This is a great question. Love it love it love it!!
The following is just what occurs to me off the top of my head.
Show a graph paper grid with a dot at the origin and a dot at (3,4). Say we want to find the distance between the dots. We could guess that it would be 3+4=7. Well, that would be right if these were city blocks, but it's not the right answer as ...
I really like the idea of a "discovery fiction" -- it gives a name to something I often try to use when teaching. Here is one suggestion.
I will try to come back and write a more elaborated version of this answer later, with diagrams and proper notation, but briefly:
(a) Don't let on that you are going to prove the Pythagorean Theorem -- don't ...
I am quite surprised by the suggestion (and apparent consensus?) that kites and rhombi are not very important for high school geometry. Here are three entirely different arguments for why they should, in fact, play a central role in the curriculum.
Argument 1: Understanding hierarchical relationships.
One of the goals of secondary Geometry is to help ...
It would be hard to say what you could get rid of without knowing what is on your state test. In my state (NY), they push quads quite a bit. I'll grant you that kites aren't interesting or important, but rhombuses are both.
One of the usual capstone problems is being given the coordinates of four points and demonstrating that the quadrilateral formed from ...
OP: "I am fairly sure that rhombuses and kites are pretty useless."
Understanding parallelograms and rhombi is quite useful in pop-up card design:
Constructed by Ian Agol.
"inspired by @divbyzero's tweet." 19Dec2020.
The observation that a rhombus's diagonals bisect its internal angles gives rise to an efficient method to find an angle bisector.
Here, a British national-exam question hints this method to the candidate: