In the previous three lessons, students explored the “how many groups?” interpretation of division. Their explorations included situations where the number of groups was a whole number or a mixed number. In this lesson, they extend the work to include cases where the number of groups is a fraction less than 1, that is, situations in which the total amount is smaller than the size of 1 group. In such situations, the question becomes “what fraction of a group?”.
Students notice that they can use the same reasoning strategies as in situations with a whole number of groups, because the structure
\((\mathrm{number}\;\mathrm{of}\;\mathrm{groups})⋅(\mathrm{size}\;\mathrm{of}\;\mathrm a\;\mathrm{group})=(\mathrm{total}\;\mathrm{amount})\)
is the same as before (MP7). They write multiplication equations of this form and for the corresponding division equations.
Throughout the lesson, students practice attending to details (in diagrams, descriptions, or equations) about how the given quantities relate to the size of 1 group.
Lesson overview
 7.1 Warmup: Estimating a Fraction of a Number (5 minutes)

7.2 Activity: Fractions of Ropes (10 minutes)
 There is a digital applet in this activity.
 7.3 Activity: Fractional Batches of Ice Cream (20 minutes)
 Lesson Synthesis
 7.4 Cooldown: A Partially Filled Container (5 minutes)
Learning goals:
 Comprehend the phrase “What fraction of a group?” (in spoken and written language) as a variation of the question “How many groups?” that is used when the quotient is less than 1.
 Create a tape diagram to represent and solve a problem asking “How many groups?” in which the quotient is a fraction less than 1.
 Write multiplication and division equations to represent a problem asking “How many times as long?”
Learning goals (student facing):
 Let’s think about dividing things into groups when we can’t even make one whole group.
Learning targets (student facing):
 I can use diagrams and multiplication and division equations to represent and answer “what fraction of a group?” questions.
 I can tell when a question is asking for the number of groups and that number is less than 1.
Required materials:
 geometry toolkits
Glossary:
 Access the complete Grade 6 glossary.
Standards
 This lesson builds on the standard: CCSS.5.NF.B.4MO.5.NF.B.7aMO.5.NF.B.7bMO.5.NF.B.7c
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).
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