Math Background

Lesson: Rates
Introducing the Concept

Your students may have encountered rates and ratios before, but they will need to do a thorough review of these concepts before proceeding to use them to solve problems.

Prerequisite Skills and Concepts: Students should have a basic understanding of ratios — how to write them and how to simplify them. Students should also have an ability to work with fractions and find equivalent fractions.

  • Say: Today we are going to look at a special type of ratio called a rate. Does anyone know what I mean by rate?
    Students may say that a rate is a ratio in which the quantities being compared use different units, such as dollars per ounce or miles per hour. If they don't give you this answer, tell them what a rate is.
  • Say: Rates are commonly found in everyday life. The prices in grocery stores and department stores are rates. Rates are also used in pricing gasoline, figuring how fast a person runs or a car moves, and in hourly wages or monthly fees.
  • Say: Two important ideas are unit rates and unit prices. Does anyone know what a unit rate is?
    Students will probably not know what a unit rate is, so provide them with the following explanation.
  • Say: When we have a unit we have one of something. A unit rate is a rate for one of something. We write the rate as a ratio with a denominator of one. For example, if you ran 70 yards in 10 seconds, you would run an average of 7 yards in 1 second. Both of the rates, 70 yards in 10 seconds and 7 yards in 1 second, are rates, but the 7 yards in 1 second is a unit rate.
  • Ask: Now that you know what a unit rate is, what do you think a unit price is?
    Students will say that it is the price of one item. If they don't, tell them what it is.
  • Ask: What is the unit price of 10 pounds of potatoes that cost $2.80?
    Help students calculate that the unit price is $0.28 cents per pound by dividing the price by 10.
  • Write the following problem on the board: “One grocery store advertises carrots on sale for $1.14 for 3 pounds, while another store has carrots on sale for $0.78 for two pounds. Which store has the better buy?”
  • Ask: What are we trying to find in this problem?
    Students should say that we are trying to find out which is the better buy for carrots.
  • Ask: What would help us find the better buy?
    Students should say that if we find the unit price for the carrots at each store, we would know which was the better buy.
  • Say: Find the unit prices for the carrots at both stores and then we will discuss what you did.
    Have a volunteer come to the board to explain what he/she did and which was the better buy.
    ($0.38 per pound is a better rate than $0.39, so the carrots at $1.14 for 3 pounds are the better buy.)

    After discussing the problem, have students solve the following problem and discuss it.

“One animal can run 60 feet in 4 seconds, while another animal can run 100 feet in 8 seconds. Which animal runs the fastest?”
(The first animal runs the fastest at 15 feet per second.)

Houghton Mifflin Math Grade 6