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?
- 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.
- Complete the tape diagram to represent the question: “How many groups of 2 are in 7?” Then answer the question.
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.
- 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.
- 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?
Lesson 6.3 Finding Number of Groups
- 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?
- Write a question that can be represented by the division equation 5 ÷ 1½ = ?. Then answer the question. Show your reasoning.
Lesson 6 Practice Problems
- We can think of 3 ÷ ¼ as the answer to the question “How many groups of ¼l are in 3?” Draw a tape diagram to represent the question. Then answer the question.
- Describe how to draw a tape diagram to represent and answer 3 ÷ ⅗ = ? for a friend who was absent.
- 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.
- 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.
- 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 = ?
- 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?
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