This unit develops students’ understanding of division of fractions by fractions. This work draws on students’ prior knowledge of multiplication, division, and the relationship between the two. It also builds on concepts from grades 3 to 5 about multiplicative situations—equal-size groups, multiplicative comparison, and the area of a rectangle—and about fractions.

Students begin by exploring meanings of division and the relationship between the quantities in division situations. They recall that we can think of dividing as finding an unknown factor in a multiplication equation. In situations involving equal-size groups, division can be used to answer two questions: “How many groups?” and “How much in each group?”

Next, students investigate ways to answer those two questions. They reason about situations in which the size of a group is known but the number of groups is not (as in, “How many s are in 1?”) and in which the number of groups is know but the size is not (as in, “What is in each bottle if there are 14 liters in bottles?”). They also explore division in situations involving multiplicative comparison.

A tape diagram with three equal parts. The first two parts are shaded and are each labeled one third, total 1. A bracket is labeled 1 group of two thirds, and contains the first two parts.

Students then apply their insights to generalize the process of finding quotients. In reasoning repeatedly to find the value of expressions such as , , and , students notice regularity: Dividing a number by a fraction is the same as multiplying that number by .

Students go on to use this algorithm to solve problems about geometric figures that have fractional length, area, or volume measurements. They also apply the concepts from the unit to solve multi-step problems involving fractions in other contexts.

Throughout the unit, students interpret and create equations and diagrams to make sense of the relationship between known and unknown quantities.

A deeper understanding of multiplication, division, and ways to represent them will support students in reasoning about decimal operations as well as in writing and solving variable equations later in the course.

A note about diagrams:

Because tape diagrams are a flexible tool for illustrating and reasoning about division of fractions, they are the primary representation used in this unit. Students may, however, create other representations to support their reasoning.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes, such as interpreting, representing, justifying, and explaining. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Interpret and Represent

• Situations involving division (Lessons 2, 3, 9, 12, and 16).
• Situations involving measurement constraints (Lesson 17).

Justify

• Reasoning about division and diagrams (Lessons 4 and 5).
• Strategies for dividing numbers (Lesson 11).
• Reasoning about volume (Lesson 15).

Explain

• How to create and make sense of division diagrams (Lesson 6).
• How to represent division situations (Lesson 9).
• How to find unknown lengths (Lesson 14).
• A plan for optimizing costs (Lesson 17).

In addition, students are expected to critique the reasoning of others about division situations and representations, and to make generalizations about division by comparing and connecting across division situations and across the representations used in reasoning about these situations. The Lesson Syntheses in Lessons 2 and 12 offer specific disciplinary language that may be especially helpful for supporting students in navigating the language of important ideas in this unit.

The table shows lessons where new terminology is first introduced in this course, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.