A number of years ago, I had a 9th grade math class where I decided we would investigate when the decimal representation of a fraction repeats, and when it terminates. We also decided to investigate to see if there was a pattern in the number of repeating digits when the decimal representation of a fraction does repeat. It took us about two weeks to conclude that we weren’t going to find a pattern without a lot more work, and so we abandoned the investigation, but I still remember the process quite keenly.

If you decide to try this investigation with your own students, you may find this arbitrary precision division calculator (we quickly ran into the limits of our computer’s calculator when doing this investigation) useful: http://davidwees.com/divider/