Illuminations: Human Conics

# Human Conics

 In this lesson students use sidewalk chalk and rope to illustrate the locus definitions of ellipses and parabolas. Kinesthetics, teamwork, and problem solving are stressed as students take on the role of focus, directrix, and point on the conic, and figure out how to construct the shape.

### Learning Objectives

 Students will: Define conic sections as a locus of points Apply locus definitions to draw conic sections Collaborate with partners to solve a problem

### Materials

 Sidewalk chalk Lightweight rope (about 10-12 feet per group) Right angle measures (e.g., 8.5×11 sheets of cardstock or right angle rulers) Compasses Overhead markers Ellipse Definition Overhead Parabola Definition Overhead Human Circle Activity Sheet Human Ellipse Activity Sheet Human Parabola Activity Sheet Human Circle Answer Key Human Ellipse Answer Key Human Parabola Answer Key Hyperbola Definition Overhead (optional)

### Instructional Plan

This 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.

Classroom Activities

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, 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.

Outdoor Activities

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.

Circle

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? [center of the circle and point on the circle] What did the rope represent? [the constant distance or radius]

Ellipse

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.

Parabola

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 parabola.
• 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 to the right, 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 parabolas.

• 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 to the line.]
• 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.]

### Questions for Students

 What effect does the length of the rope have on the shape of the conic? Is the rope ever too short? [A longer rope makes a larger conic. When the rope length is equal to the distance between the focus and directrix of the parabola or the distance between the foci of the ellipse, the conic can no longer be created.] Why can you draw the circle with fewer people than the ellipse or the parabola? [The circle only has one fixed element. This question could lead to a more general discussion about the differences between the conics.] Which conics can you draw as a continuous line, without picking up your chalk? [The circle and the ellipse were easy to draw as a continuous line. The parabola can also be drawn continuously, but it would be far more difficult because of the need to maintain the right angle to the directrix.] How could you use paper and pencil to draw or verify conics? [Be open to all student suggestions and let them try to demonstrate as many as feasible. Some ideas may prove to be more difficult than they anticipate. Ideas for drawing may include taping the ends of a string to paper, measuring each point, or using multiple rulers. The definitions can be verified by using a ruler to measure distances between points on the conic and the foci or directrix.]

### Assessment Options

 Give students a picture of an ellipse and a parabola with possible foci or directrix indicated. Ask them to use a ruler and right angle measure to determine and explain whether or not the figure is actually the named conic. Give students two thumbtacks, string, and a piece of cardboard to draw an ellipse. This is an individual reproduction of the chalk activity. Students may also be challenged to find a way to draw a parabola using these materials. Ask students to write a summary of either the ellipse or parabola construction for the benefit of a classmate who has missed the lesson. The summary should include the definition and an explanation of how the drawing technique applies the definition. Students and teachers who are comfortable with the technology may construct these conics using geometry construction software.

### Extensions

 Use the Hyperbola Definition overhead to give students the last conic definition. Discuss reasons that the hyperbola would be difficult to draw using chalk and rope. Point out the fact that the hyperbola consists of 2 distinct branches. Investigate lengths and distances in the ellipse. Using the Ellipse Definition overhead discuss the relationships between the lengths of the major and minor axis and the focal constant of the ellipse.

### Teacher Reflection

 How did students respond to the opportunity to work outside? Did you challenge the achievers? How? Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments and were these adjustments effective? To what extent were students able to do the activities without instruction? Can the circle activity be skipped for some of your students? Did the students understand the definitions before going outside? If not, did the outside activity clarify the definitions?

### NCTM Standards and Expectations

 Geometry 9-12Draw and construct representations of two- and three-dimensional geometric objects using a variety of tools. Use geometric models to gain insights into, and answer questions in, other areas of mathematics.
 This lesson was prepared by Ellen R. S. Bush as part of the Illuminations Summer Institute.

1 period

### NCTM Resources

 Focus in High School Mathematics: Reasoning and Sense Making Navigating through Geometry in Grades 9–12 (with CD-ROM)

### Web Sites

 More and Better Mathematics for All Students
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