Multiplication and Division of Fractions

Lesson 23

Math

Unit 5

5th Grade

Lesson 23 of 25

Objective


Write and evaluate numerical expressions involving operations with fractions.

Common Core Standards


Core Standards

  • 5.NF.B.3 — Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
  • 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.
  • 5.NF.B.7 — Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.
  • 5.OA.A.1 — Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
  • 5.OA.A.2 — Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation "add 8 and 7, then multiply by 2" as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

Foundational Standards

  • 5.NF.A.1
  • 5.NF.A.2

Criteria for Success


  1. Write numerical expressions based on verbal/written descriptions of calculations involving fractions (e.g., write "$${{{{1\over2}}} \times (8+7)}$$" to express the calculation “half of the sum of $$8$$ and $$7$$”) (MP.7).
  2. Write descriptions of calculations involving fractions based on numerical expressions (e.g., write “add $$8$$ and $$7$$, then multiply by $${{1\over2}}$$” to describe the expression "$${{{{1\over2}}} \times (8+7)}$$") (MP.7).
  3. Evaluate expressions with and without grouping symbols that include fractions using the order of operations (MP.6).
  4. Interpret expressions that include fractions without evaluating them.

Tips for Teachers


  • The expressions chosen for the Anchor Tasks below were intentionally designed to address all cases of fraction multiplication and division. You might decide to swap some of the expressions for others so that students have some additional practice on any computations they have not yet shown success with.
  • This lesson connects the work of adding and subtracting fractions (5.NF.A) as well as multiplying and dividing fractions (5.NF.B) with writing and interpreting numerical expressions (5.OA.A), connecting content across clusters and domains in order to address the note in the Progressions “that the numbers in expressions need not always be whole numbers” (OA Progression, p. 32).
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Anchor Tasks


Problem 1

Write an expression to represent the tape diagrams below. In each tape diagram, their units are equal. Then solve for the value represented by "?".

a.   

b.   

c.   

Guiding Questions

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References

EngageNY Mathematics Grade 5 Mathematics > Module 4 > Topic D > Lesson 10Concept Development

Grade 5 Mathematics > Module 4 > Topic D > Lesson 10 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 G > Lesson 28Concept Development

Grade 5 Mathematics > Module 4 > Topic G > Lesson 28 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

Write numerical expressions that record the following calculations. Then evaluate them. 

a.   Twice the sum of $${{{{3\over5}}}}$$ and $${{1{{{1\over2}}}}}$$

b.   Half the sum of $${{{{3\over5}}}}$$ and $${{1{{{1\over2}}}}}$$

c.   $${{1\over2}}$$ subtracted from $${{3\over4}}$$ and then divided by $$3$$

Guiding Questions

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References

EngageNY Mathematics Grade 5 Mathematics > Module 4 > Topic H > Lesson 32Concept Development

Grade 5 Mathematics > Module 4 > Topic H > Lesson 32 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

Write the following numerical expressions in words. Then evaluate them.

a.   $$\frac{3}{5} \times \left(\frac{1}{2} - \frac{2}{5} \right)$$

b.   $$\frac{1}{3} \div \left(\frac{3}{4} + 1\frac{1}{4} \right)$$

c.   $$\left(5 \div \frac{1}{2} \right) \times \frac{3}{10} $$

Guiding Questions

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References

EngageNY Mathematics Grade 5 Mathematics > Module 4 > Topic H > Lesson 32Concept Development

Grade 5 Mathematics > Module 4 > Topic H > Lesson 32 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 Set


Answer Keys

Unlock the answer keys for this lesson's problem set and extra practice problems to save time and support student learning.

Discussion of Problem Set

  • Look at #2. Why are both expressions equivalent? 
  • Look at the first and fourth rows in #3. What do you notice about these two expressions? Before evaluating, did anyone estimate which one was going to be greater? How? 
  • Look at #6. Why are both expressions equivalent? 
  • Look at the first and second rows in #7. What do you notice about these two expressions? Before evaluating, did anyone estimate which one was going to be greater? How? 
  • Look at #8. How did you decide whether the first expression was greater than, less than, or equal to the second expression?

Target Task


Problem 1

Write an equivalent expression in numerical form. Then evaluate it. 

A third as much as the sum of three-fourths and two-eighths

Problem 2

Write an equivalent expression in words.

a.   $${1{1\over2} \times \left ( 2 + {1\over4} \right )}$$

b.   $${\left ( 1 - {4\over 5 } \right ) \div 6}$$

Student Response

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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 22

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Lesson 24

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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