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The Möbius Strip; A Lab Activity

by
Kenneth Fuller
(copyright information 2006)


Teacher's preparation:

Teacher introduction:
The Moebius strip is great for the "WOW factor", it produces a "discrepant event".  That means that the result is counter-intuitive, different than the obvious expectation.  Good for a "Think about it.  Why do you suppose it came out that way instead of what you expected?"

This particular lesson plan was written during a year when we were required to use "Collaborative Learning".  So you can see how I organized my classes for group activities.  I did have the walls labeled north, east, south, west.  I think you should be able to rewrite the plan to suit your own organization, and students.

If you prefer to perform a demonstration before turning the activity over to the students, you might be interested in the accompanying story, which my mother told me when she introduced me to the strip.

It seems Paul Bunion once struck gold.  After a time the foreman came to paul and said, "Our mine tunnel has gotten so long, the men spend most of their time hauling ore to the surface.  Perhaps we should get a  conveyer belt."
    "Great idea!" Says Paul, "But I'll tell you what, so the sides will wear evenly, put a half twist in the return side of the loop."
    The foreman didn't quite understand, but he followed directions.  And it worked fine.
    Much later, he reported to Paul, "The tunnel has gone so much deeper, we need a new, longer belt.  But the vein has narrowed, so we don't need such a wide belt."
    "Oh no, not a new belt," said Paul, "Just cut the belt we have right down its center, and it will be long enough."
    "What?" said the foreman, "Oh well, you're the boss."  He followed the instruction, and lo and behold, he had a belt twice as long and half as wide.  He was impressed.
    As the vein narrowed and the mine got deeper, The foreman knew what to do this time.  He was going to impress Paul.  So, he cut the belt right down the middle again and....  Oops!


Preparing to teach: 
If you are not already familiar with the properties of the Mobius strip, be sure to practice the activity before presenting it to the class.

Topology is the study of surfaces and edges.

An ordinary loop has two surfaces and two edges.  The Mobius strip has only one surface and one edge.

Materials needed: 
For convenience I use adding machine tape cut in 60 cm lengths.  But any paper (including newspaper) which can be cut and pasted into strips will work.

Tape or glue stick for each group.

Scissors for each group.

Student Introduction:
Moebius strip lab 1 Before we try our lab activity for today, I wish to clarify a point.  Think carefully, then raise your hand.  What is meant by the word "Prediction"?  --write on screen

--call on student---anyone wish to correct, clarify, or add to that?

--write on screen--  pre- before dict- to  say    "To say before"

 

Today we are not concerned with wild guesses, nor with prophets, nor with psychics.  We are concerned with prediction in the scientific sense.  To predict is to use all available information together with past experience to tell what will most likely happen in a particular situation.

For example: We are beginning a study of meteorology.  Perhaps the most important product of meteorology is the ability to predict weather.  A forecast of "80% chance of rain" means that after checking all the measurements of weather, and looking up the records of times in the past when the measurements were about the same, 8 out of 10 times rain followed.

Some predictions have a 100% chance, such as, "Sunrise will be at __     tomorrow morning. 

One more question before we start our lab activity.  Hands please.  How can we prove that a prediction was correct?  Only by what did happen.

Now we are going to make some predictions and test them. 

Each of you will need a paper and pencil.  Put your name and period number on the paper.

At the signal __All students in these two rows, turn your chairs 180 to face those at the table behind you.  NOW 

You are now in groups of four.  As I give you your group designation,

Write it at the top of your paper.  A,B,C...

 ///place group functions on screen///

One student will write, speak for each group. <>The student on the SW of each group, raise your hand now.  Under the group designation on your paper write "speaker". 

One in each group will be responsible for getting and returning materials.  The student on the NE of each group, raise your hand now.  Under the group designation on your paper write "gofer".

One in each group will be responsible for insuring that instructions are carried out as explained and demonstrated.  The student on the SE of each group, raise your hand now.  Under the group designation on your paper write "director". 

One in each group will be responsible for seeing that every member contributes to the group effort, and within the guidelines on the screen.  The student on the NW of each group, raise your hand now.  Under the group designation on your paper write "coordinator".

 

Now speaker, if anyone in your group has a question about procedure so far, raise your hand.

As soon as all instructions given so far have been carried out by your group, I want the director to hold a hand up until all groups are ready. 

Good, hands down.  Gofers, when I give the signal, go to the front desk and get one strip of paper and the other materials for your group.  When the gofer has returned and you are ready for the next step, directors raise your hand and wait quietly. ... John, what will the gofer get?   Now gofers.

Procedure:

Directors, be sure your group does not use the tape until specifically instructed.

In each group, lay the paper strip flat, like so.....eyes this way. 

Lift the ends to make a loop, like this....eyes.  Speaker, hold the loop up so we can all see.

now directors, twist one end 180 like this so the bottom is on the top.  Speaker, hold the loop up again.

Now gofer, tape it.  When ready for the next step I want see director's hands.

///place directions for lab on the screen///

You will each turn in a paper on which you have recorded your work.  But for sharing with the class only your speaker will talk.

Now finally we are ready for the prediction:

Listen carefully to the situation.  Past experience, you have cut many pieces of paper in your life time, so you know something about how scissors and paper work.  Available facts, the loop of paper for you to observe.  Without saying anything, take 20 seconds to examine the loop very carefully.

 

Now, write on your paper, a description of the loop.  When all are finished I'll see directors hands.

Speaker how did you describe the loop?   half twist

Eyes this way.... I am going to cut the loop down the middle lengthwise all the way around.  Understood?

At the signal, write on your paper a description of this paper after I have finished cutting it.  When everyone in your group has done that, directors raise your hand.

Speaker A, what predictions have members of your group written? ... Does any other speaker have something different? ...

At the signal, the gofer will get a pair of scissors from the front desk.  While the others help keep the loop moving smoothly, the coordinator will cut all the way around near the center of the strip.  Then I will see the director's hand.

 

Now observe the result carefully and write as complete a description as you can.  I'll look for director's hands

Speaker B, did your group's results agree with your predictions? ... Any other group? ...

Note on your paper the specific differences between your predictions and your results.

Conclusion:

Now, when they ask what you learned in school, you can amaze your friends and family.

Remember, we usually learn more from predictions which are not correct than from those which are.

We will extend this activity another day.

At the signal:  Speaker- collect the papers from your group and place them in the turn-in box.  Gofer- return scissors. Coordinator- place the paper loop in the trash can. Then all return to regular seats.

Now,do it.

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