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Slide1
Investigations
of Paper Folding and Regular Polygons
Presented by:
Ed Knote
&
Bhesh
Mainali
University
of Central Florida,
Phd
.
in Education, Mathematics Education
Graduate StudentsSlide2
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Mathematical Reflections In a Room with Many Mirrors
by Peter Hilton, Derek Holton, Jean PedersenChapter 4: Paper-Folding, Polyhedra-Building, and Number TheorySlide4
Introduction
Greeks were fascinated with the challenge of constructing regular convex polygons.
They wanted to construct them with Euclidean tools: unmarked straightedge compassSlide5
Objectives
We will perform, understand, & explain: Paper-folding procedure
Paper-folding construction of regular convex octagons
Optimistically use Paper-folding to construct regular convex heptagons Slide6
Key Terms
Folding and twisting (FAT-algorithm)Optimistic strategyPrimary crease line
Secondary crease line
Dn Um-folding procedureSlide7
Prerequisite Skills
Angle relationships, Parallel Lines, and TransversalsPolygon Interior and Exterior Angle SumsDegree and Radian conversionsSlide8
Parallel Lines & TransversalsSlide9
Polygon Interior Angle SumsSlide10
Polygon Exterior Angle Sums
Quadrilateral
Pentagon
HexagonSlide11
Radian: A unit of angle, equal to an angle at the center of a circle whose arc is equal in length to the radius. Slide12
Radian
DegreeSlide13
Degree
RadianSlide14
Radian
DegreeSlide15
Regular Polygons & Radians
How did
w
e find the degree measure of each exterior angle of a regular polygon?What would that formula look like in radians? Slide16
FAT-AlgorithmSlide17
FAT-Algorithm
Fold And Twist
Assume we have a nice strip of paper with straight parallel edges
Mark your first vertex (near the left side)Construct your angle (where b is the number of sides for your polygon)Fold this angle in half and mark itThen repeat process at equally spaced verticesSlide18
FAT-Algorithm
What is the significance of the angle ? What are some angles in this form we can easily construct?What polygons do they relate to?
What are some angles that we can not?Slide19
General Paper Folding
Each new crease line goes in the forward (left to right) direction along the strip of paper
Each new crease line always bisects the angle between the last crease line and the edge of the tape from which it originates. Slide20
Optimistic Strategy
What is a good estimate of on a protractor?Lets take a look at our optimistic strategy.Time to fold.Slide21
General Paper Folding
Each new crease line goes in the forward (left to right) direction along the strip of paper
Each new crease line always bisects the angle between the last crease line and the edge of the tape from which it originates. Slide22
General Paper Folding
Slide23
Optimistic Strategy
Did your angle get closer to ?Why do you think this happens?Can we prove this mathematically?
How can we show this in Excel? Slide24
Optimistic Strategy
Is this perfect or just a close estimate?Is this folding procedure the same for all polygons?What would it be for a pentagon. Slide25
General Paper Folding
Slide26
Optimistic Strategy
Now you come up with the folding procedure for a 13-gon.Slide27
Webpage
Knote.pbworks.comNCTM Paper Folding 2013