Unit Rates LAUNCH (7 MIN) Before How can a ratio help you to solve this problem? During What would make the ratios easier to compare? How does writing the ratios in simplified form help you compare them? After Why are units left in the solution? What do you call a ratio that compares two quantities with different units? How do the unit rates tell you which turtle will win the race? KEY CONCEPT (5 MIN) Emphasize that a rate is special type of ratio in which the units in the terms are different. Solicit other examples of ratios, rates and unit rates from students. You can use a Venn Diagram organizer to classify examples of ratios that are and are not rates, and examples of rates that are and are not unit rates. Discuss why each one represents a ratio, rate, and/or unit rate. Give an example of a ratio whose terms have different units but can be written without units. PART 1 (6 MIN) How do you know that this problem asks for a unit rate? Jay Says (Screen 2) Use the Jay Says button to help students understand the word molt and explain why the question being asked is important. After solving the problem How is the unit rate helpful in finding the number of times a lobster molts in 6 years? PART 2 (7 MIN) During the Intro For which ratio are you given both the number for the kitten and for the adult cat? Given two plain T-shirts, how do you know which one is the better buy? How can you tell which package is a better buy? Jay Says (Screen 2) Use the Jay Says button to point out that the highest unit rate is not always best. When it comes to buying an item, the least unit rate is the best buy. When it comes to selling an item, the greatest unit rate is best. How would you choose the best unit rate if you were selling the T-shirts instead? PART 3 (7 MIN) After the Intro Why would you want to find the unit rate first? Jay Says (Screen 2) Use the Jay Says button to point out words other than per that are used to indicate a comparison between quantities. Where in the problem statement can you usually find a word that tells you to write a ratio of two quantities? CLOSE AND CHECK (5 MIN) How are rates the same as other ratios? How are they different from other ratios? How can you use unit rates to help make decisions in the real world?
Unit Rates LESSON OBJECTIVE Compute unit rates associated with ratios of fractions. FOCUS QUESTION How can you identify a rate? How can unit rates help you to solve problems? MATH BACKGROUND In the sixth grade, students learned about ratios and rates. In the previous lesson, students used their knowledge of ratios in fraction form to compare quantities with whole-number terms. They used equivalent ratios to show the same relationship between two quantities, demonstrating proportional reasoning. In this lesson, students continue to work with ratios in fraction form. They apply ratio reasoning to a specific type of ratio: rates. They write equivalent rates to find unit rates and use those unit rates to make comparisons and solve problems, such as determining which product is the better buy. Building these skills lays a foundation for solving problems with ratios and rates whose terms are fractions and mixed numbers. In the remaining lessons of this topic, students work with complex fractions to interpret and solve problems involving ratios. LAUNCH (7 MIN) Objective: Apply unit rates to solve a problem. Students explore equivalent rates and unit rates by applying what they know about equivalent ratios. This problem helps students recognize the usefulness of unit rates in making comparisons. Before How can a ratio help you to solve this problem? [I can write a ratio of distance to time for each turtle and then compare the ratios for some common amount of time.] During What would make the ratios easier to compare? [Sample answer: They would be easier to compare if all the ratios used the same amount of time.] How does writing the ratios in simplified form help you compare them? [Since they all have a denominator of 1, I can easily compare the numerators.] After Why are units left in the solution? What do you call a ratio that compares two quantities with different units? [The units are different and cannot be divided out. These ratios are also rates.] How do the unit rates tell you which turtle will win the race? [Sample answer: The unit rate with the greatest numerator is the fastest turtle.] Some students may divide each rate by a factor that is not the greatest common factor. Students can continue to divide by other common factors until the rate is simplified. Allow students to simplify the rates in any way they can, since the important point is for every denominator to have the same amount of time.
Students may find the unit rate for each turtle and then use the unit rate to help determine how long it will take to travel 30 feet. Others may just compare the unit rates themselves, which is more efficient. Still other students may use a different common denominator in order to compare rates, such as 2 (the GCF) or 12 (the LCM). You can also show students how to find an equivalent rate with a numerator of 30 for each turtle and compare these rates. Connect Your Learning Move to the Connect Your Learning screen. Students should realize that you cannot always cancel out units by converting them. In the Launch, students used rates involving distance and time. It is easier to compare these rates when you use equivalent ratios to describe them as a quantity per 1 unit. Use the Focus Question to discuss how unit rates were useful in the Launch. Discuss situations in which you would want to know the greatest or least rate. KEY CONCEPT (5 MIN) ELL Support Beginning Play the audio for the Key Concept. Then explain to students that a unit is one of something. (Compare to the Spanish word uno.) Ask them to name some common unit rates for different quantities: the speed of a car [miles per hour]; the cost of cheese [dollars per pound]; the amount a person earns at a job [dollars per hour]. Intermediate Play the audio for the Key Concept. Then explain that a unit rate tells how many units of one thing correspond to one unit of another. Have students brainstorm a list of unit rates that they use in everyday life or in science and math classes. [Samples: feet per second, cost per person, dollars per gallon, heartbeats per minute, interest per year; words per page, price per ounce, grams per milliliter] Advanced Play the audio for the Key Concept. Explain that unit rates are usually expressed with the word per. Ask students what operation the word per indicates. [division] Show them two ways to write a unit rate. For example, a person s heart rate can be written as beats per minute or beats/min. Ask: How would you figure out the price per ounce if you know that 10 ounces cost $3.50? [Divide $3.50 by 10 ounces to get $.35 per ounce.] Teaching Tips for the Key Concept Use the Key Concept to emphasize that a rate is special type of ratio in which the units in the terms are different. These units usually cannot be converted to the same unit of measure. (An example is the scale on a map: for example, 1 inch : 1 mile.) A unit rate is a rate of a quantity to one unit. In other words, a unit rate has 1 in the denominator when it is written as a fraction. Students can write any rate as an equivalent unit rate. Use this example and solicit other examples of ratios, rates and unit rates from students. You can use a Venn Diagram organizer to classify examples of ratios that are and are not rates, and examples of rates that are and are not unit rates. Discuss why each one represents a ratio, rate, and/or unit rate. Give an example of a ratio whose terms have different units but can be written without units. [Sample answer: A ratio of miles to feet has different units of length, but you can convert miles to feet. Once both measurements have the same units, you can divide the units to get a ratio without any units.]
PART 1 (6 MIN) Objective: Compute unit rates associated with ratios of whole numbers. Students use equivalent rates to determine unit rate. They should recognize that this problem requires a unit rate since per suggests a quantity to one unit. How do you know that this problem asks for a unit rate? [The word per indicates a rate with 1 year in the denominator.] Jay Says (Screen 2) Use the Jay Says button to help students understand the word molt and explain why the question being asked is important. After solving the problem How is the unit rate helpful in finding the number of times a lobster molts in 6 years? [Sample answer: You can multiply both the numerator and denominator of the unit rate by 6 to find an equivalent rate. It is more difficult to find a rate equivalent to 20 times 5 years with a denominator of 6.] You may want to use a Know-Need-Plan organizer to help students get started solving the problem. The organizer is formally introduced for the first time in Part 3. Have students consider how they can check to make sure the solution is correct. Students should respond that they can multiply the numerator and denominator of the unit rate by 5 to confirm that a lobster molts 20 times in 5 years. Error Prevention Students may have difficulty distinguishing between a rate and a ratio in which they must convert to the same unit of measure. Point out that the units in the terms of a rate cannot be converted because they are different types of measurement. While feet and yards are both lengths, molts are very different from years. Got It Notes Encourage students to use estimation and reasoning to make sure they have the correct solution. They may reason that if there are 219 heartbeats in 3 minutes, the number in 1 minute must be much less than 219. Therefore, students can eliminate choices C and D. If you show answer choices, consider the following possible student errors: If students use the conversion factor 1 minute, they may choose B. Students who 60 seconds choose C may be subtracting the terms. Emphasize that they should write the quantities as a ratio and then simplify. If students select D, they may be multiplying the terms. PART 2 (7 MIN) Objective: Compute unit prices to solve comparison problems like best buy and worst buy. Students extend their understanding of unit rates by using them to compare prices and solve a real-world problem. They must interpret the meaning of the unit prices while comparing them and recognize that the greatest unit rate is not always the best.
Instructional Design Unit Rates continued You can call on students to find each unit rate on the whiteboard. Once they have found all four unit rates, have a volunteer drag the tiles that represent the best and worst buys to the appropriate areas. When you click the Check button, any incorrect answers will snap back to their original positions. Use this opportunity to review, as a class, how you found each unit rate and to reread the question carefully. Note that this problem assumes the T-shirts are all of similar quality. If students ask whether the shirts are different brands or sizes, tell them to assume the only difference is the price. During the Intro Why might you want a unit price for movie tickets? [Sample answer: If you buy tickets for a group of 4, you need to know how much money each person owes you.] Given two plain T-shirts, how do you know which one is the better buy? [Sample answer: The item that costs less is usually the better buy.] The T-shirts in this problem are sold in packages. How can you tell which package is a better buy? [Sample answer: You can find the cost per T-shirt for each package and then compare to see which costs the least per item.] Jay Says (Screen 2) Use the Jay Says button to point out that the highest unit rate is not always best. When it comes to buying an item, the least unit rate is the best buy. When it comes to selling an item, the greatest unit rate is best. How would you choose the best unit rate if you were selling the T-shirts instead of buying them? [The best rate would be the highest one.] Students may use the same method as the provided solution and find the cost per T-shirt before comparing rates. Some students may want to find the unit rate of the number of T-shirts per dollar. Since the number of T-shirts per $1 would be a fractional amount, show students that this method is more challenging and less efficient. In the case of unit price, the greatest number in the numerator is the worst buy. Differentiated Instruction For struggling students: A common linguistic misconception is that best means biggest or most, and worst means smallest. Make sure students understand that best buy means the most quantity for the least amount of money. Have students find the unit rate of T-shirts per dollar to determine whether the best and worst buys are the same. Ask them to explain why they can also use this ratio to find the best and worst buys. They should recognize that the higher the rate of T-shirts per dollar, the better the buy. For advanced students: Use newspaper ads to find examples of items that are the best buys and have students write their own ads/examples of best and worst buys that can then be explained to the class. Ask students to find another method of comparing the different rates. Got It Notes As in the Example, students should recognize that the best buy is the item with the lowest price per fluid ounce. Students may instead choose to find the rate for the number of fluid ounces per $1. In that case, the best buy is the greatest number of fluid ounces per $1. The division is more complicated using this method because
finding unit rates involves dividing a whole number by a decimal. You can show the Intro again to convince students that finding unit prices is the better method. If you show answer choices, consider the following possible student errors: If students select A, they may be selecting the lowest price without considering the number of fluid ounces. Remind these students to find each unit rate, which is the cost per fluid ounce. If students choose B, they may be looking for the greatest unit rate. Emphasize that the greatest unit price is the most expensive per fluid ounce. PART 3 (7 MIN) Objective: Compute and apply unit rates to solve real-world problems. ELL Support On the Student Companion page for the Part 3 Got It, there are three tasks for students to complete and discuss: Compare your answers to this problem. Discuss what it means for an answer to be reasonable. How can you apply what you know about checking an answer for reasonableness to the answers to this problem? Beginning and Intermediate Have students discuss what they know about the answer before they begin to calculate. For example, if the satellite travels for less than 8 minutes, students may recognize that the distance that it travels would be less than 2,272 miles. Once they have discussed the problem, have students complete the three tasks. Advanced Have students share with the class what they consider to be a good model explanation. Focus on the reasonableness of the answer. Students find a unit rate and use it to make a plan for another quantity that involves an equivalent rate. This problem shows students that finding equivalent rates can be more efficient if you first find the unit rate. Instructional Design This is the first time students use a Know-Need-Plan organizer in this course. Introduce students to the organizer using the animation in the Intro. Help them understand why a problem-solving tool is helpful to figure out what you need and to form a plan. On Screen 2, you can call students to the whiteboard to fill in the blank organizer. Have one student fill in each box and make sure you follow the Plan box as you solve the problem as a class. After the Intro Why would you want to find the unit rate first? [Sample answer: It is easier to find a rate equivalent to a unit rate than to the rate cannot multiply 22 by a whole number and get 39.] 330 gallons, because you 22 min
Jay Says (Screen 2) Use the Jay Says button to point out words other than per that are used to indicate a comparison between quantities. Where in the problem statement can you usually find a word that tells you to write a ratio of two quantities? [Sample answer: The word is often used between the two quantities.] Some students may use proportional reasoning to realize that 9 is 4.5 times as much as 2, so the answer is 4.5 times as much as 18. Show students that using the unit rate is generally more useful by asking them how many milligrams the astronaut s diet should contain for 37 days. Got It Notes If you show answer choices, consider the following possible student errors: Students who choose A may be finding the unit rate for 3 minutes and then multiplying by 8. If students select C, they may have found the unit rate for 2,272 miles in 3 minutes. Students who select D chose the unit rate and did not multiply by 3. CLOSE AND CHECK (8 MIN) Focus Question Sample Answer A ratio is a rate if it compares two quantities measured in different units. Because unit rates give you a rate per one item, they are easier to compare and make sense of. You can use unit rates to quickly find equivalent rates. Focus Question Notes Look for answers that indicate a rate is a ratio with 1 in the denominator. By comparing a quantity to one unit, it is easier to determine the greatest or least amount per unit from several rates. Such comparisons can help determine which product is a better buy or which speed is the greatest. Stress that rate problems are typically faster when you find the unit rate first and then use the unit rate to find an equivalent rate that contains the term that you know and the term that you need. Essential Question Connection The Focus Question asks how to identify rates and how unit rates help solve problems, which refers to the Essential Question: How do you distinguish the different kinds of rates? What kind of real-world relationships are rates? Use the questions that follow to establish this connection. How are rates the same as other ratios? How are they different from other ratios? [Sample answer: A rate compares one quantity to another, just like any other ratio. A rate is different because its terms have different units that cannot cancel out.] How can you use unit rates to help make decisions in the real world? [Sample answer: I can compare the prices of two sizes of the same item by comparing the unit prices to decide which one costs less per unit.]