lesson presents analogous approaches for the locus definition of three
conics, the circle, the ellipse, and the parabola. The circle may be
too easy for some students and the parabola may be too difficult for
some. The overall lesson is divided into two parts, classroom and
outdoor. Plan the conics you will use ahead of time so you can do the
classroom sections together before going outdoors to do the associated
activities. Also in preparation for the lesson, mark the midpoint of
each rope with a permanent marker or a piece of tape.
Give each student a compass and the Human Circle Activity Sheet.
- What is the definition of a circle?
[A set of points equidistant from a given point, called the center.]
- What do the parts of the compass represent in this definition?
[Needle = center, pencil marks = circle, and distance from needle to pencil point = radius.]
For a large group demonstration, replace the pencil in a compass
with an overhead marker and demonstrate the use of the compass on a
transparency. Emphasize the importance of not squeezing the compass so
that the radius is maintained. Let students practice by having each of
them draw a circle with a radius equal to the length of their index
finger. Discuss how the construction is related to the locus
definition. Remind students that the circle is just the locus of
points, not its interior.
Hand out the other activity sheets:
Show students the Ellipse Definition Overhead, covering the title, and ask what shape they see. If students
say oval, explain that an oval and an ellipse may look alike, but the
shapes we deal with in our study of conics are called ellipses.
Indicate the foci and simply state that these are called focal points
or foci. Give students the definition of an ellipse. Point out that
"foci" is the plural of focus. Use different colors to illustrate the
definition by selecting points on the ellipse and drawing lines to the
foci. Do not answer questions or engage students in discussion so that
students may ponder the definition as they work outside.
Show students the Parabola Definition Overhead, covering the title, and ask what shape they see. If students
have previously worked with parabolas, explain that considering the
graph of a quadratic function is only one way of looking at it. Give
students the locus definition. Remind students that the distance from a
point to a line is the perpendicular to that line. Use different colors
to illustrate this for several other points on the parabola.
Before going outside, separate students into groups of three.
Three students are needed for the ellipse and parabola, two to
represent foci or directrix and one to draw. For the circle, only two
students are required, the center and the chalk, but it is usually
easier not to rearrange groups mid-activity. Explain to students that
they will be working in groups to draw a circle, an ellipse, and a
parabola. Each group will have one piece of chalk and one piece of
rope. Do not give instructions or hints on how to draw the conics until
students have had ample opportunity to experiment.
Instruct students to draw a perfect circle using the chalk and
the rope. If students need a hint, suggest that they consider
themselves to be a human compass.
If students need further instruction: Fold the rope in half.
One student puts the ends together, and holds them on the ground to be
the center of the circle. The second student stretches the rope and
puts the chalk in the bend at the midpoint. The second student then
drags the chalk along the ground, while pulling the rope taut. Note
that students figuring out the activity independently may not fold the
rope. This is not a problem.
As you observe students, ask how many students were actually
needed to draw the circle? [Two.] What were their roles? [One person represents the center of the
circle, and the other represents a point on the circle.] What did the rope represent? [The
constant distance, called the radius.]
If students need hints, tell them that the fact that there are
three people in the group is significant and to consider what they did
to draw the circle.
If students need further instruction: Two students are human
foci, holding the ends of the rope at fixed points on the ground. These
students should not hold the rope taut. The third student uses the
chalk to pull the rope taut and sweeps out the locus of points.
As students finish, ask them to consider and discuss the questions on the activity sheet.
Groups who finish the ellipse should begin experimenting with
the parabola. If possible, suggest that students use an existing
straight line such as a parking space or sidewalk crack as the
directrix. Some students may need to see a demonstration before
effectively drawing the parabola. Gather those students needing an
outdoor demonstration. Have one group of students demonstrate the
parabola drawing as you direct them through the instructions:
- Draw a focus approximately 2 feet from the directrix. This does not
need to be precise. You are just looking for a distance that will allow
students room to maneuver and will produce an easily recognizable
- Assign roles to the three students in the group: F, D, and A.
Student D will be responsible for the directrix and will need a right
angle measure, such as cardstock or a right angle ruler, to approximate
right angles. Student F will be responsible for the focus of the
parabola. Student A will mark points on the parabola.
- Assign each student a point on the rope. Student A is at the
marked midpoint of the rope. At equal distances from her, measured by
folding the rope, are F and D.
- F should hold her point of the rope at the focus on the
ground. D should place the right angle measure on the directrix and
guide the rope along the side of the measure. She should move the card
and rope along the directrix while A pulls the rope taut. When the rope
is taut and perpendicular to the directrix, A should mark the point on
the parabola. In the figure above, person D is correctly
positioned perpendicular to the directrix, but person B is not.
Students do not need to mark the congruent lengths on the ground,
although some will naturally do this to clarify their thinking.
- Students use the same rope length and repeat the procedure to
draw a point on the other side of the parabola. Then, change lengths
and repeat for a total of at least six points.
The following questions should be asked of students while they are
still outside completing their constructions. These questions serve as
the closure to the lesson, and should be used to ensure that all
students have a conceptual understanding of the locus definitions of
- What is the shortest segment from the focus to the directrix?
[The perpendicular segment that goes through the vertex.]
- What is the midpoint of this segment?
[The vertex of the parabola.]
- Why is it important to keep the rope perpendicular to the directrix?
[The distance between a point and a line is the perpendicular line that goes through both.]
- How can you find the vertex of the parabola using your rope, right angle measure, and group members?
[Use your materials to find the perpendicular segment from
the focus to the directrix. The vertex is the midpoint of the segment.]