Lesson 6 | Multiplication and Division of Fractions | 5th Grade Mathematics

Objective

Relate multiplication of a fraction by a whole number to multiplication of a whole number by a fraction and use this to develop a general method to multiply any fraction by any whole number (or vice versa).

Common Core Standards

Core Standards

The core standards covered in this lesson

5.NF.B.4.A — Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

Number and Operations—Fractions

5.NF.B.4.A — Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
5.NF.B.5 — Interpret multiplication as scaling (resizing), by:

Number and Operations—Fractions

5.NF.B.5 — Interpret multiplication as scaling (resizing), by:

Foundational Standards

The foundational standards covered in this lesson

4.NF.B.4

Number and Operations—Fractions

4.NF.B.4 — Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Understand that multiplication is commutative, even when one factor is a fraction (MP.7).
Use commutativity to show that $${{a\over b}\times q=q\times{a\over b}={a\over b}+{a\over b} \space+ ... +\space {a\over b} \space(q \space \mathrm{times})={q\times a\over b}}$$.
Use this generalized method to compute products of whole numbers and fractions or fractions and whole numbers.
Simplify computations before performing them and understand that this is more efficient than simplifying afterward.
Before computing a product of a whole number and a fraction, predict whether the product will be greater than, equal to, or less than the whole number (as an introduction to the idea of scaling).

Tips for Teachers

Suggestions for teachers to help them teach this lesson

In Lessons 4 and 5, students “reason[ed] repeatedly from many examples (MP.8), using [a set model], tape diagrams, and number line diagrams” (NF Progression, p. 18) to multiply a fraction times a whole number. Students will “recognize numerical instances from” their repeated reasoning, though they “need not express the formula in [a] general algebraic form” Thus, after establishing commutativity, students will consolidate that repeated reasoning into a few strategies that they can use to solve any problem involving multiplication of a fraction times a whole number (or vice versa).
As a supplement to the Problem Set, students can play “Parts of a Whole," from the North Carolina Department of Public Instruction. Note that the directions prompt students to think of the multiplication they perform as “3 groups with 1/4” and to use fraction circles to solve, which you may choose not to have students do.

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Anchor Tasks

Tasks designed to teach criteria for success of the lesson, and guidance to help draw out student understanding

Problem 1

Makayla said, “I can represent $${3\times {{{{{2\over3}}}}}}$$ with $$3$$ rectangles each of length $${{{{{2\over3}}}}}$$.”

Connor said, “I know that $${{{{{2\over3}}}}}\times3$$ can be thought of as $${{{{2\over3}}}}$$ of $$3$$. Is $$3$$ copies of $${{{{2\over3}}}}$$ the same as $${{{{2\over3}}}}$$ of $$3$$?"

a. Draw a diagram to represent $${{{{2\over3}}}}$$ of $$3$$.

b. Explain why your picture and Makayla’s picture together show that $$3\times {2\over 3} = {{{{{2\over3}}}}} \times 3$$.

c. What property of multiplication do these pictures illustrate?

Guiding Questions

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References

Illustrative Mathematics Connor and Makayla Discuss Multiplication

Connor and Makayla Discuss Multiplication, accessed on April 24, 2018, 12:44 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.

Modified by Fishtank Learning, Inc.

Problem 2

a. Solve. Draw a model if it is helpful.

$${8\times {1\over2}}$$
$${{3\over4} \times 18}$$
$${10\times{3\over5}}$$
$${2\over18} \times 24$$

b. Soheil says that to multiply a whole number by a fraction or a fraction by a whole number, he’d multiply the whole number and the numerator and use the same denominator. Do you agree with Soheil? Explain your reasoning.

Guiding Questions

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Problem 3

When solving $$10 \times \frac35$$, Guy computed like so:

$$10 \times \frac35 = \frac{10\times3}{5} = \frac{\overset{2}{\cancel{10}}\times3}{\underset{1}{\cancel{5}}} = \frac{2\times3}{1} = 6$$

He said, “I simplified my fraction before multiplying which made it easier!"

a. Explain why Guy's strategy works.

b. Use Guy's strategy to find the value of the other expressions in Part (a) in Anchor Task 2 above. Decide whether your answers are equivalent.

Guiding Questions

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Problem 4

Before solving, predict whether each product will be less than, equal to, or greater than $${12}$$. Then solve.

a. $${5\over8} \times {12}$$

b. $${9\over8} \times {12}$$

c. $${8\over8} \times {12}$$

Guiding Questions

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Problem Set

Problem Set

Answer Keys

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Discussion of Problem Set

Look at #1 and #2. How do the two methods compare with diagrams? How do the two methods compare with computations?
Look at #3. Which expressions are equivalent to $$4\times \frac{2}{3}$$? What does that tell you about the various ways you can compute the product of a whole number and a fraction?
Was anyone able to solve #4 without needing to multiply? How?
Look at #5. Why was the product in #5b less than 35? Why was the product in #5c greater than 35?
Look at #7. We now know how to multiply a whole number and a fraction. What are some division problems for which we can use our multiplication skills to check our work? (Note that if a quotient is rewritten as a mixed number, they should still be able to solve based on what they learned in Grade 4, but we haven’t explored that concept yet in Grade 5.)
What is the most efficient way to solve a problem that involves multiplication of a fraction and a whole number? Does it depend on the numbers involved?

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Solve. Show or explain your work.

a. $${{3\over5}\times20}$$

b. $${10\times{5\over4}}$$

Student Response

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Additional Practice

The Extra Practice Problems can be used as additional practice for homework, during an intervention block, etc. Daily Word Problems and Fluency Activities are aligned to the content of the unit but not necessarily to the lesson objective, therefore feel free to use them anytime during your school day.

Extra Practice Problems

Extra Practice Problems

Answer Keys

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Word Problems and Fluency Activities

Help students strengthen their application and fluency skills with daily word problem practice and content-aligned fluency activities.

Lesson 5

Lesson 7

Lesson Map

Topic A: Fractions as Division

Relate equal shares of objects to division expressions and visual representations of fractions.

5.NF.B.3

Write division expressions that represent fractions and vice versa.

5.NF.B.3

Solve division problems when the quotient is a fraction or mixed number, including cases with larger values.

5.NF.B.3

Topic B: Multiplying a Fraction by a Whole Number

Multiply a fraction by a whole number where the product is a whole number.

5.NF.B.4.A 5.NF.B.6

Multiply a fraction by a whole number where the product is a whole number or a fraction.

5.NF.B.4.A 5.NF.B.6

5.NF.B.4.A 5.NF.B.5

Solve real-world problems involving multiplication of fractions and whole numbers and create real-world contexts for expressions involving multiplication of fractions and whole numbers.

5.NF.B.4 5.NF.B.6 5.OA.A.2

Topic C: Multiplying a Fraction by a Fraction

Multiply a fraction by a fraction without subdivisions using tape diagrams and number lines.

5.NF.B.4 5.NF.B.5 5.NF.B.6

Multiply a fraction by a fraction with subdivisions using tape diagrams and number lines.

5.NF.B.4 5.NF.B.5 5.NF.B.6

Multiply a fraction by a fraction with more complicated subdivisions using an area model.

5.NF.B.4 5.NF.B.5 5.NF.B.6

Develop a general method to multiply a fraction by a fraction.

5.NF.B.4 5.NF.B.5 5.NF.B.6

Solve real-world problems involving multiplication of fractions with fractions and create real-world contexts for expressions involving multiplication of fractions with fractions.

5.NF.B.4 5.NF.B.6 5.OA.A.2

Topic D: Multiplying with Mixed Numbers

Multiply mixed numbers by whole numbers.

5.NF.B.4 5.NF.B.5 5.NF.B.6

Multiply mixed numbers by fractions.

5.NF.B.4 5.NF.B.5 5.NF.B.6

Multiply mixed numbers by mixed numbers.

5.NF.B.4 5.NF.B.5 5.NF.B.6

Develop a general method to multiply with mixed numbers.

5.NF.B.4 5.NF.B.5 5.NF.B.6

Solve real-world problems involving multiplication with mixed numbers and create real-world contexts for expressions involving multiplication with mixed numbers.

5.NF.B.4 5.NF.B.6 5.OA.A.2

Interpret multiplication as scaling.

5.NF.B.5 5.NF.B.5.A 5.NF.B.5.B

Topic E: Dividing with Fractions

Divide a unit fraction by a whole number.

5.NF.B.7.A 5.NF.B.7.C

Divide a whole number by a unit fraction.

5.NF.B.7.B 5.NF.B.7.C

Solve real-world problems involving division with fractions and create real-world contexts for expressions involving division with fractions.

5.NF.B.7.C 5.OA.A.2

Topic F: Fraction Expressions and Real-World Problems

Solve real-world problems involving multiplication and division with fractions.

5.NF.B.3 5.NF.B.6 5.NF.B.7

Write and evaluate numerical expressions involving operations with fractions.

5.NF.B.3 5.NF.B.4 5.NF.B.5 5.NF.B.6 5.NF.B.7 5.OA.A.1 5.OA.A.2

Topic G: Line Plots

Create line plots.

5.MD.B.2

Solve problems involving information presented in a line plot (dot plot).

5.MD.B.2

Multiplication and Division of Fractions

Objective

Common Core Standards

Core Standards

Number and Operations—Fractions

Number and Operations—Fractions

Foundational Standards

Number and Operations—Fractions

Criteria for Success

Tips for Teachers

Anchor Tasks

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Problem 4

Guiding Questions

Problem Set

Discussion of Problem Set

Target Task

Student Response

Additional Practice

Extra Practice Problems

Word Problems and Fluency Activities

Lesson Map

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