Ratios and Proportionality

Transcription

1 Ratios and Proportionality MDULE? ESSENTIAL QUESTIN How can you use rates and proportionality to solve real-world problems? LESSN.1 Unit Rates 7.RP.1 LESSN. Constant Rates of Change 7.RP., 7.RP.a, 7.RP.b, 7.RP.c LESSN.3 Proportional Relationships and Graphs 7.RP.a, 7.RP.b, 7.RP.c, 7.RP.d Image Credits: cean/corbis Real-World Video You can use rates to describe lots of real-world situations. A cyclist can compute rates such as miles per hour or rotations per minute. Math n the Spot Animated Math Personal Math Trainer 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. 113

2 Are YU Ready? Complete these exercises to review skills you will need for this module. perations with Fractions EXAMPLE Divide _ 5 8 = 3 10 _ 8 5 = = 1 5 Multiply by the reciprocal of the divisor. Divide by the common factors. Simplify. Personal Math Trainer nline Practice and Help 1. 3_ _ 5. 5_ _ 8 1_ _ 9 rdered Pairs EXAMPLE -5 5 y 5 x To write the ordered pair for A, start at the origin. Move units right. Then move units down. The ordered pair for point A is (, -). -5 A Write the ordered pair for each point. y 5. B 6. C 7. D 8. E 9. F 10. G -5 B E 5 F G C D 5 x Unit

3 Reading Start-Up Visualize Vocabulary Use the words to complete the graphic. You can put more than one word in each bubble. _ = inches 1 foot Uses of Ratios Understand Vocabulary 36% 6 to 1 Match the term on the left to the definition on the right. Vocabulary Review Words constant (constante) conversion factor (factor de conversión) equivalent ratios (razones equivalentes) percent (porcentaje) rate (tasa) ratio (razón) Preview Words complex fraction (fracción compleja) constant of proportionality (constante de proporcionalidad) proportion (proporción) proportional relationship (relación proporcional) rate of change (tasa de cambio) unit rates (tasas unitarias) 1. rate of change A. Statement that two rates or ratios are equivalent.. proportion B. A rate that describes how one quantity changes in relation to another quantity. 3. unit rate C. Rate in which the second quantity is one unit. Active Reading Three-Panel Flip Chart Before beginning the module, create a three-panel flip chart to help you organize what you learn. Label each flap with one of the lesson titles from this module. As you study each lesson, write important ideas like vocabulary, properties, and formulas under the appropriate flap. Module 115

4 GETTING READY FR Ratios and Proportionality Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this module. 7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. What It Means to You Given a rate, you can find the equivalent unit rate by dividing the numerator by the denominator. EXAMPLE 7.RP.1 Key Vocabulary rate (tasa) A ratio that compares two quantities measured in different units. unit rate (tasa unitaria) A rate in which the second quantity in the comparison is one unit. Lisa hikes 1_ 3 mile every 1_ 6 hour. How far does she hike in 1 hour? 1_ 3_ 1_ 6 = 1_ 3 1_ 6 = 1_ = miles 7.RP.b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. Key Vocabulary constant (constante) A value that does not change. constant of proportionality (constante de proporcionalidad) A constant ratio of two variables related proportionally. What It Means to You You will determine the constant of proportionality for proportional relationships. EXAMPLE 7.RP.b The graph shows the distance a bicyclist y travels over time. How fast does the 75 bicyclist travel? 5 slope (speed) = rise (distance) 15 run (time) 1 15 = The bicyclist travels at 15 miles per hour. Time (h) The bicyclist's speed is a unit rate. It is indicated on the graphed line by the point (1, 15). Distance (mi) x Image Credits: David Epperson/PhotoDisc/Getty Images Visit to see all CA Common Core Standards explained. 116 Unit

5 ? LESSN.1 Unit Rates ESSENTIAL QUESTIN How do you find and use unit rates? 7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. EXPLRE ACTIVITY 7.RP.1 Exploring Rates Commonly used rates like miles per hour make it easy to understand and compare rates. Jeff hikes 1_ mile every 15 minutes, or 1_ hour. Lisa hikes 1_ mile every 3 10 minutes, or 1_ hour. How far do they each hike in 1 hour? hours? 6 A Use the bar diagram to help you? miles determine how many miles Jeff hikes. How many 1_ 1 1 -hours are in hour hour 1 hour? How far does Jeff hike in 1 hour? 1 mile 1 hour 1 hour B Complete the table for Jeff s hike. Distance (mi) 1_ Time (h) 1_ 1_ 3_ 1 C D Complete the bar diagram to help you determine how far Lisa hikes. How many miles does she hike in 1 hour? Complete the table for Lisa s hike hour 6 hour 6 hour 6 hour 6 hour 6 hour Distance (mi) 1_ 3 Time (h) 1_ 6 1_ 3 1_ 1 Lesson.1 117

6 EXPLRE ACTIVITY (cont d) Reflect 1. How did you find Jeff s distance for 3 _ hour?. Which hiker walks farther in one hour? Which is faster? Math n the Spot Finding Unit Rates A rate is a comparison of two quantities that have different units, such as miles and hours. Ratios and rates can be expressed as fractions. A rate or ratio with a denominator of 1 unit is called a unit rate. 60 miles hours = 30 miles 1 hour This means 30 miles per hour. When one or both of the quantities being compared in the rate is a fraction, the rate is expressed as a complex fraction. 15 miles 1_ hour This means 15 miles per 1_ hour. EXAMPLE 1 7.RP.1 While remodeling her kitchen, Angela is repainting. She estimates that she paints 55 square feet every half-hour. How many square feet does Angela paint per hour? STEP 1 STEP STEP 3 Determine the units of the rate. The rate is area in square feet per time in hours. Find Angela s rate of painting in area painted per time. area painted: 55 sq ft time: 1 hour area painted 55 = square feet time 1_ hour The fraction represents area in square feet per time in hours. Find Angela s unit rate of painting in square feet per hour. 55 square feet 1_ hour = 55 1_ Rewrite the fraction as division. = 55 1 _ Multiply by the reciprocal. 1 Image Credits: Howard Barlow/Alamy = 110 square feet 1 hour The unit rate has a denominator of Unit Angela paints 110 square feet per hour.

7 YUR TURN 3. Paige mows 1_ 6 acre in 1_ hour. How many acres does Paige mow per hour?. Greta uses 3 cups of pasta for every _ 3 cup of sauce. How much pasta Personal Math Trainer nline Practice and Help does she use per cup of sauce? Using Unit Rates You can use unit rates to simplify rates and ratios that appear complicated, such as those containing fractions in both the numerator and denominator. EXAMPLE 7.RP.1 Math n the Spot Two pools are leaking. After 15 minutes, pool A has leaked _ gallon. After 3 0 minutes, pool B has leaked _ 3 gallon. Which pool is leaking faster? My Notes STEP 1 Find the rate in volume (gallons) per time (hours) at which each pool is leaking. First convert minutes to hours. Pool A 3 gal _ 15 min = 3 gal 1_ h 15 min = 1 h Pool B 3 gal 3_ 0 min = gal 1_ 3 h 0 min = 1 3 h STEP To find the unit rates, first rewrite the fractions. Pool A Pool B 3 gal 1_ = _ h 3 1_ 3 gal 1_ = _ 3 3 h 1_ 3 STEP 3 STEP To divide, multiply by the reciprocal. Pool A _ 3 1_ = _ 3 _ 1 = _ 8 3, or _ gal per h 3 Compare the unit rates. Pool A Pool B _ 3 > 1 Pool B 3_ 1_ 3 = 3 _ 3 _ 1 = _ 9, or 1_ gal per h Math Talk Mathematical Practices How do you compare mixed numbers? So, Pool A is leaking faster. Lesson.1 119

8 YUR TURN Personal Math Trainer nline Practice and Help 5. Jaylan makes limeade using _ 3 cup water per 1_ 5 cup lime juice. Wanchen makes limeade using _ 3 cup water per 1_ 6 cup lime juice. Find the unit rates of water (cups) per lime juice (cups). Whose limeade has a weaker lime flavor? Explain. Guided Practice 1. Brandon enters bike races. He bikes 8 1_ miles every 1_ hour. Complete the table to find how far Brandon bikes for each time interval. (Explore Activity) Distance (mi) 8 1_ Time (h) 1_ 1 1 1_ 1_ Find each unit rate. (Example 1). Julio walks 3 1_ miles in 1 1_ hours. 3. Kenny reads 5_ 8 page in _ 3 minute.. A garden snail moves 1_ 6 foot in 1_ 3 hour. 5. A fertilizer covers 5 _ 8 square foot in 1_ hour. Find each unit rate. Determine which is lower. (Example )? 6. Brand A: 0 mg sodium for 1_ 3 pickle or Brand B: 35 mg sodium for 1_ pickle ESSENTIAL QUESTIN CHECK-IN 8. How can you find a unit rate when given a rate? 7. Ingredient C: 1_ cup for _ 3 serving or Ingredient D: 1_ 3 cup for 3_ serving 10 Unit

9 Name Class Date.1 Independent Practice 7.RP.1 9. The information for two pay-as-you-go cell phone companies is given. Personal Math Trainer nline Practice and Help b. If you needed to triple the recipe, would the ratio change? Explain. n Call 3.5 hours: $10 1 Talk Time hour: $1.5 a. What is the unit rate in dollars per hour for each company? c. What is the unit rate of the ingredients in your recipe? b. Analyze Relationships Which company offers the best deal? Explain your answer. 11. A radio station requires DJs to play commercials for every 10 songs they play. What is the unit rate of songs to commercials? c. What If? Another company offers a rate of $0.05 per minute. How would you find the unit rate per hour? d. Draw Conclusions Is the rate in part c a better deal than n Call or Talk Time? Explain. 10. Represent Real-World Problems Your teacher asks you to find a recipe that 1_ includes two ingredients with a ratio of cup 1_ cup. 8 a. Give an example of two ingredients in a recipe that would meet this requirement. 1. Multistep Terrance and Jesse are training for a long-distance race. Terrance trains at a rate of 6 miles every half hour, and Jesse trains at a rate of miles every 15 minutes. a. What is the unit rate in miles per hour for each runner? b. How long will each person take to run a total of 50 miles at the given rates? c. Sandra runs at a rate of 8 miles in 5 minutes. How does her unit rate compare to Terrance s and to Jesse s? Lesson.1 11

10 13. Analyze Relationships Eli takes a typing test and types all 300 words in 1 10 hour. He takes the test a second time and types the words in 1 1 hour. Was he faster or slower on the second attempt? Explain. FCUS N HIGHER RDER THINKING Work Area 1. Justify Reasoning An online retailer sells two packages of protein bars. Package 10-pack of.1 ounce bars 1-pack of 1. ounce bars Cost ($) a. Which package has the better price per bar? b. Which package has the better price per ounce? c. Which package do you think is a better buy? Justify your reasoning. 15. Check for Reasonableness A painter painted about half a room in half a day. Coley estimated the painter would paint 7 rooms in 7 days. Is Coley s estimate reasonable? Explain. 16. Communicate Mathematical Ideas If you know the rate of a water leak in gallons per hour, how can you find the number of hours it takes for 1 gallon to leak out? Justify your answer. 1 Unit

11 LESSN. Constant Rates of Change 7.RP. Recognize and represent proportional relationships between quantities. Also 7.RP.a, 7.RP.b, 7.RP.c? ESSENTIAL QUESTIN How can you identify and represent proportional relationships? EXPLRE ACTIVITY 7.RP.a, 7.RP.b Discovering Proportional Relationships It takes a giant tortoise 3 seconds to travel 10.5 inches. A The diagram shows the rate at which one tortoise moves. Use the diagram to determine if the tortoise is moving at a steady pace. Explain. B What unit rate is represented in the diagram? 10.5 in. 1 sec 1 sec 1 sec Image Credits: blickwinkel/alamy C D E Complete the table. 3.5 in. 3.5 in. Time (sec) Distance (in.) 10.5 For each column of the table, write a rate that compares the distance and the time. Put distance in the numerator and time in the denominator. Divide to write the rate as a decimal. = = = = = What do you notice about the decimal forms of the rates?? F Conjecture How do you think the distance the tortoise travels is related to the time? Lesson. 13

12 EXPLRE ACTIVITY (cont d) Reflect 1. Suppose the tortoise travels for 1 seconds. Explain how you could find the distance the tortoise travels.. How would you describe the rate of speed at which a tortoise travels? Math n the Spot Proportional Relationships A proportion is a statement that two rates or ratios are equivalent. For example, 6 mi h = 3 mi 1 h, or _ = 1_. A rate of change is a rate that describes how one quantity changes in relation to another quantity. A proportional relationship between two quantities is one in which the rate of change is constant or one in which the ratio of one quantity to the other is constant. Proportional relationships are often described using words such as per or for $1.5 each. For example, the rate could be described as $1.5 per pound or 1 pound $1.5 for each pound. My Notes EXAMPLE 1 7.RP.a, 7.RP.b Callie earns money by dog sitting. Based on the table, is the relationship between the amount Callie earns and the number of days a proportional relationship? Number of Days Amount Earned ($) Math Talk STEP 1 Mathematical Practices How can you use the constant rate to find how much Callie earns for 10 days of dog sitting? STEP Write the rates. Amount earned Number of days = $16 1 day $3 days = $16 1 day $8 3 days = $16 1 day $6 days = $16 1 day $80 5 days = $16 1 day Put the amount earned in the numerator and the number of days in the denominator. Each rate is equal to $16, or $16 per day. 1 day Compare the rates. The rates are all equal. This means the rate is constant, so the relationship is proportional. 1 Unit The constant rate of change is $16 per day.

13 YUR TURN 3. The table shows the distance Allison drove on one day of her vacation. Is the relationship between the distance and the time a proportional relationship? Did she drive at a constant speed? Explain. Time (h) Distance (mi) Personal Math Trainer nline Practice and Help Writing an Equation for a Proportional Relationship If there is a proportional relationship between x and y, you can describe that relationship using the equation y = kx. The variable k is called the constant of proportionality, and it represents the constant rate of change or constant ratio between x and y. The value of k is represented by the equation k = y _ x. Math n the Spot EXAMPLE 7.RP.c, 7.RP.b Two pounds of the cashews shown cost $19, and 8 pounds cost $76. Show that the relationship between the number of pounds of cashews and the cost is a proportional relationship. Then write an equation for the relationship. Describe the proportional relationship in words. STEP 1 Make a table relating cost in dollars to pounds. Number of Pounds 3 8 Cost ($) STEP Write the rates. Put cost in the numerator and pounds in the denominator. Write each rate as a decimal. STEP 3 Cost Number of Pounds 19 = = = 9.50 The rates are all equal to $9.50 per pound. They are constant, so the relationship is proportional. The constant rate of change is $9.50 per pound. To write an equation, first tell what the variables represent. Let x represent the number of pounds of cashews. Let y represent the cost in dollars. Use the decimal form of the constant rate of change as the constant of proportionality. The equation for the relationship is y = 9.5x. Math Talk Mathematical Practices How can you use your equation to find the cost of 6 pounds of cashews? The cost is $9.50 per pound. Lesson. 15

14 YUR TURN Personal Math Trainer nline Practice and Help. For a school field trip, there must be 1 adult to accompany 1 students, 3 adults to accompany 36 students, and 5 adults to accompany 60 students. Show that the relationship between the number of adults and the number of students is a proportional relationship. Then write an equation for the relationship. Number of students Number of adults Guided Practice 1. Based on the information in the table, is the relationship between time and the number of words typed a proportional relationship? (Explore Activity and Example 1) Time (min) 1 3 Number of words Number of words : Minutes 5 1 = = = = The relationship is / is not proportional. Find the constant of proportionality k. Then write an equation for the relationship between x and y. (Example )?. x x y y 6 8 ESSENTIAL QUESTIN CHECK-IN. How can you represent a proportional relationship using an equation? 16 Unit

15 Name Class Date. Independent Practice 7.RP., 7.RP.a, 7.RP.b, 7.RP.c Personal Math Trainer nline Practice and Help Information on three car-rental companies is given. 5. Write an equation that gives the cost y of renting a car for x days from Rent-All. 6. What is the cost per day of renting a car from A-1? 7. Analyze Relationships Which company offers the best deal? Why? Rent-All Days Total Cost ($) A-1 Rentals The cost y of renting a car for x days is $10.99 for each half day Car Town The cost of renting a car from us is just $19.5 per day! 8. Critique Reasoning A skydiver jumps out of an airplane. After 0.8 second, she has fallen 100 feet. After 3.1 seconds, she has fallen 500 feet. Emtiaz says that the skydiver should fall about feet in 1.5 seconds. Is his answer reasonable? Explain. Steven earns extra money babysitting. He charges $31.5 for 5 hours and $50 for 8 hours. 9. Explain why the relationship between how much Steven charges and time is a proportional relationship. 10. Interpret the Answer Explain what the constant rate of change means in this context. 11. Write an equation to represent the relationship. Tell what the variables represent. 1. How much would Steven charge for 3 hours? Lesson. 17

16 A submarine dives 300 feet every minutes, and 6,750 feet every 5 minutes. 13. Find the constant rate at which the submarine dives. Give your answer in feet per minute and in feet per hour. 1. Let x represent the time of the dive. Let y represent the depth of the submarine. Write an equation for the proportional relationship using the rate in feet per minute. 15. Draw Conclusions If you wanted to find the depth of a submarine during a dive, would it be more reasonable to use an equation with the rate in feet per minute or feet per hour? Explain your reasoning. FCUS N HIGHER RDER THINKING Work Area 16. Make a Conjecture There is a proportional relationship between your distance from a thunderstorm and the amount of time that elapses between the time you see lightning and the time you hear thunder. If there are 9 seconds between lightning and thunder, the storm is about 3 kilometers away. If you double the amount of time between lightning and thunder, do you think the distance in kilometers also doubles? Justify your reasoning. 17. Communicate Mathematical Ideas A store sells 3 ears of corn for $1. They round prices to the nearest cent as shown in the table. Tell whether you would describe the relationship between cost and number of ears of corn as a proportional relationship. Justify your answer. Ears of corn 1 3 Amount charged ($) Unit

17 ? LESSN.3 Proportional Relationships and Graphs ESSENTIAL QUESTIN 7.RP.a Decide whether two quantities are in a proportional relationship, e.g., by...graphing on a coordinate plane and observing whether the graph is a straight line through the origin. Also 7.RP., 7.RP.b, 7.RP.c, 7.RP.d, 7.RP.3 How can you use graphs to represent and analyze proportional relationships? EXPLRE ACTIVITY Graphing Proportional Relationships The equation y = 5x represents the relationship between the number of gallons of water used (y) and the number of minutes (x) for most showerheads manufactured before 199. A 7.RP.a, 7.RP.b, 7.RP.d Explain why the relationship is proportional. How does the equation represent the constant of proportionality? B Complete the table. Time (min) Water used (gal) 5 35 Each minute, 5 gallons of water are used. So for minutes, 5 gallons are used. C Write the data in the table as ordered pairs (time, water used). Water Use D E F (1, 5), (, ), (3, ), (, 35), (10, ) Plot the ordered pairs. If the showerhead is used for 0 minutes, how many gallons of water will be used? What ordered pair represents this situation? What is this location called? If the showerhead is used for 1 minute, how many gallons of water will be used? What ordered pair represents this situation? What does this point represent? Water used (gal) Time (min) G Draw Conclusions If you continued the table to include 3 minutes, would the point (3, 15) be on this graph? Why or why not? Lesson.3 19

18 Math n the Spot Identifying Proportional Relationships In addition to using a table to determine if a relationship is proportional, you also can use a graph. A relationship is a proportional relationship if its graph is a straight line through the origin. EXAMPLE 1 7.RP.a, 7.RP.b The table shows the relationship between the amount charged by a housecleaning company ($) and the amount of time worked (hours). Is the relationship a proportional relationship? Explain. Time (h) Total cost ($) Animated Math STEP 1 STEP Write the data in the table as ordered pairs (time, cost). (1, 5), (, 90), (3, 135), (5, 5), (8, 360) Graph the ordered pairs. Total cost ($) Time (h) Place time on the x-axis and total cost on the y-axis. Plot each point. Notice that the points are on a line. The graph is a line that goes through the origin. The relationship is proportional. The point (1, 5) on the graph shows that the constant of proportionality, or unit rate, is $5 for 1 hour. The housecleaning company charges $5 per hour. YUR TURN 1. Jared rents bowling shoes for $6 and pays $5 per bowling game. Is the relationship a proportional relationship? Explain. Games 1 3 Total cost ($) Personal Math Trainer nline Practice and Help Total cost ($) Games 130 Unit

19 Analyzing Graphs Recall that you can describe a proportional relationship with the equation y = kx. The constant of proportionality k tells you how steep the graph of the relationship is. The greater the absolute value of k, the steeper the line. EXAMPLE The graph shows the relationship between time in minutes and the number of miles Damon runs. Write an equation for this relationship. STEP 1 STEP Choose a point on the graph and tell what the point represents. The point (5,.5) represents the distance (.5 miles) that Damon runs in 5 minutes. What is the constant of proportionality? Distance (mi) RP., 7.RP.3 Math n the Spot The points appear to form a line through the origin, so the relationship is proportional. STEP 3 Because distance =.5 mi time 5 min = 1, the constant 10 Time (min) of proportionality is Write an equation in the form y = kx. y = 1 10 x Reflect. What does the point (0, 0) on the graph represent? (1, 1 10 )? 3. What If? Suppose you drew a graph representing the relationship y = 1_ x between time in minutes and the number of miles Esther runs. 8 How would the graph compare to the one for Damon? Explain.. Use your equation to find how far Damon runs in 0 minutes. How long would it take him to run 7.5 miles? YUR TURN 5. The graph shows the relationship between the distance a bicyclist travels and the time in hours. a. What does the point (, 60) represent? Distance (mi) Math Talk Mathematical Practices What is the meaning of the point on the graph in Exercise 5 with x-coordinate 1? b. What is the constant of proportionality? c. Write an equation in the form y = kx for this relationship. 6 8 Time (h) Personal Math Trainer nline Practice and Help Lesson.3 131

20 Guided Practice For each situation, tell whether the relationship is a proportional relationship. Explain why or why not. (Explore Activity) 1. The table shows the number of pages a student reads in various amounts of time. Time (h) The amount a babysitter earns is given by the equation y = 7.5x where x is the number of hours and y is the amount earned in dollars. Pages Tell whether the relationship is a proportional relationship. Explain why or why not. (Explore Activity and Example 1) 3. Chores. Movie Rentals Number of chores Cost ($) Age (years) Number of movies Write an equation of the form y = kx for the relationship shown in each graph. (Example )? Balloon height (ft) Time (s) Number of items ESSENTIAL QUESTIN CHECK-IN 7. How does a graph show a proportional relationship? Cost ($) Unit

21 Name Class Date.3 Independent Practice 7.RP., 7.RP.a, 7.RP.b, 7.RP.c, 7.RP.d, 7.RP.3 For Exercises 8 1, the graph shows the relationship between time and distance run by two horses. Distance (mi) Time (min) 8. Explain the meaning of the point (0, 0). B A Personal Math Trainer nline Practice and Help 13. A bullet train can travel at 170 miles per hour. Will a graph representing distance in miles compared to time in hours show a proportional relationship? Explain. 1. Critical Thinking When would it be more useful to represent a proportional relationship with a graph rather than an equation? 9. How long does it take each horse to run a mile? 10. Multiple Representations Write an equation for the relationship between time and distance for each horse. 15. Multiple Representations Bargain DVDs cost $5 each at Mega Movie. a. Graph the proportional relationship that gives the cost y in dollars of buying x bargain DVDs. 11. Draw Conclusions At the given rates, how far would each horse run in 1 minutes? 1. Analyze Relationships Draw a line on the graph representing a horse than runs faster than horses A and B. Cost ($) DVDs b. Give an ordered pair on the graph and explain its meaning in the real world context. Lesson.3 133

22 The graph shows the relationship between distance and time as Glenda swims. 16. How far did Glenda swim in seconds? 17. Communicate Mathematical Ideas Is this a proportional relationship? Explain your reasoning. Distance (ft) Multiple Representations Write an equation that shows the relationship between time and distance. Use your equation to find how long it would Time (s) take in minutes for Glenda to swim 1_ mile at this rate. FCUS N HIGHER RDER THINKING Work Area 19. Make a Conjecture If you know that a relationship is proportional and are given one ordered pair, how can you find another pair? The tables show the distance traveled by three cars. Car 1 Car Car 3 Time (h) Distance (mi) Time (h) Distance (mi) Time (h) Distance (mi) Communicate Mathematical Ideas Which car is not traveling at a constant speed? Explain your reasoning. 1. Make a Conjecture Car is traveling at twice the rate of speed of car. How will the table values for car compare to the table values for car? 13 Unit

23 MDULE QUIZ Ready.1 Unit Rates Find each unit rate. Round to the nearest hundredth, if necessary. Personal Math Trainer nline Practice and Help 1. $10 for 18 ft. 1 lb for $.99 Circle the better deal in each pair. Then give the unit rate for the better deal. 3. $56 5 gal $3.05 or 15 gal $ g $315 or 9 g. Constant Rates of Change 5. The table shows the amount of money Tyler earns for mowing lawns. Is the relationship a proportional relationship? Why or why not? Number of Lawns Amount Earned ($) n a recent day, 8 euros were worth $9 and euros were worth $7. Write an equation of the form y = kx to show the relationship between the number of euros and the value in dollars., where y is dollars and x is euros.3 Proportional Relationships and Graphs 7. The graph shows the number of servings in different amounts of frozen yogurt listed on a carton. Write an equation that gives the number of servings y in x pints. 8. A refreshment stand makes large servings of frozen yogurt from 3 pints. Add the line to the graph and write its equation. ESSENTIAL QUESTIN 9. How can you use rates to determine whether a situation is a proportional relationship? Servings Frozen Yogurt Pints Module 135

24 MDULE MIXED REVIEW Assessment Readiness Personal Math Trainer nline Practice and Help 1. Consider each ratio. Is the ratio equivalent to a unit rate of 1_ cup of milk per cup of flour? Select Yes or No for expressions A C. A. 1_ cup of milk per cups of flour Yes No B. 1 1_ cups of milk per cups of flour Yes No C. 1 3 _ cup of milk per 7 cups of flour Yes No. Consider the number Choose True or False for each statement. A. The number is rational. True False B. The number can be written as a repeating decimal. True False C. The number is less than -0.. True False 3. The graph shows the relationship between the gallons of water a sprinkler system uses and the number of minutes the system is used. Write an equation for this relationship. Explain how you determined your answer. Amount of water (gal) y 6 Time (min) x. A blimp travels 765 feet in 1_ minute, 1,530 feet in 1_ minute, and 3,060 feet in 1 minute. Is there a proportional relationship between the distance the blimp travels and the time it travels? Justify your answer. 136 Unit