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.

Some students may have trouble finding the general expression for the amount owed after years. Suggest that they work with expressions that use the original $450 for each year of the loan until a pattern becomes apparent.

The discussion should aim to clarify the path toward the final expression . Here is one way to clarify how the process of finding the amount owed for any year could be generalized, and how the generalized expression simplifies the calculation:

Discuss questions (for the Year 2 calculations) such as:

• “In Step 2, what does the 450 (1.18)(0.18) represent?” (It represents the amount of interest added in the second year.)
• “In Step 3, how does the (1 + 0.18) come about?” (Because both things being added had , the expression could be rewritten using the distributive property with that part multiplied by the sum .)
• “In Step 4, how can we interpret the  ?” (This means that the new amount owed after 2 years is 100% of the initial amount as well as 18% interest applied twice.)

Make sure students understand the meaning of each part of the final expression in terms of the context:

• 450 is the loan amount in dollars.
• 1.18 is how much the amount owed on the loan changes after each year. Because no other payments are made, the initial balance is unchanged (this is the 1) and the 18% interest is applied on that initial balance (this is the 0.18).
• represents the number of years the 18% interest has been applied.

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.

Students may attempt to write very long expressions in the last two rows of the table. Encourage them to think back to other notation that can be used to represent multiplying by the same value multiple times.

Select groups to share their responses. To involve more students in the discussion, ask if others obtained the same results or reasoned the same way. If not, invite them to share their responses or approaches.

Review the meaning of the expressions for the loan balance after years without payment. Make sure students understand that, for instance, the balance of the 12% loan after years is because the initial $1,000 is multiplied by 1.12 times, once for each year that the 12% interest is charged.

Display the graphs for all to see. Use them to highlight the effect that the different interest rates have on the owed amount, especially when the loan is left unpaid for a long period of time. This will be a recurring theme in the next several lessons. Also make sure that students can use the graphs to estimate the time that it takes each loan balance to double.

Select students to share their calculations for the average rates of change for each loan. Invite students to share their observations about how the different average rates of change compare. If not mentioned by students, ask them to quantify how the average rate of change increased between the two intervals for the different loans, such as by calculating how many times larger the value for the second interval is when compared to the first.

Consider using technology to graph the functions and visually represent the average rates of change by connecting the specified points with a secant line and examining the slope of that secant line.