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: 2 grad. sem. hrs. 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.