The relationship in Table D may not be obvious to students. Encourage students who get stuck to look at the equations in the last question and to try to figure out which equation describes the relationship between the numbers in the table.

Invite previously identified students to share how they thought about one of the relationships in the first question. Start with students who reasoned only in terms of numerical operations, and move toward those who interpreted the quantities in context (as shown in the Activity Narrative). If possible, record and display their descriptions for all to see, and highlight the connections between the different responses. 

Discuss with students whether or how their ways of thinking about each relationship affected the work of matching the tables and equations. If not brought up in students' comments, point out that some ways of describing a relationship could make it easier to identify or write a corresponding equation. To really understand what's happening in the situation, however, often requires carefully interpreting the operations that relate the two quantities.

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Invite 2–3 students to share the norm they chose and how it supported the work of the group or a realization they had about a norm that would have worked better in this situation. Provide these sentence frames to help students organize their thoughts in a clear, precise way:

• “I picked the norm ‘.’ It really helped me/my group because .”
• “I picked the norm ‘.’ During the activity, I realized the norm ‘’ would be a better focus because .”

If students assume that the relationships must be expressed as equations and they get stuck, clarify that verbal descriptions, tables, or other representations are just as welcome.

When answering the first question, students may look only at the relationship between the first few rows of the table and say that the -values are decreasing by 8 each time, not noticing that this is not always the case. Encourage them to look farther down the table and to also look at the relationships between the values in the columns.

Some students may have trouble getting started on the question about the volume of milk or setting up a table. Ask students which units are given in the problem, or suggest the headings “gallons,” “cups,” and “fluid ounces.” Then, ask them to use the given information to complete a row in the table. This might involve trying a different unit to start with. (For example, if they start with gallon and struggle to find the equivalent amount in cups and fluid ounces, try starting with "4 cups" or "8 cups.") A more direct hint is to suggest finding the number of fluid ounces in 8 cups.

Invite previously selected students to share their responses and thinking. Sequence the discussion of the strategies by the order listed in the Activity Narrative. If possible, record and display their work for all to see. Remind students to specify what the variables represent if they use equations.

If no students considered using tables to make sense of the pairs of quantities, ask how these would help, and show an example. A table, for instance, can be particularly helpful for reasoning about the last two relationships.

Display the equations that can represent the three situations. Highlight that writing equations is an efficient way to capture the constraints in a situation.

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

• “As the number of winners increases, how does the amount of the winnings change? How can you see that in each representation?” (The winnings decrease as the number of winners increases. This can be seen in the table by going down the rows or in the equation because the number of winners is in the denominator.)
• “How many cups are in 1 gallon? How was this used for the different representations?” (16 cups in a gallon. This was used when multiplying 32 fluid ounces by 4 to be equivalent to 16 cups.)
• “How would you use each representation to find the number of fluid ounces in 5 gallons?” (Draw another row on the table for 5 gallons and multiply 128 fluid ounces by 5. Substitute 5 for in the equation.)
• If equivalent equations are given for the same relationship, ask, “How can we see that these equations are equivalent?” (Dividing each side by shows that is equivalent to .)

To help students connect the equations to prior work, ask students which quantities vary and which remain constant in each equation. Point out that these equations are also equations in two variables, but unlike the equations we saw in the previous activity, not all of these represent linear relationships.