Divide and Conquer

Divide and Conquer is a game played by subtracting proper divisors from the initial number. One is always a proper divisor and no number can be divided by itself. The game is won when a player presents his/her opponent with one.
As we worked with 100, which has several proper divisors, we tried strategies that included making a list and drawing a diagram on graph paper. In the list we found a proper divisor of 100, and then subtracted the divisor from 100. We repeated the process with each new difference until one player could present his opponent with one. Similarly, we drew a 10 x 10 grid on graph paper and took turns crossing off proper divisors until one square remained.
We discussed that many middle school students will learn how to play this game with the initial number of 30, They will memorized the specific steps for 30 and attempt to generalize those exact steps to other numbers. The conclusion was that in order for students to truly demonstrate their understanding of factors in this game, they need to play it with other numbers.
I shared this game with my summer school students. They were thrilled to be able to beat the teacher.