Lesson 8

By now students have written many division equations based on verbal descriptions of situations. This warm-up prompts them to go in the other direction: to interpret a division expression and write a fitting question the expression could help answer. Then, they trade descriptions with a partner and reason about the value of the same expression in someone else’s story. 

As students write their descriptions, look out for scenarios that are unclear and ask their authors clarifying questions. Select 2–3 students whose descriptions clearly match the expression to share later.

Students may struggle to answer their partner’s question because the descriptions are confusing or do not match the given expression. Encourage them to ask the questions they have about the description, but still try to find the answer to the given expression.

Arrange students in groups of 2. Explain to students the two halves of this activity: to write a situation and a question that can be represented by the equation \(12 \div \frac23 = ?\), and to trade papers with their partner and find the answer to their partner’s question.

Give 2 minutes of quiet think time for the first question, another minute for the second question, and follow with a whole-class discussion.

Some students may struggle writing a context. Ask them to reflect on the work they have done and what the division expression means for them, or suggest that they write related multiplication expressions to help them understand the division expression.

Ask selected students to share their stories with the whole group. After each student shares, have another student explain how the story matches the division equation. Afterward, ask these and other students for the value of the given expression, and write and display the completed division equation for all to see.

If not already mentioned by students, highlight that the answer is the same for all scenarios because they are all based on the same division expression.

In this task, students explore division situations (in the context of baking cookies) where the number of groups and a total amount are given, but the size of 1 group is unknown. They write multiplication equations in which the missing factor answers the question “how much in each group?” instead of “how many groups?”

Students continue to use equations, diagrams, and the connection between multiplication and division in their reasoning. No grid is provided here, but allow students to use graph paper to support their diagram drawing if desired or needed.

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 to share later.

Arrange students in groups of 3–4. Give groups 5–6 minutes of quiet work time and 2 minutes to share their responses in their group. Provide access to geometry toolkits (especially graph paper and colored pencils).

When writing multiplication equations, some students might simply use the smaller number as a factor and the larger number as the product. Or, when writing division equations, they might simply divide the larger number by the smaller number without attending to what the numbers mean. Encourage them to think about what the numbers represent. To help them see the connection between verbal statements and mathematical equations, support their reasoning with simple examples, e.g., “To make 2 batches of cookies, we need 6 cups of flour. How many cups are needed for 1 batch?” This can be interpreted as “2 groups of what equals 6?” and then \(2 \boldcdot {?} = 6\).

Select previously identified students to share their diagrams and reasoning, or display the ones here for all to see. If other students used alternative approaches, invite a couple of them to share.

To highlight some key ideas in this lesson and in the unit, discuss:

• “How are these diagrams like those in previous lessons? How are they different?” (Alike: We can use them to show a total amount, the number of groups, the size of 1 group. Different: We don't know the value of 1 group here. In past diagrams, we were given the size of 1 group, but needed to find the number of groups.)
   
• “How are these equations and the ones in previous lessons alike and different?” (Alike: The multiplication equations have one unknown factor. The division equations have the same setup. Different: The unknown factor here represents a different quantity than in past lessons.)
   
• “How are the diagrams for the second and third questions different from the one in the first question?” (In the first one, the given amount is greater than the size of 1 batch. In the other two, the given amount is less than the size of 1 batch or is a fraction of a batch.)​

In this activity, students continue to reason about the size of 1 group in situations involving equal-sized groups. In the first part, the given total amount is a whole number. In the second, the given amount is a fraction. The visual representations for both parts are very similar, allowing students to notice the structure of the relationships and generalize their observations (MP7).

Display the two sets of images in the activity. Give students a minute to observe them. Ask them to be prepared to share at least one thing they notice and one thing they wonder. Then, solicit a few observations and questions from students.

Keep students in groups of 3–4. Give students 3–4 minutes of quiet think time for the first set of questions (about water in containers) and ask them to pause for a brief group discussion before continuing on to the second set (about highways). Tell students to compare their responses and discuss any questions or discrepancies until they reach an agreement. Afterward, give them another 4–5 minutes of quiet think time to complete the remaining questions and, if possible, time to discuss their responses in groups.

Most of the discussions are to take place in small groups. Reconvene as a class to discuss any unresolved disagreements or common misconceptions, and to highlight the following points:

• “In both sets of questions, what information was missing?” (The size of 1 group.)
• “How were the division equations in the two contexts different?” (In the water context, the dividend, which is the given amount of water, is a whole number and the quotients are greater than 1. In the highway context, the dividend is a fraction and the quotient could be greater or less than 1.)