Quote of the Day:
"Mathematics — the subtle fine art."
— Jamie Byrnie Shaw"
Objectives:
The student will apply the delta-epsilon definition of a limit to making a Hexaflexagon.
1. Collect Homework.
2. Distribute sheets of adding machine paper to each student — they should be three feet in length. Begin folding the sheets to form triangles — eventually they will become equilateral triangles.
Take the sheet and fold down and make a crease. Then unfold the paper and fold up along the crease, making another crease. Then unfold and fold down along the last crease.
Continue this process until you have 25 triangles.
The last 19 triangles are the ones we will use — they should be very close to equilateral!
3. The purpose of this exercise is to see that the angles in the triangle are approaching 60 degrees.
Here is a proof (making the triangles equilateral):
Represent the first angle in triangle #1 by 60 + ε degrees, where ε is the difference between the measure of the angle and 60 degrees. Then using alternate interior angles, there is an angle in triangle #2 which has this same measure.
Since the sum of the angles along a line equal 180 degrees, and the transversals represent the folds, the middle angle at the top of triangle #2 must equal 60 - ε/2 degrees. Then continue using alternate interior angles and the sum of the angles along a line equaling 180 degrees to discover the measures of the remaining angles.
Notice that the angles are approaching 60 degrees: from (60 + ε) to (60 - ε/2) to (60 + ε/4) to (60 - ε/8) to (60 + ε/16), etc.
If the original angle was 80 degrees, so that it was off by 20 degrees (ε = 20), then by triangle #5, ε = 20/16 = 1.5 degrees!!!! So the angle would equal 61.5 degrees. And the next ones would be closer.
Click here for instructions on folding and flexing the Hexaflexagon
5. Assignment:
Finish decorating all 6 sides of the Hexaflexagon
p. 121 (5, 10, 21)
p. 140 (10, 16, 22(proof))
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