The protocol my partner and I came up with was very straight forward. We immediately recognized that each shape could be placed in each position; first, second, and third. We would have three sets of patterns; Circle in the first position, Square in the first position, and Triangle in the first position. The same procedure for the second and third positions. To identify all possible patterns, we started with a shape such as a circle. Then place a circle in the next two positions: c-1, c-2, c-3, then change out position 3 with another shape: c-1, c-2, s-3 and c-1, c-2, t-3. Next return to the starting pattern but change out the shape in the second position: c-1, s-2, c-3; c-1, s-2, s-3; c-1, s-2, t-3, etc. Continue this process until all possible combinations for a circle in position 1 have been created.
The protocol, aka algorithm, my partner and I completed was the easy part. The difficulty I experienced was that I wanted to make the idea of a number system much too complicated. I was thinking how do I create a number system of shapes circles, squares, and squares in terms of addition, subtraction, etc. Although, I doubt I’ll I have many students who think in terms of closed number systems, sets, and groups, I do think some students may over- think the problem to be solved. Suggestions on how to gently guide students to the simpler concept would be greatly appreciated.