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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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## Investigations of Paper Folding and Regular Polygons

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**Investigations of Paper Folding and Regular Polygons**Presented by: Ed Knote& BheshMainali University of Central Florida, Phd. in Education, Mathematics Education Graduate Students**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**Mathematical ReflectionsIn a Room with Many Mirrors**• by Peter Hilton, Derek Holton, Jean Pedersen • Chapter 4: Paper-Folding, Polyhedra-Building, and Number Theory**Introduction**• Greeks were fascinated with the challenge of constructing regular convex polygons. • They wanted to construct them • with Euclidean tools: • unmarked straightedge • compass**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**Key Terms**• Folding and twisting (FAT-algorithm) • Optimistic strategy • Primary crease line • Secondary crease line • DnUm-folding procedure**Prerequisite Skills**• Angle relationships, Parallel Lines, and Transversals • Polygon Interior and Exterior Angle Sums • Degree and Radian conversions**Polygon Exterior Angle Sums**• Quadrilateral • Pentagon • Hexagon**Radian: A unit of angle, equal to an angle at the center of**a circle whose arc is equal in length to the radius.**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?**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**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?**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.**Optimistic Strategy**• What is a good estimate of on a protractor? • Lets take a look at our optimistic strategy. • Time to fold.**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.**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?**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.**Optimistic Strategy**• Now you come up with the folding procedure for a 13-gon.**Webpage**• Knote.pbworks.com • NCTM Paper Folding 2013

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