Math Background

Lesson: Rates
Developing the Concept

Now that students know how to find a unit rate, they will learn how to find an equivalent ratio using unit rates. Finding equivalent ratios is similar to finding equivalent fractions.

  • Say: You have learned how to find a unit rate. Today you are going to learn how to use a unit rate to solve problems. Look at this problem.

    Write the following on the board: “Yesterday Richard ran 18 laps around the track in 12 minutes. If he runs at that rate for 30 laps, how long will it take him?”

  • Ask: What are we trying to find in this problem?
    We are trying to find out how long it will take Richard to run 30 laps.
  • Ask: What information do we know that will help us solve this problem?
    We know that Richard can run 18 laps in 12 minutes. We also know he is to run at that same rate for 30 laps.
  • Ask: How far does Richard run in one minute?
    Richard runs 1.5 laps in one minute.
  • Say: Let's make a table to list the information we know.
    Make the following table and fill it in by soliciting class input.

    Discuss the table and how to fill in each row.

  • Ask: Does anyone see another way we could have found the answer of 20 minutes without completeing the whole table?
    Some students might recognize that if they divided 30 by 1.5, they would get 20 minutes. Discuss this strategy with the class.
  • Write the following problem on the board: “Maria wanted to buy a pencil for everyone in her class. If 3 pencils cost $0.78, how much would Maria have to spend if she bought a pencil for each of her 24 classmates?”
  • Say: I'd like you to solve this problem at your desks and then we'll discuss what you did.
    Have students come to the board and share their solutions. Many of the students may have solved it by finding the unit price. Others may have solved it by saying if 3 pencils cost $0.78, and 24 = 3 x 8, we could multiply $0.78 by 8 and get $6.24. If they don't do it that way, suggest it as an alternative method for them to solve this problem. Here again, making a table may prove helpful.

Wrap-Up and Assessment Hints
Rates are basic to students' use of mathematics in everyday life. You can ensure that students understand the concept and can apply their understanding to real-world situations, such as figuring the unit cost of items packaged in multiples, calculating comparable cost of different quantities, and comparing rates of speed measured in different units. Provide ample practice for students to express rates as ratios of different units and make comparisons between them.

Houghton Mifflin Math Grade 6