Lesson 23

The purpose of this algebra talk is to remind students that expressions can be written in different, equivalent ways, and to give them an opportunity to recall relevant terminology like “distributive property.”

Display one problem at a time. Give students 30 seconds of quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate _____’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to _____’s strategy?”
• “Do you agree or disagree? Why?”

The purpose of this task is to encourage students to represent each situation using an expression with a variable, so that the calculation methods can be shown to be equivalent in a straightforward way using familiar properties. This approach will be useful for tackling the next activity. Students may find other ways of explaining why the calculation methods are equivalent, but the synthesis should highlight explaining why two expressions that use a variable are equivalent.

Depending on the time available, ask each student to respond to 1, 2, or all of the situations. Note that the last situation is a bit more difficult than the first two.

For students who have trouble getting started, ask them to first calculate using a few specific values. For example, calculate the temperature in Fahrenheit if the temperature in celsius is 0 degrees, 5 degrees, 10 degrees, and then use the same operations to write an expression for \(x\) degrees. Another option would be to provide a bank of expressions for students to choose from, rather than asking them to generate the expressions.

For each situation, invite at least one student to share their reasoning for why the two calculation methods are equivalent. Be sure to include, for each situation, an approach that involves writing expressions that contain a variable and using properties of operations to show that they are equivalent.

Spend a little time on the last situation, emphasizing that 20% off an amount is the same as 80% of the amount. That is, if you want to compute \(x\) decreased by 20%, you can write \(0.8x\), because \(x-0.2x\) is equivalent to \(0.8x\). This insight will help students write less-complicated expressions in the next activity.

The purpose of this activity is to use expressions with a variable to represent applying coupons in a different order. By analyzing these expression, you learn that no matter how much you spend, you always pay $6 less if the 20% off coupon is applied first.

Monitor for how students are approaching the problem. If, once students have an answer, they have not written expressions with a variable to show the best way to apply the coupons, consider asking them to try writing an expression using a variable to represent the purchase amount.

Students should be familiar with the idea of coupons and discounts from their work in an earlier unit. If this is not the case, more time may be needed for the launch to familiarize students with the context.

Students may notice that if you spend less than $30, the store probably won’t let you take $30 off. This is a possible constraint, and students who include this constraint are engaging in aspects of mathematical modeling (MP4).

Display the image of two coupons for all to see. Ask students to think of some things they notice and some things they wonder. Things students might notice:

• There are two coupons to the same store.
• One coupon is for 20% off and one coupon is for $30 off.

Things students might wonder:

• Can you use both coupons on the same purchase?
• Do the coupons expire?
• Is there a minimum or maximum amount you have to spend?

If no students wonder this, ask, “What if you could use both coupons on the same purchase? Should you deduct 20%, and then $30 from the result? Or the other way around? Does it matter?”

If students have trouble getting started, suggest that they calculate which order is better for some specific purchase amounts.

If any students only computed their resulting bill using a specific dollar amount, ask them to present their solution first. For example, on a $100 purchase, this might look like:

• 20% off is $80. Then $30 off of $80 is $50.
• $30 off is $70. Then 20% off of $70 is $56.

So if your purchase was $100, it’s better to apply the 20% off coupon first. What about other purchase amounts? (If students tried other purchase amounts, consider also having them demonstrate. It is helpful if students see a few different examples that always result in a $6 difference.)

Select a student to present who wrote an expression using a variable for the purchase amount. If no students did so, demonstrate this approach.

• Let \(x\) represent the amount of the purchase.
• 20% off is \(0.8x\) or equivalent. $30 off that is \(0.8x-30\).
• $30 off is \(x-30\). 20% off that is \(0.8(x-30)\). By the distributive property, this is equivalent to \(0.8x-24\).
• Comparing \(0.8x-30\) to \(0.8x-24\), your resulting bill will always be $6 less if you use the 20% off coupon first.