This activity guides students through a process of writing an equation to solve a problem with three unknown quantities. While this activity has students write their equation in terms of the number of hats, there are other ways to approach this situation. In this activity students reason abstractly and quantitatively when writing expressions and finding values (MP2).

Monitor for students who compute the profit in different ways, such as:

• Finding the total income and total cost, and then subtracting them.
• Finding the profit for each item, and then adding them together.

Invite students to share their answers to the last two questions (and others, if they wish). Consider asking questions like the following:

• “Is it a good idea to check your solution by substituting values into the expressions and equations you wrote?” (No, you might have written or simplified them incorrectly. The best option is to go back to the original problem.)
• “Why was the number of hats chosen as the unknown quantity to represent with a variable? Would you have chosen differently? How would that change the solution process?” (This is just one choice. A different one could be made.)
• “How did you compute the profit?” (Compute the total income minus total cost for all items, or add item by item.)
• “I saw that many of you wrote this expression . What is the meaning of the different parts of the expressions in this situation?” ( is the cost for the hats,  is the cost for the postcards, and is the cost for the magnets.)

In this activity, students write an equation to represent a situation and solve a problem. Most of the supports for the solution process are removed. Students need to think about how to choose which quantity to represent with a variable, how to represent the other two quantities with expressions in terms of the variable, and how to write an expression for a total. The only guidance offered is the reminder to write the expression with as few terms as possible. Students build on the structure they used in the earlier activities as they write and solve their expressions (MP7).

The goal of this discussion is to show that the same situation can be represented mathematically in different ways while arriving at the same answer. Invite several students to share their expressions that represent the amount of yarn Jada needs to crochet a scarf. Here are some questions for discussion:

• “What do the representations have in common? How are they different?”

• “Are there any benefits or drawbacks to one representation compared to another?”

• “How does the inch less yarn show up in each representation?”

• “How could you check your answer?” (Use the information from the problem stem to find the length of the other two crochet stitches and use the number of each type of stitch to see if the total length matches the 5540 inches of yarn Jada uses.)

If it does not come up during the discussion, ask students about the purpose of the different forms of the expression. For example, the longest expression is helpful for seeing how each term connects to each part of the situation, while the equivalent expression with less terms is helpful for solving for the value of the variable.