Fraction Operations

Lesson 17

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.

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Anchor Tasks

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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.

Guiding Questions

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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?

Guiding Questions

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Problem 3

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

Guiding Questions

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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 $${{3\over4}+{3\over4}+{3\over4}}$$?

Target Task

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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.

Mastery Response

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Additional Practice

Unit Practice

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