Using Geometer’s Sketchpad in a Euclidian Geometry class
What use is Geomter’s Sketchpad, a dynamic geometry software package, in a traditional Eucildian geometry class?
In the Spring of 2002, I
student-taught the Euclidian Geometry section of Dr. Bernard Feigenbaum’s class at
The constraints on the course I was teaching were significant. I was expected to use Geometry: A Contemporary Approach (2nd ed) by Harry Lewis. This text is not currently available—it was last published in 1968. Someone at Stuyvesant arranged for a private printing of selected chapters so they would have enough copies to continue to use this text. The original printing is two-color, the edition the students received was photocopied and bound. The institutional prejudice appears to be that this text does the most logically consistent and complete treatment of Euclidian geometry of any text they have reviewed. The textbook, however, has no examples. It is as far as one can get from the current crop of colorful, application filled, technology-enhanced texts now available.
As a student teacher, I felt I did not have the liberty to spend time off-topic investigating dynamic issues not central to traditional definition-postulate-theorem-proof geometry. Furthermore, I did not feel comfortable dedicating significant time to learning how to use the computer package.
The course I was taking at NYU introduced me to many of the features that are offered by Geometer’s Sketchpad. However, most of the work focused on dynamic geometry problems little addressed in a first course in Euclidian Geometry. Daniel Shier’s website, http://www15.addr.com/~dscher/, has many examples of the types of problems we studied in the class. However, little of what was done in the NYU class seemed directly appropriate for use in the class I was teaching.
Nevertheless, I found three ways in which the dynamic geometry software package could be integrated into a traditional geometry curriculum, without significant loss of time. The first is actually a static use of the program. It does allow the teacher to draw properly proportioned figures, and thus I found it helpful for preparing overheads and exams. Second, I did a couple of interactive proofs using Geometer’s Sketchpad and an LCD projector in the classroom. These required a good deal of preparation on my part, but allowed students to see completely worked-out proofs by the end of the class. Third, I designed three lessons that allowed the students to use the software in the computer lab in lieu of a classroom lesson on these topics.
Overheads
The following were printed on overhead transparencies to be used on a regular projector in the classroom. The advantage these have over sketches on the blackboard or overhead is the diagrams are to-scale and thus are more accurate than other illustrations. One place where the dynamic nature of the software was helpful was in the Chapter 2 file, in the overhead titled: “perpendicular, complementary”. Here, I have illustrations where there are assumptions that students might draw looking at the top illustrations that are not based on the information given. The bottom pictures were made by simply dragging points in the top pictures.
Overheads - Chapter 3 and 4.gsp
Exams
Similar to the overheads, I made use of Geometer’s Sketchpad in making my exams. Note that some parts of the exam were not done using GSP.
Interactive Proofs
These proofs were done in class using a computer and an LCD projector. I selected a student in each class to operate the computer while I wandered about the classroom and directed the proofs. Note that these proofs must proceed according to the prescribed pattern. Of course, students can complete the proofs in different but nevertheless correct ways. This was the first introduction of the program itself into the classroom. However, many students have previous exposure to the program. There is a Freshman class at Stuyvesant in mathematical explorations. The class investigates a variety of problems using different techniques, and thus many students had already used the software.
Interactive Proof - Isosceles Triangle.gsp
Interactive Proof - Thm 16 (points equidistant...).gsp
Lessons for Computer Lab
Finally, I had the students use the software in the computer lab. In the first lesson, students open files with four-sided figures and are asked to drag points in the picture to see what the figures are. In the other two, students are asked to prove various theorems in class with the aid of the software.
Lesson - Four Sided Polygons.doc
Lesson - Four Sided Polygons pg 2-3.xls
Lesson - Four Sided Polygons Figure 1 - 7.gsp
Lesson - Angles of a Polygon.doc
Lesson - Angles of a Polygon continued.doc
Lesson - Toward Similar Triangles.doc