Saw this question from John Golden on Twitter a few minutes ago:
My immediate idea was two prior Zometool projects that we’ve done that touch on rotation groups, but they require a Zometool set.
My two next thoughts were a bit more technical – Galois Theory and Elliptic curves. On reflection, though, I feel like both are pretty tough tasks for one class.
So my next idea related to three things I’ve seen on Twitter recently.
(1) Start by watching the first 10 minutes or so of this wonderful public lecture by Jacob Lurie from last year’s Breakthrough Prize:
In the first part of the talk he discusses rings and touches on Emmy Noether’s work on the subject in the early 1900s.
Here’s how I used this video with my kids last week (we did not explicitly dive into abstract algebra, but we did talk about clock arithmetic):
Using Jacob Lurie’s Breakthrough Prize Lecture to inspire kids
(2) Next check out this video linked by Steven Strogatz last week:
In this video you learn about a few incredible ideas related to abstract algebra. For example, when you adjoin i to the integers, you get new primes, but you still have unique factorization. However, when you adjoin the square root of 5, you lose unique factorization. These ideas are just one step removed from what Lurie touched on in his lecture.
Oh, and the punchline of this video about the square root of 163 is pretty amazing!
(sorry not TeX-ing this, I’m writing in a hurry)
So, even just stopping with the ideas in this video you’ve got some neat facts that are pretty accessible (and cool!).
(3) Finally, if you have time, take a look at this “new to me” proof that e is irrational that Dave Radcliffe tweeted about last week:
https://twitter.com/daveinstpaul/status/669205374034034688%5Bembed%5D
Essentially this proof looks at numbers of the form A + B*e where A and B are integers. This set of numbers isn’t a ring, but it is at least another example of expanding a number system. For a one day lecture it seems close enough to what’s going on in part (2) above to keep the class flowing. Plus, it is sort of fun to see this proof that e is irrational.
It is also easy to skip of the first two parts take longer than expected.
Anyway, that’s my “pondering this Twitter question for 20 minutes” idea.
Mike's Math Page 
The video was great for us – linking Gauss and Ramanujan in a math history class?
The Lurie talk with Dave’s link will really be a neat tie together day. So amazing! I’ll report results.