Graph paper: 1 sheet per student
Discussion of Lesson:
The teacher will begin the lesson by placing
the list of knowledge from the previous lesson on the overhead and (very
briefly) reviewing it. Two students will be chosen to share with the rest
of the class their solutions to the two homework exercises from the previous
lesson. The teacher will provide feedback on the solutions before moving
into a discussion on ellipses.
Particular attention will be paid to developing definitions for ellipse
(including the terms: center, vertices, axes and foci in relation to the
ellipse). The teacher will use the thumbtacks (foci), string and pencil
apparatus to draw an ellipse. See illustration below. This demonstration
will be used by the teacher to demonstrate the properties of the ellipse
and its foci. The teacher will use this discussion to guide students toward
discovering how an ellipse is constructed by the intersection of a double
right circular cone and a plane. Before beginning activity one, the teacher
may want to use a flashlight to demonstrate the creation of an ellipse by
the intersection of a cone and a plane. By holding the flashlight slightly
off of the perpendicular to the wall, an ellipse is formed by the light
(the cone) being intersected by the plane (the wall).
In activity
one the students will use a model
similar to the one used in the previous lesson to discover that an ellipse
is formed when a plane intersects only one nappe ("half") of a
double right circular cone and that plane is parallel to neither the base
nor an edge of the cone. The teacher will provide guidance to the groups
as they construct their ellipses and will clear up any misconceptions or
misunderstandings the students have concerning the construction process.
Once the constructions are complete, the students will move on to exploring
the equation of an ellipse.
This exploration will begin with the teacher explaining that the standard
equation for an ellipse like that for the circle is derived from the distance
formula. The teacher will present to the students a given ellipse with given
foci, vertices, center and point on the ellipse. The students will be asked
to recall their definitions of an ellipse (particularly that the sum of
the distances from the foci to any point on the ellipse remains constant
no matter where the point is located on the ellipse). In activity two,
the students will work in their groups to derive the standard equation for
this given ellipse using the distance formula. The students will then generalize
this equation to that of an arbitrary ellipse with arbitrary foci, vertices
and center and a horizontal or vertical major axis.
In activity three the students will work on several examples in
which they are either asked to determine the standard equation for an ellipse
with given foci and vertices or determine the foci and vertices of an ellipse
given the standard equation.
In activity
four the students will work on
several examples in which they are either asked to determine the general
equation for an ellipse with given foci and vertices or determine the foci
and vertices of an ellipse given the general equation.
To end the discussion, the teacher will place the list of knowledge of ellipses
back on the overhead this time having the students add to that list the
knowledge they gained from this lesson. To assess the students
progress, they will be asked to write in their journals about their knowledge
of ellipses. As a minimum they should give their own definition of an ellipse,
describe how to obtain the equation of an ellipse given its foci and center
and explain why the general equation of an ellipse is a special case of
the general second degree (quadratic) equation Ax^2+Bxy+Cy^2+Dx+Ey+F=0.
Also, students will be given a problem to solve to help them further explore
ellipses.
Activities:
Activity
1: Constructing An Ellipse By "Slicing"
a Double Right Circular Cone
1. Flatten one nappe of your double right circular cone.
2. Find the line that is perpendicular to the axis of the cone and intersects the cone in a single point (i. e. the vertex of the double right circular cone). Select a point on this line.
3. Using your compass construct an arc
on the cone with this selected point as its center. Make sure this arc intersects
the two edges of the flattened cone.
4. Cut out the marked arc.
5. Open your cone back up and place the sheet of construction paper between
the two sections of the cone. Mark the intersection between the plane (i.
e. the construction paper) and the cone. What figure do you get? Will the
plane intersect the other half of the double right circular cone? Why or
why not? Is this the only ellipse that can be created by intersecting the
cone with a plane?
6. What do you think would happen if a plane perpendicular to the base of the cone intersected the cone at the vertex? This figure is called a degenerate ellipse.
Activity
2: Deriving the Standard Equation for an Ellipse using the Distance
Formula
Rotate roles before beginning this activity.
1. Determine the sum of the distances between the foci (point A and point
B) and the point on the ellipse (point C) of the ellipse below. What will
happen to this sum if you choose another point on the ellipse? Use the formula
for the sum of the distances to determine the standard equation of the ellipse
below.
2. Generalize the standard equation for an arbitrary ellipse such as the one below. What would happen if the major axis of the ellipse was the y-axis? What would happen if the center of the ellipse was not located at the origin?
Activity
3: Determining the standard equation
of an ellipse/ Determining the foci and vertices of an ellipse
Rotate roles before beginning this activity.
1. Determine the standard equation for the ellipse with foci (2, 0) and
(5, 0) and vertices (0,0) and (7, 0).
2. Determine the standard equation for the ellipse with foci at (3, 4),
and (3, -4) and vertices (3, 5) and (3, -5).
3. Determine the standard equation for the ellipse with foci at (-3, 2)
and (9, 2) and a major axis of 15.
4. Determine the standard equation for the ellipse with foci at (2/3, 4)
and (2/3, -6) and a major axis of 49/4.
5. Determine the foci and vertices for the ellipse with standard equation x^2/25+y^2/36=1.
6. Determine the foci and vertices for the ellipse with standard equation (x-1)^2/9+(y-2)^2/16=1.
Activity 4: Determining the general equation of an ellipse/ Determining the foci and vertices of an ellipse
Rotate roles before beginning this activity.
1. Determine the general equation for the ellipses in activity three.
2. Determine the foci and vertices for the ellipse with general equation 2x^2+y^2+8x-8y-48.
3. Determine the foci and vertices for the ellipse with general equation 4x^2+9y^2-48x+72y+144=0.
4. Does the graph of every equation of
the form Ax^2+Cy^2+Dx+Ey+F=0 determine an ellipse? If yes, why? If no, give
a counterexample.
Assessment:
Journal Assignment: In your journal write about what you have learned
about ellipses. Give your own definition of an ellipse. Describe the construction
of an ellipse using a plane and a double right circular cone (i. e. what
is the mathematics behind the construction we did in class?). Describe how
to obtain the equation of an ellipse given its foci and vertices. Explain
why the general equation of an ellipse is a special case of the general
second degree (quadratic) equation Ax^2+Bxy+Cy^2+Dx+Ey+F=0.
Exercises: Complete the following exercise using your knowledge of ellipses.
1. A special pool table in the shape of an ellipse has only one hole in
the surface as pictured. There are only two balls in the game, a cue ball
and a target ball. The object of the game is to hit the target ball with
the cue ball and deposit the target ball into the hole after one bounce
off the elliptical cushion. The cue ball can be placed anywhere on the table.
Describe a technique that will ensure that the target ball will go into
the hole every time given proper placement of the cue ball.
Did you know?
The great rotunda between the Senate and House Chambers of the U.S. Capitol Building is elliptical. Standing at one of the foci of this or any elliptical room, you can hear the conversations of anyone speaking at the other focus. This is due to the reflective nature of the ellipse. Sound waves traveling through the path of one focus are reflected from the perimeter of the ellipse to the other focus. Such buildings as the Mormon Tabernacle Choir building in Salt Lake City, the Red Rock Theater near Denver, Carnegie Hall in New York and the Hollywood Bowl in Los Angeles are other examples of where this elliptical property is used in construction.
This same reflective property is also used in the nonsurgical treatment of kidney stones called lithotripsy. In lithotripsy, an ultrasonic emitter is placed at one focus point within an elliptical reflector. When the ultrasound waves are emitted, they strike the walls of the reflector at different points but are all reflected back to a single point, the other focus. The patient is positioned so that the location of the kidney stone is at this focus.
In 1609, Johannes Kepler discovered that each planet in our solar system has an elliptical orbit with the sun being one of the foci. This is known as Kepler's first law of planetary motion. Sir Isaac Newton later used this idea in formulating his theory of universal gravitation. Besides the planets, moons and artificial satellites around the planets have elliptical orbits. Of the 610 comets identified, 245 have elliptical orbits. Halley's comet is among this list. It's elliptical orbit helps predict it's return every 75 years.
In grammar, the word ellipsis means "marks used to show an omission in writing or printing, usually shown as ...". How do you think this relates to the word ellipse in mathematics.