Students investigate the definition of a parabola through one of its geometric definitions. They study conic sections. They examine an ellipse as a locus of points such that the sum of distances from the foci to the traced path is constant.

Before the Activity

Install the Cabri Jr.™ App on the students' graphing calculators using one of these two methods:

TI-Connect™, a TI Connectivity Cable, and the Unit-to-Unit Link Cable

See the attached PDF file for detailed instructions for this activity

Print pages 1 - 6 from the attached PDF file for your class

During the Activity

Distribute the pages to your class.

Follow the Activity procedures:

Exploration 1:

Draw a segment to connect a line (directrix) with a point (focus) not on it

Construct a perpendicular to the directrix at the point D where the segment intersects it

Construct a perpendicular bisector of the original segment

Mark Point P as the point of intersection of the two perpendiculars

Draw a circle using P as center P passing through the focus

Drag point D along the directrix

Observe that the path traced by point P is a parabola

Exploration 2:

Draw a circle

Draw a segment connecting a point A on the circle to a point B within the circle

Construct a perpendicular bisector of the segment

draw a line to connect the point A to the center of the circle

Label the point of intersection of the radius and the bisector as P

Drag the point A on the circle along the circumference

Observe that the path traced by P is an Ellipse

Drag point B outside the circle and observe the path traced is a hyperbola

Exploration 3:

Draw a circle and mark a point B outside the circle

Draw a second circle with a point on the first circle as its center, and passing through B

Drag the center of the second circle around the circumference of the first circle

Observe that the path traced by point B is a limacon with a loop when the point is outside the first circle, a limacon without a loop if it is inside the circle, and a cardioid if it is on the circle

After the Activity

Review student results:

As a class, discuss questions that appeared to be more challenging