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Provide class time for group discussion and charting of comparisons made concerning 4-gon, 8-gon, and 20-gon polygons.

- Propose that not all polygons have even-numbered sides. Given 20 points on a percentage circle, can an odd-numbered regular polygon be drawn? How many sides could it have?
This question leads students to explore ways of dividing 100 by odd and even numbers in order to construct polygons that may or may not have inscribed sides of equal length line segments. Factoring 100 by a succession of odd and even numbers dramatizes the limits of the circumference as a ruler with a fixed division of numbered intervals. Some odd- and even-numbered divisions require approximations.

- Just as the students have explored the number of times a square can be inscribed (rotated) in a percentage circle of 20 points, propose that they can solve this for other polygons (odd- or even-sided).

Students should be encouraged to go on to develop inscribed designs of their own, and to seek out regular polygons of odd and even numbers of sides. When these designs are developed and presented to the class, they provide an opportunity for comparing numbers and configurations of diagonal lines.

Propose the question whether there might be some general principle or geometric property concerning diagonals in all odd- and even-numbered polygons.The teacher can explore how ready students are to discuss geometry as a mathematics of axioms, theorems, and proofs. It may be that they will perceive axioms as the givens and a theorem as a set of constructions demonstrating certain geometric properties.Efforts to present such materials as demonstrations or "proofs" will enable teacher and students to DO geometry inventively, even as it was done by artists and geometers in times past. Just as students explore the number of times a square can be inscribed (rotated) in a percentage circle of 20 points, propose that they can solve this question for any polygon, odd- or even-sided.Discuss how valid their conclusions are, and how well they might demonstrate them.This will lead to making a series of rotational designs with regular polygons having 4, 5, 8, and 10 sides. Given 200 points of interval on the circumference, students are asked to determine what if any other polygons they could construct and rotate without having to use approximations.

## Constructions

- Make a set of inscribed regular polygons with points that coincide with points of whole and half degrees on the percentage ruler, including both odd- and even-sided polygons.

- Construct a regular hexagon by means of percentage approximation.

- Use approximation to inscribe geometric shapes having 7, 9, and 11 sides.

- Employ constructions of regular polygons to investigate relationships that can be discovered concerning the number of sides of inscribed polygons and the chord lengths that make up their perimeters.

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