## Thursday, May 29, 2008

### Schlope and PMAN

Mathematics is the art of giving the same name to different things. - Henri Poincare
So in this case it is not the quite the same name, but I have sort of "invented" some terminology. The idea of "schlope". Schlope is quite similar to slope. The slope of a line tells you how much to go up for every 1 that you move to the right. Schlope tells you what to multiply by for every one step to the right. Schlope is very important for exponential functions. Generally, in exponential situations we talk about the "rate of growth". If the rate of growth is 8% then the function that models that growth has "1.08 to the t" in it. That 1.08 is what I call "schlope". So, schlope is the number that you actually use in the formula.

In fact, you can create parallels for all your favorite formulas and constructs from linear functions. This parallel construction highlights another construct of mine that I like to call the hierarchy. The hierarchy is simply the recognition that the fundamental ways of putting numbers together come in an order. At the bottom level we have Addition and Subtraction. In the middle we have Multiplication and Division, and at the top powers, roots, and logarithms. Another teacher at my school refers to this as "PMAN" for powers, multiply, add, nothing. It is very helpful to think about PMAN when doing calculations with exponents and logs. If you are taking powers of powers, "P", then you "M"ultiply the powers. If you are "M"ultiplying powers with the same base then your "A"dd the exponents. If you are "A"dding different powers of the same base together then you do "N"othing.

To move from Linear functions to Exponential functions you simply must move up in the hierarchy. Moving from repeatedly adding a number to the height every time to multiplying that height by some fixed number. The formulas can be written quite similarly: