In this activity, students encounter two situations in which performing the same calculation with different values of one quantity gives the values of another quantity. Students interpret a situation with the support of tables. Through repeated reasoning, students see that they can summarize the calculations they performed with an algebraic expression (MP8) and use these expressions to answer questions about specific values. As students relate quantities in situations to numerical and variable expressions, they practice reasoning quantitatively and abstractly (MP2).

Monitor for students who use different strategies to answer the last question about the number of stickers Noah sold. Here are some likely strategies, listed from more concrete to more abstract:

• Guess a number and multiply it by 0.5 to see if it gives 127.5.
• Reason about a relationship seen in the table but is not explicit in the expression. (For example, the number of stickers is twice the number of dollars.)
• Write and solve an equation.

The goal of the discussion is to make it explicit to students that we can write a mathematical expression to represent a calculation, even if we do not know what one of the numbers is in the calculation. 

Invite students to share their expression for finding the score of an esport player who scored points fewer than Priya did, and to explain their reasoning. If no students mentioned that the same calculation—subtracting the difference in points from 473—was done to find the other students’ scores, emphasize this idea. (Consider displaying , , and for all to see.)

Next, ask previously selected groups to share their reasoning for the last question. 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 by asking questions, such as:

• “How are these strategies alike? How are they different?”
• “How does each strategy use the relationship between the number of stickers and the number of dollars?”

If no students wrote and solved the equation (or equivalent), display the equation for all to see. Ask students to explain how this equation represents the question and why and how it can be used to answer the question. Discuss questions such as:

• “What does represent?” (the amount of money collected for selling stickers) 
• “What does represent?” (the amount of money collected for selling stickers)
• ”How can we find the value of ?” (Divide each side by 0.5.) 

During the discussion, emphasize the use of terms, such as “coefficient,” “variable,” and “solution,” to reinforce students’ knowledge of mathematical language.

This activity prompts students to write expressions to find unknown values in a situation and to describe the calculation process more generally.  Then students use the expression with a variable to write and solve an equation involving the same situation. 

The activity is structured in a way that allows students to progress gradually from concrete reasoning toward abstraction and to notice regularity.  For the first three questions, students reason about numbers repeatedly before they write an expression that uses a variable (MP8).

The goal of this discussion is to make sure students see the connection between the expressions they write and the value given in the last part of each problem.

Invite students to share the variable expression they wrote for each situation and the equation in the last part of each question. Ask students how they know that each equation represents the question and can be asked to find the answer. If not mentioned by students, point out that the expression and the given value both represent the same quantity. (Consider displaying the pairs of quantities, as shown.) An equation can be written to describe that equality, and then solved to find the answer.

Then ask students to share how they solved each equation.

Continue to highlight the use of “expression,” “variable,” and “coefficient” as students discuss expressions and equations in context.