The Moebius Strip is a subset within Topology another branch of Mathematics, that is often useful but really looked at or understood!
As a start you might like to show the "Mr Bean at the Seaside" and then see if the students can replicate his antics.
Secondly the handcuff problem is a great way to get students cooperating with each other, as they wont be able to do it on their own. WARNING it will bring out some levels of frustration
Third, the scissor problem could be a "Math's Table Activity" Have a couple of students set it up.
Finally an indepth exploration with strips of paper, twists and none twists, predictions and checking. Starting by reading the Paul Bunyan Story
Mr Bean at the Seaside https://www.youtube.com/watch?v=ZWCSQm86UB4
In “At the Seaside” Mr Bean managed to put his togs on without taking off his trousers first. Explore ways of putting your togs on without taking your jeans off first Be prepared to show, and/or explain how this can be done.
Handcuffs
Find a partner and two lengths of string, each about 1 metre long. Tie one length of string to your partner’s wrists.
Now tie the other piece of string to your wrist, loop it past your partner’s string and then tie it to your other wrist.
Without cutting, untying, breaking the string, disconnect yourself from you partner. Explain how you became “unjoined”.
Scissor Puzzle
Make a scissor puzzle.
Find a way to remove the string without cutting or unfastening.
GOING LOOPY!
Paul Bunyan versus the Conveyor Belt
By WILLIAM HAZLETT UPSON*
"One of Paul Bunyan's most brilliant successes came about not because of brilliant thinking, but because of Paul's caution and carefulness. This was the famous affair of the conveyor belt.
Paul and his mechanic, Ford Fordsen, had started to work a uranium mine in Colorado. The ore was brought out on an endless belt which ran half a mile going into the mine and another half mile coming out - giving it a total length of one mile. It was four feet wide. It ran on a series of rollers, and was driven by a pulley mounted on the transmission of Paul's big blue truck "Babe." The manufacturers of the belt had made it all in one piece, without any splice or lacing, and they had put a half-twist in the return part so that the wear would be the same on both sides.
After several months' operation, the mine gallery had become twice as long, but the amount of material coming out was less. Paul decided he needed a belt twice as long and half as wide. He told Ford Fordsen to take his chain saw and cut the belt in two lengthwise.
"That will give us two belts," said Ford Fordsen. "We'll have to cut them in two crosswise and splice them together. That means I'll have to go to town and buy the materials for two splices."
"No," said Paul. "This belt has a half-twist -- which makes it what is known in geometry as a Moebius strip."
"What difference does that make?" asked Ford Fordsen.
"A Moebius strip," said Paul Bunyan, "has only one side, and one edge, and if we cut it in two lengthwise, it will still be in one piece. We'll have one belt twice as long and half as wide."
"How can you cut something in two and have it still in one piece?" asked Ford Fordsen.
Paul was modest. He was never opinionated. "Let's try this thing
out," he said.
They went into Paul's office. Paul took a strip of gummed paper about two inches wide and a yard long. He laid it on his desk with the gummed side up. He lifted the two ends and brought them together in front of him with the gummed sides down. Then he turned one of the ends over, licked it, slid it under the other end, and stuck the two gummed sides together. He had made himself an endless paper belt with a half-twist in it just like the big belt on the conveyor.
"This," said Paul, "is a Moebius strip. It will perform just the way I said - I hope."
Paul took a pair of scissors, dug the point in the centre of the paper and cut the paper strip in two lengthwise. And when he had finished sure enough - he had one strip twice as long, half as wide, and with a double twist in it.
Ford Fordsen was convinced. He went out and started cutting the big belt in two. And, at this point, a man called Loud Mouth Johnson arrived to see how Paul's enterprise was coming along, and to offer any destructive criticism that might occur to him. Loud Mouth Johnson, being Public Blow-Hard Number One, found plenty to find fault with.
"If you cut that belt in two lengthwise, you will end up with two belts, each the same length as the original belt, but only half as wide."
"No," said Ford Fordsen, "this is a very special belt known as a Moebius strip. If I cut it in two lengthwise, I will end up with one belt twice as long and half as wide.'.
"Want to bet?" said Loud Mouth Johnson.
"Sure," said Ford Fordsen.
They bet a thousand dollars. And, of course, Ford Fordsen won. Loud Mouth Johnson was so astounded that he slunk off and stayed away for six months. When he finally came back he found Paul Bunyan just starting to cut the belt in two lengthwise for the second time.
"What's the idea?" asked Loud Mouth Johnson.
Paul Bunyan said, "The tunnel has progressed much farther and the material coming out is not as bulky as it was. So I am lengthening the belt again and making it narrower."
"Where is Ford Fordsen ?
Paul Bunyan said, "I have sent him to town to get some materials to splice the belt. When I get through cutting it in two lengthwise I will have two belts of the same length but only half the width of this one. So I will have to do some splicing."
Loud Mouth Johnson could hardly believe his ears. Here was a chance to get his thousand dollars back and show up Paul Bunyan as a boob besides. "Listen," said Loud Mouth Johnson, "when you get through you will have only one belt twice as long and half as wide."
"Want to bet?"
"Sure."
So they bet a thousand dollars and, of course, Loud Mouth Johnson lost again. It wasn't so much that Paul Bunyan was brilliant. It was just that he was methodical. He had tried it out with that strip of gummed paper, and he knew that the second time you slice a Moebius strip you get two pieces - linked together like an old fashioned watch chain.
*Reprinted from the Ford Time.v, by permission of the Ford Motor Company.
Investigating The Moebius Strip
Materials required: Sellotape, strips of paper (30cm x 3cm), scissors, coloured pens.
•Take a number of strips of paper and then in each strip make a loop by joining the ends. Try to make each loop different by twisting the strip a different number of times before joining the ends.
•Take a pair of scissors and cut each loop - lengthwise, predict what is likely to happen before you cut. Try by cutting down the middle, or cutting the strip in 1/3 or 2/3 sections length ways.
Make a table of your predictions and record your results.
Take one strip of paper, make a half twist before joining the ends to make a loop.
How many sides does this loop have? (Remember that a side is a flat surface -not an edge).
Find a way to prove the number of sides that your loop has?
* Take two strips of paper and lay them on top of each other. Now connect the ends after making 1 twist.
* Predict: What do you have? A double Moebius strip? A single etc.?
* Now separate the two loops and compare with your results as in your table
* Repeat with three strips of paper.
Investigations Without Loops
* Make two loops of paper without twists.
Sellotape the loops at right angles to each other.
Predict - what do you expect to get if you cut around the middle of each strip?
Now check it out.
Investigate what happens when you use :
* 3 Loops at right angles
* 4 loops at right angles
* 2 loops joined at 60° to each other
* 3 loops joined at 60° to each other
* 4 loops joined at 60° to each other
* All of the above if each loop is a Moebius strip
* Explain how you get 5 squares from 4 joined loops.
For Excellence
You have been investigating ideas that are part of the area of mathematics called “Topology”.
Other areas of topology include: Map colouring
Inside and outside closed curves
The maths of knots
The Klein Bottle, Donuts
Choose an area of topology. Use the internet and other resources to make your own report on that aspect of topology.
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