Lesson plan

# Lesson 10: Dividing by Unit and Non-Unit Fractions

teaches Common Core State Standards MP8 http://corestandards.org/Math/Practice/MP8
teaches Common Core State Standards MP7 http://corestandards.org/Math/Practice/MP7
teaches Common Core State Standards 6.NS.A.1 http://corestandards.org/Math/Content/6/NS/A/1

# Lesson 10: Dividing by Unit and Non-Unit Fractions

This is the first of two lessons in which students pull together the threads of reasoning from the previous six lessons to develop a general algorithm for dividing fractions. Students start by recalling the idea from grade 5 that dividing by a unit fraction has the same outcome as multiplying by the reciprocal of that unit fraction. They use tape diagrams to verify this.

Next, they use the same diagrams to look at the effects of dividing by non-unit fractions. Through repetition, they notice a pattern in the steps of their reasoning (MP8) and structure in the visual representation of these steps (MP7). Students see that division by a non-unit fraction can be thought of as having two steps: dividing by the unit fraction, and then dividing the result by the numerator of the fraction. In other words, to divide by $$\frac25$$ is equivalent to dividing by $$\frac15$$, and then again by 2. Because dividing by a unit fraction $$\frac15$$ is equivalent to multiplying by 5, we can evaluate division by $$\frac25$$ by multiplying by 5 and dividing by 2.

Lesson overview

• 10.1 Warm-up: Dividing by a Whole Number (10 minutes)
• 10.2 Activity: Dividing by Unit Fractions (15 minutes)
• 10.3 Activity: Dividing by Non-unit Fractions (15 minutes)
• Includes "Are you Ready for More?" extension problem
• Lesson Synthesis
• 10.4 Cool-down: Dividing by $$\frac13$$ and $$\frac35$$ (5 minutes)

Learning goals:

• Interpret and critique explanations (in spoken and written language, as well as in other representations) of how to divide by a fraction.
• Use a tape diagram to represent dividing by a non-unit fraction $$\frac ab$$ and explain (orally) why this produces the same result as multiplying the number by $$b$$ and dividing by $$a$$.
• Use a tape diagram to represent dividing by a unit fraction $$\frac1b$$ and explain (orally and in writing) why this is the same as multiplying by $$b$$.

Learning goals (student facing):

• Let’s look for patterns when we divide by a fraction.

Learning targets (student facing):

• I can divide a number by a unit fraction $$\frac1b$$ by reasoning with the denominator, which is a whole number.
• I can divide a number by a non-unit fraction $$\frac ab$$ by reasoning with the numerator and denominator, which are whole numbers.

Required materials:

• geometry toolkits

Glossary:

• reciprocal - Dividing 1 by a number gives the reciprocal of that number. For example, the reciprocal of 12 is $$\frac1{12}$$, and the reciprocal of $$\frac25$$ is $$\frac52$$.
• Access the complete Grade 6 glossary.

Standards:

• This lesson builds on the standard: CCSS.5.NF.B

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.