Section 8.2 of AoPS Introduction to Algebra introduces (i) the idea that a linear equation looks like $Ax+By=C$  and (ii) the slope of the line.  With some students I get in a pinch because sometimes they are already somewhat familiar with graphing so in an effort to be efficient we rush into the definition of slope and then subsequently backtrack because of confusion.

So this week I was prepared for my student.

We work the problem where we first calculate the slope of a line (Problem 8.7), using $2$ pairs of points to get the idea that the slope is the same everywhere on the line, no matter which $2$ points you choose.  But I was more careful.  Before getting into $\frac{y_1-y_2}{x_1-x_2}$  I give him a heads up.  “Now I know this formula is going to look like it’s coming from outer space, but just bear with me for a second while we do these calculations.”  Then we calculate this unnamed ratio for 2 pairs of points and we find they have the same result, $\frac{1}{2}$.  Then I ask him:

“What do you think?  Do you think this formula will be $\frac{1}{2}$ no matter which $2$ points we choose?”

Thoughtful silence.  We had carefully traced the path of  Hopsalot the rabbit, so I reminded that between every consecutive point, we go right $2$ units and up $1$ unit.  Eyes light up.  He gets it, that the ratio is always going to be the same.

Even better, we didn’t actually finish the problem, or name that this ratio is called slope.  I like leaving him with this epiphany for a while so he can internalize it a bit.  Then we can review this once more before proceeding.

Since we use the slope to generate the point-slope form of the line, it’s imperative that students understand that the slope is the same everywhere on the line, no matter which 2 points we choose.  We use this property to generate the equation because it’s one of the few things that consistently describes every point on the line.    I’m going to give him this heads up and reiterate as much as possible that the slope is an inherent property of a line, no matter which 2 points you choose.