The Moebius Strip

Arnita Newton                  Kenwood Academy
                               5015 Blackstone Avenue
                               Chicago IL 60615
                               312-535-1350

Objective:

To investigate mathematical patterns using the Moebius Strip.

Materials Needed:

Strips of paper 10 inches long and 2 inches wide.  Adding machine tape, 
construction paper or graph paper.  Allow 10 strips for each student.
Markers or colored pencils optional.  Students will also need Scotch "Magic" 
tape and scissors.

Strategy:

Students are asked to examine their strips of paper to determine that each strip                                   
has two edges and two sides.   Students are told to make a loop, turn one end 
over (in a half-twist) and tape the ends together.  Make a second loop without 
twisting.

Students are asked to draw a lengthwise line in the center of each strip, 
continuing until they reach their starting point.  Students will report all of 
their observations.

Students will note that their pencils never crossed over the edge.  This surface 
does not have a top and a bottom or a front and a back.  Instruct students to 
cut the Moebius strip along the line they drew.  Ask students, "What did you 
get, how many sides does it have, how many half-twists does it have?" 
 
Students are asked to make another Moebius strip and cut along a path that is 
about one-third of the distance from one edge to the other.  Describe the 
results.  Other Moebius strips may be constructed in the same way, cutting one-
fourth of the way in and cutting one-sixth of the way in.  Students are asked 
to describe their results after cutting each strip. 

Students are encouraged to investigate strips with different numbers of half-
twists, cutting each strip down the middle as they did with the first Moebius 
strip.

Make a Moebius strip and draw lines to divide it in thirds lengthwise.  Shade 
the middle third.  Cut along the edge of the shaded third.  Describe your 
results.
   
Conclusion:
  
The Moebius Strip is an interesting topological figure.  The investigation also
provides a good exercise in having students derive a generalization from their  
empirical observations.