Dihedral Group of order 3 for 13 Year-olds

A young student of mine told me that they had started a new topic in class, namely symmetries of plane figures. These were translations, reflections and rotations of polygons and some irregular shapes. This was a good opportunity to do an exercise involving symmetries that I’ve done a few times with young high-school students (maybe once with a primary school student) which is to fill in the Cayley table for the dihedral group. It took us an hour, starting with drawing a perfect equilateral triangle (a good problem in its own right); defining the expression ‘symmetry preserving transformation’ and then finding them all; learning that such a transformation can be described by the action on the vertices; discussing the idea that ‘doing nothing’ is a transformation; and finally drawing a table with seven rows and columns (also a non-trivial task apparently).

We then started to compose them, doing for example two rotations and then a flip, and noticing that this is the same a doing a flip and then a single rotation. It’s a joy to watch a young person work these out, since with the right framework they are able to comfortably handle group algebra with a non-commutative operation. For example my student had no problem at all with the reasoning $RF + FR = R + FF + R = R + e + R = RR$ (since it could so easily be verified) and was pleased when I pointed out that we were doing algebra not with numbers but with symmetry preserving rotations. The final test is to ask them to look for patterns in the table as they fill it in, giving them the opportunity to notice the ‘sudoku’ pattern, i.e. that every row and column must contain every element.