This past week I spent part of a class period on a spherical geometry activity, and it went well. Beforehand, I went to the store and bought some 4"-5" diameter soft/squishy balls (tennis balls will do, but not really large 12" diameter ones, I think, because of the rubber bands). I bought enough for one per table (4 kids). I also had 3 rubber bands per table.
As always we had a quick discussion about these things being "math tools" and not annoying distractions that you throw or bounce or snap people with, etc. MATH TOOLS. Then I had a brief introduction on how we've been studying Euclidean geometry where everything happens on a plane and everything is defined on a plane, and what if we defined things on a sphere where our objects had to all live on the surface of the sphere? With that I handed out this worksheet:
and told them to do the bold questions, 1,5,7,9,11,14. We discussed the answers, then I walked them through 2,3,4. Then I set them on their own to explore the remaining questions, one at a time, and making sure that everyone had a chance with the ball. And we discussed the answers after each problem.
There are a ton of resources on the web that have answers, and I love the surprising results. They're brain hurters. For their homework, they had to find at least 2 more cool facts about spherical geometry, and then they had to find 1 new fact about a geometry on another type of surface.
This all took about 45-50 minutes with my advanced class. They came back with some interesting facts, and it was a nice addition to our normal routine of "every day geometry".