Lesson 1

Ratios, Recipes, and Proportions

Est. Class Sessions: 2–3

Developing the Lesson

Part 1: Find Equivalent Fractions and Ratios

Review Ratios. Read the Peanut Brittle section in the Student Guide to study ratios and proportions. Compare the two recipes for Peanut Brittle on the page.

Ask students these or similar questions:

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  • What is the ratio of peanuts to sugar in Recipe No. 83? (Possible response: 1 : 3 or 1/3 )
  • Let's double Recipe No. 83. What strategies can you use to find out how many cups of roasted peanuts and how many cups of sugar are used in the doubled recipe? (See Figure 1 for possible responses.)
  • Let's double Recipe No. 84. What ratio shows how many cups of roasted peanuts and how many cups of sugar are used in the doubled recipe? (Possible response: 2 cups of peanuts to 4 cups of sugar; 2 : 4; 2/4)
  • What strategies did you use to find this answer? (Possible responses: I used what I know about equivalent fractions to multiply both the top and bottom number by 2; I used the equivalent fraction number line to show that 1/2 and 2/4 are equal ratios; I used circle pieces to show that 2/4 is equivalent to 1/2 by covering a pink piece [1/2] with two yellow pieces [2/4].)
  • Is the proportion of peanuts to sugar equivalent in Recipe No. 83 and Recipe No. 84? Will each peanut brittle taste as sweet as the other one? (Possible response: No, 1/3 and 1/2 are not equivalent fractions, so the proportion of peanuts to sugar is different in each recipe. Since there is more sugar in Recipe No. 84, it will taste sweeter.)

For Questions 1–2, students write ratios using two different types of notation. For example, the ratio of peanuts to sugar in a peanut brittle recipe is 1 cup peanuts to 2 cups sugar. Students write this ratio as 1 cup/2 cups and also as 1 : 2. They use the ratios to solve problems in Questions 3–4. Discuss students' strategies. Two possible strategies include using multiplication and using equivalent fractions and ratios. Display the Finding Equivalent Fractions and Ratios Menu in the Student Guide Reference section as you highlight various strategies.

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Colon Notation. Since ratios are sometimes written using colon notation, it is introduced here so students will be able to recognize ratios written this way. However, the fraction notation is used more often since it is usually easier to use when solving problems.

In solving these ratio problems, students need to keep track of which ratio they are writing:

cups of peanuts : cups of sugar, or

cups of sugar : cups of peanuts.

One way to do this is to use the units. In other words, write 1 cup peanuts/2 cups sugar. Another method is to use variables; for example, the letter P for cups of peanuts and S for cups of sugar. Then we write, P/S = 1 c/2 c. Remind students that this equation is just a shorthand way of saying “the ratio of cups of peanuts to cups of sugar is 1 cup to 2 cups.”

Introduce Proportion. Use the text before Questions 5–6 to introduce the term proportion as a statement of equality between ratios. The following questions encourage students to use what they have learned about equal ratios to solve problems involving the ratios in two peanut brittle recipes. For example, to solve Question 6, students refer to a recipe that calls for 1 cup peanuts and 2 cups sugar. They must find the number of cups of sugar to mix with 5 cups of peanuts. Students can use either of these proportions:

So the amount of sugar is 10 cups.

They can solve the proportions using the same strategies and procedures they have used to find equivalent fractions.

Student Guide: 518 519
Answers
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SG_Mini
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SG_Mini
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Student Activity Book:
Master:
Possible strategies to double 1/3
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