Curriculum  /  Math  /  4th Grade  /  Unit 5: Fraction Operations  /  Lesson 5

Fraction Operations

Lesson 5

Math

Unit 5

4th Grade

Lesson 5 of 21

Objective

Decompose non-unit fractions less than 2 as a sum of unit fractions, as a sum of non-unit fractions, and as a whole number times a unit fraction.

Common Core Standards

Core Standards

  • 4.NF.B.3.B — Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
  • 4.NF.B.4.A — Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

Foundational Standards

  • 3.NF.A.1
  • 3.NF.A.2

Criteria for Success

  1. Decompose a fraction less than 2 into a sum of unit fractions, recording the decomposition with an equation.
  2. Decompose a fraction less than 2 into a sum of fractions with the same denominator in more than one way, recording each decomposition with an equation.
  3. Decompose a fraction less than 2 into a multiple of a unit fraction, recording the decomposition with an equation.
  4. Use these decompositions to show why a fraction greater than one is equivalent to a mixed number and vice versa.
  5. Justify decompositions with a visual model, such as a tape diagram or number line.

Tips for Teachers

  • In Teaching Student-Centered Mathematics, 3-5, vol. 2, John A. Van de Walle states, "The term ‘improper’ can be a source of confusion because it implies that this representation is not acceptable, which is false. Instead it is often the preferred representation in algebra. Avoid using this term and instead use ‘fraction’ or ‘fraction greater than one’” (p. 217). Thus, this lesson and all following avoid the use of the term with students. 
  • Before the Problem Set, you could have students play around with manipulatives to decompose fractions into a sum of unit fractions or non-unit fractions, similar to Joe Schwartz’s blog post “Building Towers." If you don’t have fraction towers or even fraction tiles, you can create them from paper.
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Anchor Tasks

25-30 minutes

Problem 1

Lin’s family recipe for cookies also calls for $$\frac64$$ cups chopped nuts. Write an addition equation and a multiplication equation to show how she can use the $$\frac14$$-cup measure to measure the amount of chopped nuts.

Guiding Questions

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Student Response

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References

Illustrative Mathematics Grade 4 Unit 3 Lesson 7 Activity 1

Grade 4 Unit 3 Lesson 7 Activity 1 ("Activity 1: Barley Soup"), accessed on March 18, 2022, 10:07 a.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.

Modified by Fishtank Learning, Inc.

Problem 2

a.   On each number line, draw two “jumps” to show how to use thirds to make a sum of $$\frac53$$. Then, write an equation to represent each combination of jumps.

b.   Noah draws the following number line and writes: $$\frac53 = \frac33 + \frac23$$ and $$\frac53= 1+\frac23$$. Which equation is correct? Explain your reasoning.

c.   Write $$\frac64$$ as a sum of a whole number and a fraction. Use a number line if it would help you. 

Guiding Questions

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Student Response

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References

Illustrative Mathematics Math Grade 4 Unit 3 Lesson 8 Activity 1 — "Sum of Jumps"

Math Grade 4 Unit 3 Lesson 8 Activity 1, accessed on March 18, 2022, 10:17 a.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.

Modified by Fishtank Learning, Inc.

Problem 3

a.   On each number line, draw “jumps” to show how to use eighths to make a sum of $$1\frac38$$. Then, write an equation to represent each combination of jumps.

b.   Explain why $${1{3\over8}}={11\over8}$$ using a number line or words.

Guiding Questions

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Student Response

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References

Illustrative Mathematics Math Grade 4 Unit 3 Lesson 8 Activity 1 — "Sum of Jumps"

Math Grade 4 Unit 3 Lesson 8 Activity 1, accessed on March 18, 2022, 10:17 a.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.

Modified by Fishtank Learning, Inc.

Problem Set

15-20 minutes

Discussion of Problem Set

  • In #1–#4, why do we need to label 1 on our tape diagrams and number lines? What would happen if we didn’t?
  • What do you notice about #5(a) and #5(c)? 
  • What is the advantage of representing fractions using multiplication? 
  • In our lesson, when we expressed $${{5\over3}}$$ as $${\left(3\times{1\over3}\right)+\left(2\times{1\over3}\right)}$$, what property were we using? 

Target Task

5-10 minutes

Decompose the following fractions (i) as a sum of unit fractions, (ii) as a multiple of a unit fraction, and (iii) in at least one other way.

a.   $${{7\over6}}$$

b.   $${1{3\over4}}$$

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

Add and subtract fractions that require regrouping where the total is less than 2.

Lesson 6
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Lesson Map

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

Topic A: Building, Adding, and Subtracting Fractions Less Than or Equal to 1

Topic B: Building, Adding, and Subtracting Fractions Less Than 2

Topic C: Building, Adding, and Subtracting Fractions Greater Than or Equal to 2

Topic D: Multiplication of Fractions

Topic E: Line Plots

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