Author:
Chris Adcock
Material Type:
Lesson Plan
Level:
Middle School
Grade:
6
Provider:
Pearson
Tags:
6th Grade Mathematics, Dividing Fractions
License:
Creative Commons Attribution Non-Commercial
Language:
English
Media Formats:
Text/HTML

Education Standards

Divide a Fraction by a Fraction

Divide a Fraction by a Fraction

Overview

Students explore methods of dividing a fraction by a unit fraction.

Key Concepts

In this lesson and in Lesson 5, students explore dividing a fraction by a fraction.

In this lesson, we focus on the case in which the divisor is a unit fraction. Understanding this case makes it easier to see why we can divide by a fraction by multiplying by its reciprocal. For example, finding 34÷15 means finding the number of fifths in 34. In this lesson, students will see that this is 34 × 5.

Students learn and apply several methods for dividing a fraction by a unit fraction, such as 23÷14.

  • Model 23. Change the model and the fractions in the problem to twelfths: 812÷312. Then find the number of groups of 3 twelfths in 8 twelfths. This is the same as finding 8 ÷ 3.
  • Reason that since there are 4 fourths in 1, there must be 23 × 4 fourths in 23. This is the same as using the multiplicative inverse.
  • Rewrite both fractions so they have a common denominator: 23÷14=812÷312. The answer is the quotient of the numerators. This is the numerical analog to modeling.

Goals and Learning Objectives

  • Use models and other methods to divide fractions by unit fractions

Mia’s Method

Lesson Guide

Have students work with a partner for a few minutes to read and discuss Mia’s method before discussing with the entire class.

Mathematics

Have students explain how Mia’s method is similar to the ones she used in Lessons 2 and 3. How has it been modified?

ELL: Modeling is essential for ELL students. It helps them clarify difficult concepts and directions. It also gives students the opportunity to ask questions and it gives you, as instructor, the opportunity to model effective learning strategies. Some students may not recognize a general pattern for dividing fractions.

SWD: These types of concrete and basic examples will be helpful for students with more literal thinking skills.

Opening

Mia's Method

Mia uses the following method to find 23÷14:

“To find 23÷14, I need to find the number of fourths in 23. I can make a model of 23, but I think it would be difficult to figure out the number of fourths in the model.

“I think the problem would be easier if both fractions had the same denominator. I can change the denominator of each fraction to 12 and rewrite the problem as 812÷312.

“Now I can make a model of 812 and then find the number of groups of 3 twelfths in my model.

“In my model there are 2 groups of 3 twelfths, with 23 of a group of 3 twelfths left over.
So, 23÷14=223.”

  • Discuss Mia’s method with your partner and then with the class.

Math Mission

Lesson Guide

Discuss the Math Mission. Students will explore methods for dividing a fraction by a unit fraction.

Opening

Explore methods for dividing a fraction by a unit fraction.

Explore Dividing Fractions by Unit Fractions

Lesson Guide

Students should work with a partner on the first two problems. Help students who have difficulty. As students work, encourage them to look for a general pattern they can use to divide any fraction by a unit fraction.

SWD: When listening to students' responses, give students with disabilities advance notice of when they will be presenting their work on a specific problem during the Ways of Thinking section. This will give them ample time to prepare a thoughtful response.

Interventions

Student has difficulty starting 45÷13.

  • How did Mia start her method?
  • What does the problem 45÷13 mean? What are you trying to find?
  • Try drawing a model for 45. How could you find the number of thirds inside of 45?

Student finds the wrong solution.

  • How can you check to make sure your answer is correct?
  • Think about the sizes of the numbers in the problems. Does your answer seem reasonable?

Answers

  • 45÷13

    Change the denominator of each fraction:

    45×33=1215     13×55=515

    Rewrite the problem: 1215÷515

    Now make a model of 1215, then find the number of groups of 5 fifteenths in the model:

In this model there are 2 groups of 5 fifteenths, with 25 of a group of 5 fifteenths left over, so:

45 ÷ 13=225

  • Check: 225×13=125×13=1215=45

Work Time

Explore Dividing Fractions by Unit Fractions

  • Use Mia’s method to find 45÷13.
  • Check your answer.

Ask yourself:

  • Did you change 45 and 13 to fractions that have the same denominator?
    • How many five-fifteenths (515) are in 1215?

Carlos’s Method

Lesson Guide

Students learn Carlos's method of dividing a fraction by a unit fraction by multiplying the reciprocal or multiplicative inverse of a fraction.

Interventions

If students are having difficulties:

  • What was Carlos dividing by and what are you dividing by?
  • What did Carlos do?
  • What is the inverse of 15?

Answers

  • 83÷15=83×5=403=1313

Work Time

Carlos’s Method

Carlos uses the following method to find 72÷14:

“I can change this problem to a multiplication problem by multiplying by the inverse. The multiplicative inverse, or reciprocal, of 14 is 4.”

Here is Carlos’s solution:

72×4=282 or 14

So, 72÷14 = 14.

  • Use Carlos’s method to find 83÷15.

How can you rewrite the division problem as a multiplication problem using the reciprocal of 15?

Prepare a Presentation

Preparing for Ways of Thinking

Choose solutions that use a variety of models and methods to be presented during Ways of Thinking. Choose both incorrect and correct solutions. You can use the incorrect solutions to clear up misconceptions students may have.

Select students who did the Challenge Problem to present during Ways of Thinking. If possible, choose one student for each method.

Challenge Problem

Answers

  • Students will choose either Method 1 or Method 2. Explanations will vary. Possible explanations: Method 1: For 45÷13, the denominator of 13, which is 3, is the number of thirds in 1 whole. The number of thirds in 45 is 45 of the number of thirds in 1 whole, or 45 of 1 x 3. This is the same as (4×35). We are multiplying the numerator of the fraction by the denominator of the unit fraction. Method 2: We can rewrite 45÷13 as 1215÷515. We must find the number of groups of 5 fifteenths in 12 fifteenths. This is just 12 ÷ 5.

Work Time

Prepare a Presentation

Explain how you divided a fraction by a unit fraction. Use your work to support your explanation.

Challenge Problem

Below are two methods for dividing a fraction by a unit fraction.

Method 1
To divide a fraction by a unit fraction, multiply the numerator of the fraction by the denominator of the unit fraction.

Method 2
To divide a fraction by a unit fraction, rewrite both fractions so they have a common denominator. The answer is the quotient of the numerators.

  • Choose one of the methods. Use the division problem 45÷13 to explain why the method works.

Make Connections

Lesson Guide

Have students present a variety of models and methods for solving the problems.

ELL: When giving directions for this discussion, be sure that students understand that they can complement their oral explanation with pictures, number lines, and drawings. Provide sentence frames that will help ELLs explain their thinking.

Mathematical Practices

Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.

Encourage students to clearly explain the steps in their solutions using correct mathematical language. Other students can ask questions and critique the presenter’s method and reasoning, using the hints as prompts.

Mathematical Practice 6: Attend to precision.

If possible, have at least one student explain each method from the Challenge Problem. If no students did the Challenge Problem, present and explain the methods yourself.

Performance Task

Ways of Thinking: Make Connections

As your classmates present, take notes to clarify your understanding of how to divide a fraction by a unit fraction.

As your classmates present, ask questions such as:

  • Do all of the methods make sense?
  • Which methods did you find easier to use? Why?
  • Which methods did you find more difficult to use? Why?
  • How are the two models you just looked at alike? How are they different?
  • Where do you see the unit fraction in your model?
  • Why can multiplication be used to solve a division problem?
  • Is there a general pattern you can use to divide any fraction by a unit fraction?

Servings of Rice

Lesson Guide

Challenge students to use a method that they had not used in the Work Time problems. For students who are fluent in multiple methods, challenge them to create real-world problems. Have students work alone.

Mathematics

This problem has a dividend that is a mixed number. You may need to remind students that a mixed number can be rewritten as a fraction.

Answers

  • 73÷12
  • Methods will vary. 143 or 423

Possible methods:

73÷12=73×2=143

73÷12=146÷36=143

 

The model shows 213÷12=423

 

  • Check: 12×143=73

 

Work Time

Servings of Rice

A pot contains 213 cups of rice. How many 12-cup servings does the pot contain?

  • Write a division problem to represent this situation.
  • Solve the problem using any method you wish.
  • Check your answer.

Ask yourself:

  • What is the total amount of rice in the pot? You should divide the total amount of rice into equal groups of what size? How many equal-sized groups are there?
  • Did you write 2 1 3 as a fraction? Would rewriting both fractions so they have a common denominator help you find the answer?

Divide a Fraction by a Unit Fraction

A Possible Summary

To divide a fraction by a unit fraction, you must first determine the number of unit fractions inside the fraction. One way to do this is to draw a model. Sometimes it is hard to see the number of unit fractions inside the model, so you need to change both fractions (and the model) so they have a common denominator.

We learned two shortcut methods:

  • Multiply the numerator of the fraction (the dividend) by the denominator of the unit fraction.
  • Rewrite the fractions in the problem so they have a common denominator. The answer will be the quotient of the numerators.

Formative Assessment

Summary of the Math: Divide a Fraction by a Unit Fraction

Write a summary about how to divide a fraction by a unit fraction.

Check your summary.

  • Do you describe at least two methods for dividing a fraction by a unit fraction?

Reflect On Your Work

Lesson Guide

Have students write a brief reflection before the end of class. Review the reflections to find out their understanding of fraction division.

Work Time

Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

One way I can figure out how to solve a division problem when I am dividing by a unit fraction is...