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Important definitions:

Polygon: closed plane figure bounded by straight-line segments (sides) intersecting at points called vertices.

From The Concise Columbia Encyclopedia, Third Edition


Recursion: (noun) Mathematics. 1. An expression, such as a polynomial, each term of which is determined by application of a formula to preceding terms.

2. A formula that generates the successive terms of a recursion.

From the American Heritage Dictionary of the American Language.

Time Estimate


Main Idea

25 min


Add triangle to a polygon to see what happens, adding to the non-regular shapes created earlier.

Do this with concave polygons as well.

Further justification! Add 2 sides and lose 1 side (net gain is 1 side). Adds 180 degrees from the triangle. Recursion idea needs to be discussed

10 min


What about the bow tie?

Exploring the definition of a polygon by looking at non-polygons.

Step-by-step guide

1. Fine the formula for the angle sum of the next polygon after you add another triangle onto one side. S(3) = 180; S(n) = S(n-1) + 180. See the definition of recursion at the left.

2. Discuss the bow tie. Present the shape with polystrips or GSP to provide visual support for the shape. Why does this not work? Appears to create 2 triangles with 4-sided shape, but it is really two polygons connected. That is, lines of a polygon cannot cross. If they do, it is no longer a polygon.

Possible Problems and Concerns:
Recursion idea needs to be discussed. S(3) = 180; S(n) = S(n-1) + 180. We could use this formula to generate the next polygon. The formula is recursive because it uses preceding terms to define itself.
Internet Resources:

mste.illinois.edu/m2t2/algebra/concave.gsp| mste.illinois.edu/m2t2/algebra/nonpoly.gsp

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