Applying Ratios MDULE and Rates? ESSENTIAL QUESTIN How can you use ratios and rates to solve real-world problems? LESSN. Comparing Additive and Multiplicative Relationships..A LESSN. Ratios, Rates, Tables, and Graphs.5.A LESSN.3 Solving Problems with Proportions.5.A LESSN. Converting Measurements..H Image Credits: Bravo/ Contributor/Getty Images Real-World Video Chefs use lots of measurements when preparing meals. If a chef needs more or less of a dish, he can use ratios to scale the recipe up or down. Using proportional reasoning, the chef keeps the ratios of all ingredients constant. 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. 0
Are YU Ready? Complete these exercises to review skills you will need for this chapter. Graph rdered Pairs (First Quadrant) Personal Math Trainer nline Assessment and Intervention EXAMPLE 0 y A To graph A(, 7), start at the origin. Move units right. Then move 7 units up. Graph point A(, 7). 0 x Graph each point on the coordinate grid above.. B(9, ). C(0, ) 3. D(, 0). E(3, ) Write Equivalent Fractions EXAMPLE = = = 7 7 = _ 3 Write the equivalent fraction. Multiply the numerator and denominator by the same number to find an equivalent fraction. Divide the numerator and denominator by the same number to find an equivalent fraction. 5. = 3. = 7. = 5. 9 = 9. 5 9 = 5 0. Multiples 5 = 0. EXAMPLE List the first five multiples of. = = 3 = = 5 = 0 3 5 =. 0 3 = 0 Multiply by the numbers,, 3,, and 5. List the first five multiples of each number. 3. 3. 7 5. 0 Unit 3
Reading Start-Up Visualize Vocabulary Use the words to complete the graphic. Single item Ratio of two quantities that have different units Numbers that follow a rule Understand Vocabulary Comparing Unit Rates Complete the sentences using the preview words. Rate in which the second quantity is one unit Vocabulary Review Words equivalent ratios (razones equivalentes) factor (factor) graph (gráfica) pattern (patrón) point (punto) rate (tasa) ratio (razón) unit (unidad) unit rate (tasa unitaria) Preview Words conversion factor (factor de conversión) hypotenuse (hipotenusa) legs (catetos) proportion (proporción) scale drawing (dibujo a escala) scale factor (factor de escala). A is a rate that compares two equivalent measurements.. The two sides that form the right angle of a right triangle are called Active Reading Tri-Fold Before beginning the module, create a tri-fold to help you learn the concepts and vocabulary in this module. Fold the paper into three sections. Label one column Rates and Ratios, the second column Proportions, and the third column Converting Measurements. Complete the tri-fold with important vocabulary, examples, and notes as you read the module.. The side opposite the right angle in a right triangle is called the. Module 03
MDULE Unpacking the TEKS Understanding the TEKS and the vocabulary terms in the TEKS will help you know exactly what you are expected to learn in this module...h Convert units within a measurement system, including the use of proportions and unit rates. Key Vocabulary unit rate (tasa unitaria) A rate in which the second quantity in the comparison is one unit. What It Means to You You will convert measurements using unit rates. UNPACKING EXAMPLE..H The Washington Monument is about 5 yards tall. This height is almost equal to the length of two football fields. About how many feet is this? 5 yd 3 ft yd = 5 yd = 555 ft 3 ft yd The Washington Monument is about 555 feet tall..5.a Represent mathematical and real-world problems involving ratios and rates using scale factors, tables, graphs and proportions. Key Vocabulary ratio (razón) A comparison of two quantities by division. rate (tasa) A ratio that compares two quantities measured in different units. What It Means to You You will use ratios and rates to solve real-world problems such as those involving proportions. UNPACKING EXAMPLE.5.A The distance from Austin to Dallas is about 00 miles. How far apart will these cities appear on a map with the scale of in.? 00 = 50? = inches 50 mi? Image Credits: Getty Images Visit to see all the unpacked. 0 Unit 3
? LESSN. Comparing Additive and Multiplicative Relationships ESSENTIAL QUESTIN How do you represent, describe, and compare additive and multiplicative relationships? Proportionality..A Compare two rules verbally, numerically, graphically, and symbolically in the form of y = ax or y = x + a in order to differentiate between additive and multiplicative relationships. EXPLRE ACTIVITY Discovering Additive and Multiplicative Relationships A..A Every state has two U.S. senators. The number of electoral votes a state has is equal to the total number of U.S. senators and U.S. representatives. The number of electoral votes is the number of representatives. Complete the table. Image Credits: Medioimages Photodisc/Getty Images B Representatives 5 5 Electoral votes 3 Describe the rule: The number of electoral votes is equal to the number of representatives plus / times. Frannie orders three DVDs per month from her DVD club. Complete the table. Months 3 DVDs ordered 3 Describe the rule: The number of DVDs ordered is equal to the number of months plus / times. Reflect. Look for a Pattern What operation did you use to complete the tables in A and B? Lesson. 05
Math n the Spot Graphing Additive and Multiplicative Relationships To find the number of electoral votes in part A of the Explore, add to the number of representatives. We call this an additive relationship. To find the number of DVDs Frannie has ordered after a given number of months, multiply the number of months by 3. We call this a multiplicative relationship. EXAMPLE..A My Notes A 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. The total weight is equal to the weight of the items plus the weight of the lunchbox. The relationship is additive. STEP Make a table relating the weight of the items to the total weight. Weight of items (oz) 3 5 Total weight (oz) 7 9 0 To find the total weight, add the weight of the items and the weight of the lunchbox. Total weight = Weight of items + Weight of lunchbox 9 = + 5 STEP STEP 3 Total Weight (oz) List the ordered pairs from the table. The ordered pairs are (, ), (, 7), (3, ), (, 9), and (5, 0). Graph the ordered pairs on a coordinate plane. 0 0 Weight of Items (oz) To plot (,), go right unit from the origin and then up units. The points of the graph form a straight line for an additive relationship. A line drawn through the points would not go through the origin. 0 Unit 3
B skar sells bracelets for two dollars each and donates the money he collects to a charity. Graph the relationship between the number of bracelets sold and the total donation. STEP Complete the table. Bracelets sold 3 5 Total donation ($) 0 To find the total donation, multiply the number of bracelets sold by the donation per bracelet. Total donation = Bracelets sold Donation per bracelet His donation is equal to the number of bracelets sold times the donation for each bracelet. The relationship is multiplicative. 0 = 5 STEP List the ordered pairs from the table. The ordered pairs are (, ), (, ), (3, ), (, ), and (5, 0). STEP 3 Graph the ordered pairs on a coordinate plane. YUR TURN Donation ($) 0 0 Bracelets Sold. Ky is seven years older than his sister Lu. Graph the relationship between Ky s age and Lu s age. Is the relationship additive or multiplicative? Explain. Lu s age 3 5 Ky s age The points of the graph form a straight line for a multiplicative pattern. A line drawn through the points would intersect the origin. The line is steeper than the line in part A. Ky s Age (years) 0 Math Talk Mathematical Processes How are the graphs in part A and part B the same? How are they different? 0 Lu s Age (years) Personal Math Trainer nline Assessment and Intervention Lesson. 07
Guided Practice. Fred s family already has two dogs. They adopt more dogs. Complete the table for the total number of dogs they will have. Then describe the rule. (Explore Activity) Dogs adopted 3 Total number of dogs. Graph the relationship between the number of dogs adopted and the total number of dogs. (Example ) Number of Dogs 0 0 Dogs Adopted 3. Frank s karate class meets three days every week. Complete the table for the total number of days the class meets. Then describe the rule. (Explore Activity) Weeks 3 Days of class. Graph the relationship between the number of weeks and the number of days of class. (Example ) Days of class 30 Weeks 0? 5. An internet café charges ten cents for each page printed. Graph the relationship between the number of pages printed and the printing charge. Is the relationship additive or multiplicative? Explain. (Example ) ESSENTIAL QUESTIN CHECK-IN. How do you represent, describe, and compare additive and multiplicative relationships? Printing Charges ($) 0.50 0.0 0.30 0.0 0.0 0 Pages Printed 0 Unit 3
Name Class Date. Independent Practice..A The tables give the price of a kayak rental from two different companies. Personal Math Trainer nline Assessment and Intervention The graph represents the distance traveled by a car and the number of hours it takes. Raging River Kayaks Hours 3 Cost ($) 9 7 5 7 Paddlers Hours 5 0 Cost ($) 5 50 Distance (mi) 00 0 30 0 0 0 Time (h) 7. Is the relationship shown in each table multiplicative or additive? Explain. 0. Persevere in Problem Solving Based on the graph, was the car traveling at a constant speed? At what speed was the car traveling?. Yvonne wants to rent a kayak for 7 hours. How much would this cost at each company? Which one should she choose?. Make a Prediction If the pattern shown in the graph continues, how far will the car have traveled after hours? Explain how you found your answer. 9. After how many hours is the cost for both kayak rental companies the same? Explain how you found your answer.. What If? If the car had been traveling at 0 miles per hour, how would the graph be different? Lesson. 09
Use the graph for Exercises 3 5. 3. Which set of points represents an additive relationship? Which set of points represents a multiplicative relationship? 0. Represent Real-World Problems What is a real-life relationship that might be described by the red points? 5. Represent Real-World Problems What is a real-life relationship that might be described by the black points? 3 5 FCUS N HIGHER RDER THINKING Work Area. Explain the Error An elevator Time (s) 3 leaves the ground floor and rises three feet per second. Lili makes Distance (ft) 5 7 the table shown to analyze the relationship. What error did she make? 7. Analyze Relationships Complete each table. Show an additive relationship in the first table and a multiplicative relationship in the second table. A 3 A 3 B B 3 Use two columns of each table. Which table shows equivalent ratios? Name two ratios shown in the table that are equivalent.. Represent Real-World Problems Describe a real-world situation that represents an additive relationship and one that represents a multiplicative relationship. 0 Unit 3
? LESSN. Ratios, Rates, Tables, and Graphs ESSENTIAL QUESTIN Proportionality.5.A Represent mathematical and real-world problems involving ratios and rates using tables, graphs, How can you represent real-world problems involving ratios and rates with tables and graphs? EXPLRE ACTIVITY.5.A Finding Ratios from Tables Students in Mr. Webster s science classes are doing an experiment that requires 50 milliliters of distilled water for every 5 milliliters of solvent. The table shows the amount of distilled water needed for various amounts of solvent. Solvent (ml) 3 3.5 5 Distilled water (ml) 00 00 50 A Use the numbers in the first column of the table to write a ratio of distilled water to solvent. B How much distilled water is used for milliliter of solvent? Use your answer to write another ratio of distilled water to solvent. C D E The ratios in A and B are equivalent/not equivalent. How can you use your answer to B to find the amount of distilled water to add to a given amount of solvent? Complete the table. What are the equivalent ratios shown in the table? 00 = = 3 3.5 = 00 = 50 5 Reflect. Look for a Pattern When the amount of solvent increases Math Talk Mathematical Processes Is the relationship between the amount of solvent and the amount of distilled water additive or multiplicative? Explain. by milliliter, the amount of distilled water increases by milliliters. So milliliters of solvent requires distilled water. milliliters of Lesson.
EXPLRE ACTIVITY Graphing with Ratios A.5.A Copy the table from Explore Activity that shows the amounts of solvent and distilled water. Solvent (ml) 3 3.5 5 Distilled water (ml) 00 00 50 B C Write the information in the table as ordered pairs. Use the amount of solvent as the x-coordinates and the amount of distilled water as the y-coordinates. (, ) (3, ), (3.5, ), (, 00), (5, 50) Graph the ordered pairs and connect the points. Describe your graph. For each ordered pair that you graphed, write the ratio of the y-coordinate to the x-coordinate. Distilled Water (ml) 300 (5, 50) 00 00 Solvent (ml) D E The ratio of distilled water to solvent is. How are the ratios in C related to this ratio? The point (.5, 5) is on the graph but not in the table. The ratio of the y-coordinate to the x-coordinate is. How is this ratio related to the ratios in C and D? F.5 milliliters of solvent requires milliliters of distilled water. Conjecture What do you think is true for every point on the graph? Reflect. Communicate Mathematical Ideas How can you use the graph to find the amount of distilled water to use for.5 milliliters of solvent? Unit 3
Representing Rates with Tables and Graphs You can use tables and graphs to represent real-world problems involving equivalent rates. EXAMPLE The Webster family is taking an express train to Washington, D.C. The train travels at a constant speed and makes the trip in hours. A Make a table to show the distance the train travels in various amounts of time..5.a Math n the Spot Animated Math STEP STEP Write a ratio of distance to time to find the rate. distance = time 0 miles hours Use the unit rate to make a table. = 0 miles = 0 miles per hour hour Time (h) 3 3.5 5 Distance (mi) 0 0 0 0 300 B Graph the information from the table. STEP STEP YUR TURN Write ordered pairs. Use Time as the x-coordinates and Distance as the y-coordinates. (, 0), (3, 0), (3.5, 0), (, 0), (5, 300) Distance (mi) Water used (gal) 300 0 0 0 0 Graph the ordered pairs and connect the points. 3. A shower uses gallons of water in 3 minutes. Complete the table and graph. Time (min) 3 3.5.5 Water used (gal) 0 0 3 y (, 0) x 3 5 Time (h) 0 Time (min) Personal Math Trainer nline Assessment and Intervention Lesson. 3
Guided Practice. Sulfur trioxide molecules all have the same ratio of oxygen atoms to sulfur atoms. A number of molecules of sulfur dioxide have oxygen atoms and sulfur atoms. Complete the table. (Explore Activity ) Sulfur atoms 9 xygen atoms What are the equivalent ratios shown in the table?. Graph the relationship between sulfur atoms and oxygen atoms. (Explore Activity ) xygen Atoms 90 7 5 3 30 Sulfur Atoms 3. Stickers are made with the same ratio of width to length. A sticker inches wide has a length of inches. Complete the table. (Explore Activity ) Width (in.) 7 Length (in.) What are the equivalent ratios shown in the table?. Graph the relationship between the width and the length of the stickers. (Explore Activity ) Length (in.) 0 0 Width (in.)? 5. Five boxes of candles contain a total of 0 candles. Each box holds the same number of candles. Complete the table and graph the relationship. (Example ) Boxes 5 Candles 0 ESSENTIAL QUESTIN CHECK-IN. How do you represent real-world problems involving ratios and rates with tables and graphs? Candles 0 9 7 0 Boxes Unit 3
Name Class Date. Independent Practice.5.A Personal Math Trainer nline Assessment and Intervention The table shows information about the number of sweatshirts sold and the money collected at a fundraiser for school athletic programs. For Exercises 7, use the table. Sweatshirts sold 3 5 Money collected ($) 0 0 7. Find the rate of money collected per sweatshirt sold. Show your work.. Use the unit rate to complete the table. 9. Explain how to graph information from the table. 0. Write the information in the table as ordered pairs. Graph the relationship from the table.. What If? How much money would be collected if sweatshirts were sold? Show your work.. Analyze Relationships Does the point (5.5, 0) make sense in this context? Explain. Money Collected ($) 0 0 00 0 0 0 0 0 Sweatshirts Sold Lesson. 5
3. Communicate Mathematical Ideas The table shows the distance Randy drove on one day of her vacation. Find the distance Randy would have gone if she had driven for one more hour at the same rate. Explain how you solved the problem. Time (h) 3 5 Distance (mi) 55 0 5 0 75 Use the graph for Exercises 5.. Analyze Relationships Does the relationship show a ratio or a rate? Explain. 5. Represent Real-World Problems What is a real-life relationship that might be described by the graph? Time (days) 70 5 0 Time (weeks) FCUS N HIGHER RDER THINKING Work Area. Make a Conjecture Complete the table. Then find the rates distance time and time distance. Time (min) 5 Distance (m) 5 00 distance = time time distance = time a. Are the rates equivalent? Explain. distance b. Suppose you graph the points (time, distance) and your friend graphs (distance, time). How will your graphs be different? 7. Communicate Mathematical Ideas To graph a rate or ratio from a table, how do you determine the scales to use on each axis? Unit 3
LESSN.3 Solving Problems with Proportions Proportionality.5.A Represent mathematical and real-world problems involving ratios and rates using proportions.? ESSENTIAL QUESTIN How can you solve problems with proportions? Using Equivalent Ratios to Solve Proportions A proportion is a statement that two ratios or rates are equivalent. _ 3 and _ are equivalent ratios. _ 3 = _ is a proportion. Math n the Spot EXAMPLE.5.A Sheldon and Leonard are partners in a business. Sheldon makes $ in profits for every $5 that Leonard makes. If Leonard makes $0 profit on the first item they sell, how much profit does Sheldon make? STEP Write a proportion. Sheldon s profit is unknown. Sheldon s profit Leonard s profit $ $5 = $0 Sheldon s profit Leonard s profit STEP Use common denominators to write equivalent ratios. YUR TURN $ $5 = $0 $ $0 = $0 = $ 0 is a common denominator. Equivalent ratios with the same denominators have the same numerators. If Leonard makes $0 profit, Sheldon makes $ profit.. The PTA is ordering pizza for their next meeting. They plan to order cheese pizzas for every 3 pepperoni pizzas they order. How many cheese pizzas will they order if they order 5 pepperoni pizzas? Math Talk Mathematical Processes How do you know 0 = _ is a proportion? 5 Personal Math Trainer nline Assessment and Intervention Lesson.3 7
Using Unit Rates to Solve Proportions You can also use equivalent rates to solve proportions. Finding a unit rate may help you write equivalent rates. Math n the Spot My Notes EXAMPLE The distance Ali runs in 3 minutes is shown on the pedometer. At this rate, how far could he run in 0 minutes?.5.a STEP Write a proportion. time distance 3 minutes = 0 minutes 3 miles miles time distance 0 is not a multiple of 3. STEP Find the unit rate of the rate you know. You know that Ali runs 3 miles in 3 minutes. 3 3 3 3 = minutes = mile 0 minutes miles STEP 3 Write equivalent rates. Math Talk Mathematical Processes Compare the fractions 3 3 and 0 5 using <, > or =. Explain. YUR TURN Think: You can multiply 5 = 0. So multiply the denominator by the same number. 5 5 = 0 0 5 = 0 = 5 miles Equivalent rates with the same numerators have the same denominators. At this rate, Ali can run 5 miles in 0 minutes. Personal Math Trainer nline Assessment and Intervention. Ms. Reynold s sprinkler system has 9 stations that water all the parts of her front and back lawn. Each station runs for an equal amount of time. If it takes minutes for the first stations to water, how long does it take to water all parts of her lawn? Unit 3
Using Proportional Relationships to Find Distance on a Map A scale drawing is a drawing of a real object that is proportionally smaller or larger than the real object. A scale factor is a ratio that describes how much smaller or larger the scale drawing is than the real object. Math n the Spot A map is a scale drawing. The measurements on a map are in proportion to the actual distance. If inch on a map equals miles actual distance, the scale factor is miles inch. EXAMPLE 3.5.A The distance between two schools on Lehigh Avenue is shown on the map. What is the actual distance between the schools? STEP STEP Write a proportion. miles inch = miles 3 inches The scale factor is a unit rate. Use common denominators to write equivalent ratios. R Park Blvd. Lehigh Ave. 3 in. Broad St. North St. Eighth St. T 3 3 = 3 3 is a common denominator. Scale: inch = miles miles 3 inches = 3 inches Equivalent ratios with the same denominators have the same numerators. = miles The actual distance between the two schools is miles. YUR TURN 3. The distance between Sandville and Lewiston is shown on the map. What is the actual distance between the towns? Sandville Traymoor.5 in. Sloneham Lewiston Baymont Scale: inch = 0 miles Personal Math Trainer nline Assessment and Intervention Lesson.3 9
Guided Practice Find the unknown value in each proportion. (Example ). 3 5 = 30. 0 = 5 3 = 5 30 = 0 5 Solve using equivalent ratios. (Example ) 3. Leila and Jo are two of the partners in a business. Leila makes $3 in profits for every $ that Jo makes. If Jo makes $0 profit on the first item they sell, how much profit does Leila make?. Hendrick wants to enlarge a photo that is inches wide and inches tall. The enlarged photo keeps the same ratio. How tall is the enlarged photo if it is inches wide? Solve using unit rates. (Example ) 5. A person on a moving sidewalk travels feet in 7 seconds. The moving sidewalk has a length of 0 feet. How long will it take to move from one end to the other?. In a repeating musical pattern, there are 5 beats in 7 measures. How many measures are there after 0 beats? 7. Contestants in a dance-a-thon rest for the same amount of time every hour. A couple rests for 5 minutes in 5 hours. How long did they rest in hours?. Francis gets paychecks in weeks. How many paychecks does she get in 5 weeks?? 9. What is the actual distance between Gendet and Montrose? (Example 3) ESSENTIAL QUESTIN CHECK-IN 0. How do you solve problems with proportions? Gendet.5 cm Montrose Scale: centimeter = kilometers Gravel 0 Unit 3
Name Class Date.3 Independent Practice.5.A Personal Math Trainer nline Assessment and Intervention. n an airplane, there are two seats on the left side in each row and three seats on the right side. There are 90 seats on the right side of the plane. a. How many seats are on the left side of the plane? b. How many seats are there altogether?. The scale of the map is missing. The actual distance from Liberty to West Quall is 7 miles, and it is inches on the map. Abbeville Liberty Foston Mayne a. What is the scale of the map? West Quall a. How many cups of punch does the recipe make? b. If Wendell makes 0 cups of punch, how many cups of each ingredient will he use? cups pineapple juice cups orange juice cups lemon-lime soda c. How many servings can be made from 0 cups of punch?. Carlos and Krystal are taking a road trip from Greenville to North Valley. Each has their own map, and the scales on their maps are different. a. n Carlos s map, Greenville and North Valley are.5 inches apart. The scale on his map is inch = 0 miles. How far is Greenville from North Valley? b. Foston is directly between Liberty and West Quall and is inches from Liberty on the map. How far is Foston from West Quall? Explain. b. The scale on Krystal s map is inch = miles. How far apart are Greenville and North Valley on Krystal s map? 5. Multistep A machine can produce 7 inches of ribbon every 3 minutes. How many feet of ribbon can the machine make in one hour? Explain. 3. Wendell is making punch for a party. The recipe he is using says to mix cups pineapple juice, cups orange juice, and cups lemon-lime soda in order to make servings of punch. Lesson.3
Marta, Loribeth, and Ira all have bicycles. The table shows the number of miles of each rider s last bike ride, as well as the time it took each rider to complete the ride.. What is Marta s unit rate, in minutes per mile? Distance of Last Ride (in miles) Time Spent on Last Bike Ride (in minutes) Marta 0 Loribeth Ira 5 75 7. Whose speed was the fastest on their last bike ride?. If all three riders travel for 3.5 hours at the same speed as their last ride, how many total miles will all 3 riders have traveled? Explain. 9. Critique Reasoning Jason watched a caterpillar move 0 feet in minutes. Jason says that the caterpillar s unit rate is 0. feet per minute. Is Jason correct? Explain. FCUS N HIGHER RDER THINKING Work Area 0. Analyze Relationships If the number in the numerator of a unit rate is, what does this indicate about the equivalent unit rates? Give an example.. Multiple Representations A boat travels at a constant speed. After 0 minutes, the boat has traveled.5 miles. The boat travels a total of 0 miles to a bridge. a. Graph the relationship between the distance the boat travels and the time it takes. b. How long does it take the boat to reach the bridge? Explain how you found it. Distance (mi) 0 0 0 00 Time (min) Unit 3
LESSN. Converting Measurements Proportionality..H Convert units within a measurement system, including the use of proportions and unit rates.? ESSENTIAL QUESTIN How do you convert units within a measurement system? EXPLRE ACTIVITY..H Using a Model to Convert Units The two most common systems of measurement are the customary system and the metric system. You can use a model to convert from one unit to another within the same measurement system. STEP Use the model to complete each statement below. feet 3 9 yard = 3 feet yards yards = feet 3 3 yards = feet yards = feet STEP Write each rate you found in Step in simplest form. feet yards = 3 feet yard(s) 9 feet 3 yards = 3 feet yard(s) feet yards = 3 feet yard(s) Since yard = 3 feet, the rate of feet to yards in any measurement is always 3 _. This means any rate forming a proportion with 3 _ can represent a rate of feet to yards. 3_ =, so feet = yards. 3_ = 5, so feet = yards. Reflect. Communicate Mathematical Ideas How could you draw a model to show the relationship between feet and inches? Lesson. 3
Math n the Spot Converting Units Using Proportions and Unit Rates You can use rates and proportions to convert both customary and metric units. Use the table below to convert from one unit to another within the same measurement system. ft = in. yd = 3 in. yd = 3 ft mi = 5,0 ft mi =,70 yd km =,000 m m = 00 cm cm = 0 mm Customary Measurements Length Weight Capacity lb = oz T =,000 lb c = fl oz pt = c qt = pt qt = c gal = qt Metric Measurements Length Mass Capacity kg =,000 g g =,000 mg L =,000 ml My Notes EXAMPLE A What is the weight of a 3-pound human brain in ounces? Use a proportion to convert 3 pounds to ounces. ounces Use to convert pounds to ounces. pound STEP Write a proportion. ounces pound = ounces 3 pounds..h B STEP Use common denominators to write equivalent ratios. 3 3 = 3 3 = 3 = ounces The weight is ounces. 3 is a common denominator. Equivalent rates with the same denominators have the same numerators. A moderate amount of daily sodium consumption is,000 milligrams. What is this mass in grams? Use a proportion to convert,000 milligrams to grams.,000 mg Use g to convert milligrams to grams. Unit 3
STEP STEP Write a proportion.,000 mg =,000 mg g g Write equivalent rates. Think: You can multiply,000 =,000. So multiply the denominator by the same number.,000 =,000,000 =,000 Equivalent rates with the same numerators have the same denominators. YUR TURN = grams The mass is grams.. The height of a doorway is yards. What is the height of the doorway in inches? Math Talk Mathematical Processes How would you convert 3 liters to milliliters? Personal Math Trainer nline Assessment and Intervention Converting Units by Using Conversion Factors Another way to convert measurements is by using a conversion factor. A conversion factor is a rate comparing two equivalent measurements. EXAMPLE Elena wants to buy gallons of milk but can only find quart containers for sale. How many quarts does she need? You are converting to quarts from gallons. STEP Find the conversion factor. Write quarts = gallon as a rate: quarts gallon..h Math n the Spot STEP Multiply the given measurement by the conversion factor. gallons quarts gallon = quarts gallons quarts gallon = quarts Cancel the common unit. Elena needs quarts of milk. Lesson. 5
Personal Math Trainer nline Assessment and Intervention YUR TURN 3. An oak tree is planted when it is 50 centimeters tall. What is this height in meters? Guided Practice Use the model below to complete each statement. (Explore Activity ). cups quarts Use unit rates to solve. (Example ) 3 _ = 3, so cups = quarts. _ =, so cups = quarts 3. Mary Catherine makes gallons of punch for her party. How many cups of punch did she make?. An African elephant weighs tons. What is the weight of the elephant in pounds? 5. The distance from Jason s house to school is 0.5 kilometer. What is this distance in meters?. The mass of a moon rock is 3.5 kilograms. What is the mass of the moon rock in grams? Use a conversion factor to solve. (Example )? 7..75 grams,000 mg =. 7 millimeters cm g 0 mm = 9. A package weighs 9 ounces. What is the weight of the package in pounds? ESSENTIAL QUESTIN CHECK-IN. How do you convert units within a measurement system? 0. A jet flies at an altitude of 5,00 feet. What is the height of the jet in miles? Unit 3
Name Class Date. Independent Practice..H Personal Math Trainer nline Assessment and Intervention. What is a conversion factor that you can use to convert gallons to pints? How did you find it? 3. Three friends each have some ribbon. Carol has inches of ribbon, Tino has.5 feet of ribbon, and Baxter has.5 yards of ribbon. Express the total length of ribbon the three friends have in inches, feet and yards. inches = feet = yards. Suzanna wants to measure a board, but she doesn t have a ruler to measure with. However, she does have several copies of a book that she knows is 7 centimeters tall. a. Suzanna lays the books end to end and finds that the board is the same length as books. How many centimeters long is the board? b. Suzanna needs a board that is at least 3.5 meters long. Is the board long enough? Explain. Sheldon needs to buy gallons of ice cream for a family reunion. The table shows the prices for different sizes of two brands of ice cream. Price of small size Price of large size Cold Farms $.50 for pint $.50 for quart Sweet Dreams $.5 for quart $9.50 for gallon 5. Which size container of Cold Farm ice cream is the better deal for Sheldon? Explain.. Multistep Which size and brand of ice cream is the best deal? Lesson. 7
7. In Beijing in 00, the Women's 3,000 meter Steeplechase became an lympic event. What is this distance in kilometers?. How would you convert 5 feet inches to inches? FCUS N HIGHER RDER THINKING 9. Analyze Relationships A Class truck weighs between,000 and,000 pounds. a. What is the weight range in tons? b. If the weight of a Class truck is increased by tons, will it still be classified as a Class truck? Explain. Work Area 0. Persevere in Problem Solving A football field is shown at right. a. What are the dimensions of a football field in feet? 53 yd 3 0 yd b. A chalk line is placed around the perimeter of the football field. What is the length of this line in feet? c. About how many laps around the perimeter of the field would equal mile? Explain.. Look for a Pattern What is the result if you multiply a number of cups by ounces cup and then multiply the result by cup? Give an example. ounces. Make a Conjecture hour = 3,00 seconds and mile = 5,0 feet. Make a conjecture about how you could convert a speed of 5 miles per hour to feet per second. Then convert. Image credits: Michael Steele/ Getty Images Unit 3
MDULE QUIZ Ready. Comparing Additive and Multiplicative Relationships Complete each table and describe the rule for the relationship. Personal Math Trainer nline Assessment and Intervention. Meal time :00 :30 :00 Swim time :5 :5. Sets of pens 3 5 Number of pens 9 5. Ratios, Rates, Tables, and Graphs 3. Charlie runs laps around a track. The table shows how long it takes him to run different numbers of laps. How long would it take Charlie to run 5 laps? Number of laps 0 Time (min) 0 0 30 0 50.3 Solving Problems with Proportions. Emily is entering a bicycle race for charity. Her mother pledges $0.0 for every 0.5 mile she bikes. If Emily bikes 5 miles, how much will her mother donate?. Converting Measurements Convert each measurement. 5. meters = centimeters. 5 pounds = ounces 7. quarts = fluid ounces. 9 liters = milliliters ESSENTIAL QUESTIN 9. Write a real-world problem that could be solved using a proportion. Module 9
Personal Math Trainer Module MIXed REVIEW Texas Test Prep Selected Response. The table below shows the number of babies and adults at a nursery. Babies 0 Adults 3 5. The table below shows the number of petals and leaves for different numbers of flowers. Petals 5 0 5 0 Leaves Which represents the number of babies? How many petals are present when there are leaves? A adults A 5 petals B adults B 30 petals C C adults + D adults + 0 5. A recipe calls for 3 cups of sugar and 9 cups of water. If the recipe is reduced, how many cups of water should be used with cups of sugar? A 3 cups B cups C cups D cups 0 Time (h) Which is an ordered pair on the line? A (.5, ) C (.5, ) B (.5, 5) D (.5, 9) 3. n a map of the city, inch represents.5 miles. What distance on the map would represent miles? Gridded Response. Janice bought oranges for $3.0. What is the unit price?. + 0 0 0 0 0 0 A inches B inches 3 3 3 3 3 3 5 5 5 5 5 5 7 7 7 7 7 7 9 9 9 9 9 9 C inches D inches Unit 3 - Distance (mi) 35 petals D 3 petals. The graph represents the distance Manuel walks over several hours. 30 nline Assessment and Intervention