The Biggest Hole; A Lab Activity

by
Kenneth Fuller
(copyright information 2006)


Teacher's preparation:


This is activity has the "WOW" factor of a discrepant event, that is the result is "impossible".  In my experience most students will desire to repeat the activity for themselves.

 Topology is the study of surfaces and edges.  Our piece of paper has two surfaces and one edge.  We generally refer to the surfaces as the front and back of the paper.  And we divide the edge into the sides, top and bottom of the paper.  In topology we don't worry about the shape of a surface or edge.  When we cut a hole in the paper, we add a second edge.  One edge around the outside of the surface, and one around the inside of the surface (the outside of the hole).  Student will intuitively know that the inside edge can not be longer than the outside edge, and will estimate the largest diameter of their hole accordingly.  What student almost never recognize is that we can make the outside edge of the paper longer.

Materials needed: 
Ordinary lined paper, preferably without holes, 8.5 X 11 inches is what I used.  But any paper (including newspaper)  will work.

Scissors for each group.
Student Introduction:

Here we have an ordinary piece of writing paper.  It has two surfaces, and one edge. If we cut a hole in the paper, it will have two edges.  Remember, a diameter is the length of a straight line drawn across an area, in this case the area of the hole.

Without actually cutting the paper, I want each team to estimate the longest diameter they will be able to get for a hole cut in the paper.  Who can cut the largest hole in the paper without separating the paper so that it has to be taped back together or anything like that.  I’ll give you 3 minutes for your team to think about it, then I will ask for your estimates.

(Request estimates from volunteers.)

Now that we have your best thoughts on the subject, I am telling you that I am going to cut a hole in this sheet of paper with a diameter of more than 5 feet.  Watch!

(Perform demonstration.)

John, please stand and hold this part of the paper.  Jane, please stand over here and hold this corner carefully, it tears easily.  Ok, hold it higher.  There, a hole you could ride your bicycle through!

Now you each have a chance to try it for yourself.

Directions:

  1. Take a standard piece of 8.5 X 11 inch lined paper, preferably without holes.
  2. Fold it in half lengthwise, and crease it.
  3. From the side opposite the fold, beginning with the second line from the bottom, cut both thicknesses of paper along the line to about 0.5 inch from the fold.
  4. Continue to cut similarly on alternate lines.  Be sure leave a strip at the top at least as wide as the others.
  5. From the fold side, cut along each uncut line to about 0.5 inch of the edge.
  6. Being sure that you do not cut either of the end strips, cut along the fold.
  7. Gently unfold and stretch the paper out.  The diameter of hole will be many times larger than the diameter of the whole paper.
    8.

Today when asked what you learned in school, you will have an amazing answer.



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