# Illustrative Mathematics Unit 6.4, Lesson 6: Using Diagrams to Find the Number of Groups

Learning Targets:

• I can use a tape diagram to represent equal-sized groups and find the number of groups.

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Illustrative Math

### Lesson 6: Using Diagrams to Find the Number of Groups

Let’s draw tape diagrams to think about division with fractions.

Illustrative Math Unit 6.4, Lesson 6 (printable worksheets)

### Lesson 6 Summary

The following diagrams show how to use a tape diagram to represent equal-sized groups and find the number of groups.  ### Lesson 6.1 How Many of These in That?

1. We can think of the division expression 10 ÷ 2½ as the answer to the question: “How many groups of 2½ are in 10?” Complete the tape diagram to represent the question. Then answer the question.
• Show Tape Diagram 2. Complete the tape diagram to represent the question: “How many groups of 2 are in 7?” Then answer the question.
• Show Tape Diagram ### Lesson 6.2 Representing Groups of Fractions with Tape Diagrams

To make sense of the question “How many ⅔s are in 1?,” Andre wrote equations and drew a tape diagram.

• Show Image 1. In an earlier task, we used pattern blocks to help us solve the equation 1 ÷ ⅔ = ?. Explain how Andre’s tape diagram can also help us solve the equation.
2. Write a multiplication equation and a division equation for each of the following questions. Draw a tape diagram to find the solution. Use the grid to help you draw, if needed.
a. How many ¾s are in 1?
b. How many ⅔s are in 3?
c. How many 3/2s are in 5?
• Show Grid #### Lesson 6.3 Finding Number of Groups

1. For each question, draw a diagram to show the relationship of the quantities and to help you answer the question. Then, write a multiplication equation or a division equation for the situation described in the question. Be prepared to share your reasoning.
a. How many ⅜-inch thick books make a stack that is 6 inches tall?
b. How many groups of ½ pound are in 2¾ pounds?
2. Write a question that can be represented by the division equation 5 ÷ 1½ = ?. Then answer the question. Show your reasoning.

#### Lesson 6 Practice Problems

1. We can think of 3 ÷ ¼ as the answer to the question “How many groups of &frac14l are in 3?” Draw a tape diagram to represent the question. Then answer the question.
2. Describe how to draw a tape diagram to represent and answer 3 ÷ ⅗ = ? for a friend who was absent.
3. How many groups of ½ days are in 1 week?
a. Write a multiplication equation or a division equation to represent the question.
b. Draw a tape diagram to show the relationship between the quantities and to answer the question. Use graph paper, if needed.
4. Diego said that the answer to the question “How many groups of ⅚ are in 1?” is ⅚ or 1⅕. Do you agree with his statement? Explain or show your reasoning.
5. Select all equations that can represent the question: “How many groups of ⅘ are in 1?”
A. ? · 1 = ⅘
B. 1 · ⅘ = ?
C. ⅘ ÷ 1 = ?
D. ? · ⅘ = 1
E. 1 ÷ ⅘ = ?
6. Calculate each percentage mentally.
a. What is 10% of 70?
b. What is 10% of 110?
c. What is 25% of 160?
d. What is 25% of 48?
e. What is 50% of 90?
f. What is 50% of 350?
g. What is 75% of 300?
h. What is 75% of 48?

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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