Standard for Mathematical Practice number 8 is probably the hardest for people to wrap their heads around:

MP8. Look for and express regularity in repeated reasoning.

There are too many words in there: regularity, repeated, reasoning. I’ve seen a lot of people latching onto one or two of these. If it’s regular, it’s MP8! If it’s repeated, it’s MP8! If it’s both regular and repeated, it must really be MP8!! One thing that is fairly regular and repeated is generating coordinate pairs from an equation in two variables. So there are lots of fake MP8 lessons out there about generating points from a linear equation in two variables to draw the graph of the equation, a straight line. The more points, the better—it’s more repeated that way. And regular.

But that word reasoning is also important. There’s precious little reasoning involved in generating coordinate pairs from an equation. But if we turn the question around, there’s lots of reasoning. Instead of going from an equation to a line, let’s go from a line to an equation. Consider a line through two points in the coordinate plane, say (2,1) and (5,3). How do I tell if some randomly chosen third point, say (20,15), is on this line or not? Given any two points on a line in the coordinate plane, I can construct a right triangle with vertical and horizontal legs, using the line to form the hypotenuse, as shown here.

It is a wonderful geometric fact that all of these triangles are similar. (Exercise: prove this!) So, if (20,15) is on my line, then the triangle formed by (20,15) and (2,1) should be similar to the triangle formed by (5,3) and (2,1). If these two triangles are similar, the ratio of their vertical to horizontal legs should be equivalent:

$$

\frac{15-1}{20-2} = \frac{3-1}{5-2}?

$$

Oops. Not true. So (20,15) is not on the line. Let’s try (20,13) instead. If (20,13) is on the line, then the triangle formed by (20,13) and (2,1) should be similar to the triangle formed by (5,3) and (2,1). If these two triangles are similar, the ratio of their vertical to horizontal legs should be equivalent:

$$

\frac{13-1}{20-2} = \frac{3-1}{5-2}?

$$

Yes! Both sides are equal to $\frac23$. And in fact, to confirm, the reasoning works the other way: if the ratios are equivalent, then the triangles are similar, then the base angles are the same, so the hypotenuses of these two triangles are on the same line. (Exercise: prove all this, too!)

So we have a way of testing whether points lie on the same line. (This is Al Cuoco’s point tester; google it.)

After testing a lot of points, we look for some regularity in our repeated reasoning. Every one of our calculations looks the same. We can express the regularity by a general statement: to test whether a point $(x,y)$ is on the line, we check whether

$$

\frac{y-1}{x-2} = \frac{3-1}{5-2}.

$$

By our reasoning, every point on the line satisfies this equation, and no point off the line satisfies it. We have discovered the equation for the line by expressing regularity in our repeated reasoning.

All the words in MP8 are important: reasoning, repeated, regularity, and also express and look for. See this post by Dev Sinha for more discussion.