Experiment with the constructions below. Put the best examples into your journal, along with comments that describe and explain what is going on. Be careful to make your examples large enough to illustrate clearly the symmetries that are present. Also make sure that your cuts are interesting enough so that extra symmetries do not creep in. Concentrate on creating a collection of examples that will get across clearly what is going on, and include enough written commentary to make a connected narrative.
Fold a sheet of paper in half, and then unfold. Cut along the fold to the center of the paper. Now wrap the paper into a conical shape, so that the cut edge lines up with the uncut half of the fold. Continue wrapping, so that the two cut edges line up and the original sheet of paper wraps two full turns around a cone. Now cut out any pattern you like from the cone. Unwrap and lay it out flat. The resulting pattern should have two-fold rotational symmetry.
Try other examples of this technique, and also try experimenting with rolling the paper more than twice around a cone.
Make or round up a strip of paper which is long compared to its width (perhaps made from ribbon, computer paper, adding-machine rolls, or formed by joining several shorter strips together end-to-end). Coil it around several times around in a Möbius band pattern. Cut out a pattern along the edge of the Möbius band, and unroll.