Make a Mobius Strip!
step 1 Cut a long (8 inches or more) rectangular strip of paper (about 1 inch wide) and draw circles and squares as shown.
step 2 Take one end of the strip and twist once (a half-turn, actually)
step 3 Put the ends together so that squares meet squares and circles meet circles. Tape the ends of the strip together, making a loop. It should look something like the drawing on the right.

Before testing the loop for how many sides it has, first cut another strip of paper, except this time, DO NOT TWIST it before taping the ends together (see drawing ==>)

Experiment #1

See what happens with the untwisted loop first. Take a pencil or pen, and holding the loop on the table, draw a little star or square to indicate where you will start drawing. Then hold your pencil in place--do not lift the pencil off the table--as you drag the loop around. Keep going until you reach the little star or square you drew in the beginning.

Now look at the paper. Do all sides of the strip have your pencil markings? No. This, according to topology, proves that the loop is two-sided. To get the pencil marking on the other side of the paper, you would have had to lift the pencil point off the table and turn the loop over.

OK. Now pick up the loop of paper that you twisted (this one is the Mobius strip). Go through the same procedure: draw a little star or square so that you can always see where you started. Keeping the pencil down and moving the loop under the pencil, keep going until you come back to the little star or square. Now look at the loop. Is there a side with no pencil marking? If not, then the strip has only one side!!

Experiment #2

Take the two-sided loop of paper and carefully cut it in half. Perform the sided-ness test on each half. What is the result?

Now take the one-sided loop and carefully cut it in half. Perform the sided-ness test on each half. What is the result?

If you want to keep going, make another Mobius strip only this time, cut it in thirds. What is the result?

Neat, huh?