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13. The graph shows the relationship between the distance that a snailcrawls and the time that it crawls.10Time (min)a. Use the points on the graph to make a table.Snail CrawlingDistance (in.)Time (min)b. Write the equation for the relationship and tell what eachvariable represents.8642O10 20 30 40 50Distance (in.)c. How long does it take the snail to crawl 85 inches?Work AreaFOCUS ON HIGHER ORDER THINKING14. Communicate Mathematical Ideas Explain why all of the graphs in thislesson show the first quadrant but omit the other three quadrants.15. Analyze Relationships Complete the table.Length of side of square12345Perimeter of squarea. Are the length of a side of a square and the perimeter of the squarerelated proportionally? Why or why not?b. Are the length of a side of a square and the area of the square relatedproportionally? Why or why not?16. Make a Conjecture A table shows a proportional relationship where k isthe constant of proportionality. The rows are then switched. How doesthe new constant of proportionality relate to the original one?76Unit 2 Houghton Mifflin Harcourt Publishing CompanyArea of square

LESSON3.2?Rate of Changeand SlopeCOMMONCORE8.F.4 Determine the rate ofchange of the functionfrom two (x, y) values,including reading these froma table or from a graph .ESSENTIAL QUESTIONHow do you find a rate of change or a slope?Investigating Rates of ChangeA rate of change is a ratio of the amount of change in the dependent variable,or output, to the amount of change in the independent variable, or input.EXAMPL 1EXAMPLECOMMONCORE8.F.4Math On the Spotmy.hrw.comEve keeps a record of the number of lawns she has mowed and the moneyshe has earned. Tell whether the rates of change are constant or variable.Day 1Day 2Day 3Day 4Number of lawns1368Amount earned ( )154590120STEP 1Identify the input and output variables.Input variable: number of lawnsSTEP 2Output variable: amount earnedFind the rates of change.change in - 1530 45 15Day 1 to Day 2:3-12change in lawnsMath TalkMathematical Practices Houghton Mifflin Harcourt Publishing Companychange in - 4545Day 2 to Day 3: 90 156-33change in lawnsWould you expect the ratesof change of a car’s speedduring a drive through acity to be constant orvariable? Explain.change in - 9030Day 3 to Day 4: 120 158-62change in lawnsThe rates of change are constant: 15 per lawn.YOUR TURN1.The table shows the approximate height of afootball after it is kicked. Tell whether therates of change are constant or variable.Find the rates of change:The rates of change are constant / variable.Time (s)Height (ft)000.5181.531226PersonalMath TrainerOnline Assessmentand Interventionmy.hrw.comLesson 3.277

EXPLORE ACTIVITYCOMMONCORE8.F.4Using Graphs to Find Rates of ChangeYou can also use a graph to find rates of change.The graph shows the distance Nathan bicycled over time.What is Nathan’s rate of change?change in distance 30 change in time12-1miles per hour(4,60)50(3,45)4030(2,30)2010OB Find the rate of change from 1 hour to 4 hours.change in distance 60   change in time4-Distance (mi)A Find the rate of change from 1 hour to 2 hours.60(1,15)246Time (h)miles per hourC Find the rate of change from 2 hours to 4 hours.change in distance 60     change in time4-miles per hourReflect2.Make a Conjecture Does a proportional relationship have a constantrate of change?3.Does it matter what interval you use when you find the rate of changeof a proportional relationship? Explain.78Unit 2 Houghton Mifflin Harcourt Publishing CompanyD Recall that the graph of a proportional relationship is a line throughthe origin. Explain whether the relationship between Nathan’s time anddistance is a proportional relationship.

Calculating Slope myWhen the rate of change of a relationship is constant,any segment of its graph has the same steepness.The constant rate of change is called the slope of the line.Run 3Rise 2xOMath On the Spotmy.hrw.comSlope FormulaThe slope of a line is the ratio of the change in y-values (rise) for asegment of the graph to the corresponding change in x-values (run).y2 - y 1m x -x21EXAMPL 2EXAMPLECOMMONCORE8.F.4My NotesFind m the slope of the line.STEP 1Choose two points on the line.P1(x1, y1)P2(x2, y2)STEP 2Find the change in y-values (rise y2 - y1)and the change in x-values (run x2 - x1)as you move from one point to the other.run x2 - x1 -6 - (-3) -3rise y2 - y1 4-2 2Rise 2P1 (-3, 2)Oy -yrise12m run x - x122 -3 - 23YOUR TURN4. The graph shows the rate at which wateris leaking from a tank. The slope of theline gives the leaking rate in gallons perminute. Find the slope of the line.Rise Slope Run yAmount (gal) Houghton Mifflin Harcourt Publishing CompanySTEP 3P2 (-6, 4)If you move up or right,the change is positive.If you move down or left,the change is negative.5Run -3Leaking tank642Ox2468 10Time (min)PersonalMath TrainerOnline Assessmentand Interventionmy.hrw.comLesson 3.279

Guided PracticeTell whether the rates of change are constant or variable. (Example 1)1. building measurements2. computers soldFeet3122775Week24920Yards14925Number Sold61225603. distance an object falls4. cost of sweatersDistance (ft)1664144256Number2479Time (s)1234Cost ( )3876133171Erica walks to her friend Philip’s house. The graph shows Erica’s distancefrom home over time. (Explore Activity)Distance (ft)10005. Find the rate of change from 1 minute to 2 minutes.400 changein distance change in time2-ft per min800600400200O246Time (min)6. Find the rate of change from 1 minute to 4 minutes.Find the slope of each line. (Example 2)8.yy5-5O5xx5-5-5slope ?O-5slope ESSENTIAL QUESTION CHECK-IN9. If you know two points on a line, how can you find the rate of change ofthe variables being graphed?80Unit 25 Houghton Mifflin Harcourt Publishing Company7.

NameClassDate3.2 Independent PracticeCOMMONCOREPersonalMath Trainer8.F.4my.hrw.comOnlineAssessment andIntervention10. Rectangle EFGH is graphed on a coordinate plane with vertices atE(-3, 5), F(6, 2), G(4, -4), and H(-5, -1).a. Find the slopes of each side.b. What do you notice about the slopes of opposite sides?c. What do you notice about the slopes of adjacent sides?11. A bicyclist started riding at 8:00 A.M. The diagram below shows thedistance the bicyclist had traveled at different times. What wasthe bicyclist’s average rate of speed in miles per hour?8:00 A.M.4.5 miles8:18 A.M.7.5 miles8:48 A.M. Houghton Mifflin Harcourt Publishing Company12. Multistep A line passes through (6, 3), (8, 4), and (n, -2). Find the value of n.13. A large container holds 5 gallons of water. It begins leaking at a constantrate. After 10 minutes, the container has 3 gallons of water left.a. At what rate is the water leaking?b. After how many minutes will the container be empty?14. Critique Reasoning Billy found the slope of the line through the2 - (-2)points (2, 5) and (-2, -5) using the equation 25. What mistake5 - (-5)did he make?Lesson 3.281

15. Multiple Representations Graph parallelogramABCD on a coordinate plane with vertices at A(3, 4),B(6, 1), C(0, -2), and D(-3, 1).10y6a. Find the slope of each side.2-6b. What do you notice about the slopes?-2 O-226x10-6c. Draw another parallelogram on the coordinateplane. Do the slopes have the same characteristics?-10FOCUS ON HIGHER ORDER THINKINGWork Area16. Communicate Mathematical Ideas Ben and Phoebe are finding theslope of a line. Ben chose two points on the line and used them to findthe slope. Phoebe used two different points to find the slope. Did theyget the same answer? Explain.18. Reason Abstractly What is the slope of the x-axis? Explain.82Unit 2 Houghton Mifflin Harcourt Publishing Company17. Analyze Relationships Two lines pass through the origin. The lines haveslopes that are opposites. Compare and contrast the lines.

LESSON3.3?Interpreting the UnitRate as SlopeCOMMONCORE8.EE.5Graph proportionalrelationships, interpreting theunit rate as the slope of thegraph. Compare two differentproportional relationshipsrepresented in different ways.Also 8.F.2, 8.F.4ESSENTIAL QUESTIONHow do you interpret the unit rate as slope?EXPLORE ACTIVITYCOMMONCORE8.EE.5, 8.F.4Relating the Unit Rate to SlopeA rate is a comparison of two quantities that have different units, such asmiles and hours. A unit rate is a rate in which the second quantity in thecomparison is one unit.A Find the slope of the graph using thepoints (1, 2) and (5, 10). Remember thatthe slope is the constant rate of change.Misty Mountain StormSnowfall (in.)A storm is raging on Misty Mountain. The graphshows the constant rate of change of the snowlevel on the mountain.105 Houghton Mifflin Harcourt Publishing Company Image Credits: Cavan Images/Getty ImagesO510Time (h)B Find the unit rate of snowfall in inches per hour. Explain your method.C Compare the slope of the graph and the unit rate of change in thesnow level. What do you notice?D Which unique point on this graph can represent the slope of the graph andthe unit rate of change in the snow level? Explain how you found the point.Lesson 3.383

Graphing Proportional RelationshipsMath On the SpotYou can use a table and a graph to find the unit rate and slope that describea real-world proportional relationship. The constant of proportionality for aproportional relationship is the same as the slope.my.hrw.comEXAMPLE 1COMMONCORE8.EE.5Every 3 seconds, 4 cubic feet of water pass over a dam. Draw a graphof the situation. Find the unit rate of this proportional relationship.Make a table.Time (s)3Volume (ft )AnimatedMathSTEP 2my.hrw.comSTEP 3Math TalkIn a proportional relationship,how are the constant ofproportionality, the unit rate,and the slope of the graphof the relationshiprelated?69121548121620Water Over the DamDraw a graph.Find the slope.8riseslope run 6Mathematical Practices3 43Amount (cu ft)STEP 1208106O1020Time (sec)The unit rate of water passing over the dam and the slope of thegraph of the relationship are equal, 43 cubic feet per second.ReflectWhat If? Without referring to the graph, how do you know that thepoint ( 1, 43 ) is on the graph?YOUR TURNPersonalMath TrainerOnline Assessmentand Interventionmy.hrw.com84Unit 2Tomas rides his bike at a steady rate of2 miles every 10 minutes. Graph the situation.Find the unit rate of this proportionalrelationship.Tomas’s Ride10Distance (mi)2.5O5Time (min)10 Houghton Mifflin Harcourt Publishing Company1.

Using Slopes to Compare Unit RatesYou can compare proportional relationships presented in different ways.EXAMPL 2EXAMPLECOMMONCOREQuantity (barrels)STEP 2Amount (barrels)Well B Pumping RateUse the equation y 2.75x to makea table for Well A’s pumping rate, inbarrels per hour.Time (h)Math On the Spotmy.hrw.comThe equation y 2.75x represents the rate, inbarrels per hour, that oil is pumped from Well A.The graph represents the rate that oil is pumpedfrom Well B. Which well pumped oil at a faster rate?STEP 18.EE.5, 8.F.212342.755.58.25112010O1020Time (h)Use the table to find the slope of the graph of Well A.- 2.75 2.75 2.75 barrels/hourslope unit rate 5.512-1STEP 3Use the graph to find the slope of the graph of Well B.rise10slope unit rate run 4 2.5 barrels/hour Houghton Mifflin Harcourt Publishing Company Image Credits: Tom McHugh/Photo Researchers, Inc.STEP 4Compare the unit rates.2.75 2.5, so Well A’s rate, 2.75 barrels/hour, is faster.Reflect3.Describe the relationships among the slope of the graph of Well A’s rate,the equation representing Well A’s rate, and the constant of proportionality.YOUR TURN4.The equation y 375x represents the relationship between x, the timethat a plane flies in hours, and y, the distance the plane flies in miles forPlane A. The table represents the relationship for Plane B. Find the slopeof the graph for each plane and the plane’s rate of speed. Determinewhich plane is flying at a faster rate of speed.Time (h)Distance (mi)123442585012751700PersonalMath TrainerOnline Assessmentand Interventionmy.hrw.comLesson 3.385

Guided PracticeGive the slope of the graph and the unit rate. (Explore Activity and Example 1)1. Jorge: 5 miles every 6 hours2. AkikoJorge481216Distance (mi)5101520Akiko10O10Distance (mi)Distance (mi)20Time (h)20Time (h)2010O1020Time (h)3. The equation y 0.5x represents the distance Henry hikes, in miles,over time, in hours. The graph represents the rate that Clark hikes.Determine which hiker is faster. Explain. (Example 2)ClarkDistance (mi)2010O?Time (x)1246Distance (y)153060905.Time (x)16324864Distance (y)6121824ESSENTIAL QUESTION CHECK-IN6. Describe methods you can use to show a proportional relationshipbetween two variables, x and y. For each method, explain how you canfind the unit rate and the slope.86Unit 220 Houghton Mifflin Harcourt Publishing CompanyWrite an equation relating the variables in each table. (Example 2)4.10Time (h)

NameClassDate3.3 Independent PracticeCOMMONCOREPersonalMath Trainer8.EE.5, 8.F.2, 8.F.4my.hrw.comOnlineAssessment andIntervention7. A Canadian goose migrated at a steady rate of 3 miles every 4 minutes.a. Fill in the table to describe the relationship.Time (min)4820Distance (mi)912b. Graph the relationship.c. Find the slope of the graph and describewhat it means in the context of thisproblem.Migration FlightDistance (mi)2010O1020Time (min)8. Vocabulary A unit rate is a rate in which thefirst quantity / second quantityin the comparison is one unit.9. The table and the graph represent the rate at which two machines arebottling milk in gallons per second.Machine 2Machine 1Time (s)Amount (gal)12340.61.21.82.4Amount (gal) Houghton Mifflin Harcourt Publishing Company2010O1020Time (sec)a. Determine the slope and unit rate of each machine.b. Determine which machine is working at a faster rate.Lesson 3.387

10. Cycling The equation y 19 x represents the distance y, in kilometers, thatPatrick traveled in x minutes while training for the cycling portion of atriathlon. The table shows the distance y Jennifer traveled in x minutes inher training. Who has the faster training rate?Time (min)40648096Distance (km)581012Work AreaFOCUS ON HIGHER ORDER THINKING11. Analyze Relationships There is a proportional relationship betweenminutes and dollars per minute, shown on a graph of printing expenses.The graph passes through the point (1, 4.75). What is the slope of thegraph? What is the unit rate? Explain.13. Critical Thinking The table showsthe rate at which water is beingpumped into a swimming pool.Time (min)25Amount (gal)36907126 216Use the unit rate and the amount of water pumped after 12 minutesto find how much water will have been pumped into the pool after13 12 minutes. Explain your reasoning.88Unit 212 Houghton Mifflin Harcourt Publishing Company12. Draw Conclusions Two cars start at the same time and travel at differentconstant rates. A graph for Car A passes through the point (0.5, 27.5), anda graph for Car B passes through (4, 240). Both graphs show distance inmiles and time in hours. Which car is traveling faster? Explain.

MODULE QUIZReadyPersonalMath Trainer3.1 Representing Proportional Relationships1. Find the constant of proportionalityfor the table of values.Online Assessmentand Interventionmy.hrw.comx2345y34.567.52. Phil is riding his bike. He rides 25 miles in 2 hours,37.5 miles in 3 hours, and 50 miles in 4 hours.Find the constant of proportionality and write anequation to describe the situation.3.2 Rate of Change and SlopeFind the slope of each line.3.y4.y4422xx-4-2O24-4-2O-2-2-4-424 Houghton Mifflin Harcourt Publishing Company3.3 Interpreting the Unit Rate as Slope5. The distance Train A travels is represented byTime (hours) Distance (km)d 70t, where d is the distance in kilometers and2150t is the time in hours. The distance Train B travels4300at various times is shown in the table. What is the5375unit rate of each train? Which train is going faster?ESSENTIAL QUESTION6. What is the relationship among proportional relationships, lines, rates ofchange, and slope?Module 389

MODULE 3 MIXED REVIEWPersonalMath TrainerAssessmentReadinessmy.hrw.com5. What is the slope of the line below?Selected Responsey1. Which of the following is equivalent to 5–1?C2D -5x2. Prasert earns 9 an hour. Which tablerepresents this proportional relationship?AHoursEarnings ( )436654872BHoursEarnings ( )436645854CHoursEarnings ( )29318427DHoursEarnings ( )2183274543. A factory produces widgets at a constantrate. After 4 hours, 3,120 widgets havebeen produced. At what rate are thewidgets being produced?A 630 widgets per hourB 708 widgets per hourC4- 15780 widgets per hourD 1,365 widgets per hour4. A full lake begins dropping at a constantrate. After 4 weeks it has dropped 3 feet.What is the unit rate of change in the lake’slevel compared to its full level?A 0.75 feet per week-4-2D 1.33 feet per week12C1B -2D 26. Jim earns 41.25 in 5 hours. Susan earns 30.00 in 4 hours. Pierre’s hourly rate is lessthan Jim’s, but more than Susan’s. What ishis hourly rate?A 6.50CB 7.75D 8.25 7.35Mini-Task7. Joelle can read 3 pages in 4 minutes,4.5 pages in 6 minutes, and 6 pages in8 minutes.a. Make a table of the data.MinutesPagesb. Use the values in the table to find theunit rate.c. Graph the relationship betweenminutes and pages read.y642Ox246Minutes90Unit 24A -2Pages 0.75 feet per week2-2B 1.33 feet per weekCO Houghton Mifflin Harcourt Publishing CompanyA 41B5OnlineAssessment andIntervention

my.hrw.com Math On the Spot 3 Get immediate feedback and help as you work through practice sets. Personal Math Trainer Interactively explore key concepts to see how math works. Animated Math Go digital with your write-in student edition, accessible on any device. Scan with your smart phon