 For the most part, math is actually pretty easy. And that's because once you've reduced a problem to a mathematical statement, there's a rather limited number of things you can do at that point. The difficult part of the problem is phrasing it as a mathematical question, and a lot of that centers around this notion of constructing a function for a given situation. If you're not given a function, you might have to construct a function. To construct a function, you will need to rely on what you know about algebra. Geometry, life, the universe, and everything. In the process of constructing a function, it's useful to distinguish between variables, constants, and parameters. A variable quantity changes its value during the story of a problem. In contrast, constants and parameters never change their value during the story of the problem. The only time a constant or parameter would change is if our story changed. And again, variable, constant, and parameters are mathematical terms, and they have fairly clear meanings. However, the determination of whether something is a variable, constant, or parameter, requires you to rely on everything you know about algebra, geometry, life, the universe, and everything. So, for example, if a cell is changing shape, for example, an amoeba crawling along a surface, then the volume is constant, but its surface area will change. Or let's say a city is growing, its population is variable, but its area will remain a constant. And you might wonder how we know these things, and again the answer to that is we have to rely on what we know about the universe. As a cell moves, the laws of biology and physics say that the content of the cell, mathematically its volume, remain constant. And as the city grows, the laws of man say that its size, as defined by its city limits, don't change. Again, it's possible that in a different story, these quantities may change, the amoeba may engulf something and its volume might increase, or the town council may change the boundaries of a city. And so the reason that mathematical problems about the real world are difficult is not the mathematics, but it's that the real world is difficult. The mathematics is the easy part. So now let's talk about writing functions. The first important rule of writing functions is to let variables vary. In other words, be sure to represent variable quantities with a variable. So let's take a look at an actual problem. So say we have a 25 foot ladder resting against one side of a wall, the base of the ladder is 5 feet away from the slide of the wall, but it begins sliding away from the wall at 3 feet per second. To begin with, let's identify the variables and the constants and parameters. So you might imagine our ladder here beginning to slide away from the wall, and so some of the things we might notice here, the length of the ladder never changes. It's always 25 feet. We also suspect that the rate at which the ladder slides away from the wall also seems to be constant 3 feet per second. Now if you're a physicist or an engineer, you might question this assumption of constancy. And so we'll add a second factor. Even if this rate was changing, nothing in this problem tells us how to find what the rate will be. And that's an important idea, because maybe in a different problem we will need to be able to figure out what that rate of change is. Finally, we notice that the distance of the base of the ladder from the wall will change, so this quantity is a variable quantity. And in fact we also see that the height of the ladder changes, so that too is variable. An important idea to remember is that you don't have to write down the function right away. It's often enough to start by writing down any relationship at all between the variables and constants. So in this case we determined that x, the distance of the base of the ladder from the wall, and y, the top of the height of the ladder, were both variables, while the length of the ladder was a constant 25. So ignoring what we're actually trying to find, let's see if we can write down any relationship at all among these variables. And sometimes a change in view point is helpful. And here, if I look at our situation, we see that the distance of the base of the ladder from the wall and the height of the top of the ladder and the ladder itself form three sides of a right triangle. And this means I can write down the Pythagorean relationship. And this gives me a nice relationship between x and y. Well, let's take it to the next step. Let's actually write a function. So we've already determined that we can set up our variables x, the distance of the base of the ladder from the wall, y, the height of the top of the ladder, and we know that x squared plus y squared equals 25 squared. And the important thing to note here is if you don't play, you can't win. And if we don't have this equation, we can't do anything. But once we have this relation, we can try to solve for any of the variables. And since we'd like to find the function giving the height of the top of the ladder, we'll solve our equation for y. Now, strictly speaking, if we take the square root of something, we do have to include this plus or minus to allow for the possibility of a negative answer. But since the height y should be positive, we'll take the positive square root only and get our function y equals square root 25 squared minus x squared.