# Fraction Operations

## Objective

Multiply a whole number by a non-unit fraction.

## Common Core Standards

### Core Standards

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• 4.NF.B.4.B — Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

## Criteria for Success

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1. Extend multiplication of whole numbers to the case where one number is not a whole number (MP.7).
2. Multiply a fraction by a whole number using a visual model, unit form, or repeated addition.
3. Generate a general method for multiplying a whole number by a non-unit fraction, i.e., ${n\times{{a\over b}}={{n\times a}\over b}}$ (MP.8).
4. Understand a multiple of ${a\over b}$ as a multiple of ${1\over b}$.

## Tips for Teachers

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#### Remote Learning Guidance

If you need to adapt or shorten this lesson for remote learning, we suggest prioritizing Anchor Task 2 (benefits from worked example). Find more guidance on adapting our math curriculum for remote learning here.

#### Fishtank Plus

• Problem Set
• Student Handout Editor
• Vocabulary Package

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### Problem 1

1. Write a story problem that can be solved by finding ${5\times 4}$.
2. Draw two different diagrams that show that ${{{5\times4=20}}}$. Explain how your diagrams represent ${{{5\times4=20}}}$.
3. Which of the diagrams you used to represent ${{{5\times4=20}}}$ can be used to represent ${5\times {2\over3}}$? Draw the diagram if possible.

#### References

Illustrative Mathematics Extending Multiplication from Whole Numbers to Fractions

Extending Multiplication from Whole Numbers to Fractions, accessed on July 18, 2018, 12:48 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

### Problem 2

1. Solve.

a.  ${4\times {3\over5}}$

b.  ${2\times{6\over7}}$

c.  ${7\times{5\over8}}$

1. What do you notice about #1? What do you wonder?

### Problem 3

Show or explain how you know ${3\times {5\over6}=15\times{1\over6}}$.

## Problem Set & Homework

#### Discussion of Problem Set

• What did Angie do to solve? Will her strategy work all the time?
• How did you record your solutions to #6?
• For which problems in #6 would repeated addition be a particularly inefficient strategy? Why?
• Look at your answers for #6(c) and #6(d). Convert each answer to a mixed number. What do you notice? How are the expressions in #6(c) and #6(d) similar?
• How did Goldie solve in #7? How is this similar to showing ${{3\over4}+{3\over4}+{3\over4}}$?

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### Problem 1

Which of the following is equivalent to ${7\times {3\over8}}$? There are three correct answers.

A.   ${{7+3}\over 8}$

B.   ${{7 \times 3} \over 8}$

C.   ${21\times{1\over8}}$

D.   ${8\times {3\over7}}$

E.   ${3\times {7\over8}}$

F.   ${10\times {1\over8}}$

### Problem 2

Explain how to find ${3 \times {4\over5}}$ using a number line. Find the product.

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