Much discussion takes place between partners. Invite students to share how they connected the representations of the situation. Once all groups have completed the matching, discuss the following:

• “Which matches were tricky? Explain why.”
• “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?”
• “How did you determine what would be the coefficient of in the equations?” (Because each situation involved items being purchased, the cost for each item is multiplied by .)
• “The equations are written from the store’s perspective (‘How much should we charge this person for all of these items?’). From the buyer’s perspective, it may be better to think of it as something like, ‘I have $120 to spend on shirts. How many shirts can I get?’ How can you use the equation you matched with the shirts equation to answer this question?” (I can substitute $120 for in the equation to get . Then I can solve that equation to know that the person can buy 20 shirts.)
• “The equations and are somewhat similar. What in the situation determines the difference between them?” (Since represents the total cost to the customer, 3 is subtracted when there is a discount or coupon being used, and 3 is added when there is an extra expense.)
• “Describe any difficulties you experienced and how you resolved them.” (I originally thought about the situation with the socks as being $2 for each pair of socks, but I didn’t see any equations involving the number 2. I then realized that the equation was written to include the information about $6 for 3 pairs.)

In this activity students rearrange equations to isolate a variable. In the associated Algebra 1 lesson students rewrite equations to find the inverse function. This activity allows students to focus on the mechanics of the rearrangement so they can consider the solutions in a broader context for the Algebra 1 lesson. As students rearrange the equations they reason abstractly and quantitatively (MP2).

The purpose of the discussion is to share the methods that students use to rearrange the equation. Select students to share the solutions, including any intermediate steps they used to arrive at the solution. After each solution is shared, ask if there are other methods students used to arrive at the solution. For example, with the equation some students may distribute the 2 first and then work to isolate , while others may immediately divide by 2 then solve for .

If there is additional time, these questions can help students connect their solutions to the context:

• “One of these equations represents a situation in which each customer gets 1 item free then must buy additional items for $2 each. Which equation matches that situation?” ( )
• “Using the equation , how much does each item cost?” (50 cents since the equation can also be written as which means it costs half of a dollar for each item.)
• “The equation might represent a situation in which each item costs $1 and the customer has a $2 coupon for the total order. In terms of the situation, explain the meaning of the addition in the rewritten equation .” (The number of items the person can purchase is equal to the number of dollars they have plus 2 additional items they can get from the coupon.)