Unit 4

Dividing by Unit and Non-Unit Fractions

Lesson 10

Dividing Fractions

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Expressions and Equations

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Let’s look for patterns when we divide by a fraction.

Unit 4 ● Lesson 10

Learning

Goal

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Dividing by a Whole Number

Unit 4 ● Lesson 10 ● Activity 1

Work with a partner. One person solves the problems labeled “Partner A” and the other person solves those labeled “Partner B.” Write an equation for each question. If you get stuck, consider drawing a diagram.

• Partner A:

How many 3s are in 12? Division equation:

How many 4s are in 12? Division equation:

How many 6s are in 12? Division equation:

Partner B:

What is 12 groups of ⅓ ? Multiplication equation:

What is 12 groups of ¼ ? Multiplication equation:

What is 12 groups of ⅙ ? Multiplication equation:

• What do you notice about the diagrams and equations? Discuss with your partner.
• Complete this sentence based on what you noticed: Dividing by a whole number produces the same result as multiplying by ________.

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Warm-up

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Dividing by a Whole Number

Unit 4 ● Lesson 10 ● Activity 1

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Warm-up

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Dividing by Unit Fractions

Unit 4 ● Lesson 10 ● Activity 2

To find the value of , Elena thought, “How many s are in 6?” and then she drew this tape diagram. It shows 6 ones, with each one partitioned into 2 equal pieces.

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• For each division expression, complete the diagram using the same method as Elena. Then, find the value of the expression.

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Value of the expression: ________________

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Dividing by Unit Fractions

Unit 4 ● Lesson 10 ● Activity 2

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Value of the expression: ________________

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c.

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Value of the expression: ___________________

• Examine the expressions and answers more closely. Look for a pattern. How could you find how many halves, thirds, fourths, or sixths were in 6 without counting all of them? Explain your reasoning.

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Dividing by Unit Fractions

Unit 4 ● Lesson 10 ● Activity 2

• Use the pattern you noticed to find the values of these expressions. If you get stuck, consider drawing a diagram.

• Find the value of each expression.

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Dividing by Unit Fractions

Unit 4 ● Lesson 10 ● Activity 2

• How is the division by a unit fraction depicted in the diagrams?
• Where in the diagrams do we see the multiplication?
• How are the two—the division by a unit fraction and the multiplication—related?
• How do we find the value of or using a diagram?
• Would you use diagrams to find these quotients? Why or why not?
• When working on the task, did you stop partitioning the tape diagrams at some point? If so, why?
• Why do we use diagrams? When can they be helpful?

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Dividing by Non-unit Fractions

Unit 4 ● Lesson 10 ● Activity 3

• To find the value of , Elena started by drawing a diagram the same way she did for .

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• Complete the diagram to show how many s are in 6.
• Elena says, “To find , I can just take the value of and then either multiply it by or divide it by 2.” Do you agree with her? Explain your reasoning.

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Dividing by Non-unit Fractions

Unit 4 ● Lesson 10 ● Activity 3

• For each division expression, complete the diagram using the same method as Elena. Then, find the value of the expression. Think about how you could find that value without counting all the pieces in your diagram.

Value of the expression: ________________

Value of the expression: ________________

Value of the expression: ________________

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Dividing by Non-unit Fractions

Unit 4 ● Lesson 10 ● Activity 3

• Elena examined her diagrams and noticed that she always took the same two steps to show division by a fraction on a tape diagram. She said:

“My first step was to divide each 1 whole into as many parts as the number in the denominator. So if the expression is , I would break each 1 whole into 4 parts. Now I have 4 times as many parts.

My second step was to put a certain number of those parts into one group, and that number is the numerator of the divisor. So if the fraction is , I would put 3 of the s into one group. Then I could tell how many s are in 6.”

Which expression represents how many s Elena would have after these two steps? Be prepared to explain your reasoning.

6 ÷ 4 • 3 6 • 4 ÷ 3

6 ÷ 4 ÷ 3 6 • 4 • 3

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Dividing by Non-unit Fractions

Unit 4 ● Lesson 10 ● Activity 3

• Use the pattern Elena noticed to find the values of these expressions. If you get stuck, consider drawing a diagram.

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Dividing by Non-unit Fractions

Unit 4 ● Lesson 10 ● Activity 3

• What expression is represented by this tape diagram?
• Does Elena’s pattern work to find the value of the expression?

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Dividing by Unit and Non-Unit Fractions

Unit 4 ● Lesson 10

• What did we notice about the result of dividing a number by a unit fraction? Can you explain with an example?
• What observations did we make when dividing a number by a non-unit fraction? Can you explain with an example?
• Suppose we are finding . How might these observations help us find this quotient?

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Lesson Synthesis

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Unit 4 ● Lesson 10

• I can divide a number by a non-unit fraction by reasoning with the numerator and denominator, which are whole numbers.
• I can divide a number by a unit fraction by reasoning with the denominator, which is a whole number.

Learning Targets

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Dividing by ⅓ and ⅕

Unit 4 ● Lesson 10 ● Activity 4

• Explain or show how you could find . You can use this diagram if it is helpful.

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• Find . Try not to use a diagram, if possible. Show your reasoning.

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Cool-down

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

This slide deck is copyright 2020 by Kendall Hunt Publishing, https://im.kendallhunt.com/, and is licensed under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0), https://creativecommons.org/licenses/by-nc/4.0/.

All curriculum excerpts are under the following licenses:

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.

Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.

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