See the September 2003 issue of Teaching K-8 for some great activities with Moebius strips.  Here's some additional information on these wonderful and curious objects.

The Story

The Moebius strip is named for August Ferdinand Moebius (1803-1868), a German mathematician who achieved minor fame during his life.  After his death, some notes were discovered wherein he described the properties of a twisted band of paper.  The notes were published and he became famous for it, albeit too late to enjoy the glory in his lifetime.
 Today, the Moebius strip is part of a branch of math called Topology, which is the study of properties of objects that don’t change under transformations.  Topologically speaking, a donut and a coffee cup are the same with regard to the number of holes in their surfaces; one could be transformed into the other without breaking the surface.  There is a joke about topology:  A topologist is a person who can’t tell the difference between a coffee cup and a hole in his head, but he can tell the difference between a coffee cup and two holes in his head!
 Other properties of interest in topology include number and relative positions of faces, edges, and vertices of solids.  Many topics in topology are interesting and accessible ? and some of them are very complex and takes years of study to understand.  The moebius strip is a puzzling but relatively easy to understand topological curiosity.
 What is it good for?  Some people have suggested applications for a moebius strip, as a conveyer belt or a fan belt that wears out evenly on both sides.  Mostly, the Moebius strip makes us wonder and has inspired great art (most notably some of the works of M.C.Escher, such as his woodcut entitled “Moebius II”, which shows ants crawling around the outside of a moebius strip.)
 

In the classroom

The activities in the September 2003 issue of Teaching K-8 could easily use up an hour of class time.  If you don’t have a family math night or a school math club, you might want to mete out these activities ? have you students do one or two of the variations in those odd 5 or 10 minutes that sometimes pop up.  Slicing strips in a paper cutter in advance can save a lot of time.
 There will probably never be a question on a standardized test involving Moebius strips, but these activities encourage spatial reasoning, which will help your students in all kinds of ways, be it mathematics, sports, art, or what-have-you.  And by gum, Moebuis strips are just really cool.  They’re something that everyone should know about to be a cultured and complete person.  When students realize that these fascinating objects are part of mathematics, you may see a positive change in some students attitudes towards math.
 An interesting idea is to have students write a story on a Moebius strip.  It can be a story that never ends such as:

“It was a dark and stormy night.  The crew had gathered below deck to tell stories.  The captain went first.  He began by saying, “It was a dark and stormy night.  The crew had gathered below deck. . .

 A colleague of mine gave a writing assignment to his class and asked that the essay be limited to one side of a piece of paper.  One of his students found a loophole, and turned in the assignment written on a moebius strip!
 Another idea is to make moebius strips from zippers.  They can be unzipped and rezipped again and again, and can be a curious addition to your math centers.

As a magic trick

Cut two sheets of paper in half the long way.  Tape the four pieces together to make a long strip.  Make a cut in the center of one of the short sides of the strip, and cut down the middle about 3/4 of the way down one of the sheets of paper.  Give one of these thin ends a half twist and tape it to the other end of the strip, and give the other end a full twist and tape it next to the first end.
 The idea is to have a big loop that you can cut apart to end up with half-twist and a full-twist moebius strips.  If you make the twists right, the whole thing looks like a fancy strong-man belt.  In fact, there’s a very nice story that goes with it...

 “Before I was a teacher, I was a clown in the circus... One day, the strong man pulled a muscle in his little finger and decided it was time to retire.  He gave his strong-man belt to his best friends, the giant and the Siamese twins.  He tore the belt in half and gave one half to the giant, and the other half to the twins.”
 [tear belt in half]
 “The giant, however, was disappointed that he couldn’t wear the belt -- he was too big.  So I took my magic pair of scissors, and said ‘Maybe if I cut it in two, it will be twice as long.”  [cut in half -- surprise!]
 “The Siamese twins were disappointed, too, because even though they were connected together, there were still two of them, and the each wanted their own belt.  So I offered to cut theirs in half with my magic scissors, too.”  [cut in half -- surprise!]
 “Now everyone was pleased and the strong-man could retire happily.”

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