The Warm-up prompts students to think about equivalent ways to express a compounded interest calculation. The given expressions describe the same initial investment for the same period of time, but one expression counts the number of months while the other counts the number of years. This work prepares students to look at different compounding intervals in this lesson and upcoming ones.

If time is limited, focus only on the first question.

Invite groups to share their explanations for why Tyler's and Elena's expressions both represent the account balance after 3 years. If not already mentioned in students explanations, highlight the following:

• In Elena's expression, correctly represents the 1% interest applied or compounded every month for 36 months, which is 3 years.
• In Tyler's expression, represents the 1% interest compounded every month for 12 months or a year. So for 3 years, we need to multiply the initial balance by that yearly rate 3 times, or , which is equivalent to.
• By the power of powers property, we also know that .
• (If time permits:) Evaluating gives approximately 1.1268. This means that 12.68 is the effective annual rate, which is the rate that Kiran used in his expression.

Building on their work in the Warm-up, students make several compound-interest calculations in a credit card context. They revisit and explore how nominal and effective interest rates are used by credit institutions.

In a borrowing context, the nominal annual percentage rate, or the nominal APR, is 12 times the monthly interest rate. The effective annual percentage rate is the compounded interest rate if no payments are made over a year. As in other compounding situations, the nominal APR is lower than the effective annual rate (the actual rate cardholders pay). For this reason, credit card companies usually report the nominal APR rather than the effective annual rate to make the card more appealing.

Students also practice writing different expressions to represent the same quantity in an exponential situation. Look for these variations in students' work, and ask them to share later:

• about

This task prompts students to build mathematical models to compare two different interest options. Along the way, they need to make assumptions, most notably about the length of the investment because the better investment option may depend on what they assume to be true.

The given rates and compounding intervals are chosen so that it is not clear how the investments would pay for an investment time that is not a multiple of the compounding interval. For example, if the investment is left for 9 months, the option that compounds interest every 3 months will make 3 payouts, but what about the option that compounds interest every 4 months? Students might assume that each method pays only at the end of the compounding interval. In that event, the option that compounds interest every 4 months would pay interest only twice.

Notice how students deal with the length of the investment. Some students may recognize the structural similarity between the situation here and that in the previous activity (that a 1% monthly rate leads to a higher interest than a 12% annual rate does) and reason accordingly.

In asking "What if?" questions, stating their assumptions, and applying what they know about exponential growth to solve a real-world problem, students are engaging in aspects of mathematical modeling (MP4). While the investment context is authentic and practical, the rates in this task are theoretical, because they are chosen to enable students to find and compare effective annual rates.

In this activity, students interpret an exponential equation in context. The equation describes college tuition cost as a function of time in years. Students are invited to examine how the tuition changes each decade.

Along the way, students observe that the rate of change is the same for any decade and that the time it takes tuition cost to double remains consistent. (If desired, this may be an opportunity to introduce the Rule of 72, a method for estimating the doubling time of a quantity that grows exponentially. In an earlier lesson, when comparing the graphs of the owed amounts at 12%, 24%, and 30% interest rates, students made estimates of when each loan would double. If the Rule of 72 is introduced, consider referring back to that activity to test the rule.)  Students may also notice that the current rate of growth for tuition is likely unsustainable.

For the second question, students may reason inductively (by evaluating the expressions for certain 10-year periods) or deductively (by using the structure of the expressions and properties of exponents). Identify students who use each approach so they can share later.