In this section, students investigate the quantities in a division situation, recall the relationship between multiplication and division, and revisit ways to represent this relationship.

First, students look at how the numbers in a division situation are related. Students observe—in concrete and abstract situations—how the size of the dividend and the divisor affect the size of the quotient. Next, students reason about division in terms of equal-size groups. They are reminded that:

• Multiplication is a way to find the total amount when given the number of groups and the size of each group.
• Division is a way to find an unknown factor, which can either be the number of groups or the size of one group, when given a total amount. This means division can be interpreted in two ways.

For instance, here are two ways to think about :

How many groups of 6 are in 12?
 

How much is in one group if there are 6 groups in 12?
 

The last lesson in the section is optional. It offers additional practice in interpreting and representing situations that involve equal-size groups, encouraging students to pay close attention to what the parts and numbers mean. 

A note about the term “group”:

Students may be most familiar with the idea of a group as a collection of people or objects. Clarify that the term is used more broadly here. A “group” can refer to a part, a batch, a bag, a section, or another quantity with a particular value. So “equal-size groups” can refer to collections with the same number of items or people in each, as well as parts with the same value, sections of equal length, bags of the same weight, and so on. As students reason about various multiplication and division situations, the meaning of “group” in each situation will become more intuitive.

A note about notation:

When writing multiplication equations to represent, for instance, “How many groups of 4 are in 12?” students may write either or as long as they understand what each factor represents. Because we tend to say “ groups of ” in these materials, we follow that order in writing the multiplication equation:

In this section, students make sense of division situations involving fractions, gradually building an understanding that can later be generalized into an algorithm.

The first four lessons explore “How many groups?” questions in various contexts. In the first lesson, the size of a group is a fraction and the number of groups is a whole number. This lesson is optional because it reinforces grade 5 work on multiplication and division of whole numbers by unit fractions.

Then, students investigate situations in which the number of groups is a fraction greater than 1 or less than 1. For the latter, it makes sense to ask “What fraction of a group?” instead of “How many groups?” To represent and reason about different situations, students use tape diagrams and equations. For instance, “How many s are in 2?” can be represented in these ways:

Next, students explore cases in which the size of 1 group is unknown. They learn that questions such as “If there are 3 pounds in bags, how much is in 1 bag?” and “ of what number is 4?” can also be answered by dividing and can be represented with diagrams and equations.

The section concludes with an optional lesson to practice finding the amount in 1 group and solving various division problems in context.

In this final section, students have the opportunity to apply their thinking from throughout the unit. As this is a short section followed by an End-of-Unit Assessment, there are no section goals or checkpoint questions. The first lesson in this section is optional because it consolidates the concepts from this unit with those of an earlier course. It offers additional opportunities to build fluency in interpreting, representing, and solving problems about situations that involve all four operations with fractions.

In this section, students apply their insights about multiplication and division, as well as prior knowledge about area and volume, to solve geometric problems.

Students first solve multiplicative comparison problems in contexts that involve fractional lengths in one dimension. They use their understanding of division to answer questions such as “How many times as tall or as far is this as that?”

Next, students reason about measurements of two-dimensional figures. They find the area of rectangles and triangles given base-height pairs and find an unknown base or height given the area and a known height or base.

Students then go on to reason about three-dimensional figures. They multiply and divide to find the volume of rectangular prisms with fractional edge lengths and to find an edge length when the volume of a prism and two other edge lengths are known. The work in this section builds on geometric concepts from grade 5, during which students found the area of rectangles with fractional side lengths and the volume of rectangular prisms with whole-number edge lengths.

In this section, students develop a general algorithm for dividing a fraction by a fraction, pulling together the thread of reasoning from earlier lessons.

First, students use tape diagrams to reason repeatedly about division of a whole number by a unit fraction. They look for structure and regularity in the process. Then, they do the same with division by a non-unit fraction. Students generalize that to divide a number by a non-unit fraction is to figure how many unit fractions are in (find ) and then see how many groups of unit fractions can be formed (find ).

For example, to divide 6 by is to figure how many s are in 6 (find ) and see how many groups of three s can be formed (find ).

Fraction bar diagram. 24 equal parts. Every fourth part has a solid line. All other lines are dashed. First part labeled the fraction 1 over 4. Total labeled 6.

Next, students extend their generalized steps to divide a fraction by a fraction. They conclude that to find is to find , or . Finally, students practice applying this algorithm and using it strategically to calculate quotients.