Multiplication and Division of Fractions

Lesson 11

Math

Unit 5

5th Grade

Lesson 11 of 25

Objective


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

Common Core Standards


Core Standards

  • 5.NF.B.4 — Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
  • 5.NF.B.5 — Interpret multiplication as scaling (resizing), by:
  • 5.NF.B.6 — Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

Foundational Standards

  • 3.MD.C.7
  • 4.NF.B.4

Criteria for Success


  1. Deduce the generalized method for multiplying a fraction times a fraction (MP.8).
  2. Use this generalized method to compute products of fractions and fractions. 
  3. Simplify computations before performing them and understand that this is more efficient than simplifying afterward. 
  4. 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. 
  5. Solve word problems that involve the multiplication of a fraction by a fraction (MP.4).

Tips for Teachers


  • In Lessons 8-10, 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 by a fraction. While “Grade 5 students need not express the formula in [a] general algebraic form, but rather recognize numerical instances from” their repeated reasoning, this lesson serves to consolidate that repeated reasoning into a few strategies that they can use to solve any problem involving multiplication of fractions.
  • As a supplement to the Problem Set, students can play the game outlined in Level C of the Problem of the Month “Got Your Number“ by Inside Mathematics (having students just use multiplication rather than using a spinner to determine the operation).
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Anchor Tasks


Problem 1

Below is a model of the fraction $${{{{4\over5}}}}$$

Explain how to find $${{6\over7}}$$ of $${{{{4\over5}}}}$$. Try not to make additional drawings on the model above. 

Guiding Questions

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References

John A. Van de Walle Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades 3-5 (Volume II)Page 246

Van de Walle, John A. Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades 3-5 (Volume II). Pearson, 2nd edition, 2013.

Problem 2

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

  1. $${{3\over5}\times{1\over2}}$$
  2. $${{1\over12}\times{8\over9}}$$
  3. $${{5\over9}\times{3\over5}}$$
  4. $${{15\over4}\times{8\over5}}$$

b.   Connor says that to multiply a fraction by a fraction, he’d multiply the numerators of the factors together to find the numerator of the product and multiply the denominators of the factors together to find the denominator of the product. Do you agree with Connor? Explain your reasoning.

Guiding Questions

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Problem 3

When solving $$\frac{1}{12} \times \frac89$$, Almira computed like so:

$$\frac{1}{12}\times\frac{8}{9}=\frac{1\times \overset{2}{\cancel{8}}}{\underset{3}{\cancel{12}}\times9} = \frac{2}{27}$$

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

a.   Explain what Almira did in your own words.

b.   Use Almira’s strategy to solve Parts (iii) and (iv) in Anchor Task 2. Determine 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 $${{2\over7}}$$. Then solve.

a.   $${3\over10}\times{{{2\over7}}}$$

b.   $${14\over10}\times{{{2\over7}}}$$

c.   $${10\over10}\times {{{2\over7}}}$$

Guiding Questions

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Problem Set


Answer Keys

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Discussion of Problem Set

  • Look at #1. Did anyone simplify any of these computations before multiplying? Which ones?
  • Was anyone able to solve #3 without needing to multiply each expression? How? 
  • Look at #4b. Before computing, would you have expected the product to be more or less than $$\frac{4}{5}$$? Why?
  • Look at #4c. Before computing, would you have expected the product to be more or less than 1? Why?
  • Look at #4d. I got 1 as my answer, which I think is wrong because I think the product of a fraction and a fraction should be a fraction, not a whole number. Do you agree or disagree? 
  • Look at #5. How did you find the number of tons of corn Farmer Dave took to market? I solved it by turning 18 into a fraction and using the strategy we learned today. Can I do that? Why or why not? 
  • Look at #6. How did you solve for the missing number? 
  • Look at #7. Is there more than one correct answer? 

Target Task


Solve. Show or explain your work.

a.   $${{2\over3} \times {5\over6}}$$

b.   $${{4\over9} \times { 3\over 8}}$$

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

Answer Keys

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Word Problems and Fluency Activities

Word Problems and Fluency Activities

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Lesson 10

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Lesson 12

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 Expressions and Real-World Problems

Topic G: Line Plots

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