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 (i.e. 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 discussing 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.
What do we mean by a geometric construction?
A geometric construction is a finite succession of compass and straightedge constructions that begins with a given geometric figure G (e.g. a line segment AB, a triangle ABC or an angle ABC), and ends with a desired geometric figure F (e.g. the circle F centered at the midpoint of the given line segment AB with AB as its diameter, the circle F passing through the vertices of the given triangle ABC, or the trisector of the given angle ABC).
The mathematical heart of this module is that the finite succession of straightedge and compass constructions used to construct a required geometric figure F from a given figure corresponds exactly to a finite expanding succession of subfields of the field of all real numbers. In each successive step of the geometric construction, a given subfield F of the field R of real numbers is extended to a subfield F[c] which contains F as well as all numbers of the form p + q Sqrt(c) where p, q and c are numbers in F.
Some familiarity with the concept of fields and subfields of the real number system should be considered prerequisite background for this module. Teachers who have taken a course in abstract algebra or other college level courses that discussed number fields should have the necessary background for this module. The content and homework of the module will give such teachers an opportunity review, apply and extend their previous knowledge of number fields.
Credit: 2 graduate semester hrs.
Common Core Standards for Mathematical Practice that are emphasized include:
- 3. Construct viable arguments.
- 4. Model with mathematics.
- 5. Use appropriate tools strategically.
- 7. Look for and make use of structure.
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.