Lesson Plan on Ellipses

Objectives: Students will be able to

  1. draw an ellipse and state its parts
  2. know how to represent parts of an ellipse algebraically and geometrically
  3. use Pythagorean Theorem to find the length of the semi-minor axis.
  4. know and use the formula for eccentricity
  5. write an equation of an ellipse
  6. graph the elliptical orbits of planets using characteristics of an ellipse

Grade Levels: 10-12

Background Information:

An ellipse is a conic section because it is formed by slicing through a cone with a plane at various angles. In studying the solar system it is important to understand characteristics of an ellipse because planets, as well as other planetary objects, have elliptical orbits (Kepler's first law of planetary motion). An ellipse is a figure drawn around two points of reference called the foci ( foci is plural of focus) such that the distance from one focus to any point on the ellipse back to the other focus equals a constant measure. This constant is the measure of the long diameter of the ellipse, called the major axis. The short diameter is the minor axis. The diameters are also two axes of symmetry for an ellipse. Algebraically, we often represent the distance of the major axis as 2a and the minor axis as 2b.

[Geometric diagram of ellipses]

In the previous diagram, an ellipse is geometrically represented showing that wherever P is located, PF1 + PF2 = 2a.

If P is located on the minor axis (the y-axis on the right diagram) then PF1 = PF2 = a.

The measure of half of the minor axis (or the semi-minor axis), b, can be found using the Pythagorean Theorem, a2=b2+c2. Solving that equation for b, we get that b=sqrt(a^2-c^2).

A number called the eccentricity of an ellipse is used to measure the amount of its elongation. It is defined as follows:

e=c/a

In the following illustrations, the left diagram shows an ellipse with its major axis lying horizontal and its center at the origin. The equation for this type of ellipse is below the diagram. The one on the right shows an ellipse with its major axis vertical as well as the accompanying equation. In both cases, the endpoints of the major axis are called the vertices of the ellipse.

[Diagram of Ellipses and Standard Equations]

Activities

Activity: Graphing the Orbits of Planetary Objects

No Frames Table of Contents

[LMGFP home page] Contact Karen Krupinsky (kgurley@gsfc.nasa.gov) or
Tammy Seergae (tseergae@umd.edu) for further information.