Elizabeth Lord & Kerrie Lynch
C & I 303
September 11th, 2001

Lesson Activity For The Conic Sections
The Ellipse & The Hyperbola

Today students will explore the meaning of an ellipse and a hyperbola using both a hands-on activity with wax paper and a computer activity on Geometer's Sketchpad.  Teachers will pass out wax paper and explain the process of folding two sheets of wax paper (one to construct an ellipse and the other to construct a hyperbola; see directions below).  As they fold their paper, teachers will ask students for input on what they are seeing and if they have seen these figures before in mathematics.  Following the folding activity, more discussion will take place between the students and teachers about what the definition of these shapes (an ellipse and a hyperbola) is.  The "formal" definitions and properties will come out of that discussion (see definitions and properties below).  In order to further understand these definitions, students will then construct an ellipse and hyperbola on Geometer's Sketchpad using the direction below (see also the linked sketches).  As they construct and explore these sketches, some students may want to attempt to formally prove that their sketches are actually ellipses and hyperbolas by our definitions.  Such formal proofs are included in this paper.

We do not think that this activity could be finished in one or two days.  We as college students are still discovering the meanings of ellipse and hyperbola and related terms and theorems.  High school students who are being introduced to these concepts for the first time need extra time for exploration and understanding.  We would plan on at least one week of instruction to include all different aspects of exploration and discussion.  We also realize that there may be problems or confusion when using Geometer's Sketchpad and other technology, so we would allow plenty of time (at least a full day using only Sketchpad) for those activities.  We hope that students will come away from this lesson with knowledge and understanding about the conic sections as well as increased experience using Geometer's Sketchpad and working with each other.

How To Construct An Ellipse Using Wax Paper:

Start out with a piece of uncrumbled wax paper about the size of half a sheet of standard notebook paper (8.5 x 11).  Draw a circle in the center of the wax paper without writing off the paper.  Then draw a point anywhere inside the circle excluding on the circle and the center.  The next step is to fold up the circle so it touches the point inside.  When this portion of the circle is aligned with the point, crease the paper and fold it accordingly.  Choose another part of the circle and align this with the point, creasing the wax paper.  Repeat this step several times until the majority of the circle has touched the point inside or until an ellipse is visible.

How To Construct A Hyperbola Using Wax Paper:

Start out with a piece of uncrumbled wax paper about the size of half a sheet of standard notebook paper (8.5 x 11).  Draw a circle in the center of the wax paper without writing off the paper.  Then draw a point anywhere outside the circle excluding on the circle.  The next step is to fold up the point onto the circle so they are touching.  When this point is aligned onto the circle, crease the paper and fold it accordingly.  Choose another part of the circle and align this with the point, creasing the wax paper.  Repeat this step several times until the point has touched the majority of the circle or until a hyperbola is visible.

Formal Definition Of An Ellipse:

An ellipse is the set of all points that are equidistant (the same distance) from a point which is inside a circle and the circle itself.  The sum property of an ellipse states that the sum of the distances from two points inside the ellipse to any point on the ellipse is constant.  These two points are called the foci of the ellipse.

Formal Definition Of A Hyperbola:

A hyperbola is the set of all points that are equidistant (the same distance) from a point outside a circle and the circle itself.  The difference property of a hyperbola states that the difference of the distance from two certain points to any point on the hyperbola is constant.  These points are, again, called the foci of the hyperbola.

How to Construct an Ellipse Using Geometer's Sketchpad:

1. On a new sketch, construct a circle with center A using the circle tool.
2. Using the point tool, draw a point B inside the circle (but not on it and not at the center).
3. Select the point B and make a line segment between it and a point C on the circle.
4. Place a point D at the midpoint of that segment.
5. Construct the perpendicular bisector of line segment BC and while this line is still selected, choose Trace Line from the Display menu.
6. To simulate the process of folding, select both the point C and the circle and under Edit, choose Action Button and select Animation.
7. Double click on the animation button and watch the ellipse form!

How to Construct a Hyperbola Using Geometer's Sketchpad:

1. On a new sketch, construct a circle with center A using the circle tool.
2. Using the point tool, draw a point B outside the circle (but not on it).
3. Select the point B and make a line segment between it and a point C on the circle.
4. Place a point D at the midpoint of that segment.
5. Construct the perpendicular bisector of line segment BC and while this line is still selected, choose Trace Line from the Display menu.
6. To simulate the process of folding, select both the point C and the circle and under Edit, choose Action Button and select Animation.
7. Double click on the animation button and watch the hyperbola form!

Proofs

Statement: The sum of the distances from two points (the foci) inside an ellipse to any point on the ellipse is constant.
Proof: The foci on our sketch are the points A (the center of the circle) and B (the point inside the circle).  We will again use Geometer's Sketchpad to visualize our proof.  Construct a line segment from A to C and place a point of intersection E where this new line segment intersects the perpendicular line passing through point D.  This point E traces out the ellipse.  Construct another line segment between points E and B.  You should now have a triangle BCE with a perpendicular bisector ED.  Because ED is the perpendicular bisector, BD is congruent to DC and the angles BDE and CDE are both 90 degrees.  Of course, DE is congruent to itself, and thus we have two triangles with two congruent sides with an included congruent angle.  By SAS, the triangles are congruent and therefore EB is congruent to EC.  The radius of the circle is AC = AE+EC = AE+EB.  So since the radius is constant, the sum AE+EB is always constant.

Statement: The difference of the distances from two points (the foci) inside a hyperbola to any point on the hyperbola is constant.
Proof: The foci on our sketch are again the points A (the center of the circle) and B (the point outside the circle).  We will again use Geometer's Sketchpad to visualize our proof.  Construct a line through points A and C and place a point of intersection E where this new line intersects the perpendicular line passing through point D.  This point E traces out the hyperbola.  Construct a line segment between points E and B.  You should now have a triangle BCE with a perpendicular bisector ED.  Because ED is the perpendicular bisector, BD is congruent to DC and the angles BDE and CDE are both 90 degrees.  Of course, DE is congruent to itself, and thus we have two triangles with two congruent sides with an included congruent angle.  By SAS, the triangles are congruent and therefore EB is congruent to EC.  The radius of the circle is AC and AE = AC+EC.  Then AE-EC = AC and therefore AE-BE=AC.  So since the radius is constant, the difference AE-BE is always constant.