# 2-1 Proportional Relationships And Tables - Pearson Education

15d ago
10 Views
7.36 MB
26 Pages
Last View : 8d ago
Transcription

2-1LaunchProportional Relationships andTablesCommon Core State Standards: 7.RP.2: Recognize and represent proportional relationships .7.RP.2.a: Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalentratios in a table or graphing on a coordinate plane and observing whether the graph is a straight linethrough the origin.A company manufactures custom car paints. A manager receives ordersfor 25, 35, and 105 gallons of their Powerful Purple paint shade. Thepaint manager panics and says, “None of the orders match our mixingchart. How can we make these orders?”Provide a solution for the panicked paint man.Powerful PurpleMixing GuideGallons ofRed PaintGallons ofBlue Paint234620304060ReflectHow do you know if two ratios are equivalent?Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 2MGM12 ANC07 CPN L02.indd 1212Lesson 2-126/03/11 4:31 PM

Close and CheckFocus QuestionWhat does it mean for two quantities to have a proportional relationship? How canyou tell if a table shows a proportional relationship between two quantities?1. The table shows a proportionalrelationship between the number ofteachers and the number of students.Complete the table.Teachers35Students4. Writing Explain how you determinedthe proportional ratios in Exercise 3.1075120Ratiostudents/teachers5. Writing Give a real-world example ofwhen you might use proportionalrelationships. Explain why it isproportional.2. Tell whether the relationship betweenx and y shown in the table isproportional or not proportional.xy12340612183. Circle the ratios that are proportionalto 279.93634     1365418    129Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 2MGM12 ANC07 CPN L02.indd 1313Lesson 2-115/03/11 9:35 PM

2-2LaunchProportional Relationships andGraphsCommon Core State Standards: 7.RP.2.a: Decide whether two quantities are in a proportionalrelationship, e.g., graphing on a coordinate plane and observing whether the graph is a straight linethrough the origin. Also, 7.RP.2.d.The manager of the car paint company prepares orders for its GrannyApple Green paint.Gallons of BlueTell how the graph can help you find the amounts of blue and yellowpaint needed to make 8 gallons, 24 gallons, and 56 gallons of GrannyApple Green paint.35302520151050Granny Apple GreenMixing Graphyx0 3 6 9 12 15 18 21Gallons of YellowReflectDo you think a graph or a table makes it easier for the company to find the correct mix ofblue and yellow paint to make Granny Apple Green?Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 2MGM12 ANC07 CPN L02.indd 1414Lesson 2-226/03/11 4:32 PM

Close and CheckFocus QuestionHow can you tell if a graph shows a proportional relationship between twoquantities?1. The relationship between time x anddistance y can be represented by theequation y 2x. Complete the tableand graph.y 2xxyDistance01234. Writing Does the graph in Exercise 1represent a proportional relationship?Explain how you know.10 y987654321x00 1 2 3 4 55. Reasoning Do all linear graphsrepresent proportional relationships?Explain.Time456. Error Analysis A classmate says thatnot all proportional relationships arelinear. Do you agree? Explain.2. What is the distance when time is equalto 15?units3. What is the unit rate of the graph?Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 2MGM12 ANC07 CPN L02.indd 1515Lesson 2-231/03/11 6:23 PM

2-3LaunchConstant of ProportionalityCommon Core State Standards: 7.RP.2: Recognize and represent proportional relationships betweenquantities. 7.RP.2.b: Identify the constant of proportionality (unit rate) in tables, graphs, equations,diagrams, and verbal descriptions of proportional relationships. Also, 7.NS.2.d.The car paint company provides a perplexing pay graph for newmanagers.What does the graph show? Which point—A, B, or C—might be the mostuseful? Explain.Money Earned ( )Manager Pay GraphyC807060B50403020A10x00 1 2 3 4 5 6 7 8Hours WorkedWhat the Graph Shows:Which Point Might Be Most Useful:ReflectWhich point may be the least useful? Explain.Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 2MGM12 ANC07 CPN L02.indd 1616Lesson 2-312/05/11 10:16 AM

Close and CheckFocus QuestionWhat is a constant of proportionality? What does the constant of proportionalitytell you?1. Each bus carries 24 passengers. Thenumber of buses needed for a field tripdepends on the number of studentsgoing on the trip. What is the constantof proportionality for this situation?4. Writing Which variable in Exercise 3represents the independent variableand which represents the dependentvariable? Explain.per2. Your class collects cans for a local foodbank. On Monday, 7 students collect63 cans. Using the constant ofproportionality, find the number ofstudents who collect 90 cans onTuesday.5. Reasoning How can you use therelationship between the independentand dependent variables to write aunit rate?students3. The table showsthe number ofconcert tickets soldbased on thenumber hours thetickets areavailable. What isthe constant ofproportionality forthis situation?Ticket SalesTime(hr)Tickets324054009720151200Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 2MGM12 ANC07 CPN L02.indd 1717Lesson 2-326/03/11 4:35 PM

2-4LaunchProportional Relationships andEquationsCommon Core State Standards: 7.RP.2: Recognize and represent proportional relationships betweenquantities. 7.RP.2.b: Identify the constant of proportionality (unit rate) in equations of proportionalrelationships. 7.RP.2.c: Represent proportional relationships by equations .The paint company manager grows tired of answering a particular payquestion from employees. He mulls over two equations to help theemployees.What question could employees keep asking? Which equation wouldhelp them? Explain.ReflectIf you could have had one more piece of information before starting to answer the problem,what would it be and why?Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 2MGM12 ANC07 CPN L02.indd 1818Lesson 2-426/03/11 4:37 PM

Close and CheckFocus QuestionHow can you tell if an equation shows a proportional relationship between twoquantities? How can you identify the constant of proportionality in an equation thatrepresents a proportional relationship?1. The equation q 12c represents thequantity q of t-shirts in any number ofcartons c.4. Writing Can setting up a proportionhelp you find the constant ofproportionality in a relationship?Explain.a.  What is the constant ofproportionality?b. How many shirts are in 8 cartons?shirts2. A car manufacturer completes 81 carsevery 180 seconds. Write an equationto represent the total number of cars yfor x seconds of production.5. Error Analysis Assume 130 out of150 students buy lunch each day.There are 180 school days in a year.A classmate writes an equation to findhow many lunches will be sold in oneschool year. Is he correct? Explain.3. Use the table to write an equation tofind how much money y is received forx ounces of silver on the open market.130x180150Silver Exchange RateSilver (oz)Price ( )5912151.35272.43363.24Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 2MGM12 ANC07 CPN L02.indd 1919Lesson 2-415/03/11 9:56 PM

2-5LaunchMaps and Scale DrawingsCommon Core State Standards: 7.G.1: Solve problems involving scale drawings of geometric figures,including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at adifferent scale.A peculiar professor wants a perfectly proportioned poster of a photo ofher pet paper cup. She brings the 4-inch by 6-inch photo to your storeand says the poster must be at least three feet tall.Identify the side lengths of two possible posters.4 in.6 in.ReflectWhat will the pet paper cup look like if the poster is not proportional to the photo? Explain.Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 2MGM12 ANC07 CPN L02.indd 2020Lesson 2-512/05/11 10:28 AM

Close and CheckFocus QuestionHow can you use proportional relationships to solve problems that involve maps andscale drawings?1. A replica of a popular car is built to ascale of 1 in. 24 in. The length of thereplica car is 5.6 inches. What is thelength, in inches, of the actual car?4. Writing Can understanding scaledrawings help you make decisions?Explain.2. The official ratio of length to width ofthe U. S. flag is 1.9 1. If the width of aflag is 3 ft, what is the area of the flag?3. A company designs a billboard toadvertise their grand opening. Theyused a scale of 1 in. 10 ft. What wouldbe the dimensions of the billboard inthe drawing if they use a scale of2 in. 25 ft instead?5. Compare and Contrast How are scaleand ratio related?10 in.8 in.Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 2MGM12 ANC07 CPN L02.indd 2121Lesson 2-531/03/11 6:28 PM

2-6LaunchProblem SolvingCommon Core State Standards: 7.G.1: Solve problems involving scale drawings of geometric figures,including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at adifferent scale. Also, 7.RP.2, 7.RP.2.a, 7.RP.2.b, 7.RP.2.c, and 7.RP.2.d.The peculiar professor plans a room additionfor her pet paper cup and poster. The scaleof the model is 1 in. 3 ft.Draw a net for the walls and ceiling of theroom on the inch grid. Label the dimensions.9 ft9 ft12 ftReflectIs the ratio of the length to the width of the ceiling in the actual room the same as the ratioof the length to the width of the ceiling in the scale drawing? Explain.Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 2MGM12 ANC07 CPN L02.indd 2222Lesson 2-631/03/11 7:35 PM

Close and CheckFocus QuestionIn this topic you have studied different ways to represent proportional relationships.In what ways can you represent proportional relationships? How can knowing howto represent proportional relationship in different ways be useful in solvingproblems?1. Circle the situation(s) that describesproportional relationships.4. Writing Explain how you decidedwhich relationships were proportionalin Exercise 1 and which ones were not.A.  The number of roller coaster ridersis 12 more than the number of seats.B.  There are 2 pieces of pizza for everyfriend.C.  There are 3 times as many books asthere are children.2. An office is drawn to a scale of3 in. 10 ft. The drawing measures4.5 in. by 7.5 in. How many square feetof carpet are needed to carpet theoffice?5. Reasoning The scale drawing inExercise 2 is redrawn to a scale of4 in. 9 ft. Is this second drawing largeror smaller than the first? Explain.3. The Statue of Liberty is 151 ft tall fromthe base to the top of the torch. Youmake a scale drawing of the monumentusing a scale of 1 in. 20 ft. Your frienduses a scale of 1 in. 25 ft.a. Whose drawing is larger?b. How much larger?Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 2MGM12 ANC07 CPN L02.indd 2323Lesson 2-615/03/11 10:08 PM

The Percent Equation3-1LaunchCommon Core State Standards: 7.RP.2.b: Identify the constant of proportionality (unit rate) in tables,graphs, equations, diagrams, and verbal descriptions of proportional relationships. 7.RP.2.c: Representproportional relationships by equations . Also, 7.RP.2.A down-in-the-dumps drummer scours the internet for a new kit toimprove his mood and swing. Based on buyers’ reviews, which kit shouldhe get? Show how you decided.Kit 1: 9 of 13buyers love this kit!Kit 2: 15 of 21buyers love this kit!ReflectDo you like the method of showing buyers’ reviews (e.g., 9 of 13)? Explain.Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 3MGM12 ANC07 CPN L03.indd 2424Lesson 3-112/05/11 10:36 AM

Close and CheckFocus QuestionHow do percents and the percent equation help describe things in the real world?1. Is the question “What percent of 8 is5?” looking for the part, the percent,or the whole?4. Writing The total bill at a restaurantequals 65.38. The waiter typicallyreceives a tip equal to 15% of the totalbill. He is given a 13 tip. Should he behappy with the amount of the tip?Explain.2. Read each statement. Write T if it istrue or F if it is false.9 is 15% of 60.12% of 120 is 10.32 is 40% of 80.90% of 180 is 162.3. A shoe store is having a closeout sale.All the prices are 80% of the originalprice. The shoes you want originallycost 40. Find the sale price of theshoes.5. Vocabulary Explain how to solve “72is 18% of what number?” by using theterms part, percent, and whole.Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 3MGM12 ANC07 CPN L03.indd 2525Lesson 3-118/02/11 4:51 PM

3-2LaunchUsing the Percent EquationCommon Core State Standards: 7.RP.2: Recognize and represent proportional relationships betweenquantities. 7.RP.3: Use proportional relationships to solve percent problems. Examples: tax, gratuitiesand commissions .A movie studio sets two offers in front of a movie star to act in ablockbuster upcoming action flick. The star can choose only one offer.Make a case for accepting each offer. Then explain which offer youwould choose. 3 million as aone-time payment3% of allticket salesReflectWhich offer has a constant of proportionality? Explain.Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 3MGM12 ANC07 CPN L03.indd 2626Lesson 3-226/03/11 4:24 PM

Close and CheckFocus QuestionIn what situations are fixed numbers better than percents of an amount? In whatsituations are percents better than fixed numbers?1. You pay 3.50 per gallon includingtaxes for 15 gallons of gas. Federal andstate gas taxes make up 14% of thetotal cost. How much do you pay ingas taxes?4. Error Analysis A basketball playermade 88% of 125 free throw attempts.Your friend calculates how many freethrows the player made below. Explainher error and find the correct number.total 88% 125 88.0 125 11,0002. A new car depreciates (loses value) by9% immediately after it is purchasedand driven from the lot. If a new carcosts 28,400, how much is it worthright after it is driven off the car lot?3. A company expects a new salesemployee to work 40 hours per week.The company offers the employeeeither 25 per hour or a salary of 500per week plus a 7% sales commission.Each sales employee averages 7,800 insales each week. Which offer paysmore?5. Writing Your uncle offers to sell yourguitar at his auction. He offers you 75or 50% of the final selling price. Howwould you choose? Explain.Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 3MGM12 ANC07 CPN L03.indd 2727Lesson 3-218/02/11 4:51 PM

3-3LaunchSimple InterestCommon Core State Standards: 7.RP.2: Recognize and represent proportional relationships betweenquantities. 7.RP.3: Use proportional relationships to solve multistep percent problems. Examples: simpleinterest .Two banks offer different incentives to make a deposit at their bank.Bank A offers a 10 gift card for making a minimum initial deposit. Bank Boffers cash back equal to 2% of your deposit at the end of the year.Tell which line represents each bank’s offer. Explain your reasoning.Value of Bank OfferLine 1Line 2Deposit AmountReflectWhat would you need to know to decide which offer to choose? Explain.Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 3MGM12 ANC07 CPN L03.indd 2828Lesson 3-322/03/11 10:53 PM

3-4Compound InterestCommon Core State Standards: 7.NS.3: Solve real-world and mathematical problems involving the fouroperations with rational numbers.LaunchA bank manager shows your friend a graph of different types of interestthe bank can pay on her 100 deposit. The manager advises your friend toopt for 2% simple interest “because it’s easy to compute and a good rate.”Savings Balance ( )Tell which line represents simple interest. Explain whether your friendshould listen to the bank manager.Line 1114112110108106104102100Line 212345Time (years)ReflectWhat makes simple interest simple? How is the other interest line different? Explain.Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 3MGM12 ANC07 CPN L03.indd 3030Lesson 3-428/03/11 6:22 PM

Close and CheckFocus QuestionHow is compound interest different from simple interest? When do you useeach kind?1. You invest 750 in an account thatearns 4.5%, compounded annually.Find the balance of the account after2 years. Round your answer to thenearest cent.4. Writing Explain how can you use whatyou have learned about compoundinterest to develop a savings orinvestment plan.2. A 10-year investment at 3.5%,compounded annually, totals 4,250.Find the amount of the original depositto the nearest cent.5. Error Analysis Your friend can get a3-year car loan for 3,700 at 7.5%annual interest compounded quarterly.Explain the error she makes incalculating the amount of interest.3. You qualify for a 5-year loan of 2,000.The interest rate is 9.5% with no latefees. How much more or less would youowe if interest is compoundedquarterly rather than compoundedannually?B 3,700(1 0.01875)43,985.40I 3,985.40 3,700285.40Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 3MGM12 ANC07 CPN L03.indd 3131Lesson 3-418/02/11 4:51 PM

3-5LaunchPercent Increase and DecreaseCommon Core State Standards: 7.RP.2: Recognize and represent proportional relationships betweenquantities. 7.RP.3: Use proportional relationships to solve multistep percent problems. Examples: percent increase and decrease .Two friends argue about which of their little town’s populations grewthe most between 2000 and 2009.Write an argument to support each friend’s point of view. Explain yourreasoning.U.S. Census Bureau DataPopulation dataLittle Falls, MNLittle Falls, WI2009 population8,0671,5402000 population7,7191,334ReflectWhich do you think tells more about the growth of a town — percent increase inpopulation or increase in number of people? Explain.Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 3MGM12 ANC07 CPN L03.indd 3232Lesson 3-522/03/11 10:28 PM

Close and CheckFocus QuestionHow can you use a percent to represent change?1. The banded ribbon worm is acarnivorous aquatic worm. It measuresabout 2.5 ft when contracted and up to4. Reasoning Explain how you knowwhether a percent of change is apercent increase or a percent25 ft when expanded. Find the percentincrease between the contracted andexpanded length of a banded ribbonworm.decrease.2. The oceans’ tide levels vary based onthe phases of the moon. Find thepercent decrease in tide levels whenhigh tide is 4.25 ft and low tide is0.75 ft above sea level.5. Writing Give a real-world example ofwhen it might be useful to calculate apercent of change. Would you expectthe percent of change to be a percentincrease or a percent decrease?3. The national debt in 2000 was about 5.7 trillion. In 2010, the nation debthad risen to about 13.6 trillion. Findthe approximate percent of change inthe national debt during that ten-yearperiod.Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 3MGM12 ANC07 CPN L03.indd 3333Lesson 3-518/02/11 4:51 PM

3-6Markups and MarkdownsCommon Core State Standards: 7.RP.3: Use proportional relationships to solve multistep percentproblems. Examples: markups and markdowns .LaunchA sporting goods store holds a storewide clearance sale. You comb theads for boxing items.Order the ads from best deal to worst deal. Explain your reasoning.ReflectShould you always choose the item with the greatest change in price when shopping?Explain.Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 3MGM12 ANC07 CPN L03.indd 3434Lesson 3-622/03/11 10:28 PM

Close and CheckFocus QuestionWhen are percent markups and percent markdowns used? How are they similar?How are they different?1. A car dealership pays 26,215 for a newcar. They sell the car for 28,265. Findthe percent markup on the car.4. Reasoning Does a seller make a profitif the percent markdown on an item isequal to the percent markup? Explainusing an example.2. A computer is purchased by a store for 756 and offered to the customer for 1,200. At the end of the year, thestore discounts the computer for quicksale. Find the greatest percentmarkdown possible without losingmoney.5. Vocabulary A 70 jacket is markeddown to 52.50. Your friend says themarkdown is 25%. You say themarkdown is 17.50. Explain how bothof you can be correct.3. Gaming Unlimited buys a gamingsystem for 249. It sells the system for 385. This week the system is on salefor 30% off. Find the amount of profitmade on each gaming system sold.Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 3MGM12 ANC07 CPN L03.indd 3535Lesson 3-626/03/11 4:27 PM

Problem Solving3-7LaunchCommon Core State Standards: 7.RP.3: Use proportional relationships to solve multistep percentproblems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percentincrease and decrease .Two friends agree to pay half each for their mostly excellent meatloafdinner and most excellent 20% tip. Each friend has 8.50.Do they have enough money to complete their excellent plan? If so, byhow much? If not, tell what they should do.ouseMeatloaf H5.50Dinner Meatloaf5.50Dinner Meatloaf 1.00Soda 1.00Soda 13.00Subtotal 1.30Tax 14.30TotalReflectHow do you use percents in your life outside of school? Provide one example.Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 3MGM12 ANC07 CPN L03.indd 3636Lesson 3-730/03/11 3:50 PM

Close and CheckFocus QuestionHow do percents help you compare, predict, and make decisions?1. You and 3 friends combine a4. Writing A researcher finds that 11 outsnowboard purchase to save onshipping fees. Each board costs 82.50after tax. The total bill is 392.70.Express the shipping fee as a percentrate to the nearest whole percent.of 15 people work within 30 miles oftheir home. He says 73% of peoplework within 30 miles of their home.Which statement do you find moreuseful? Explain.2. The number 450 is increased by 75%.The result is then decreased by 90%.What is the final number?3. You receive a 50 gift card to yourfavorite store. If sales tax is 7.5%, whatis the greatest amount you can spendwithout using any additional money?5. Reasoning A bookstore special-ordersbooks for a fee of 1.50 per book.Online orders have a shipping chargeof 10% of the total order. If 5 bookscost 70.96, which is the better offer?Copyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.Topic 3MGM12 ANC07 CPN L03.indd 3737Lesson 3-718/02/11 4:50 PM

The official ratio of length to width of the U. S. flag is 1.9 1. If the width of a flag is 3 ft, what is the area of the flag? 3. A company designs a billboard to advertise their grand opening. They used a scale of 1 in. 10 ft. What would be the dimensions of the billboard in the drawing