Mbius Bands

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Möbius Bands
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There Is Only One Edge!

• If you use your finger to feel the edge of the
  moebius strip you will notice that your finger
  will end up right where it started without
  lifting it up!
• Pivture from www.abstractart.com/robertson/ro4_rob
  ertson

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Walking Proof
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There Is No Inside or Out!

• If you put your pencil on the strip (dont lift
  if off) and then draw a line you will see that
  one line covers the whole strip! That could only
  happen if there was one surface!
• Picture frodenicola.free.fr/old/images/Moebius.gif
  

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• "An endless ring-shaped band usually has two
  distinct surfaces, one inside and one outside.
  Yet on this strip nine red ants crawl after each
  other and travel the front side as well as the
  reverse side. Therefore the strip has only one
  surface." -- M. C. Escher on Moebius Strip

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Reduce, Reuse, and RECYCLE!

• A symbol we all know is the recycling symbol.
• This symbol is a moebius strip because it has one
  twist in it, it only has one side and it only has
  one edge!

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Ferdinand Möbius
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History of Möbius Bands

• Möbius band or strip is a one sided surface
  formed by giving a rectangular strip a half twist
  before joining the two ends together. It was
  named after Augustus Ferdinand Möbius
  (1790-1868), a German astronomer and
  Mathematician, in 1858.

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History of Möbius Bands

• This band was used during the Industrial
  Revolution, when all factories had a single
  source of power (steam or water wheel).
  Individual pieces were connected to turning
  shafts by belts and wheels and these belts needed
  to be replace frequently due to wear.
• By using the Möbius band, the bands wore
  evenly and lasted twice as long. They are used
  as fan belts in cars for example.

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Task

• What kind of cuts do you want to try?
• After testing your plans, what patterns did you
  discover?
• Which band and cut, in your opinion, gave the
  most fascinating result? Explain.
• Prepare your group report.

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Lesson Plan

• Materials
• Paper Strips (See Management below)
• Transparent tape
• Scissors
• Crayons or colored pencils
• Worksheets ( Attached)
• Management
• 1. Möbius bands can be made with any reasonable
  size or type of paper. Adding machine tape
  18-24 inches long is easy for students to handle.
  Cut at least five strips for each student.
• 2. When forming the bands, always tape
  completely across both sides.
• 3.To make comparisons more easily, have students
  label the bands (after cutting) with the number
  of half twists and kind of cut.

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Procedure

• 1. Have each students label both ends of strip as
  seen below

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Procedure

• 2. Direct students to use the first strip to make
  a band with no twists. The letters should meet
  on the same side. Tape the ends together
  completely across both sides of the strips
• 3. Instruct students to take a crayon and draw
  down the center of the band until they connect to
  the starting point. Then ask the students to
  take another color and do the same on the other
  side. Ask, How many sides does this band have
  ?

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Procedure

• 4. Have students predict what will happen if you
  cut along the line just drawn.
• 5. Show students how to slightly pinch the band,
  snip down the center, and cut down the middle.
  Have them cut their own bands and record the
  results by describing and drawing.

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Procedure

• 6. Have students make a second band. Before
  taping, give one end a half twist. A blank end
  will meet a lettered end.
• 7. Challenge the students to show the number of
  sides by again drawing a line with their crayons.
  Ask, What did you discover or notice?

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Procedure

• 8. Hand out Data Sheets and Instruct students to
  make a prediction about the results of cutting
  down the center of the band. They should
  complete and record their experiment results on
  their data sheet.
• 9. Students should make another band with a half
  twist. Have them predict what will happen if
  they cut one-third of the way from the edge
  instead of along the center.

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Data Sheet Example
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Procedure

• 10. Invite students to try different numbers of
  half twist and cuts on their own and document the
  results on their individual data sheet (encourage
  intra-group discussion).
• 11. Have students prepare a formal written report
  (1 per group (format in handout)) and present to
  class. Put up an overhead of Scientific Method
  and have them read out their results (Each
  student in group must participate).

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Inquiry Based Activity

• Day one
• -Introduce a Möbius band and encourage students
  to ask What if questions that can be
  explored. As students meet in planning groups,
  guide them with the following questions
• What are the variables you are going to test?
• What kind of cuts do you want to try?
• How are you going to test for one sidedness?
• Set up a way to record their data, including
  predictions
• After carrying out your plans, what pattern did
  you discover?
• Which band and cut, in your opinion did you find
  gave the most fascinating result?
• -Students will receive a data sheet, where they
  will determine which factors they will impose
  when conducting the experiments. (See attached)
• -Student will receive handout with scientific
  method which will guide them in writing their
  reports. (See attached)
• Day Two
• -Students will give presentations of their
  experiments