Lesson 10: Dividing by Unit and NonUnit Fractions
About this lesson
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 nonunit 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 nonunit 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 Warmup: Dividing by a Whole Number (10 minutes)
 10.2 Activity: Dividing by Unit Fractions (15 minutes)

10.3 Activity: Dividing by Nonunit Fractions (15 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 10.4 Cooldown: 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 nonunit 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 nonunit 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 20172019 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/mathcurriculum/.
Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.
This site includes public domain images or openly licensed images that are copyrighted by their respective owners. Openly licensed images remain under the terms of their respective licenses.