Lesson 2 | Fraction Operations | 4th Grade Mathematics

Objective

Decompose fractions as a sum of fractions with the same denominator in more than one way.

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

Number and Operations—Fractions

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).

Number and Operations—Fractions

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

The foundational standards covered in this lesson

3.NF.A.1

Number and Operations—Fractions

3.NF.A.1 — Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
3.NF.A.2

Number and Operations—Fractions

3.NF.A.2 — Understand a fraction as a number on the number line; represent fractions on a number line diagram.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Decompose a fraction less than or equal to 1 into a sum of fractions with the same denominator in more than one way, recording each decomposition with an equation.
Justify decompositions with a visual model, such as a tape diagram or number line.

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Anchor Tasks

Tasks designed to teach criteria for success of the lesson, and guidance to help draw out student understanding

Problem 1

Lin later found a $$\frac34$$-cup measuring cup. Show how she could use the $$\frac34$$ and $$\frac14$$ measuring cups to measure the right amount of each ingredient below.

a. $$\frac34$$ cups sugar

b. 1 cup butter

Guiding Questions

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References

Illustrative Mathematics Grade 4 Unit 3 Lesson 7 Activity 1

Grade 4 Unit 3 Lesson 7 Activity 1, accessed on March 17, 2022, 3:49 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.

Modified by Fishtank Learning, Inc.

Problem 2

On each number line, draw two “jumps” to show how to use sixths to make a sum of $$\frac66$$. Then, write an equation to represent each combination of jumps.

Number line ranging from 0 to 1.

Guiding Questions

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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. Use different combinations of tenths to make a sum of $$\frac{9}{10}$$.

$$\frac{9}{10}$$ = ___ + ___ + ___ + ___ + ___
$$\frac{9}{10}$$ = ___ + ___ + ___ + ___
$$\frac{9}{10}$$ = ___ + ___ + ___
$$\frac{9}{10}$$ = ___ + ___

b. Write different ways to use fifths to make a total of $$\frac 55$$. How many can you think of? Write an equation for each combination.

c. Is it possible to write any fraction with a denominator of 8 as a sum of other eighths? Explain or show your reasoning.

Guiding Questions

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References

Illustrative Mathematics Grade 4 Unit 3 Lesson 7 Activity 2

Grade 4 Unit 3 Lesson 7 Activity 2, accessed on March 17, 2022, 3:56 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.

Modified by Fishtank Learning, Inc.

Problem Set

Problem Set

Answer Keys

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Discussion of Problem Set

How else could you have decomposed any of the number lines in #2? (You could choose to discuss specific parts based on what’s best for your students.)
Why does #3(a) start with “$${1 =}$$” as opposed to a fraction? What equivalent fraction could we use instead of $$1$$?
What other decompositions could have been possible answers in #4?
In #5, how would you record the fraction’s decomposition into unit fractions? How can you draw parentheses around a certain number of unit fractions to model another decomposition you wrote (e.g., $$\left({1\over8}+{1\over8}\right)+\left({1\over8}+{1\over8}\right) + {1\over8}$$ to model $${2\over8}+{2\over8}+{1\over8}$$)?
Did anyone write a multiplication equation for any part of #5? What is the equation?
What relationship does the unit fraction have with the number of units in a whole?
How did you decompose numbers in earlier grades? How is that similar to and different from how you decomposed fractions today?

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Decompose the following fraction in three different ways. Record the decompositions with equations.

$${{7\over8}}$$

Student Response

An example response to the Target Task at the level of detail expected of the students.

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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

Extra Practice Problems

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Word Problems and Fluency Activities

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Lesson 1

Lesson 3

Lesson Map

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

Decompose fractions as a sum of unit fractions and as a multiple of a unit fraction.

4.NF.B.3.B 4.NF.B.4.A

Decompose fractions as a sum of fractions with the same denominator in more than one way.

4.NF.B.3.B 4.NF.B.4.A

Add and subtract fractions within 1 with the same units.

4.NF.B.3.A 4.NF.B.3.C

Solve word problems that involve the addition and subtraction of fractions where the total is less than or equal to one.

4.NF.B.3.D

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

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.

4.NF.B.3.B 4.NF.B.4.A

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

4.NF.B.3.A 4.NF.B.3.C

Add two fractions where one denominator is a multiple of the other using the denominators 2, 3, 4, 5, 6, 8, 10, and 12.

4.NF.B.3.A 4.NF.B.3.C

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

Decompose and compose non-unit fractions greater than two as a sum of unit fractions, as a sum of non-unit fractions, and as a whole number times a unit fraction.

4.NF.B.3.B

Convert fractions greater than 1 to mixed numbers.

4.NF.B.3.B 4.NF.B.3.C

Convert mixed numbers to fractions greater than 1.

4.NF.B.3.B 4.NF.B.3.C

Compare and order fractions greater than 1 using various methods.

4.NF.B.3.C

Add fractions and mixed numbers where the total is greater than or equal to 2.

4.NF.B.3.C

Subtract fractions and mixed numbers where the total is greater than or equal to 2.

4.NF.B.3.C

Add and subtract mixed numbers using a variety of mental strategies.

4.NF.B.3.C

Solve word problems involving addition and subtraction of fractions.

4.NF.B.3.D

Topic D: Multiplication of Fractions

Multiply a whole number by a fraction.

4.NF.B.4.B

Multiply a whole number by a mixed number.

4.NF.B.4.B

Solve word problems involving multiplication of fractions.

4.NF.B.4.C

Solve word problems involving addition, subtraction, and multiplication of fractions.

4.NF.B.3.D 4.NF.B.4.C

Topic E: Line Plots

Make a line plot (dot plot) representation to display a data set of measurements in fractions of a unit.

4.MD.B.4

Solve problems using information presented in line plots.

4.MD.B.4

Fraction Operations

Objective

Common Core Standards

Core Standards

Number and Operations—Fractions

Number and Operations—Fractions

Foundational Standards

Number and Operations—Fractions

Number and Operations—Fractions

Criteria for Success

Anchor Tasks

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

References

Problem 3

Guiding Questions

References

Problem Set

Discussion of Problem Set

Target Task

Student Response

Additional Practice

Extra Practice Problems

Word Problems and Fluency Activities

Lesson Map

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