It's hard to coordinate with a group of other volunteers (especially over the summer) and I haven't quite setup a real meeting yet to discuss next year but I did receive another confirmation at least one of the group is committed. I also received the first notification about next year from the school coordinator and based on the last emails I'm going to tell her that we will be running 2 classrooms. Making this growth work is probably going to be the most challenging piece of next year. I don't know yet whether one of the volunteers will act as the primary leader of the fourth grade group or if we'll do a rotating schedule of facilitator/coaches or even if I'll swap things up and work in both rooms form time to time. I'm leaning towards concentrating on the fifth graders. There's also a bit of an issue with several of the volunteers really having younger children. I still think based on experience that we need to age segregate kids. What tends to happen if you don't is a tipping point reached where the older kids start assuming that a club is only for younger kids and stop signing up: "That's for babies". This is probably bound up in the general status issues associated with Mathematics and the fear of being vulnerable in front of a younger schoolmate. With all the other social issues to focus on, gender imbalances come to mind, this is one thing I'd rather not tackle.
I have two books on hold from the library. The Math book by Clifford Pickover looks particularly interesting and hopefully I'll have enough for a short write up some time in the future.
The first one I originally saw @five_triangles post. Given the following triangle what's the area of the missing piece. I draw it inside the larger rectangle because these type problems almost always are easier when conceptualized that way. There's a solution using similar triangles. But you can also work with equal areas to solve. For instance the other triangle formed using the missing diagonal will have an equal area to the existing one. Its interesting how this and several of the other variants relate to each other. This is also an excellent P.O.T.W. candidate.
The second one comes from https://solvemymaths.com/. I really should watch this site more often because it often has interesting material.
@srcav did a fun write up which pretty much matches how I initially solved it as well and does a good job of walking through the thought process:
I was curious afterwards (well really I suspected I would find something) and after adding some more structure noticed the following:
The solution as the construction above shows is that the line from the center of the 2 circles goes through the center of the in-circle of the original triangle. I'm fairly sure this is a general property although I haven't looked into yet or how to prove it (which would be a fun extension). It also may mean there is likely a way to solve the original problem via the in-circle rather than the Pythagorean theorem.
I could never do this with my kids since for one we don't have computers but geogebra does make a great tool for investigations like above. I could easily see structuring a lesson within it if I had infrastructure.