Department of Mathematics Education
Dr. J. Wilson, EMAT 6690

Instructional Unit for a Virtual Classroom:

Investigating Properties of Parallelograms

by David Wise

Introduction

I have used the Discovering Geometry textbook by Michael Serra in my past teaching experiences and have become a proponent of the text and the discovery approach. Most of the lessons of the textbook are designed to investigate one or more properties of a geometric figure, and at the end of most lessons, a section titled "Take Another Look" suggests confirming new conjectures using Geometer's Sketchpad (GSP). Key Curriculum Press, who publishes Discovering Geometry, also publishes many supplementary materials that provide excellent GSP investigations. The two that I have used and highly recommend are Exploring Geometry with the Geometer's Sketchpad and Discovering Geometry with the Geometer's Sketchpad, which both provide blackline activity lessons that complement the textbook. Another resource that I am not familiar with, but suspect is outstanding, is Rethinking Proof with the Geometer's Sketchpad by Michael de Villars.

My purpose is to develop investigations that can be pursued in a virtual classroom using GSP as the tool for discovering and proving a conjecture. I am using Discovering Geometry as the foundation for the GSP investigations, but these investigations could be used with any textbook with minor modifications. In a virtual setting, I believe these investigations will provide students with a solid understanding of the basic properties of parallelograms, help develop and strengthen the concept of proof, and help students to analyze the similarities and differences between special parallelograms. In a more traditional classroom environment, these investigations can be used as a lesson, part of a lesson, or supplemental materials to be used with students who are having difficulty understanding a particular conjecture.

The most common criticism of Discovering Geometry that I have heard from other teachers is the de-emphasis of proofs. Many students develop the false belief that a conjecture is a proof. Similarly, students often misinterpret a GSP sketch as a proof. The word conjecture tends to lose its true meaning. Therefore, a major aspect of these investigations is to use GSP to outline the approach to a proof of each conjecture.

For more information on Discovering Geometry, supplemental resources, and/or GSP, contact the publisher, Key Curriculum Press. To find instructions on setting up GSP as a helper application click here.

Table of Contents: I recommend that each page be printed out, so that the instructions are easier to follow.

1. The Virtual Classroom
A Discussion
2. Parallelograms
a. Discovering some special properties (C-52 through 55)
b. Construction
c. Outline of Proof 1 (C-52)
d. Outline of Proof 2 (C-53)
e. Outline of Proof 3 (C-54)
f. Outline of Proof 4 (C-55)
3. Rhombuses
a. Discovering some special properties (C-57 and 58)
b. Construction
c. Outline of Proof 1 (C-57)
d. Outline of Proof 2 (C-58)
4. Rectangles
a. Discovering some special properties (C-59 and 60)
b. Construction
c. Outline of Proof 1 (C-59)
d. Outline of Proof 2 (C-60)
5. Squares
Discussion of special properties and Construction
6. Parallelogram Relationships
Venn Diagram and Family Tree
7. Bibliography

If you have any questions while trying to complete this investigation, or suggestions that would be useful, especially for use at the high school level, please send e-mail to esiwdivad@yahoo.com.