Department of Mathematics Education
J. Wilson, EMAT 6680

Using Geometer's Sketchpad to Help Students Visualize
Parabolas, Ellipses and Hyperbolas
by: David Wise

Geometer's Sketchpad (GSP) is a powerful tool in helping students to discover geometric relationships, make conjectures, and develop proofs. In addition, GSP should be utilized by teachers to create more effective demonstrations. GSP is a dynamic program that can help make geometric and algebraic concepts more concrete. The purpose of this page is to provide alternative demonstration techniques using animation, trace and locus functions of GSP to generate parabolas, ellipses, and hyperbolas.

For more information on GSP, contact the publisher, Key Curriculum Press. To find instructions on setting up GSP as a helper application click here.

A Parabola is the set of points equidistant from a line, called the directrix, and a fixed point, called the focus.

To better understand the basic construction process needed to generate a parabola, click here for the script.

There are many options in creating a demonstation and/or investigation.

1. In the script and sketch above, a trace is set for point P, the constructed point that is always equidistant from the directrix and focus. To generate the parabola, click and drag point D along the directrix. Therefore, the parabola is manually generated by the user.
2. An action button can be created that will animate the generation of the parabola using the trace of point P. In this case, the user only needs to double-click the action button to view the dynamic generation of the parabola. Click here for the sketch.
3. An action button can be created that will animate the generation of the parabola using the trace of the tangent line at point P. Again, the user only needs to double click the action button to view the generation of the parabola. However, tracing the tangent line provides a dramatically different picture than in option 2 (and option 1). Click here for the sketch.
4. The locus of points, using point P, can be constructed to generate the parabola. In this case, the parabola has already been generated through the construction. The user can click and drag point D along the directrix to view point P moving along its locus of points. Click here for the sketch.
5. The locus of lines, using the tangent line at point P, can be constructed to generate the parabola. The parabola has already been generated through the construction, but provides a significantly different picture. Again, the user can click and drag point D along the directrix to view the tangent line moving along its locus of lines. Click here for the sketch.

Each option provides a variation in the presentation of the generated parabola. In the first three options, the user views the generation of the parabola, but the parabola is "lost" because it is generated through the trace function. Constructing a locus does not allow the user to see the generation of the parabola, but the parabola is not "lost". Option 1 is the most manual of all the options. In addition, focusing on point P or the tangent line, provides a different picture of the same parabola. So, the question arises, "Is one better than the other?". I feel the answer is yes and no. Yes, in the sense that some options are better than others, depending upon the specific educational goal(s). No, in the sense using any of these options is better than not utilizing GSP at all. In addition, I feel that using two or more of these options would be best designing a lesson to reach the specified educational goals. I firmly believe in the old adage that the more you have in your bag of tricks, the better prepare you are to help your students reach success. Therefore, I feel it is important to know as many options as possible for creating demonstrations and/or investigations.

If understanding how a parabola is generated, I feel any of the first three options are best. In all options, the user can move the focus in relation to the directrix to view what effect the change will have on the parabola. To understand how a parabola changes based upon the distance between the focus and directrix, I feel that options 4 and 5 are best.

Ellipses and hyperbolas can be generated using the same techniques as above. I will provide a script for the basic construction of each. I will also provide a sketch for each option used to generate the conic section.

An Ellipse GSP construction is the set of points equidistant from a circle, called the directrix, and a fixed point, called the focus. The focus must be inside the directrix.

1. Manual generation of an ellipse. Click and drag point D along the directrix to generate the ellipse. A trace is set on point P. Script and sketch.
2. Action button set to animate point D along the directrix for the trace of point P. Sketch.
3. Action button set to animate point D along the directrix for the trace of tangent line at point P. Sketch.
4. A construction of the locus of points, using point P, as point D moves along the directrix. Sketch.
5. A construction of the locus of lines, using the tangent line at point P, as point D moves along the directrix. Sketch.

Use any of the sketches (I recommend option 4 or 5) to view how the ellipse is effected when the distance between the focus and directrix is changed. What happens when the focus is placed on the center of the circle (directrix)? What happens when the focus is placed on the circle (directrix)? What happens when the focus is place outside the circle (directrix)?

A Hyperbola GSP construction is the set of points equidistant from a circle, called the directrix, and a fixed point, called the focus. The focus must be outside the directrix.

1. Manual generation of a hyperbola. Click and drag point D along the directrix to generate the ellipse. A trace is set on point P. Script and sketch.
2. Action button set to animate point D along the directrix for the trace of point P. Sketch.
3. Action button set to animate point D along the directrix for the trace of tangent line at point P. Sketch.
4. A construction of the locus of points, using point P, as point D moves along the directrix. Sketch.
5. A construction of the locus of lines, using the tangent line at point P, as point D moves along the directrix. Sketch.

Use any of the sketches (I recommend option 4 or 5) to view how the hyperbola is effected when the distance between the focus and directrix is changed.

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