Backpropagation has been one of the most popular learning algorithms for neural networks since the publication by Rumelhart, Hinton, and Williams in 1986. Below there are links to two pages with exercises. The first one leads to a page with models that delve into the type of representations backpropagation can develop, starting with thos of the infamous XOR function. The second link leads to four networks that learn a tiny fraction of the MNIST data set of handwritten digits. I have made this exercise optional because if you only possess a slow computer, these exercises might take many hours to run on the current incarnation of WalnutJS, which is not (yet) optimized for speed.
Reference
Rumelhart, D. E., Hinton, G. E., & Williams, R. J. (1986). Learning internal representations by error propagation. In D. E. Rumelhart & J. L. McClelland (Eds.), Parallel distributed processing. Volume 1: Foundations (pp. 318-362). Cambridge, MA: MIT Press.
Exercise 9: Hidden-layer representations
Exercise 10: Recognizing hand-written digits (optional but bonus points!)