This activity prompts students to solve a problem involving division of fractions in a less-scaffolded way. Students can see two relevant numbers to work with but need to interpret the context, the visual information, and the written question to decide whether the unknown value is the size of one group, the number of groups, or the given amount. As they coordinate various pieces of information, students practice making sense of a problem and persevere in solving it (MP1).

Monitor for the different strategies that students use to answer the question about the amount in the dispenser. Here are some approaches that students may take, from more concrete to more abstract:

• Annotating on the images to make sense of the quantities.
• Drawing a tape diagram or another type of diagram to represent the partially-filled water container.
• Reasoning verbally about the relationship between known and unknown quantities.
• Writing equations to describe the relationships between quantities.

One goal of the discussion is to draw students’ attention to any structure in the different reasoning strategies. Another goal is to clarify why the equations and can represent the situation.

Invite previously selected students to share their solutions and reasoning. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.

Connect the different responses to the learning goals by asking questions such as:

• “Many of the strategies involved dividing liter by 2. Where can we see that step in each strategy?”
• “What information did that step give us?” (The amount of water in one-fifth of a dispenser, which is liter)
• “Another common step was to multiply liter by 5. Where can we see that step in each strategy?”
• “What did that step do?” (It gave the amount in five-fifths of the dispenser, or the whole dispenser.)

Next, invite students to share their explanations on why Elena’s multiplication equation represents the question and situation. Highlight explanations that connect to the question “ of what number equals ?” which is what the question is asking. 

• “What division equation can we write to represent the question?” ()
• “How can we check if or liters is the quotient and the amount that can fit in a dispenser?” (We can find and see if it gives .)

In this activity, students practice reasoning about the amount in one group in division situations. They continue to write equations and draw diagrams to support their reasoning. In two problems (odd-numbered), the given number of groups is greater than 1. In the other two problems (even-numbered), a fraction of a group is given. Though this does not affect the structure of the equations that students write, students need to take care to reflect this information correctly in their diagrams. In doing so, students practice reasoning abstractly and quantitatively (MP2).

For each question, invite previously selected 1–2 groups to share their responses. Display students’ diagrams and reasoning for all to see. Ask the class to observe how the presented strategies compare to one another and to their own. Discuss students’ observations.

If doing a Gallery Walk, arrange for groups who are assigned the same question to present their visual displays near one another. Give students a few minutes to visit the displays and to see how others reasoned about the same two questions that they chose. Ask students to observe how their strategies are the same as or different from others’. Invite students to share their observations afterward.

Highlight that we can find the size of 1 group when given the amount in a larger number of groups (as in the odd-numbered questions), or when given the amount in a fraction of a group (as in the even-numbered questions).

This open-ended activity gives students a chance to choose a situation and a question that can be represented by a division expression, find the value of that expression, and make sense of the value in context.

By now students will have seen a variety of situations in which a division means finding “How many groups of this in that?” or finding “How much in each group?” and can refer to these two interpretations of division to get started. As students work, monitor for the interpretations chosen, the range of attributes involved (such as length, volume, or weight), and the different types of diagrams used. Select a couple of students who interpret the expression in different ways, and ask them to share later.

Invite two previously selected students to each share the situation they invented, the question, and their reasoning strategy. Record or display their work for all to see to highlight the two interpretations of . Or consider using the following examples to illustrate them:

• Finding the size of 1 group: If liter fills of a bottle, how many liters fill 1 bottle?
• Finding the number of groups: If 1 bottle contains liter of water, how many bottles can be filled with liter of water?

In this case, students will likely find the first interpretation of division easier to represent and to solve using a diagram.

The first diagram shows the content of of a bottle, which is liter, being multiplied by 4 to get the content of 1 bottle, which is liters.

The second diagram shows that liter fills 1 bottle, so 1 liter fills 4 bottles, and liter fills a third of the 4 bottles, which is bottles. (Students might also think of as , and as , and then reason that there are groups of in .)