Proving ellipses and hyperbolas

• Grade level: 11 or 12
• Subject: Analytic geometry
• Time: 75 minute block

Objectives:

• Students will use their knowledge of the locus of points to fold (estimate) an ellipse and a hyperbola.
• Students will use geometry and congruent triangles to prove that a locus of points generated on Geometer's Sketchpad is an ellipse or hyperbola.
• Students will generate and show the "sum of the distance from a point to each of the two foci is constant" rule for ellipses and the "difference of the distances from a point to each of two foci is constant" rule for hyperbola using Geometer's Sketchpad.

Materials:

• Wax paper (2 sheets per pair)
• Crayons (1 per student)
• Circular objects to trace (lids and cans work well).
• Computers with Geometer's Sketchpad
• Geometer's Sketchpad program that generates ellipses and hyperbolas. (These links are to the GSP files. Save the link onto the hard drive to open.) Directions for creating this program can be found here. For this lesson, the program should be stopped at step 13, and all lines should be hidden.
• Worksheet

Before Class:

• Students should be arranged in pairs. This can be done randomly by counting off as they arrive.
• Be sure the ellipse and hyperbola GSP programs are installed on each computer.

Introduction:

• Remind students about the parabola activities of the past few class periods. Ask them to define a parabola, be sure they mention the directrix.
• Ask students what would happen if the directrix were a circle instead of a line. Would it matter where you put the point?
• Student pairs should trace a circle onto each piece of wax paper, and place a point inside the circle on one sheet, and outside it on another. Each student should fold one sheet, making sure the point always lies on the circle before they crease it.
• Ask students what they think the figures are. Remind them of the different types of conic sections that were introduced at the beginning of the unit.
• Ask students how can we prove that we have made ellipses and hyperbolas? Ask them to go to their computer stations and open the ellipse.gsp file.

Activity:

• Pass out the worksheet. Ask the students to look at the ellipse page.
• Students should work in pairs to answer the questions on the worksheet, while the teacher circulates to answer questions and help students.
• As students complete the ellipse activity, ask them to open hyperbola.gsp, and continue with the hyperbola activities on the worksheet.

Conclusions:

• Ask teams of students to present for the class their relationship between the distances from the ellipse or hyperbola to the two foci.
• As a class, work out two valid defintions for each of the hyperbola and the ellipse. (The locus of points definitions and the distance to the foci definitions.)
• Ask students what would happen if your point was placed on the center of the circle, or if it was placed directly on the circle.

Assessment:

• Students can be informally assessed throughout this activity through teacher observation and questioning. This would help the teacher gauge how much additional instruction was necessary, and if the activity should be lengthened.
• The worksheet and folded conic sections could be collected from each pair to assess how thoroughly the students were able to prove that they did have a particular conic section.
• Students could be asked to save their final GSP files under a new name so that the teacher can assess their work.

Further Explorations:

• If time permits, this lesson could be split into two sessions (one for the ellipse and one for the hyperbola).
• Students could be asked to use Geometer's Sketchpad to support their answers for the last question(s) in the conclusions section.