CONIC SECTIONS: Draw It, Write It, Do It - PBworks

PHOTOGRAPH BY BARBARA LEAPARD; ALL RIGHTS RESERVEDCONIC SECTIONS:

Draw It,
Write It,
Do It

Barbara B. Leapard and Joanne C. Caniglia

Have you ever considered using conic
sections to help high school students
discover connections between art and
mathematics? As students study con-
ics, they become aware that the conics
involve parabolas, circles, ellipses, and hyperbolas,
and they learn to manipulate the equations that
pertain to those figures. Students gain greater un-
derstanding of the relationships between conics and
their equations when they create art-based projects
as a culminating activity. While increasing stu-
dents’ interest in, and awareness of, the versatility
of curves, this activity also helps students identify
and describe conics and develops their mathemati-
cal communication skills as they ask classmates to
draw, write, and graph descriptions of conics.

DESCRIPTION OF ACTIVITY
Students create drawings that include parabolas,
circles, ellipses, and hyperbolas on grid paper. In
addition, students include lines in the form y =
mx + b and hyperbolas in the form xy = k. See ac-
tivity 1. On a separate sheet of paper, students
write down the equations used to create the draw-
ing and include the domains and ranges for the
equations. Students then exchange the lists of equa-
tions with classmates and, using these equations,
attempt to recreate their partners’ drawings. Stu-

152 MATHEMATICS TEACHER | Vol. 99, No. 3 • October 2005

Copyright © 2005 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

DRAW IT WRITE IT/DO IT
For this activity use two-dimensional grid paper with The ability to work flexibly within a variety of represen-
(x, y) being the general point according to the x-y axes. tations is an important part of mathematics. This activity
All other variables represent real numbers. The center (or tests your ability to communicate effectively with another
vertex, in the case of the parabola) of the conic is (h, k) student in writing by having your partner graph a draw-
and does not have to use the same numbers for each curve ing from equations you formulated.
in the drawing. Create a drawing on grid paper that in-
cludes at least one of each of the following conic sections: 1. Upon completion of your conic section drawing, use a
table like the one shown here to list the equations used in
1. A parabola in the form y = a(x – h)2 + k creating the drawing with the domains and ranges of
those equations. Your instructor will pass your list of
2. An ellipse in the form (x − h)2 + ( y − k)2 = 1 equations to another student in the room.
a2 b2
2. The other student will attempt to create the drawing
3. A hyperbola in the form (x − h)2 − ( y − k)2 = 1 using each of the equations from the list provided. Only
a2 b2 the equations and sets of domains and ranges may be used.

4. A circle in the form (x – h)2 + (y – k)2 = r2 3. Compare the newly constructed drawing with the origi-
nal one. Discuss and resolve any discrepancies.
In addition, include at least one each of the following
equations:

1. A hyperbola in the form xy = k Equation Domain Range

2. A line in the form y = mx + b

List the domain and range for each equation used. You can Activity 2
create abstract drawings or drawings of objects such as
basketballs, violins, company logos, or cartoon characters.

Activity 1

dents compare the newly created drawing with the discussing and resolving discrepancies in the draw-
original drawing and discuss and resolve any dis- ings. In addition, students can be given a teacher-
crepancies. See activity 2. created set of equations to graph.

TEACHER NOTES As a further enhancement for the activity, stu-
This activity was implemented in an AP Calculus dents can enter the equations and domains in
class during the latter part of the school year, fol- graphing calculators to see if the graphs correspond
lowing the AP exam. Because of block scheduling, to their original graphs. Students enter the equa-
students had approximately two hours to complete tions and domains as piecewise functions and set
the activity. They spent approximately an hour and the calculator to Dot Mode. Figure 3 shows several
a half making their drawings and composing their of the equations and the accompanying diagram as
lists of equations. After exchanging lists, the stu- they appear on a TI-83 calculator. Students can also
dents drew their partners’ graphs. If errors were graph the equations using a computer program such
made (usually translation errors), discussions be- as Sunburst Technology’s Green Globs or Texas In-
tween the students cleared up the discrepancies. struments’ Conics Apps program.
Students appeared to find the activity engaging and
challenging. One student’s equations and drawing BENEFITS OF THE ACTIVITY
are shown in figures 1 and 2, respectively. One major benefit of this activity is that students
appear to understand more thoroughly the purpose
If block scheduling is not available, this activity of each of the variables in the formulas after they
can be completed during two class periods. Another have completed their drawings. Even reluctant
alternative is for students to bring their completed mathematics students, particularly those with an
drawings and lists of equations to class in anticipa- artistic bent, experience success because of this
tion of exchanging them. activity. In addition, the ability to communicate
about their graphs gives students a sense of em-
Assessment of the activity includes such possi- powerment and ownership of their mathematical
bilities as giving points for each correctly graphed understanding.
equation and for creativity or for a written report

Vol. 99, No. 3 • October 2005 | MATHEMATICS TEACHER 153

Head: y = − 1 x2 + 18 Domain: (–8.5, 8.5) Range: (0, 18)
4
Domain: (–12, 12) Range: (–12, 2)
( )2
Domain: (–4, 4) Range: (–2, 0)
Muzzle: x2 + y + 5 = 1
144 49 Domain: (–9, 9) Range: (–8, –6.5)
Domain: (–4.2, –2) Range: (14.5, 18)
Nose: y = − 1 x2 Domain: (2, 4.2) Range: (14.5, 18)
8 Domain: (–3 –13, –1.3) Range: (8.3, 11.7)
Domain: (1.3, 3 + 13) Range: (8.3, 11.7)
Mouth: y = x2 − 8 Domain: (–3 –13, – 2) Range: (9.5, 11)
50
Domain: (2, 3 + 13) Range: (9.5, 11)
Left Ear: y = –2(x + 2.7)2 + 18
Right Ear: y = –2(x – 2.7)2 + 18
Left Eye: (x – 3)2 + (y – 10)2 = 3
Right Eye: (x + 3)2 + (y – 10)2 = 3
Slant in Left Eye: y = x + 25

22
Slant in Right Eye: y = x + 25

22

Fig. 1 Equations for sample drawing

Fig. 2 Sample drawing Fig. 3 A sample set of equations and diagram graphed on
154 MATHEMATICS TEACHER | Vol. 99, No. 3 • October 2005 a TI-83 calculator

PHOTOGRAPHS BY DICK SCHWARZE; ALL RIGHTS RESERVED

Fig. 4 Fig. 5

ACTIVITY EXTENSIONS
Students can exchange their drawings only and
have their partners determine the equations based
on the drawing. This activity is very challenging
for most students, since they must use many alge-
braic manipulations in order to find the original
equations.

As an interdisciplinary endeavor, this activity
can encourage students to create conic art for
their school’s art show, thereby showing the con-
nection between art and mathematics. Students
who may not consider themselves to be artistically
inclined have an opportunity to show that mathe-
matics is an artistic language. Conversely, stu-
dents who may not consider themselves to be
mathematically talented have a chance to show-
case their artistic designs. Samples of conic art-

work are shown in figures 4–6. ∞

BARBARA LEAPARD, bleapard@ Fig. 6
emich.edu, and JOANNE CANIGLIA,
[email protected], teach K–12 Editor’s Note: For another example of chal-
mathematics education courses at lenging students to develop their mathematical
Eastern Michigan University, communications skills, see Julianna Csongor
Ypsilanti, MI 48197. Leapard is inter- and Carolyn Craig, “Say What You Mean and
ested in improving preservice math- Mean What You Say,” page 181.
ematics education at all levels, and
Caniglia is interested in adapting professional
development models for urban school districts.

Vol. 99, No. 3 • October 2005 | MATHEMATICS TEACHER 155