Multiplication and Division of Fractions

Lesson 10

Math

Unit 5

5th Grade

Lesson 10 of 25

Objective


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

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. Multiply a fraction by a fraction in which further subdivision is needed (in more complex cases) using an area model. 
  2. Demonstrate that multiplication is commutative, even when dealing with two fractions as factors (MP.7). 
  3. Solve word problems that involve the multiplication of a fraction by a fraction in which further subdivision is needed (in more complex cases), including those involving area (MP.4). 
  4. Estimate the size of a product of two fractions in comparison to the size of its factors.

Tips for Teachers


  • As the Progressions state, “For more complicated examples, an area model is useful, in which students work with a rectangle that has fractional side lengths, dividing it up into rectangles whose sides are the corresponding unit fractions” (Progressions for the Common Core State Standards in Mathematics (Number and Operations - Fractions, 3–5), p. 13). Thus, for more complicated examples in today’s lesson, students will trade the tape diagrams and number lines they’ve been working with for much of the unit thus far for an area model in this lesson and subsequent ones. 
  • When shading an area model to represent fraction multiplication, there are two ways of doing it, but one of them is somewhat problematic. For example, $$\frac{1}{3}\times \frac{1}{4}$$ can be represented in the following ways:
Preferable representation Problematic representation

The area model on the right is problematic because “the red shading extended horizontally over the entire [whole, which] implies that the whole for [one third is the same as the whole for one-fourth]-the entire whole” (emphasis mine) (Webel, Krupa, and McManus, “Using Representations of Fraction Multiplication,” NCTM Journal Teaching Children Mathematics). When students are computing $$\frac{1}{3}\times \frac{1}{4}$$, they are finding one third of one fourth, and thus the unit for one third is one fourth, so shading should correspond to this distinction. 

Lesson Materials

  • Rectangular piece of paper (3 per student) — These should measure about 4.25" by 3.75"
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Anchor Tasks


Problem 1

a.   Jan has $${{1\over3}}$$ of a pan of brownies. She sends $${{1\over4}}$$ of the brownies to school with her children. What fraction of a pan of brownies does Jan send to school? 

b.   Jan has $${{7\over8}}$$ of a pan of brownies. She sends $${{1\over2}}$$ of the brownies to school with her children. What fraction of a pan of brownies does Jan send to school?

c.   Jan has $${{3\over4}}$$ of a pan of brownies. She sends $${{5\over8}}$$ of the brownies to school with her children. What fraction of a pan of brownies does Jan send to school?

Guiding Questions

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References

EngageNY Mathematics Grade 5 Mathematics > Module 4 > Topic E > Lesson 15Concept Development

Grade 5 Mathematics > Module 4 > Topic E > Lesson 15 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.
EngageNY Mathematics Grade 5 Mathematics > Module 4 > Topic E > Lesson 14Concept Development

Grade 5 Mathematics > Module 4 > Topic E > Lesson 14 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.
EngageNY Mathematics Grade 5 Mathematics > Module 4 > Topic E > Lesson 13Concept Development

Grade 5 Mathematics > Module 4 > Topic E > 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 2

a.   There are two design proposals for a new rectangular park in town.

  • In Design 1, $${{{{{{{{{{3\over4}}}}}}}}}}$$ of the area of the park is going to be a rectangular grass area and $${{{{{{{{1\over2}}}}}}}}$$ of the grass area will be a rectangular soccer field.
  • In Design 2, only $${{{{{{{{1\over2}}}}}}}}$$ of the park is going to be a rectangular grass area and $${{{{{{{{{{3\over4}}}}}}}}}}$$ of the grass area will be a rectangular soccer field.

Which design (1 or 2) will have a bigger soccer field? Explain your answer. Draw a diagram that can be used to compare the size of the soccer field in the two designs. Label the values $${{{{{{{{1\over2}}}}}}}}$$ and $${{{{{{{{{{3\over4}}}}}}}}}}$$ on the diagram.

b.   Presley and Julia are cutting $$1$$ ft. square poster board to make a sign for the new park. Presley cut her poster so that the length of the top and bottom are $${{{{{{{{1\over2}}}}}}}}$$ ft. and the length of the sides are $${{{{{{{{{{3\over4}}}}}}}}}}$$ ft. Julia cut her poster so that the lengths of the top and bottom are $${{{{{{{{{{3\over4}}}}}}}}}}$$ ft. and the length of the sides are $${{{{{{{{1\over2}}}}}}}}$$ ft.

Draw a diagram of each poster board. Label the values on the diagram.

How are their poster boards similar and different? Justify your reasoning.

Guiding Questions

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References

Illustrative Mathematics New Park

New Park, accessed on April 24, 2018, 1:51 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 3

Solve. Show or explain your work.

a.   $${{1\over6} }$$ of $${{1\over4}}$$

b.   $${{{1\over4}}} \times {5\over6}$$

c.   $${{3\over4}\times{5\over8}}$$

d.   $${{2\over5} \times {7\over8}}$$

Guiding Questions

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Problem Set


Answer Keys

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Discussion of Problem Set

  • Look at #2c and #2d. Before even computing the products, is there a way to know which product will be larger than the other?
  • Look at #3. How did you find the area of Farida’s counter? 
  • Look at #6. Is there another way you could have drawn two diagrams?

Target Task


Problem 1

Solve. Show or explain your work.

a.   $${{2\over3} }$$ of $${{4\over5}}$$

b.   $${{4\over9}}\times \frac{5}{8}$$

Student Response

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Problem 2

A farmer’s land measures $${{7\over8}}$$ of a mile long and $${{4\over5}}$$ of a mile wide. What is the area, in square miles, of the farmer’s land?

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

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

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Lesson 9

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Lesson 11

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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