In this activity, students explore division situations in the context of recipes. In each situation, the number of groups and a total amount are given, but the size of 1 group is unknown.

Students write multiplication equations in which the unknown factor answers the question “How much in each group?” instead of “How many groups?” They also write division equations and create tape diagrams that represent the given situations. No grid is provided here, but students can use graph paper or refer to the fraction strips from an earlier lesson to support themselves as they create their own diagrams. In doing so, they practice choosing tools strategically (MP5).

As students work, monitor how they start their diagrams and which quantity they represent first. If they are not quite sure how to show a particular quantity, ask them to refer to earlier diagrams and notice how the number of groups, the size of each group, and the total amount were represented on a single diagram. Select students whose diagrams would be instructive to others, and ask them to share later.

When writing multiplication equations, students might simply use the smaller number as a factor and the larger number as the product without attending to what the numbers mean. Support their reasoning with an example that uses only whole numbers. For instance: “To make 2 batches of a drink, we need 6 cups of water. How many cups are needed for 1 batch?” This can be interpreted as “2 times what number equals 6?” and expressed as . Follow with an example using a non-whole-number total amount. Ask students what multiplication equation can represent this question: “To make 2 batches of a drink, cup of lemon juice is needed. How many cups are needed for 1 batch?” ()

Select previously identified students to share their diagrams and reasoning, or display the ones in the Student Response for all to see.

Next, invite students to compare an example from an earlier lesson and one from this activity. For instance, display the following:

• How many groups of are in 1?

• Two tablespoons of cornstarch make batch of glue. How many tablespoons of cornstarch are needed to make 1 batch?

Discuss questions such as:

• “How are the situations alike?” (They both involve and 1. They are both about how many of something are in 1.)
• “How are the situations different?” (In the first situation, the is the amount in 1 group. We are looking for the number of groups in a total amount of 1. In the second, the is the fraction of a group with 2 tablespoons in it. We are looking for how much is in 1 group.)
• “How are the diagrams alike?” (The same fraction of the diagram is shaded. Both diagrams show of 1.)
• “How are the diagrams different?” (The shaded portion represents different quantities. In the first diagram, it represents 1 group. In the second, it represents of a group. What is unknown is different.)

Emphasize that attending to the meaning of given numbers and their relationships can help us see what we are looking for and how to find an unknown value.

This activity allows students to find the size of 1 group in a concrete and familiar context and in which the total amount is a whole number. Students are prompted to match each of three descriptions with a diagram that represents the same situation. In all three situations, the context is about filling water in containers, and the given total amount is 15 cups. This amount, however, represents different numbers of groups, so students need to make sense of each problem situation carefully (MP1).

After matching the given representations, students write equations and answer the question for one situation of their choice.

Display the three diagrams for all to see. For each diagram, invite a student to share the description that matches the diagram and their reasoning. Ask another student who chose that situation to share the corresponding equations and the answer to the question. Record students’ responses near the diagrams.

Discuss any unresolved disagreements about the matches, equations, or answers. To highlight the common structure across the situations and their representations, ask students:

• “In all three situations, what information was unknown?” (The size of one container)
• “What do the multiplication equations have in common?” (They all have 15 as the product and one unknown factor.)
• “What do the division equations have in common?” (They all have 15 as the dividend or the number being divided.)
• “How can we tell if our answer to each question is correct?” (We can record the value in the diagram and see if the relationship of the numbers makes sense. We can substitute the value for the “?” in the multiplication equation and see if it gives a true equation.)

This activity prompts students to find the size of 1 group in the context of length and in which the total amount is a fraction less than 1. Students are asked to match each of three descriptions with a diagram that represents the same situation. In all three situations, the context is about cleaning sections of a highway, and the given total amount is mile. This amount, however, represents different numbers of groups, so students need to make sense of each problem situation carefully (MP1).

After matching the given representations, students write equations and answer the question for one situation of their choice.

Display and read the question: “In general, how would you write a multiplication equation and a division equation to represent any situation in this activity?”

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to this question. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.

If time allows, display these prompts for feedback:

• “ makes sense, but what do you mean when you say . . . ?”
• “Can you describe that another way?”
• “How do you know . . . ? What else do you know is true?”

Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer. If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.

Here is an example for a second draft: “For multiplication, I would write: . The length of a section is not known, so I would put a ? there. For division, I would divide the total length by the number of sections to get the length of one section.”

After Stronger and Clearer Each Time, highlight that we can check if an equation accurately describes a given situation, such as by seeing whether:

• The relationship in the equation matches the relationship in a diagram that represents the same situation.

• The solution to the equation makes sense in the diagram and in terms of the situation.