Charles Kwan (ckwan@usa.net),(ckwan@dimacs.rutgers.edu)
color pens
paper
ruler
scissors
Have two students work in the group. Give each group one sheet of paper (8.5 by 11 or 8.5 by 14). Tell students to fold the paper into half(the long way, i.e. like the hot-dog). Cut the paper into two pieces. Fold each piece of paper into half (the long way). Cut both papers into half. Students should have four strips of band.
Have the students tape the two strips of paper together to form a long strip of band. Repeat this process with another set of bands.
On both strips of band, label each corner with different letters (i.e. A, B, C, D) in one direction and one side. Take one strip of paper ABCD and join it together without twisting the paper so A meets D and B meets C.
Have the students color the band with different colors for each side.
1. What kind of the curve is this?
2. How many edges does the band have?
3. How many sides does the ring have?
4. What is the circumference of the ring?
5. What is the surface area of the ring?
6. What is the area enclosed by the ring?
Have the students cut the ring along the center of the curve into half.
7. How many rings are there?
Students should be able to make some connections in finding the area and the circumference of a circle to finding the area and the perimeter of a rectangle.
Take a second strip of the band, twist one end of the strip and tape both ends together with A meeting C and B meeting D.
Have the students color the band with one color per each side.
1. How many colors are used to color the band?
2. How many edges does the band have?
3. How many sides does the ring have?
4. What kind of the curve is this?
Tell student the name of this strip is known as “Mobius strip or Mobius band or twisted cylinder”
Have the students cut along the center of the Mobius strip.
1. What kind of the curve is this?
2. How many twists are there in the ring?
Have the students cut the same Mobius strip again along the center of the band.
1. What have you discovered?
A culminating assignment might include a research paper on how Mobius strip is used in arts, engineering and science. Also, encourage students to use the Internet as a resource for their research paper on the Mobius strip.
Vocabulary:
close curve - no break in a line (continuous curve)
open curve - break in a line (discontinuous curve)
simple curve - the perimeter of a curve figure does not intersect (cross) itself
complex curve - the perimeter of a curve figure intersects (crosses) itself
circumference - distance around a circle
edge - a line segment on a space figure where two faces intersect
perimeter - the sum of the lengths of all sides
area - the region enclosed by the a plane figure
parallel lines - two lines in the same plane that do not intersect
plane - a set of points forming a flat surface that continues forever in all directions
topology - the study of the properties of figures that endure when the figures are subjected to continuous transformation
one sided figure - plane with one side only and vertices are joined by the opposite angles of a rectangle
a.) Do you have a Mobius strip?
b.) How many sides does the ring have?
2. Cut this strip along the center of the ring. What have you discovered?
2. What are the applications of the Mobius strip or Mobius band?
3. How is Mobius strip and Mobius band used in arts?
Klein bottle
Torus
Read about the life and work of August Ferdinand Möbius.
Use WWW to explore related information for Mobius strip or Mobius band.
Math project or paper in topology
2. Pappas, Theoni. “Fractals, Googols and Other Mathematical Tales”, Wide World Publishing, 1993
3. VanCleave’s, Janice. “Geometry for Every Kid”, John Wiley & Sons, 1994
4. Vorderman, Carol. “How Math Works”, Reader’s Digest, 1996
5. Henle, Michael. “A Combinatorial Introduction to Topology”, Dover, New York, 1979
6. Goodman-Strauss, Chaim. Workshop on Topology, Princeton University, June-July 1996
7. Garner, Jan. “The Mobius Strip” 8. Algebra Group WWW Server-August Ferdinand Möbius