In this activity, students show that two different ways of calculating something are equivalent. For example, and are equivalent ways of estimating a temperature in Fahrenheit when you know , the temperature in Celsius.

The calculations are presented with words, so it’s advantageous to express them with algebraic expressions to reason abstractly and quantitatively (MP2).

Students may find other ways of explaining why the calculation methods are equivalent, but the Activity Synthesis should highlight explaining why two expressions that use a variable are equivalent.

Monitor for students or groups who:

• Substitute sample values into the calculations to see if they get the same numerical result.
• Write pseudo-expressions like “” or “10% of + 5% of .”
• Write expressions with variables.
• Use plain-language descriptions of properties and moves like “switch” and “smush.”
• Name properties and moves like “distributive property,” “combine like terms,” and “commutative property.”

The goal of this discussion is to help students understand why situations can be represented by expressions with variables, and to identify properties that indicate two expressions are equal. 
 
Display 2–3 approaches from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the different representations. Here are some questions for discussion:

• “How are the expressions the same? How are they different?”
• “Did anyone solve the problem the same way, but would explain it differently?”
• “How does the distributive property/combining like terms show up in each method?”
• “How do these different representations show the same information?”

In this activity, students use expressions with a variable to represent applying coupons in different orders. By analyzing these expressions, students learn that no matter how much we spend, we 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 may notice that if we spend less than \$30, the store probably won’t let us take \$30 off. This is a possible constraint, and students who include this constraint are engaging in aspects of mathematical modeling (MP4).

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

The purpose of this discussion is to compare expressions to help students understand why the bill will always be \$6 less if we use the 20% off coupon first.

If any students only computed their resulting cost 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. (See Student Response.)