Möbius Bands Eric A. DeCuir Jr. & Janice Jones GK-12 Fellows and Teachers Presentation June 25, 2003

Anticipatory Set: Begin with a discussion using a regular untwisted paper band and ask questions such as: –Where have you seen things like this before? –How many surfaces or sides does it have? –Ask for suggestions on how one can prove it. Show geometric shapes and ask the number of surfaces?

Anticipatory set continued…. What would happen if one would cut down the center of the band? Ask what would happen if one put a half twist in the paper and tape it together? –How many surfaces does it have? Explain that there is a proper name for this figure and go into history of Möbius band:

“Walking Proof”

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 ( ), a German astronomer and Mathematician, in This band was used during the Industrial Revolution, when many 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. These bands are still used today in modern factories. They are also employed in some types of printer bands (last twice as long as a circular band. 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 ( ), a German astronomer and Mathematician, in This band was used during the Industrial Revolution, when many 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. These bands are still used today in modern factories. They are also employed in some types of printer bands (last twice as long as a circular band. ****Ask why these bands lasted longer?

Background The background knowledge and processes needed for this exercise include: Knowledge: –Basic Geometry and spatial sense topology –Counting –Fractions –Scientific Methodology Processes: –Predicting –Observing –Collecting and Recording Data –Comparing and Contrasting

Abstract This lesson will explore the Möbius band by observing the results of varying number of twist and kinds of cuts. In this experiment students will learn to plan and conduct a simple investigation using the scientific method and employ simple equipment and tools to gather data and extend the senses. The students will investigate and predict the results of combining, subdividing and changing shapes. The math employed in this exercise will involve geometry, spatial sense topology, counting, and fractions. The integrated processes include predicting, observing, collecting and recording data, comparing and contrasting. In this lesson, the fellow will briefly introduce the Möbius band (that its one sided) and encourage student to ask “what if..?” questions that can be explored. Questions to ask students to consider: How can you test for one-sidedness? 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. After student conduct experiments and prepare their group report, a good exercise would be to have the groups present their findings to the rest of the class and discuss what facts they found interesting about the Möbius band. This lesson will explore the Möbius band by observing the results of varying number of twist and kinds of cuts. In this experiment students will learn to plan and conduct a simple investigation using the scientific method and employ simple equipment and tools to gather data and extend the senses. The students will investigate and predict the results of combining, subdividing and changing shapes. The math employed in this exercise will involve geometry, spatial sense topology, counting, and fractions. The integrated processes include predicting, observing, collecting and recording data, comparing and contrasting. In this lesson, the fellow will briefly introduce the Möbius band (that its one sided) and encourage student to ask “what if..?” questions that can be explored. Questions to ask students to consider: How can you test for one-sidedness? 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. After student conduct experiments and prepare their group report, a good exercise would be to have the groups present their findings to the rest of the class and discuss what facts they found interesting about the Möbius band.

Lesson Plan Goal The goal of this lesson will be to condition students to critically think in situations which don’t always follow traditional reasoning. I.E., when certain situations seem obvious, they don’t always follow what seems to be the most obvious answer. Other goals include refining a student’s ability to use the scientific method, or critically test and record information based on those test. These tools, which will set the stage for future activities, are crucial to a student’s ability to meticulously attack a problem associated with answering scientific and mathematical problems. The goal of this lesson will be to condition students to critically think in situations which don’t always follow traditional reasoning. I.E., when certain situations seem obvious, they don’t always follow what seems to be the most obvious answer. Other goals include refining a student’s ability to use the scientific method, or critically test and record information based on those test. These tools, which will set the stage for future activities, are crucial to a student’s ability to meticulously attack a problem associated with answering scientific and mathematical problems.

Lesson Plan Content Objectives Skills targeted for this exercise include refinement of all those skills previously described in the background: Skills targeted for this exercise include refinement of all those skills previously described in the background:Knowledge: Basic Geometry and spatial sense topology CountingFractions Scientific Methodology Processes:PredictingObserving Collecting and Recording Data Comparing and Contrasting with particular emphasis on scientific methodology so as to set the stage for future exercises in both mathematics and science.

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 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.

Estimated Time 2 (50 Minute) Class Periods –1 Day for Experiment (30 min Experimentation / 20 min intra-group discussion) –1 Day for Presentations

Procedure 1. Break the class up into groups of 3 students: –(Have them count numbers out 1-3 or have a deck of cards with three of each kind and pick a card)

Procedure 2. Distribute the activity sheets, paper strips, transparent tape, scissors, and crayons. 3. Have each students label both ends of strip as seen below:

Procedure 4. 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 5. 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 6. Have students predict what will happen if you cut along the line just drawn. 7. 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 8. Have students make a second band. Before taping, give one end a half twist. A blank end will meet a lettered end. 9. 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 10. 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. 11. 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 12. 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). 13. 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).

Scientific Method Handout

Closure, Results, Follow-up As the students complete the experiments and reports, they should have a greater understanding of how to identify the problem and then utilize the scientific method to strategically develop a method to solve the problem. This exercise will encourage critical thinking and also encourage the students to develop their ability to explain in writing the world around them. Follow up lessons: As students develop a working knowledge of the scientific method, this will allow the teacher to develop experiments with higher levels of inquiry and also give the student more freedom in developing their own strategies on how to attack a problem or answer a question.

Standards Guiding Documents: Project 2061 Benchmark Mathematics is the study of many kinds of patterns, including numbers, shapes and operations on them. Sometimes patterns are studied because they help explain how the world works or how to solve practical problems, sometimes because they are interesting in themselves. NRC Standards -Plan and conduct a simple investigation -Employ simple equipment and tools to gather data and extend the senses. NCTM Standards -Investigate and predict the results of combining, subdividing, and changing shapes. -Develop spatial sense Arkansas Math Standards - GS.1.3 Make predictions based on transformation of geometrical figures -GS.1.4 Establish and apply geometric relationships through informal reasoning. -GS.2.2 Investigate geometric properties and use them to describe -GS.2.2 Investigate geometric properties and use them to describe and explain situations in society and nature.

Assessment Students will be graded on individual and group report’s totality and descriptive qualities.

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

References Websites: – /topology.html Books: –Cordel, Betty, et al. Hardhatting in a Geo- World, AIMS Education Foundation

Acknowledgements Janice Jones for her professional input and advice. The Center for Math and Science Education for reference materials