In this activity, students use the distributive property to rewrite an expression that involves a difference and turn it into an expression that uses only multiplication. They examine and explain a worked example, and then they practice using the method to rewrite expressions.

Students may have trouble making the connection between the fraction that is being subtracted and the fraction that remains when the subtraction is rewritten as multiplication. Invite these students to think through the steps in Diego's calculation. What does the number 100 represent? What about the ? After students identify the analogous quantities in the second and third questions, encourage them to revisit the subsequent steps of Diego's calculation.

Invite students to share their explanation of the steps in the first question and how they used similar reasoning for the other situations. Make sure that students understand that a decreasing quantity can be expressed both ways, using subtraction or multiplication.

Reinforce the steps used in the first question by using this example: “There were dogs at the park and  of the dogs were on a leash. How many dogs were not on a leash?” Tell students that the number of dogs not on a leash could be expressed, using subtraction, as , but that's not a handy way to express it when working with a situation that is changing by a constant factor. Instead, we can factor that expression and write it as   to more clearly show how the value is changing over time. After combining terms and using the commutative property, we have an expression using only multiplication.

Emphasize that it is important to be able to express this situation using multiplication because exponential situations involve quantities changing by a common factor (or being multiplied by a number).

In this activity, students examine a situation in which a quantity decreases by repeated multiplication by the same factor. They represent the pattern of decrease with an equation and interpret the different parts of the equation in terms of the context (the depreciation of a car). The definition of growth factor is expanded to accommodate situations involving decrease.

The purpose of the discussion is for students to understand that when a quantity is decreased by a fraction with , the quantity is multiplied by to find the reduced value.

Questions for discussion:

• “Why does it make sense to multiply an entry in one row by to get the entry for the next row?” (Each year, only of the car value remains.)
• “Why not write the expression for each year as subtraction, for instance, ?” (We could, but writing subtraction can get unwieldy quickly. For year 2, the expression would be . If we calculate the value of each term it would be simpler ( ), but it would still not allow us to see a pattern and write a general expression as easily.)
• “Does the in the last equation mean losing of the value each year? Why or why not?” (No. Multiplying by repeatedly means each time only of the value remains, so the car is losing of its value each year.)
• “Will the value of the second car (starting at $10,000 and losing of it each year) ever be worth more than the first car (starting at $18,000 and losing  of it each year)? Explain.” (Yes. After 4 years, the value of the second car will be $4,096 while the value of the first car will be $3,555.56)

Make sure that students understand that the exponential expression for the value of the cars after years came from multiplying by the growth factor (, or $\frac$) repeatedly ( times).