Multiplication and Division of Fractions

Lesson 19

Math

Unit 5

5th Grade

Lesson 19 of 24

Objective


Divide a unit fraction by a whole number.

Common Core Standards


Core Standards

  • 5.NF.B.7.A — Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
  • 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 unit fraction by a whole number using a tape diagram and/or a number line.
  2. Solve group size unknown division word problems that involve the division of a unit fraction by a whole number (MP.4).
  3. Estimate the size of a quotient of a unit fraction divided by a whole number by reasoning that dividing a fraction into some number of groups will result in smaller groups or a smaller value (e.g., we can think of $${{{{1\over3}}} \div 6}$$ as splitting a third into $$6$$ groups, which is the same as finding $${1\over6}$$ of $${{1\over3}}$$, which is $${{1\over18}}$$) (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 unit fraction by a whole number. 

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:
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Anchor Tasks

25-30 minutes


Problem 1

Nolan gives some Fruit-by-the-Foot to his $$3$$ friends to share equally. 

a.   If he has $$3$$ feet of Fruit-by-the-Foot, how many feet of Fruit-by-the-Foot will each friend receive?

b.   If he has $$1$$ foot of Fruit-by-the-Foot, how many feet of Fruit-by-the-Foot will each friend receive?

c.   If he has $${1\over2}$$ foot of Fruit-by-the-Foot, how many feet of Fruit-by-the-Foot will each friend receive?

d.   If he has $${1\over3}$$ foot of Fruit-by-the-Foot, how many feet of Fruit-by-the-Foot will each friend receive?

Guiding Questions

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Student Response

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References

EngageNY Mathematics Grade 5 Mathematics > Module 4 > Topic G > Lesson 26Concept Development

Grade 5 Mathematics > Module 4 > Topic G > Lesson 26 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

Priya and Mai used the diagrams below to find the value of $$\frac13 \div 4$$.

Priya’s diagram:

Mai’s diagram:

a.   What is the same about the diagrams?

b.   What is different?

c.   Find the value that makes the equation true.

$$\frac13 \div 4 =$$ ____

Guiding Questions

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Student Response

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References

Illustrative Mathematics Grade 5 Unit 3 Lesson 12 Activity 1

Grade 4 Unit 3 Lesson 12 Activity 1, accessed on May 5, 2022, 10:44 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.    Solve. Then check your work.

  1. $${{1\over5}\div2}$$
  2. $${{1\over6}\div4}$$
  3. $${{1\over9}\div6}$$

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

Guiding Questions

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Student Response

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References

EngageNY Mathematics Grade 5 Mathematics > Module 4 > Topic G > Lesson 26Concept Development

Grade 5 Mathematics > Module 4 > Topic G > Lesson 26 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 Set

15-20 minutes


Discussion of Problem Set

  • In #1, what is the relationship between (a) and (b), (c) and (d), and (b) and (c)? 
  • Why is the quotient of #1(c) greater than that of #1(d)? Is it reasonable?
  • In #3, what is the relationship between (c) and (d) and (b) and (f)? 
  • In #4, the values were too big to draw a model. So, how did you solve? 
  • Look at #6. What did your diagram look like? Did anyone draw an area model? Why does that work? Can we write a multiplication sentence to represent this situation?

Target Task

5-10 minutes


Problem 1

Solve. Show or explain your work.

a.   $${{1\over3} \div 2}$$

b.   $${{1\over4} \div 9}$$

Student Response

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Problem 2

Zach spends half of his workday tutoring students. If he sees 5 students and spends the same amount of time with each student, what fraction of his workday does he spend tutoring each student?

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

Divide a whole number by a unit fraction.

Lesson 20
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Lesson Map

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

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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