Lesson 20 | Multiplication and Division of Fractions | 5th Grade Mathematics

Objective

Divide a whole number by a unit fraction.

Common Core Standards

Core Standards

The core standards covered in this lesson

5.NF.B.7.B — Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.

Number and Operations—Fractions

5.NF.B.7.B — Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
5.NF.B.7.C — Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Number and Operations—Fractions

5.NF.B.7.C — Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

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.
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.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

Divide a whole number by a unit fraction using a tape diagram and/or a number line.
Solve number of groups unknown division word problems that involve the division of a whole number by a unit fraction (MP.4).
Estimate the size of a quotient of a whole number divided by a unit fraction by reasoning that dividing a whole number into groups that are the size of a unit fraction will result in a larger number of groups or a larger value (e.g., we can think of $${6\div {1\over3}}$$ as how many thirds are in $$6$$, and since there are $$3$$ thirds in $$1$$ whole, there are $$3\times6$$ or $$18$$ thirds in $$6$$) (MP.2).
Use the relationship between multiplication and division to use multiplication to check the quotient to a problem involving the division of a whole number by a unit fraction.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

There are two interpretations for division: (a) equal group with group size unknown division (also called partitive or sharing division), and (b) equal group with number of groups unknown division (also called quotitive or measurement division). In Grade 5, students apply and extend this understanding of the two types of division with whole numbers to divide unit fractions by whole numbers and whole numbers by unit fractions. To develop an understanding of the division of a unit fraction by a whole number, they use unknown group size division, such as in the problem “$$\frac12$$ meter of cloth is cut into three equal pieces. How long is each piece of fabric?”. Inversely, to develop an understanding of the division of a unit fraction by a whole number, they use unknown number of groups division, such as in the problem, “Three meters of cloth are cut into $$\frac12$$ meter strips. How many strips are cut?” That way, as Bill McCallum notes, “students can build on their understanding of whole-number division without having to grapple with fractional groups, so long as they understand both of these interpretations of division” (Mathematical Musings, Bill McCallum, “Fraction division part 2: Two interpretations of division”). Thus, students are exclusively given group size unknown division problems in Lesson 19 and number of groups unknown division problems in Lesson 20 to help them build a strong conceptual understanding of fraction division before seeing other types of division problems in Lesson 21.
Students’ work with fraction division only spans three lessons, Lessons 19–21. You may choose to give students much more time with each of these concepts, perhaps spanning each lesson over two days. If you decide to do so, here are some recommendations for where to source additional practice problems that align to this lesson:
- Illustrative Math Grade 5 Unit 3 Lessons 13 and 14
- EngageNY Grade 5 Unit 4 Lesson 25 Problem Set #1–3, 4a, 5
- EngageNY Grade 5 Unit 4 Lesson 25 Homework #1–4
- Common Core Sheets, Dividing By Unit Fractions

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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

Jenny buys $$2$$ feet of ribbon to make bracelets. Different ways to make bracelets require different amounts of string, depending on how complicated the designs are.

a. If Jenny uses $$2$$ feet of string to make a bracelet, how many bracelets can she make?

b. If she uses $$1$$ foot of string to make a bracelet, how many bracelets can she make?

c. If she uses $$\frac{1}{2}$$ foot of string to make a bracelet, how many bracelets can she make?

d. If she uses $$\frac{1}{3}$$ foot of string to make a bracelet, how many bracelets can she make?

e. If she uses $$\frac{1}{4}$$ foot of string to make a bracelet, how many bracelets can she make?

Guiding Questions

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References

EngageNY Mathematics Grade 5 Mathematics > Module 4 > Topic G > Lesson 25 — Concept Development

Grade 5 Mathematics > Module 4 > Topic G > Lesson 25 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. Solve. Then check your work.

$${3\div{1\over4}}$$
$${5\div{1\over3}}$$
$$8\div \frac{1}{7}$$

b. What do you notice in Part (a)? What do you wonder?

Guiding Questions

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Problem Set

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

In #1, what do you notice about (a) and (b), and (c) and (d)? What are the whole and the divisor in the problems?
In #3, what do you notice about (c) and (d) and (b) and (f)?
Why is the quotient in #3(c) bigger than the quotient in #3(b)?
In #4, the values were too big to draw a model. So, how did you solve?
Share your solution and compare your strategy for solving #3 with a partner.
Look at #7. How did you solve (a) and (b)? What about (c)? What made (c) different from the other problems in the Problem Set?

Target Task

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

Problem 1

Solve. Show or explain your work.

a. $${6\div {1\over2}}$$

b. $${3\div {1\over5}}$$

Student Response

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Problem 2

Mr. Shirazi has 2 feet of fabric. He wants to make bookmarks that each uses a sixth of a foot of fabric as Eid al-Fitr gifts. Does he have enough fabric to make 10 gifts?

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

Answer Keys

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Word Problems and Fluency Activities

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Lesson 19

Lesson 21

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

Number and Operations—Fractions

Foundational Standards

Number and Operations—Fractions

Number and Operations—Fractions

Operations and Algebraic Thinking

Criteria for Success

Tips for Teachers

Anchor Tasks

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

Problem Set

Discussion of Problem Set

Target Task

Problem 1

Student Response

Problem 2

Student Response

Additional Practice

Extra Practice Problems

Word Problems and Fluency Activities

Lesson Map

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