UNIT 4: POLYGONS. LESSON 1: ANGLES OF POLYGONS AND QUADRILATERALS.
Investigations of Paper Folding and Regular Polygons
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Transcript of Investigations of Paper Folding and Regular Polygons
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Investigations of Paper Folding and Regular Polygons
Presented by:Ed Knote & Bhesh Mainali
University of Central Florida,
Phd. in Education, Mathematics Education
Graduate Students
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• Mathematical Reflections In a Room with Many
Mirrors by Peter Hilton, Derek Holton, Jean Pedersen
• Chapter 4: Paper-Folding, Polyhedra-Building, and Number Theory
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Introduction
Greeks were fascinated with the challenge of constructing regular convex polygons.
They wanted to construct them with Euclidean tools:• unmarked straightedge• compass
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ObjectivesWe will perform, understand, &
explain:– Paper-folding procedure – Paper-folding construction of regular
convex octagons – Optimistically use Paper-folding to
construct regular convex heptagons
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Key Terms• Folding and twisting (FAT-algorithm)• Optimistic strategy• Primary crease line• Secondary crease line• Dn Um-folding procedure
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Prerequisite Skills• Angle relationships, Parallel Lines, and
Transversals
• Polygon Interior and Exterior Angle Sums
• Degree and Radian conversions
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Polygon Exterior Angle Sums
• Quadrilateral
• Pentagon
• Hexagon
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Radian: A unit of angle, equal to an angle at the center of a circle whose arc is equal in length to the radius.
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Radian Degree
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Degree Radian
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Radian Degree
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Regular Polygons & Radians
• How did we find the degree measure of each exterior angle of a regular polygon?
• What would that formula look like in radians?
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FAT-Algorithm
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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 it– Then repeat process at equally spaced
vertices
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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?
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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.
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Optimistic Strategy–What is a good estimate of on a
protractor?
– Lets take a look at our optimistic strategy.• Time to fold.
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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.
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General Paper Folding•
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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?
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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.
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General Paper Folding•
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Optimistic Strategy– Now you come up with the folding
procedure for a 13-gon.
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Webpage
– Knote.pbworks.com• NCTM Paper Folding 2013