Enrollment and Credit Options: Mathematics teachers who have completed an undergraduate major in mathematics or mathematics education or the equivalent can enroll in this module for either one-half unit of University of Illinois Graduate Credit or six Continuing Education Units. Within 5 working days of completion of enrollment, these teachers will be issued a log-in and password that will provide them access to the Module Working Environment (also referred to as ClassComm) where they submit required assignments and retrieve the graded results. They will also receive an 800-number for instructional and technical support.

Mathematics teachers may also enroll as MTL Guests (free). Guest registration gives them access to instructional materials but no access to instructional support, or to the Module Working Environment.

Teachers enrolled for graduate credit must complete and submit the seven assignments for the module described in Steps #1 through #7 below and an approved Final Project (See Step #8 below for more details about this requirement.)

Teachers enrolled for Continuing Educations Units are not required to complete Final Project but they are required to complete and submit the seven assignments for the module described in Steps #1 through #7 below. Continuing Education Units are not recorded on a University transcript; However, the University of Illinois Division of Academic Outreach provides a Letter of Completion for a teacher who completes a module for continuing education units. Teachers may also enroll as MTL Guests to access instructional materials.

Mathematics teachers may also enroll as MTL Guests (free). Guest registration gives them access to instructional materials but no access to instructional support, or to the Module Working Environment.

Estimated Time Requirements for completion of the module: We estimate that the work for this module will take participating teachers an average of about 50 hours to complete with a range of about 40 to 60 hours. Technically, you have a 6-month enrollment period to complete the module after you enroll in that unit. However, we recommend that you have as a goal to complete each of the two units of the module in eight weeks or less. That should be possible even when you are teaching if you set aside three or four hours each week to work on the module.

After you have registered for Module 13 proceed as follows:

Comments on Step #1: This part of the unit establishes the technical meaning of straightedge (i.e. unmarked ruler) and compass construction in Euclidean geometry. Assignment 1.1 requires participants to complete and carefully describe several compass and straightedge constructions with these classical geometric tools.

Assignment 1.1 must be submitted by fax or snail mail. All other module assignments are submitted electronically through the Math Teacher Link Module Working Environment

STEP #2: Complete Unit 1, Part 2: Geometer's Sketchpad Tutorial. Complete and submit Assignment 1.2 according to the directions at the beginning of that assignment.

Comments on Step #2: If you already able to use all of the menu capabilities of Geometer's Sketchpad, Version 3.0 or higher,proceed directly to Assignment 1.2. Be sure to follow the directions given at the beginning of the assignment for completing and submitting Assignment 1.2

Note: Facility with Geometers Sketchpad is a prerequisite for this entire module. If you do not have this prerequisite skill, you can acquire it either completing Module 4: Using the Geometer's Sketchpad for graduate or continuing education credit, or by registering as a Guest (free!) for that module and completing it on your own. If you have used Geometer's Sketchpad in the past but currently do not use it frequently, we strongly recommend that you refresh your background by completing Module 4 as an MTL Guest.

In any case, you will be expected to demonstrate skill with Geometer's Sketchpad in this module.

Comments on Step #3: Regular polygons are a standard topic in the geometry component of the middle school and high school curriculum. Although geometry books often discuss the construction of equilateral triangles, squares, hexagons and octagons, the general question of constructing regular n-gons for integers n > 2 is not considered. The purpose of this section is to discuss most of what is known about this fascinating subject. By the way, although we show in Unit 2 that a regular heptagon (7-gon) cannot be constructed with compass and straightedge alone, the remarkably close approximation displayed on the Unit 2 home page was discovered by a middle school student!

Problems 4 and 5 of Assignment 1.3 require the development of mathematical formulas that may be most easily done in a Microsoft Word document with Geometer's Sketchpad inserts.

Comments on Step #5: The constructible numbers are the real numbers whose absolute values are the lengths of line segments that can be constructed from a line segment of length 1 using compass and straightedge alone. Although this definition may seem like an odd mix of geometry and numbers, the constructible numbers turn out to be a very respectable number system. In this section, we use geometric arguments to show that the constructible numbers form a subfield of the field of real numbers that includes all of the rational numbers and is included in the subfield of algebraic numbers (i.e., the set all real numbers that can be roots of polynomial equations with rational coefficients). These constructible numbers are the key to understanding what is possible and what is not when it comes to compass and straightedge constructions.

Comments on Step #6: A geometric construction is a series of operations of the following two sorts: 1) Drawing or extending a line or line segment through two points that have already been constructed; 2) Drawing a circle centered at an already constructed point and passing through another already constructed, point, or with a radius equal to the length of an already existing line segment. Each such construction step may produce new points of intersection with existing lines and circles.

This section shows how these construction steps correspond to arithmetic operations in the field of constructible numbers. This correspondence provides a numerical counterpart to any Euclidean construction that enables us to decide if a proposed construction is possible or not. The Constructible Roots Theorem turns out to be the key to deciding that several famous proposed constructions are impossible.

Comments on Step #7: In this section, we show how the Constructible Root Theorem can be applied to show that it is not possible to trisect an angle of 60 degrees, or duplicate a cube, or square a circle with straightedge and compass alone. In Assignment 2.3 you will also conclude the impossibility of several other constructions by using the tools we have developed. This is the payoff section for the whole module!

If you are enrolled for Continuing Education Units, you have just completed the module. Congratulations!

You have been or will be sent feedback on your seven assignments by return e-mail. After all of these assignments have been completed and graded, the University of Illinois Division of Academic Outreach will send you an official letter of completion for the module. There is no transcript record for Continuing Education Units as there is with undergraduate or graduate credit, so this letter will serve as your proof of completion.

Those of you enrolled for graduate credit still need to complete an approved Final Project. Go to Step 8 to find out how to do that.

Final Project Topic: You are given a great deal of latitude in the choice of topic for this unit because we want the choice to reflect your teaching situation and your interests. We require only that the unit is based on the module. Of course, this module is written for teachers and your final project is for high school students, so the level and emphasis of your final project will likely be different than what is found in the module.

For example, your final project may focus on only one of the impossible constructions and present the solution less formally than in the module. Historical discussions of the problems based on the information in links of the module would also be useful.

Required Final Project Proposal: After you have selected the topic for your final project, compose an e-mail message describing in a paragraph or two how you want to develop your topic, and send it to us to us in an e-mail message at: geomConst@mtl.math.uiuc.edu. We will respond with suggestions or ask for further explanation. Once your final project plan is approved, you can proceed with the development of the project.

Details about the Final Project: Your final project should be a classroom unit that might require 2 - 4 days to discuss in class. It must include a lesson plan, any necessary student worksheets or handouts and any documents for classroom demonstrations that you develop for the project. The project should be documented well enough so that another teacher can use it without further explanation. If practical, we would also like you to teach and evaluate the unit as part of your report, but that is not an absolute requirement. The final project will count for at least one-fourth of your final grade in the module, so it should reflect a corresponding effort and time commitment.

When you have completed your Final Project, submit all documents electronically through the MTL Module Working Environment. Also, submit a print copy and either a floppy disk copy or a copy on a CD, of all Final Project materials to:

Professor Tony Peressini, Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801.

Also Email geomConst@mtl.math.uiuc.edu telling us that you have sent in the Final Project.

You are done! The Math Teacher Link instructional staff will review and provide feedback on your assignments and Classroom Project.