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Investigations of Paper Folding and Regular Polygons

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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Investigations of Paper Folding and Regular Polygons

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  1. Investigations of Paper Folding and Regular Polygons Presented by: Ed Knote& BheshMainali University of Central Florida,                                                                                      Phd. in Education, Mathematics Education  Graduate Students

  2. Rate this presentation on the conference app. www.nctm.org/confapp Download available presentation handouts from the Online Planner! www.nctm.org/planner Join the conversation! Tweet us using the hashtag #NCTMDenver

  3. Mathematical ReflectionsIn a Room with Many Mirrors • by Peter Hilton, Derek Holton, Jean Pedersen • Chapter 4: Paper-Folding, Polyhedra-Building, and Number Theory

  4. Introduction • Greeks were fascinated with the challenge of constructing regular convex polygons. • They wanted to construct them • with Euclidean tools: • unmarked straightedge • compass

  5. 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

  6. Key Terms • Folding and twisting (FAT-algorithm) • Optimistic strategy • Primary crease line • Secondary crease line • DnUm-folding procedure

  7. Prerequisite Skills • Angle relationships, Parallel Lines, and Transversals • Polygon Interior and Exterior Angle Sums • Degree and Radian conversions

  8. Parallel Lines &Transversals

  9. Polygon Interior Angle Sums

  10. Polygon Exterior Angle Sums • Quadrilateral • Pentagon • Hexagon

  11. Radian: A unit of angle, equal to an angle at the center of a circle whose arc is equal in length to the radius.

  12. Radian Degree

  13. Degree  Radian

  14. Radian Degree

  15. 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?

  16. FAT-Algorithm

  17. 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

  18. 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?

  19. 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.

  20. Optimistic Strategy • What is a good estimate of on a protractor? • Lets take a look at our optimistic strategy. • Time to fold.

  21. 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.

  22. General Paper Folding

  23. 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?

  24. 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.

  25. General Paper Folding

  26. Optimistic Strategy • Now you come up with the folding procedure for a 13-gon.

  27. Webpage • Knote.pbworks.com • NCTM Paper Folding 2013

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