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This module is designed to familiarize teachers with the
close connection that exists between compass and straightedge
constructions and the real number system. Most geometry teachers
are aware that certain geometric constructions such as trisecting
certain angles or squaring a circle (ie. constructing a square
whose area is equal to that of a given circle) are known to
be impossible, but they may not know why. This module explains
the relationship between geometric constructions and a set
of real numbers called the constructible real numbers, and
uses this relationship to decide the possibility or impossibility of certain
geometric constructions. The methods used in this module avoid
the intricate machinery that is sometimes used in discuusing
these problems in college level courses in abstract algebra.
This module is divided into the following units:
- Unit 1: Discovering geometry through geometric constructions.
- Unit 2: Geometric constructions and the real number system.
Although these units deal with topics directly taught high
school mathematics, their content is not intended for use
directly in the high school classroom. However, teachers will
find that the content of these units directly informs and
influences their teaching of geometry in the high school classroom.
Credit: 1/2 grad. unit (=
2 sem. hr.) or 6 CEUs.
See Step-by-Step Instructions for more
information about enrollment options and instructions for completing
this module.
The author of this module is Tony Peressini, Professor
Emeritus of Mathematics at the University of Illinois.
This module was completed in March 2003.
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