Lesson 2 | Multiplication and Division of Fractions | 5th Grade Mathematics

Objective

Write division expressions that represent fractions and vice versa.

Common Core Standards

Core Standards

The core standards covered in this lesson

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?

Number and Operations—Fractions

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?

Foundational Standards

The foundational standards covered in this lesson

4.NF.B.3

Number and Operations—Fractions

4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
4.NF.B.4

Number and Operations—Fractions

4.NF.B.4 — Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
3.OA.A.1

Operations and Algebraic Thinking

3.OA.A.1 — Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.
3.OA.A.2

Operations and Algebraic Thinking

3.OA.A.2 — Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.
3.OA.B.6

Operations and Algebraic Thinking

3.OA.B.6 — Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Criteria for Success

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

Look for and express regularity in repeated reasoning to see that $$a\div b = \frac{a}{b}$$ (MP.8).
Write a solution to a division expression with a non-whole number quotient as a fraction or mixed number, if applicable.
Check the solution to a division problem with a fractional quotient using multiplication and/or repeated addition.
(Optional) Connect simplifying fractions to the idea of simplifying a division calculation (e.g., $$40 \div 6 = \frac {40}{6} = \frac{20}{3} = 20 \div 3$$).

Tips for Teachers

Suggestions for teachers to help them teach this lesson

In Grade 5, students connect fractions with division, understanding numerical instances of $$\frac{a}{b} = a \div b$$ for whole numbers $$a$$ and $$b$$, with $$b$$ not equal to zero (MP.8)” (NF Progression, p. 6). Thus, in this lesson, students look for and express regularity in repeated reasoning through multiple examples that help them see that $$a \div b = \frac{a}{b}$$.

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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

a. Jada’s family made sandwiches to share at a family celebration. Complete the table to show how much sandwich each person gets.

Sandwiches Being Shared	Number of People Sharing Sandwiches	Division Expression	Amount Each Person Gets
1	2
1	3
1	4
1

b. Choose one row from the table and represent your thinking with a diagram.

c. What do you notice in the table? What do you wonder?

Guiding Questions

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References

Illustrative Mathematics Grade 5 > Unit 2 > Lesson 2 > Activity 1

Grade 5 > Unit 2 > Lesson 2 > Activity 1, accessed on April 29, 2022, 11:52 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. A group of athletes are sharing some number of liters of water at practice. Complete the table to show how much water each athlete gets.

Number of Liters of Water Being Shared	Number of People Sharing Water	Division Expression	Amount Each Person Gets
4	5
		$$3 \div 2$$
		$$9 \div 12$$
			$$\frac7{10}$$
			$$2\frac14$$

b. What do you notice in the table? What do you wonder?

Guiding Questions

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Problem 3

a. Fill in the blanks to match the rules in the table.

	Each person gets ____ pound(s) of blueberries.
	more than 1	exactly 1	less than 1	exactly $$\frac{1}{2}$$
_____ people share 7 pounds of blueberries	✔
_____ people share _____ pounds of blueberries		✔
_____ people share _____ pounds of blueberries			✔
_____ people share _____ pounds of blueberries				✔

b. How many pounds of blueberries did each person get when they got more than 1 pound?

c. How many pounds of blueberries did each person get when they got less than 1 pound?

Guiding Questions

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References

Illustrative Mathematics Grade 5 > Unit 2 > Lesson 5 > Activity 1 (Relate Pounds to People)

Grade 5 > Unit 2 > Lesson 5 > Activity 1, accessed on Nov. 14, 2022, 3:32 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 did you decide which values of c and d were correct in #3?
#4f was hard since the fraction was written as a mixed number. How did you determine what numbers were being divided?
Which equations in #6 were correct? Incorrect? How do you know?
How did you determine what number Lemuel was dividing by 15 in #7?

Target Task

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

a. Which expression is equivalent to $$\frac34$$?

a. $$3 \times 4$$
b. $$3 +4$$
c. $$3 \div 4$$
d. $$3-4$$

b. Explain how you can use multiplication to prove that your answer from Part A is correct.

Student Response

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References

Smarter Balanced Assessment Consortium SBAC Item #3338

Resource: Progressions for the Common Core State Standards in Mathematics

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

Answer Keys

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Word Problems and Fluency Activities

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

Lesson 1

Lesson 3

Lesson Map

Topic A: Fractions as Division

Relate equal shares of objects to division expressions and visual representations of fractions.

5.NF.B.3

Write division expressions that represent fractions and vice versa.

5.NF.B.3

Solve division problems when the quotient is a fraction or mixed number, including cases with larger values.

5.NF.B.3

Topic B: Multiplying a Fraction by a Whole Number

Multiply a fraction by a whole number where the product is a whole number.

5.NF.B.4.A 5.NF.B.6

Multiply a fraction by a whole number where the product is a whole number or a fraction.

5.NF.B.4.A 5.NF.B.6

Relate multiplication of a fraction by a whole number to multiplication of a whole number by a fraction and use this to develop a general method to multiply any fraction by any whole number (or vice versa).

5.NF.B.4.A 5.NF.B.5

Solve real-world problems involving multiplication of fractions and whole numbers and create real-world contexts for expressions involving multiplication of fractions and whole numbers.

5.NF.B.4 5.NF.B.6 5.OA.A.2

Topic C: Multiplying a Fraction by a Fraction

Multiply a fraction by a fraction without subdivisions using tape diagrams and number lines.

5.NF.B.4 5.NF.B.5 5.NF.B.6

Multiply a fraction by a fraction with subdivisions using tape diagrams and number lines.

5.NF.B.4 5.NF.B.5 5.NF.B.6

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

5.NF.B.4 5.NF.B.5 5.NF.B.6

Develop a general method to multiply a fraction by a fraction.

5.NF.B.4 5.NF.B.5 5.NF.B.6

Solve real-world problems involving multiplication of fractions with fractions and create real-world contexts for expressions involving multiplication of fractions with fractions.

5.NF.B.4 5.NF.B.6 5.OA.A.2

Topic D: Multiplying with Mixed Numbers

Multiply mixed numbers by whole numbers.

5.NF.B.4 5.NF.B.5 5.NF.B.6

Multiply mixed numbers by fractions.

5.NF.B.4 5.NF.B.5 5.NF.B.6

Multiply mixed numbers by mixed numbers.

5.NF.B.4 5.NF.B.5 5.NF.B.6

Develop a general method to multiply with mixed numbers.

5.NF.B.4 5.NF.B.5 5.NF.B.6

Solve real-world problems involving multiplication with mixed numbers and create real-world contexts for expressions involving multiplication with mixed numbers.

5.NF.B.4 5.NF.B.6 5.OA.A.2

Interpret multiplication as scaling.

5.NF.B.5 5.NF.B.5.A 5.NF.B.5.B

Topic E: Dividing with Fractions

Divide a unit fraction by a whole number.

5.NF.B.7.A 5.NF.B.7.C

Divide a whole number by a unit fraction.

5.NF.B.7.B 5.NF.B.7.C

Solve real-world problems involving division with fractions and create real-world contexts for expressions involving division with fractions.

5.NF.B.7.C 5.OA.A.2

Topic F: Fraction Expressions and Real-World Problems

Solve real-world problems involving multiplication and division with fractions.

5.NF.B.3 5.NF.B.6 5.NF.B.7

Write and evaluate numerical expressions involving operations with fractions.

5.NF.B.3 5.NF.B.4 5.NF.B.5 5.NF.B.6 5.NF.B.7 5.OA.A.1 5.OA.A.2

Topic G: Line Plots

Create line plots.

5.MD.B.2

Solve problems involving information presented in a line plot (dot plot).

5.MD.B.2

Multiplication and Division of Fractions

Objective

Common Core Standards

Core Standards

Number and Operations—Fractions

Foundational Standards

Number and Operations—Fractions

Number and Operations—Fractions

Operations and Algebraic Thinking

Operations and Algebraic Thinking

Operations and Algebraic Thinking

Criteria for Success

Tips for Teachers

Anchor Tasks

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

Problem 3

Guiding Questions

References

Problem Set

Discussion of Problem Set

Target Task

Student Response

References

Additional Practice

Extra Practice Problems

Word Problems and Fluency Activities

Lesson Map

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