This activity makes explicit the two ways of interpreting division in terms of equal-size groups: as a way to find the size of one group and a way to find the number of groups. Students explore these two ways of seeing division by connecting it to multiplication, making sense of it in the context of a situation, and drawing visual representations.

While tape diagrams are used to help students see what could mean, students may choose to draw other kinds of diagrams to represent other division expressions in the activity.

As students interpret equations and diagrams that represent the quantities in a situation and create their own representations, they practice reasoning abstractly and quantitatively (MP2).

The goal of this discussion is to make sure students recognize that a division expression such as can be interpreted as the answer to the question “4 times what number equals 12?” or the question, “What number times 4 equals 12?"

Select a few students to share their diagrams and equations for the problems in the last question. After each explanation, highlight the connections between the expression, the diagram, and the context.

Note that students may write either or for each interpretation as long as they understand what each factor represents. Because in these materials we tend to say “ groups of ,” we follow that order in writing the multiplication:

Point out that, in general, division can be interpreted as a way to find two values:

• The size of each group when we know the number of groups and a total amount
• How many groups are in a total amount given the size of one group

In this activity, students continue to investigate division problems in context, think of them in terms of equal-size groups, and represent them using diagrams and equations.

Both division problems result in quotients that are not whole numbers, but because they represent concrete quantities, students can make sense of the fractional values (number of groups and size of each group) in terms of each situation.

As students work, monitor for students who draw diagrams as they reason about the situations. Any diagram (concrete or abstract) is fine as long as it enables students to make sense of the relationship between the number of groups, the size of one group, and a total amount. Select students whose diagrams clearly show the meanings of division, and ask those students to share later. During class discussion, students will have an opportunity to look more closely at how tape diagrams can support thinking about division.

Some students may round their answers to the nearest whole number rather than including a fraction of a scoop or of a gram. Ask students to consider if it is possible to have a part of a scoop or a part of a unit of volume. Encourage them to think about how to show a part of a whole unit on a diagram.

The goals of the discussion are to solidify students' understanding of the two interpretations of division (“How many groups?” and “How much in a group?”) and to clarify that division may not result in a whole-number quotient.

Invite students to share their responses and reasoning. Ask students who drew effective diagrams to display and explain them.

If no students drew tape diagrams, display the following diagrams as another way to reason about division.

Discuss what each part of the diagram and each label represent:

• The entire length of the tape represents the total amount.
• Each segment of the tape that is the same size represents one group (a packet or a scoop).
• The number inside each rectangle represents the size of one group, which can be a fraction. 14 grams in 4 packets means grams in 1 packet.
• The number of segments tells us how many groups there are, which can also be a fraction. There are 6 full scoops and 2 extra fl oz in 26 fl oz. The 2 extra fl oz is scoop. There are scoops in 26 fl oz.

Ask students how they can tell if the results they get from dividing 14 by 4 and 26 by 4 are correct. If no students mention multiplying each result by 4 to see if the product is 14 and 26, respectively, discuss this idea.