In this activity, students see a situation in which a percent change is applied twice, first on the initial amount, and then on the amount that results from the first percent change. The multiplicative way of expressing the change is particularly helpful in such a situation. The example uses a familiar context of sales and discounts. The repeated decay results from applying a factor that is less than 1 more than once.

Students may have chosen to calculate the price after each discount separately. For instance, they may have found the cost after the first discount to be \$24 (because , or ), and then calculated , or . To help them see the repeated multiplication by a factor, consider recording their steps without evaluating any step.

Focus the discussion on different expressions that can be used to efficiently calculate the final cost of the book, for instance:

In this activity, students use exponential expressions to describe and make sense of repeated percent increase in a borrowing context. An essential point here is that, in each repetition, the value being increased by a percent is not the same as the initial value. Instead, it includes previous increases.

To express this repetition more generally, it is important for students to represent the percent increase using multiplication. For example, if a baseball card is valued at \$15 and increases in value by 10% each year, there are many ways to write the value of the card, in dollars, after one year including:

The first three expressions are not particularly helpful for finding the value after 2 years or 3 years. The last two expressions, however, can be applied any number of times, giving the value, after years:

Look for students who write expressions using multiplication, and invite them to share during the discussion. Note that the expression is a very important way to express a 10 percent increase, even though it involves addition, because it calls attention to the percent increase. It is sometimes generalized to formulas such as , although that formula is not presented in these materials. If any students write 1.18 as in this task, make sure to invite them to share during the discussion.

Students continue to explore financial situations that involve repeated percent increase. They compare the effects of repeatedly applying different interest rates to a balance and graph the balances to see the  amount owed over time. Students see that higher interest rates have an increasingly dramatic effect when compounded over many years on an unpaid balance.

When graphing the functions, choosing an appropriate window may be challenging because the functions grow at very different rates. A small domain and range may appropriately show the 12% loan, but the same domain and range can result in the graphs for the 24% and 30.6% loans leaving the screen quickly. A large domain and range may appropriately show the 30.6% loan balance but make the 12% loan balance rate look constant.

Students could experiment with different domain and range values, though that may not be efficient. Prompt them to consider calculating the values of some expressions in the table to help them set an appropriate window for showing all three graphs meaningfully.

Students will likely produce continuous graphs although in the context the number of years is a whole number. The continuous graphs help to visualize how the different interest rates influence the balance. If some students plot only the end-of-year balances, consider inviting them to share their graphs during the discussion.

In this activity, students investigate the average rates of change of the three loan options from an earlier activity. Calculating average rates of change over the same interval of time but at different points in the lending period (early and then later) further reinforces the dramatic change that can result from compounding over time.