Curriculum  /  Math  /  5th Grade  /  Unit 5: Multiplication and Division of Fractions  /  Lesson 20

Multiplication and Division of Fractions

Lesson 20

Math

Unit 5

5th Grade

Lesson 20 of 24

Objective

Divide a whole number by a unit fraction.

Common Core Standards

Core Standards

  • 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?

Foundational Standards

  • 3.NF.A.1
  • 4.NF.B.4
  • 3.OA.B.6

Criteria for Success

  1. Divide a whole number by a unit fraction using a tape diagram and/or a number line.
  2. Solve number of groups unknown division word problems that involve the division of a whole number by a unit fraction (MP.4).
  3. 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).
  4. 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

  • 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, measurement, or equal-sharing 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

25-30 minutes

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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Student Response

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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.

  1. $${3\div{1\over4}}$$
  2. $${5\div{1\over3}}$$
  3. $$8\div \frac{1}{7}$$

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

Guiding Questions

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Student Response

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Problem Set

15-20 minutes

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

5-10 minutes

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

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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

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

Lesson 21
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Lesson Map

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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 Real-World Problems and Line Plots

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