Monday, January 26, 2015

Slope: Based in part on Section 3-4 (Day 99)

Section 3-4 of the U of Chicago text is where slope -- that very important concept -- is defined. The definition in the text is simple enough and is typical for most high school math texts. Here it is, rendered in ASCII:

The slope of the line through (x_1, y_1) and (x_2, y_2), with x_1 != x_2, is (y_2 - y_1) / (x_2 - x_1).

But this isn't good enough for Common Core. Let's look at the relevant standard:

Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
[emphasis mine]

So we have to prove that slope works. It isn't good enough just to say that slope formula works just because we defined it to work. To see why, consider the following definition:

The favorite food of the class containing the student S is the food that S likes to eat the most.

And now you instantly see the problem with this "definition." Naturally, not every student in the class likes the same food. The food that student S loves to eat may be a food to which student T is indifferent and a food that absolutely disgusts student U. The "favorite food" of a class depends on which student we ask to name a favorite food. Simply writing the word "Definition:" and setting the phrase "favorite food" in bold doesn't magically force everyone to have the same favorite food.

Likewise, writing the word "Definition:" and setting the word "slope" in bold doesn't magically force the slope to be the same no matter which two points we choose. It could be that if we choose points (x_1, y_1) and (x_2, y_2), we get a different slope from if we choose  (x_3, y_3) and (x_4, y_4), just as if we choose student S we get a different favorite food (or favorite song) from if we choose student T instead of S.

In mathematics, we would say that "favorite food of a class" is not well-defined. In order for slope to be well-defined, we must prove that the slope is independent of which two points we choose to plug into the formula. We can't prove that the favorite food (or song) is independent of which student we choose -- indeed, it's trivial to find a counterexample: simply declare one student's favorite to be the class's favorite, and watch all the counterexample students call out how much they don't like the declared favorite!

And so the main theorem for today is to prove that slope is well-defined. We want to prove that slope of a line is independent of which two points we choose to plug into the formula. The trick for doing this will remind us of the proof of the Distance Formula.

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