicating any part of this book is prohibited by law. 2015 Triumph Learning, LLCpercent error:8Neil will pay one-third of a restaurant bill that is 72, including tax. He decides to tip 20% onthe amount he pays for the bill. What is the total amount Neil will pay? Show your work.9Gayle can use her 12% employee discount once a week. Last Saturday, she bought a 451sweater at 3 off and used her discount on a suit marked 160. On which purchase didshe save more money? How much more?Lesson 4: Solving Problems with Ratio and Percent 39
10This graph shows population changes in several cities.Population ChangePopulation (in 31.63044.218.520100CedarvilleSunnysideRiver CityCity1980TykestownArkington2000Duplicating any part of this book is prohibited by law. 2015 Triumph Learning, LLCWhich city or cities had a population decrease of over 25%? Justify your answer.40Domain 1: Ratios and Proportional Relationships
LESSON 4 Solving Problems with Ratio and Percent Student Edition pages 31–40 LESSON OVERVIEW Objectives Students will: Use proportions to solve multi-step percent problems about tax, tips, and discounts Solve problems using the simple interest formula Solve problems involving percent increase, percent decrease, and percent error
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