I wanted to start the year off with this, until I realized (duh) that it really uses the Pythagorean Theorem.
But it’s a great problem, and so rich for spinning off. What about triangles? What about pentagons? What about irregular triangles? Can you always gets a circle around the polygon, no matter where you start? (An honest-to-god context for drawing a circle through three points.)
More than ever, I’d like to really focus on problem posing this year, and this problem is very rich for jumping off from. I love it, and can’t wait to bring it in to class.
Arithmagons are cool, right?
(The sum of the vertices is the value of the edge.)
Are triangles special, or can you solve them for any polygon?
What determines when you can solve one of these? Why do triangles always work, but square puzzles don’t always work?
What shapes* can you make with the bendy straws?
* Do the ends have to connect? We can either require that the ends connect, or not.
You can try to make a triangle.
But can you always make a triangle out of the two straws? What requirements are there? Did you try extending the straw out all the way? Did you try scrunching it in?
I see a shape there, but I notice that the ends don’t connect perfectly. Can you make them connect perfectly?
What categories of shapes are we able to make if the ends have to connect perfectly? (And we can use a bit of tape to make sure that the ends are sticking together.)
Are there more categories? How do we know?
What’s special about these shapes? What happens if you flip one of these shapes onto its back side? What changes? Why?
I think that this might be a nice mini-investigation that could motivate some proofy concerns.