Today I went into my daughter’s first-grade classroom, full of six-year-old girls, and gave a presentation about Möbius bands.

We cut strips of paper and at first curled them into simple bands, cylinders, which we proved had two sides by coloring them one color on the outside and another color on the inside.  Next, we cut strips and curled them around, but added a twist, to make a true Möbius band.

A Möbius band

These, of course, have only one side, a fact that the children proved by coloring it one color all the way around. And we observed that a Möbius band has only one edge.

A Möbius-like band, with two twists

We explored what happens with two twists, or more twists, and also what happens when you cut a Möbius band down the center, all the way around.

Möbius band cut down the center

It is very interesting to cut a Möbius band on a line that is one-third of the way in from an edge, all the way around. What happens? Make your prediction before unraveling the pieces–how many pieces will there be? Will they be all the same size? How many twists will they have?

Overall, the whole presentation was a lot of fun. The girls were extremely curious about everything, and experimented with additional twists and additional ways of cutting.  It seemed to be just the right amount of mathematical thinking, cutting and coloring for a first-grade class.  To be sure, without prompting the girls made various Möbius earrings, headbands and bracelets, which I had to admit were fairly cool. One girl asked, “is this really mathematics?”

It seems I may be back in the first-grade classroom this spring, and I have in mind to teach them all how to beat their parents at Nim.