# Pivot Algebra Two: Payoffs

The Algebra II Experiment continues. No major updates to speak of because I ironed out the format in the fall. Now I'm just tinkering with content. How complex do I want to get with rational functions, for example. This semester I sought to improve the credibility of the financial applications that come up during exponential and logarithmic functions.

In my most recent textbook makeover, I looked at more authentic approaches to exponential growth and decay. My main criticism of textbook growth and decay comes with financial applications. Often students are to predict future value given a starting amount of money and interest rate. Sometimes they might mention compound interest.

None of those situations model how money actually grows, since anyone doing it right is going to make regular contributions.

In researching exponential/logarithmic applications, I stumbled upon two good situations: loan payoff time, and mortgage payments.

You can find dozens of websites with calculators that will tell you how many months it will take to pay off a credit card or an estimate of your mortgage payment. But how do these calculators work?

### Authentic Scenarios

In 2014, credit card statements started including new information:

In an attempt to save you from yourself, the bill sends the message that the minimum payment is a Bad Idea™. But where did 3 years comes from? And \$478? Credit education is probably number two behind "count change" in a national "what should kids learn in math" survey.

Houses are another interesting study. What's the real cost of a house? Are houses the profit center everyone claims? Are you really missing out by "throwing your money away" in a rental?

First, the two formulas you need. These were really hard for me to find. It's like banks don't want you to know? That can't be right.

Payoff time, n, for principal, P, given APR, r, and constant payment, x.

Required payment, x, to payoff principal, P, given APR, r, and number of payments, n.

The left formula I used to generate the 3 years and \$478 from the credit card bill. The right formula to generate the expected mortgage payment for a certain loan amount.

### Talking Points

To open, we discussed credit cards. Specifically, how interest is determined, and how high interest rates can be, even if you're low risk. Compound interest enters by defining the term "APR" which some students knew, but most didn't. I started with a \$500 purchase making \$10 payments. We discussed the work manually. Deduct \$10, calculate interest on the remaining balance. Deduct \$10, calculate interest. Making \$20 in payments only nets a \$7 reduction in principal. That surprised everyone.

Houses was an even better discussion. Interest rates are lower, but the idea of paying for something over 15 or 30 years was a surprise. My students were not aware the loans were that long. You want to drop some jaws, demonstrate that a \$200,000 loan at 4% costs you \$344,000 to own.

### Challenges

The math here was tricky for them. Conceptually, they understood what we were doing. Mechanically, it was tough. Lots of breaking things down into steps. We juggled intense decimals through a long calculation. Payoff time being expressed as "-n" was confusing. We went slow. From an assessment stand point, I asked a lot of question about the concept. What's the advantage of making larger payments? What do the variables stand for? That kind of thing. Exposure to HOW the calculations are done was enough for me even if they didn't become an expert. Many of them could handle the work after a while. A lot couldn't. But the goal of financial education was achieved.

### Conclusion

This unit doesn't happen without a decision 6 months ago to challenge the notion of what an on-level Algebra II student can do. The laundry list of steps to make the calculation, the nasty decimals involved, the multiple instances of log an exponential terms, it doesn't happen. Even last year this discussion disintegrates into a mechanics nightmare.

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