I use this activity with middle school students as an introduction to divisibility and to help develop number sense. The only direct instruction I provide beforehand is a quick discussion of what it means to be divisible by something and what divisibility is.
The goal of this activity is for students to develop their own “rules” for determining which numbers will divide larger numbers evenly. For this activity, I allow calculator use, not only to help those who are not fluent with multiplication and division facts, but also to speed up the learning process a little. The intent for this activity is not to teach division, but to allow students to discover how to determine where to begin when simplifying fractions or when working with larger numbers.
This activity is intended for small groups, but can also be used in a math station for independent work or modified for whole class. Depending on the time frame, all rounds can be completed during the same class period or each round can be completed on a separate day.
This is the pdf version. If you would like to modify it, please let me know and I will send you the link to the original google doc file.