Multiplication and Division of Fractions

Lesson 16

Math

Unit 5

5th Grade

Lesson 16 of 25

Objective


Develop a general method to multiply with mixed numbers.

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
  • 5.NF.A.1

Criteria for Success


  1. Deduce a generalized method for multiplying a mixed number times a mixed number by computing all of the partial products and adding them together using either an area model or the distributive property (MP.8). 
  2. Use the generalized method of computing products of fractions to deduce a generalized method for multiplying a mixed number times a mixed number by converting them to fractions greater than 1 and multiplying the numerators and denominators together. 
  3. Decide which generalized method for computing products of mixed numbers will be most efficient for a particular problem and use it to compute the product (MP.5). 
  4. Estimate the product based on the size of its factors and use that estimate to assess the reasonableness of the answer (MP.1).  
  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. 
  6. Solve word problems that involve the multiplication of a mixed number by a mixed number (MP.4).

Tips for Teachers


In Lessons 13-15, 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 with mixed numbers. 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 with mixed numbers.

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Anchor Tasks


Problem 1

Sophia and Zack are multiplying $${2{2\over3}}$$ and $${3{1\over4}}$$. Their work is shown below. 

Sophia's Work: Zack's Work: 
$$2\frac23 \times 3\frac14 = (2\times3) +\left( 2\times \frac14 \right) + \left ( \frac23 \times 3 \right) + \left ( \frac23 \times \frac14 \right)$$ $$2\frac23 \times 3\frac14 = \frac83 \times \frac{13}{4}$$

a.   Do both of these strategies result in the same product?

b.   Which method do you prefer? Why? 

Guiding Questions

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Problem 2

Solve each of the following problems using two strategies. Then determine which strategy is more efficient for each problem.

a.   $$1\frac{1}{3} \times 3\frac{3}{4}$$

b.   $$16\frac{1}{2} \times 4\frac{1}{4}$$

Guiding Questions

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References

EngageNY Mathematics Grade 5 Mathematics > Module 5 > Topic C > Lesson 13Concept Development

Grade 5 Mathematics > Module 5 > Topic C > Lesson 13 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 3

Estimate the following products. Then, solve using whichever strategy you think is most efficient.

a.   $$\frac{2}{3} \times 6\frac{1}{2}$$

b.   $$2\frac{2}{3} \times 3\frac{3}{4}$$

c.   $$2\frac{4}{5} \times 4$$

d.   $$5\frac{5}{8} \times 6\frac{1}{4}$$

Guiding Questions

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Problem Set


Answer Keys

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Discussion of Problem Set

  • Was anyone able to solve #3 without needing to multiply each expression? How? 
  • Look at #4. Which strategy did you decide to use to solve for each part? Why? Do you think you’d always use the same strategy for all problems involving multiplication of a fraction by a mixed number, like #4a? What about problems involving multiplication of a whole number by a mixed number, like #4c? 
  • Look at #5. How did you solve? Was there more than one way to find the area of the rectangle?
  • Look at #6. For the multiplication part of your solution, which strategy did you use to solve? Why? 

Target Task


Problem 1

Solve each problem in two different ways.

a.   $${3{1\over3} \times 1{1\over4}}$$

b.   $${5{2\over5} \times 3{4\over9}}$$

Student Response

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Problem 2

Which strategy was more efficient for each problem above? Explain.

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 15

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Lesson 17

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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