[Cross Posted on the One Good Thing Teach blog]

Setup: We’ve been talking about perpendicular bisectors in various contexts in geometry. But they were just making observations and working on some simple proofs.

Last night in Geometry, students were tasked with the following:

It turns out that #3 is impossible, and #4 is possible with some guess and check. This sets up the background to have kids see something neat.

Then they are asked:

 

And now they see that for a triangle, the perpendicular bisectors of the sides all meet at a point. And that is rare and weird.  They then were asked to look at the point that the perpendicular bisectors meet at and the vertices of the triangles and make a conjecture.

Only one student “saw” it. It was fascinating for me that it was so hard for everyone else to see it! Others had conjectures that might have been true for right triangles or isosceles triangles or equilateral triangles… but not that were universally true.

For the rest of the class, to get them there, I did the following:

This was a huge setup for my “one good thing.” There were gasps, and one student said, and I quite, “MIND BLOWN.”

This weekend they are going to try to figure out what the what is up?!

PS. Yes, I am fairly certain that the setup of having students see the rarity of perpendicular bisectors meeting at a point, as well as having them look and fail to see something inside a set of seemingly random points was crucial for the big reveal. In fact, the fact that they didn’t discover it on it’s own was so powerful when they ended up seeing it.

UPDATE: The file I used is here.