Lesson 9

This activity walks students through the process of defining a variable, writing an expression, writing the expression with fewer terms, estimating a reasonable solution, computing a solution, and finally checking that the solution makes sense and is correct. Note that there are two unknown quantities (prices for student and adult tickets) and students are guided to express one in terms of the other.

Monitor for students who write the expressions different ways in the first three questions and invite them to share during the discussion. There are many possible correct answers, but some forms will lead to equations that are easier to solve (i.e. \(px+q\)) and some will be harder to solve.

If your school or a nearby school has recently performed a play, consider asking if any students went to see it and have them briefly describe the experience. Alternatively, display photos from any school play, including images of the tickets or ticket booth. Invite students to share what they notice and what they wonder.

Give students 5 minutes of quiet work time followed by a whole-class discussion.

Students might write the expression for the cost of a student ticket as \(2−a\) instead of \(a−2\) because they confuse how to represent “2 less than” a quantity. Provide these students with values for the price of an adult ticket, and have them find the price of a student ticket. Help them see the structure of subtracting 2 from the cost of an adult ticket and not the other way around. 

For students struggling with the last two questions, remind them that they know how to write expressions with fewer terms, and that they know how to solve equations of the forms \(p(x+q)=r\) and \(px+q=r\). Support them in grappling with how these concepts and skills can come together to help them solve problems.

The purpose of the discussion is for students to reflect on the problem solving process. Consider asking the following questions:

• “Why was the price of an adult ticket chosen as the variable?” (This was an arbitrary choice.)
• “Could the problem be worked by choosing the price of a student ticket as the variable? How would the expressions be different? The equation? The solution?” (Yes, instead of -2 we would have +2 in the expression. The solution to the equation would be 3.75 instead of 5.75.)
• “What does each term in the expressions you wrote represent?”
• “How did you find the price of a student ticket?”
• “How did you check that your solution is correct?”

In this activity students continue to be guided through a solution process, but with fewer supports than in the previous activity. There are now three unknown quantities and students are asked to write an expression for their total, but students are not specifically guided to think about expressions for each individual quantity.

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

Give students 6–7 minutes of quiet work time followed by a whole-class discussion.

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. We could make a different one.)
• “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 \(8h+0.10(2h)+0.50(2h+100)\). What is the meaning of the different parts of the expressions in this situation?” (\(8h\) is the cost for the hats, \(0.10(2h)\) is the cost for the postcards, and \(0.50(2h+100)\) is the cost for the magnets.)

In this activity, 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.

While students are working, monitor for students who choose to have their variable represent different quantities (e.g. number of single stitches, double stitches, or triple stitches).

Arrange students in groups of 2. Give students 6–7 minutes of quiet work time followed by partner and whole-class discussion.

Display the following questions for all to see and have students discuss them with their partner:

• “How did you define your variable? Why did you make this choice?”
• “How did you come up with your expression for the total amount of yarn? Explain what each part of your expression represents.”
• “What decisions did you make while rewriting the expression with fewer terms?”
• “How did you, or could you, check your solution?”

After students have discussed with their partner, invite students to share with the class anything interesting they noticed while comparing with their partner. For example, did they both write the same exact expression, or did they write different but equivalent expressions? Does anyone think their partner had a particularly creative strategy for solving the problem?

If any students chose their variable to represent the number of double or triple crochet stitches, invite them to explain their solution strategy to the class and discuss how it is the same and different from having the variable represent the number of single crochet stitches.