## Time, symmetry, and unexpected turns

This is a story of how a kindergarten lesson on telling time turned into a lesson on symmetry.  It all started with a fun, colorful clock, with numbers 1 through 12 made out of shapes (some related to the number, others arbitrary) that could be taken out and put back in.

We were playing a game where one person would roll two 12-sided dice, another one would set the hands of the clock to point to those numbers, and a third would tell the time.  They were doing a decent job at the beginning, but very quickly the colorful shapes with numbers became too distracting and they started playing with them, taking them out and putting them back in.

Soon, it became quite hard to keep them on track, but then one student put a piece in upside down and that gave me the idea of completely shifting gears and changing topics.

I asked the students which numbers could be put in incorrectly and in how many ways.  We went through the numbers one by one, and the students tried putting them in incorrectly in all possible ways.

The number 2, for example, could be put in upside down:

But it could also be put in sideways in two different ways.  The number three, on the other hand, could not be put in fully upside down, but there were two ways of putting it in incorrectly.  After going through all the numbers, the students discovered that the 1 and the 11 were the only numbers that could not be put in incorrectly (unless you flipped them over so that you couldn’t see the number at all).  Seven, nine, and ten were the numbers that could be put in incorrectly in just one way, and the rest of the numbers had multiple ways of being disoriented.

Here is a picture of the clock with all the numbers put in incorrectly (except for the two for which this was not possible).

The students were very engaged with this new activity and it easily lasted us until the end of the lesson.  Even though this was not what I had planned originally, I was very happy with the way the class turned out.  I have recently discovered that I really enjoy when lessons take an unexpected turn, as long as it leads us to other rich and beautiful mathematics to explore.

And of course we went back to playing with telling time in future lessons, but only to the point that they were interested in and had fun with.  I believe that at this age it is much more important to instill the joy of doing mathematics rather than have the kids acquire any particular skill.