The Distance Formula is one of the biggest reasons I created this new curriculum/sequence of content. In nearly every current Geometry book I've seen, the Distance Formula is lumped in with a bunch of random gibberish in Chapter 1. It doesn't make any sense to a student, it's just a thing they're expected to memorize and know how to use. It's antithetical to modeling for sure, but it's just bad teaching. "Don't question why it works, just trust that it does." Really? And we're not even going to cover Pythagorean Theorem until Ch. 9 sometime in March? REALLY?
How about this? Instead of throwing random formulas at students to see what sticks, let's actually ensure the students understand how the formula came to be. Better yet, let's make the students derive the formula themselves!
So, I built off the activity that we used to discover the Pythagorean Theorem. Every student has a right triangle drawn on graph paper, and I asked them to draw a set of x-y axes around the triangle. Then, they need to identify the coordinates of the vertices (labeled A, B, & C).
We had a brief tangent to discuss the difference in labeling sides vs. labeling angles (lower case vs. upper case respectively), and then I asked how to determine the length of sides a & b? Most students can glean that they can simply count the grid spaces since the sides are horizontal & vertical. At this point, they can (hopefully) recognize that strategy doesn't work on c since it's on an angle. but we know Pythagorean Theorem, so they can use that. But what if we don't have a triangle? What if *gasp* we don't even have a graph?
So I lead them to the idea that the length of side a could be written as (x2-x1) and similarly, side b could be written as (y2-y1). Then, we can substitute into P.T. and while it may look atrocious, the end result is a model that can determine the distance between ANY TWO POINTS.
How is this not a better way?