Multiplication and Division of Fractions

Lesson 6

Math

Unit 5

5th Grade

Lesson 6 of 24

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

  • 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:

Foundational Standards

  • 4.NF.B.4

Criteria for Success


  1. Understand that multiplication is commutative, even when one factor is a fraction (MP.7).
  2. Use commutativity to show that $${{a\over b}\times q=q\times{a\over b}={a\over b}+{a\over b} \space+ ... +\space {a\over b} \space(q \space \mathrm{times})={q\times a\over b}}$$.
  3. Use this generalized method to compute products of whole numbers and fractions or fractions and whole numbers. 
  4. Simplify computations before performing them and understand that this is more efficient than simplifying afterward. 
  5. 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


  • In Lessons 4 and 5, students “reason[ed] repeatedly from many examples (MP.8), using strips of paper, 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

25-30 minutes


Problem 1

For each expression, explain how the model below represents the expression.

a.   $$\frac{2}{3}\times3$$

b.   $$3\times \frac{2}{3}$$

Guiding Questions

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Student Response

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Problem 2

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

  1. $${8\times {1\over2}}$$
  2. $${{3\over4} \times 18}$$
  3. $${10\times{3\over5}}$$
  4. $${2\over18} \times 24$$

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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Student Response

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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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Student Response

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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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Student Response

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Problem Set

15-20 minutes


Discussion of Problem Set

  • Look at #2. 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 #6. 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

5-10 minutes


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.

Word Problems and Fluency Activities

Word Problems and Fluency Activities

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

Next

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.

Lesson 7
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Fractions as Division

Topic B: Multiplying a Fraction by a Whole Number

Topic C: Multiplying a Fraction by a Fraction

Topic D: Multiplying with Mixed Numbers

Topic E: Dividing with Fractions

Topic F: Fraction Real-World Problems and Line Plots

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