The sample test will serve as a good reminder of all the topics we’ve covered here. It will also alert them that knowing why *really does count*. And it will give them a bit more practice (much-needed) with compound interest.

Inside Collection: Advanced Algebra II: Teacher's Guide

Summary: A teacher's guide to example exponential curves.

This is another one of those topics where the in-class exercise and the homework may take a total of two days, combined, instead of just one. This is a difficult and important topic.

We begin with a lecture something like the following:

Earlier this year, we talked about “linear functions”: they add a certain amount every time. For instance, if you gain $5 every hour, then the graph of your money vs. time will be a line: every hour, the total will add 5. The amount you gain each hour (5 in this case) is the slope.

Can a line also subtract every day? Sure! That isn’t a different rule, because adding is the same as subtracting a negative number. So if Mr. Felder is losing ten hairs a day, and you graph his hairs vs. time, the graph will be a line going down. The total subtracts 10 every day, but another way of saying that is, it adds –10 every day. The slope is –10. This is still a linear function.

So why am I telling you all this? Because “exponential functions” are very similar, except that they multiply by the same thing every time. And, just as linear functions can subtract (by adding negative numbers), exponential functions can divide (by multiplying by fractions: for instance, multiplying by

Then they can begin to work on the assignment. They will make it through the table all right. But when it comes to finding the formula for the nth day, many will fall down. Here is a way to help them. Go back to the table and say: “On day 3, let’s not write “4”—even though it is 4 pennies. It is 2 times the previous amount, so let’s just write that:

You go through the same thing on the compound interest, only harder. A lot of hand-holding. If you end one year with

Toward the end of class, put that formula,

The assignment is also meant to bring out one other point that you want to mention explicitly at the end. When we developed our definitions of negative and fractional exponents, we wanted them to follow the rules of exponents and so on. But now they are coming up in a much more practical context, and we have a new need. We want

“Homework: ‘Real life’ exponential curves”

The sample test will serve as a good reminder of all the topics we’ve covered here. It will also alert them that knowing why *really does count*. And it will give them a bit more practice (much-needed) with compound interest.

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Comments:"This is the "teacher's guide" book in Kenny Felder's "Advanced Algebra II" series. This text was created with a focus on 'doing' and 'understanding' algebra concepts rather than simply hearing […]"