A lot of nervous energy was building up while I was waiting in the wings at the 2012 TEDxNYED conference.
Luckily, I found some polygons to play around with, so I converted my anxiety into mathematical art.
Challenge History Resources
Unsolved Math Problems
This is a nice list of famous unsolved math problems from Wolfram MathWorld:
http://mathworld.wolfram.com/UnsolvedProblems.html
There are some well-known problems here, like the Goldbach Conjecture and the Collatz Conjecture, and some lesser-known open problems like finding an Euler Brick with an integral space diagonal.
It’s especially nice that several of these challenges are easy to explain to non-mathematicians. For example, the Goldbach Conjecture asks “Can every even number be written as the sum of two prime numbers?” Somewhat surprisingly, after nearly 300 years, the best answer we have is probably.
I think I’ll make this page next year’s summer homework assignment.
Resources
Catalog of Mathematical Knots
This is an amazing collection of interactive diagrams of mathematical knots:
Click on any of the hundred or so knots here to go to a flash-based interactive diagram of the knot that can be rotated and manipulated.
Teaching
A Surprising Integral
I had a fun encounter with an innocuous looking integral.
It all started with a simple directive: evaluate 
Integration is often tricky business. Although there is a large body of integration techniques, there isn’t really one guaranteed procedure for evaluating an integral. If you see what the answer is, you write it down; if you don’t, you try a technique in the hope that it makes you see what the answer is. If that technique doesn’t work, you try another.
This particular problem is interesting in that it highlights a strange phenomenon that occasionally pops up in problem-solving: sometimes making a problem look more complicated actually makes it easier to solve.
Let 
This gives us 
This actually looks a bit more difficult than the original problem, but now we can easily integrate using Integration by Parts!
After applying this technique, we’ll get 
I was surprised that this technique worked, so I actually differentiated to make sure I got the correct answer. You can take my word for it, or you can verify with WolframAlpha.
One of the best parts of being a teacher is learning (or re-learning) something new every day!
Appreciation Geometry Resources
Mathematical Image Galleries
This is an great website full of galleries of mathematical images:
http://www.josleys.com/galleries.php
There hundreds of beautiful images in many categories, like fractals, knots, spirals, and tesselations.
There’s even a gallery inspired by the techniques of M.C. Escher!