Illustrative Mathematics Unit 6.4, Lesson 7: Using Diagrams to Find the Number of Groups

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Illustrative Math
Grade 6

Lesson 7: What Fraction of a Group?

Let’s think about dividing things into groups when we can’t even make one whole group.

Illustrative Math Unit 6.4, Lesson 7 (printable worksheets)

Lesson 7 Summary

The following diagrams show how to use diagrams and multiplication and division equations to represent and answer “what fraction of a group?” questions.

Lesson 7.1 Estimating a Fraction of a Number

1. Estimate the following quantities:
   a. What is ⅓ of 7?
   b. What is ⅘ of 9⅔?
   c. What is 2 4/7 of 10 1/9?
   
2. Write a multiplication expression for each question.
   

• See Video for Whole Lesson

Lesson 7.2 Fractions of Ropes

The segments in the applet represent 4 different lengths of rope. Compare one rope to another, moving the rope by dragging the open circle at one endpoint. You can use the yellow pins to mark off lengths.
Show Applet

1. Compare the lengths of rope B, C, and D to the length of rope A and complete each statement. Then, use the measurements shown on the grid to write a multiplication equation and a division equation for each statement.
   a. Rope B is _______ times as long as rope A.
   Multiplication equation: _________________
   Division equation: _________________
   b. Rope C is _______ times as long as rope A.
   Multiplication equation: _________________
   Division equation: _________________
   c. Rope D is _______ times as long as rope A.
   Multiplication equation: _________________
   Division equation: _________________
   
2. Each equation can be used to answer a question about rope C and D. What could each question be?
   a. ? · 3 = 9 and 9 ÷ 3 = ?
   b. ? · 9 = 3 and 3 ÷ 9 = ?
   

Lesson 7.3 Fractional Batches of Ice Cream

One batch of an ice cream recipe uses 9 cups of milk. A chef makes different amounts of ice cream on different days. Here are the amounts of milk she used:

• Monday: 12 cups
  
• Tuesday: 22½ cups
  
• Thursday: 6 cups
  
• Friday: 7½ cups
  

1. How many batches of ice cream did she make on each of the following days? Write a division equation and draw a tape diagram for the question about each day. Then answer the question.
   a. Monday
   b. Tuesday
   
2. What fraction of a batch of ice cream did she make on each of the following days? Write a division equation and draw a tape diagram for the question about each day. Then answer the question.
   a. Thursday
   b. Friday
   
3. Write a division equation, and draw a tape diagram for each question. Then answer the question.
   a. What fraction of 9 is 3?
   b. What fraction of 5 is ½?
   

Lesson 7 Practice Problems

1. A recipe calls for ½ lb of flour for 1 batch. How many batches can be made with each of the following amounts?
   a. 1 lb
   b. ¾ lb
   c. ¼ lb
   
2. Whiskers the cat weighs 2⅔ kg. Piglio weighs 4 kg. For each question, write a multiplication and a division equation, decide whether the answer is greater or less than 1, and then answer the question.
   a. How many times as heavy as Piglio is Whiskers?
   b. How many times as heavy as Whiskers is Piglio?
   
3. Andre is walking from home to a festival that is 1⅝ kilometers away. He takes a quick rest after walking ⅓ kilometers. In this situation, which question can be represented by the equation: ? · 1⅝ = ⅓?
   A. What fraction of the trip has Andre completed?
   B. How many more kilometers does he have to walk to get to the festival?
   C. What fraction of the trip is left?
   D. How many kilometers is it from home to the festival and back home?
   
4. Draw a tape diagram to represent and answer the question: What fraction of 2½ is ⅘?
   
5. How many groups of ¾ are in each of the following quantities?
   a. 11/4
   b. 6½
   
6. Which question can be represented by the equation 4 &divide &frac27; = ?
   A. What is 4 groups of 2/7?
   B. How many 1/7s are in 4?
   C. What is 2/7 of 4?
   D. How many 4s are in 2/7?
   

The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.

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