Transcription
LESSON8.1?Comparing Additiveand MultiplicativeRelationshipsESSENTIAL QUESTIONProportionality—6.4.A Compare tworules verbally, numerically,graphically, and symbolicallyin the form of y ax ory x a in order todifferentiate betweenadditive and multiplicativerelationships.How do you represent, describe, and compare additiveand multiplicative relationships?6.4.AEXPLORE ACTIVITYDiscovering Additive andMultiplicative RelationshipsA Every state has two U.S. senators. The number of electoral votesa state has is equal to the total number of U.S. senators and U.S.representatives.The number of electoral votes isthe number of representatives. Houghton Mifflin Harcourt Publishing Company Image Credits: MedioimagesPhotodisc/Getty ImagesComplete the table.Representatives12Electoral votes3452541Describe the rule: The number of electoral votes is equal tothe number of representatives plus / times.B Frannie orders three DVDs per month from her DVD club.Complete the table.Months12DVDs ordered3641322Describe the rule: The number of DVDs ordered is equal tothe number of months plus / times.Reflect1. Look for a Pattern What operation did you use to completethe tables in A and B ?Lesson 8.1205
Graphing Additive and MultiplicativeRelationshipsMath On the SpotTo find the number of electoral votes in part A of the Explore, add 2 to thenumber of representatives. We call this an additive relationship.my.hrw.comTo find the number of DVDs Frannie has ordered after a given number ofmonths, multiply the number of months by 3. We call this a multiplicativerelationship.EXAMPLE 1A Jolene is packing her lunch for school. The empty lunch box weighsfive ounces. Graph the relationship between the weight of the items inJolene’s lunch and the total weight of the packed lunchbox.STEP 1The total weight is equalto the weight of theitems plus the weightof the lunchbox. Therelationship is additive.STEP 2Make a table relating the weight of the items to thetotal weight.Weight of items (oz)12345Total weight (oz)678910To find the total weight, add the weight of the itemsand the weight of the lunchbox.Totalweight Weightof items Weight oflunchbox9 4 5List the ordered pairs from the table.The ordered pairs are (1, 6), (2, 7), (3, 8), (4, 9), and (5, 10).STEP 3Graph the ordered pairs on a coordinate plane.Total Weight (oz)To plot (1,6), goright 1 unit fromthe origin andthen up 6 units.The points of the graphform a straight line foran additive relationship.1086A line drawn throughthe points would not gothrough the origin.42O2468 10Weight of Items (oz)206Unit 3 Houghton Mifflin Harcourt Publishing CompanyMy Notes6.4.A
B Oskar sells bracelets for two dollarseach and donates the money he collectsto a charity. Graph the relationshipbetween the number of bracelets soldand the total donation.STEP 1Complete the table.Bracelets sold12345Total donation ( )246810To find the total donation, multiply the number ofbracelets sold by the donation per bracelet.STEP 2Totaldonation Braceletssold Donationper bracelet10 5 2His donation is equal to thenumber of bracelets soldtimes the donation for eachbracelet. The relationship ismultiplicative.List the ordered pairs from the table.The ordered pairs are (1, 2), (2, 4), (3, 6), (4, 8), and (5, 10).STEP 3Graph the ordered pairs on a coordinate plane.The points of the graph form astraight line for a multiplicativepattern.Donation ( )1086A line drawn through the pointswould intersect the origin.42O246The line is steeper than the linein part A.8 10Math TalkMathematical ProcessesHow are the graphsin part A and part B thesame? How are theydifferent?YOUR TURN2. Ky is seven years older than his sister Lu.Graph the relationship between Ky’s ageand Lu’s age. Is the relationship additive ormultiplicative? Explain.Lu’s ageKy’s age1234512Ky’s Age (years) Houghton Mifflin Harcourt Publishing CompanyBracelets Sold108642O2468 10Lu’s Age (years)PersonalMath TrainerOnline Assessmentand Interventionmy.hrw.comLesson 8.1207
Guided PracticeDogs adopted1232. Graph the relationship between thenumber of dogs adopted and the totalnumber of dogs. (Example 1)Number of Dogs1. Fred’s family already has two dogs. Theyadopt more dogs. Complete the table forthe total number of dogs they will have.Then describe the rule. (Explore Activity)4Total numberof dogs108642O2648 10Dogs Adopted3. Frank’s karate class meets three days everyweek. Complete the table for the totalnumber of days the class meets. Thendescribe the rule. (Explore Activity)4. Graph the relationship between thenumber of weeks and the number of daysof class. (Example 1)1234Days ofclassDays of class30Weeks2418126O2468 100.500.400.300.200.10O2468 10Pages Printed?ESSENTIAL QUESTION CHECK-IN6. How do you represent, describe, and compare additiveand multiplicative relationships?208Unit 3 Houghton Mifflin Harcourt Publishing Company5. An internet café charges ten cents for each page printed.Graph the relationship between the number of pagesprinted and the printing charge. Is the relationshipadditive or multiplicative? Explain. (Example 1)Printing Charges ( )Weeks
NameClassDate8.1 Independent PracticePersonalMath Trainer6.4.Amy.hrw.comThe tables give the price of a kayak rentalfrom two different companies.OnlineAssessment andInterventionThe graph represents the distance traveled bya car and the number of hours it takes.Raging River Kayaks1368Cost ( )9275472PaddlersHours24510Cost ( )424445507. Is the relationship shown in each tablemultiplicative or additive? Explain. Houghton Mifflin Harcourt Publishing Company8. Yvonne wants to rent a kayak for 7 hours.How much would this cost at eachcompany? Which one should she choose?9. After how many hours is the cost for bothkayak rental companies the same? Explainhow you found your answer.Distance (mi)600Hours480360240120O246810Time (h)10. Persevere in Problem Solving Basedon the graph, was the car traveling at aconstant speed? At what speed was thecar traveling?11. Make a Prediction If the pattern shownin the graph continues, how far will the carhave traveled after 6 hours? Explain howyou found your answer.12. What If? If the car had been traveling at40 miles per hour, how would the graph bedifferent?Lesson 8.1209
Use the graph for Exercises 13–15.13. Which set of points represents an additive relationship? Which set ofpoints represents a multiplicative relationship?2420161214. Represent Real-World Problems What is a real-life relationship thatmight be described by the red points?84O15. Represent Real-World Problems What is a real-life relationship thatmight be described by the black points?2345Work AreaFOCUS ON HIGHER ORDER THINKING16. Explain the Error An elevatorTime (s)leaves the ground floor and risesDistance (ft)three feet per second. Lili makesthe table shown to analyze therelationship. What error did she make?11234456717. Analyze Relationships Complete each table. Show an additiverelationship in the first table and a multiplicative relationship in thesecond table.B123A12B16323Use two columns of each table. Which table shows equivalent ratios?Name two ratios shown in the table that are equivalent.18. Represent Real-World Problems Describe a real-world situationthat represents an additive relationship and one that represents amultiplicative relationship.210Unit 3 Houghton Mifflin Harcourt Publishing CompanyA
Math On the Spot my.hrw.com Jolene is packing her lunch for school. The empty lunch box weighs five ounces. Graph the relationship between the weight of the items in Jolene's lunch and the total weight of the packed lunchbox. Make a table relating the weight of the items to the total weight. Weight of items (oz) 12345 Total weight (oz) 6789 10
