Möbius Bands

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_robertson

“Walking Proof”

There Is No Inside or Out! If you put your pencil on the strip (don’t 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

"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

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!

Ferdinand Möbius

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.

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.

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.

Lesson Plan Materials Worksheets ( Attached) Management: 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.

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

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 ?”

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.

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?”

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.

Data Sheet Example:

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).

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