Möbius BandsMöbius Bands

There Is Only One Edge!There Is Only One Edge!

If you use your finger If you use your finger to feel the edge of the to feel the edge of the moebius strip you will moebius strip you will notice that your finger notice that your finger will end up right will end up right where it started where it started without lifting it up!without lifting it up!

Pivture from www.abstractart.com/robertson/ro4_robertsonPivture from www.abstractart.com/robertson/ro4_robertson

““Walking Proof”Walking Proof”

There Is No Inside or Out!There Is No Inside or Out!

If you put your pencil on If you put your pencil on the strip (don’t lift if off) the strip (don’t lift if off) and then draw a line you and then draw a line you will see that one line will see that one line covers the whole strip! covers the whole strip! That could only happen if That could only happen if there was one surface! there was one surface!

Picture frodenicola.free.fr/old/images/Moebius.gifPicture frodenicola.free.fr/old/images/Moebius.gif

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

Reduce, Reuse, and RECYCLE!Reduce, Reuse, and RECYCLE!

A symbol we all know A symbol we all know is the recycling is the recycling symbol. symbol.

This symbol is a This symbol is a moebius strip moebius strip because it has one because it has one twist in it, it only has twist in it, it only has one side and it only one side and it only has one edge!has one edge!

Ferdinand MöbiusFerdinand Möbius

History of Möbius BandsHistory of Möbius Bands

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

History of Möbius BandsHistory of Möbius Bands

This band was used during the Industrial This band was used during the Industrial Revolution, when all factories had a single Revolution, when all factories had a single source of power (steam or water wheel). source of power (steam or water wheel). Individual pieces were connected to turning Individual pieces were connected to turning shafts by belts and wheels and these belts shafts by belts and wheels and these belts needed to be replace frequently due to wear. needed to be replace frequently due to wear.

By using the Möbius band, the bands wore By using the Möbius band, the bands wore evenly and lasted twice as long. They are used evenly and lasted twice as long. They are used

as fan belts in cars for example.as fan belts in cars for example.

TaskTask

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

Lesson PlanLesson Plan

Materials Materials – Paper Strips (See Management below)Paper Strips (See Management below)– Transparent tapeTransparent tape– Scissors Scissors – Crayons or colored pencilsCrayons or colored pencils

Worksheets ( Attached)Worksheets ( Attached)

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

ProcedureProcedure

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

ProcedureProcedure

2. Direct students to use the first strip to make 2. Direct students to use the first strip to make a band with no twists. The letters should meet a band with no twists. The letters should meet on the same side. Tape the ends together on the same side. Tape the ends together completely across both sides of the stripscompletely across both sides of the strips

3. Instruct students to take a crayon and draw 3. Instruct students to take a crayon and draw down the center of the band until they connect down the center of the band until they connect to the starting point. Then ask the students to to the starting point. Then ask the students to take another color and do the same on the take another color and do the same on the other side. Ask, “ How many sides does this other side. Ask, “ How many sides does this band have ?”band have ?”

ProcedureProcedure

4. Have students predict what will happen if 4. Have students predict what will happen if you cut along the line just drawn.you cut along the line just drawn.

5. Show students how to slightly pinch the 5. Show students how to slightly pinch the band, snip down the center, and cut down the band, snip down the center, and cut down the middle. Have them cut their own bands and middle. Have them cut their own bands and record the results by describing and drawing.record the results by describing and drawing.

ProcedureProcedure

6. Have students make a second band. Before 6. Have students make a second band. Before taping, give one end a half twist. A blank end taping, give one end a half twist. A blank end will meet a lettered end. will meet a lettered end.

7. Challenge the students to show the number 7. Challenge the students to show the number of sides by again drawing a line with their of sides by again drawing a line with their crayons. Ask, “What did you discover or crayons. Ask, “What did you discover or notice?”notice?”

ProcedureProcedure

8. Hand out Data Sheets and Instruct students 8. Hand out Data Sheets and Instruct students to make a prediction about the results of to make a prediction about the results of cutting down the center of the band. They cutting down the center of the band. They should complete and record their experiment should complete and record their experiment results on their data sheet.results on their data sheet.

9. Students should make another band with a 9. Students should make another band with a half twist. Have them predict what will happen half twist. Have them predict what will happen if they cut one-third of the way from the edge if they cut one-third of the way from the edge instead of along the center. instead of along the center.

Data Sheet Example:Data Sheet Example:

ProcedureProcedure

10. Invite students to try different numbers of 10. Invite students to try different numbers of half twist and cuts on their own and document half twist and cuts on their own and document the results on their individual data sheet the results on their individual data sheet (encourage intra-group discussion). (encourage intra-group discussion).

11. Have students prepare a formal written 11. Have students prepare a formal written report (1 per group (format in handout)) and report (1 per group (format in handout)) and present to class. Put up an overhead of present to class. Put up an overhead of Scientific Method and have them read out their Scientific Method and have them read out their results (Each student in group must results (Each student in group must participate). participate).

Inquiry Based ActivityInquiry Based Activity

Day one:Day one:-Introduce a Möbius band and encourage students to ask “ What if…” questions -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 that can be explored. As students meet in planning groups, guide them with the following questions:the following questions:– What are the variables you are going to test?What are the variables you are going to test?– What kind of cuts do you want to try?What kind of cuts do you want to try?– How are you going to test for one sidedness?How are you going to test for one sidedness?– Set up a way to record their data, including predictionsSet up a way to record their data, including predictions– After carrying out your plans, what pattern did you discover?After carrying out your plans, what pattern did you discover?– Which band and cut, in your opinion did you find gave the most fascinating Which band and cut, in your opinion did you find gave the most fascinating

result?result?-Students will receive a data sheet, where they will determine which factors -Students will receive a data sheet, where they will determine which factors

they will impose when conducting the experiments. (See attached)they will impose when conducting the experiments. (See attached)-Student will receive handout with scientific method which will guide them in -Student will receive handout with scientific method which will guide them in

writing their reports. (See attached) writing their reports. (See attached)

Day Two:Day Two:-Students will give presentations of their experiments-Students will give presentations of their experiments