This is a great video of some students sharing their thinking while solving a problem posed by their teacher.
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/
This problem allows for multiple possible solutions, and will require students to think about some assumptions they will have to make to solve this problem. You can easily vary the difficulty of the problem by changing the number of people invited to the party.
Ellen is planning a party for her friends. She has invited 100 of them, but she doesn’t know exactly how many of her friends will attend. She wants to put out tables for her friends, and she wants to put enough chairs at each table so that none of her friends has to sit alone. Assume that her friends will fill up each table as they arrive. How many tables should she put out, with how many chairs at each table?