Jolanta Walukiewicz – Analysis of the Process of Teaching
Conic Sections
Fall 2001/2002
Conic Sections: Concrete Activities and Mathematical Inquiry
Teaching for Understanding – Mathematics for All
As a physicist I learned to appreciate conic shapes. Their many applications range from parabolic
reflectors and emitters (parabolic antennas, mirrors, search lights, and emitters of various waves),
to physics of planetary movements (elliptical, hyperbolic or parabolic orbits of planets and
comets), and to LORAN locating devices which trace the constant difference between registered
signals from distant transmitters- therefore implying hyperbolas.
However, I always hesitate before teaching conic sections and regard teaching this topic as a
rather difficult task.
First, there is a little motivation for the thorough study of quadratic relations, since the high
school mathematics curriculum is designed to prepare students for the study of calculus and
hence is focussed on analyzing functions.
The mathematical difficulty of the task can also be a factor. The second-degree equations of
conics are quite complicated and the technique of completing the square used to transform them
into a standard form frequently frustrates students. Also, graphing quadratic relations is
significantly harder then graphing most functions. In addition, understanding the descriptions of
conic curves and their reflective properties requires integration of geometry into advanced
algebra, which doesn’t happen too often in the high school curriculum.
Moreover, as I’ve observed, it is difficult to motivate the learning of conics. Most students lack
the concrete experience with those curves and are quite unaware of their applications (circle is an
exception), and therefore regard the learning of this unit as a purely abstract endeavor meant to
be endured, and quickly forgotten.
As a teacher I feel that students need some concrete experience to enliven the learning process
and motivate learning and retention of mathematics followed by rigorous and thorough
mathematical analysis which they could easily understand, generalize and use to construct the
mathematical understanding. Unfortunately the textbook Algebra and Trigonometry – Functions
and Applications by Paul A. Foerster doesn’t offer much help in this regard.
The California Mathematics Content Standards for the Algebra II course state:
16.0 Students demonstrate how the geometry of the graph of a conic section (e.g. asymptotes,
foci, and eccentricity) depends on the coefficients of the quadratic equation representing it.
17.0 Given a quadratic equation of the form ax 2 + by 2 + cx + dy + e = 0, students can use
the method for completing the square to put the equation into standard form and can
recognize whether the graph of the equation is a circle, ellipse, parabola, or hyperbola.
Students can then graph the equation.
Again, I do not find much guidance in these standards. They seem to emphasize abstract
manipulations and do not even mention the reflective properties of conics.
So when I started planning how to teach these curves in a way that would engage all the students
I knew, that I needed guidance and help. Luckily, last year (fall 2000) I took a course at UC
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Jolanta Walukiewicz – Analysis of the Process of Teaching
Conic Sections
Fall 2001/2002
Berkeley “High School Mathematics from the Advanced Standpoint”. This course was designed
to deepen teachers’ understanding of the mathematics they teach.
One of our assignments called for developing a consistent and rigorous unit that can be taught in
high school. I immediately suggested to Ming Ho and John Holmes that we should consider
working on conic sections and motivate the unit by the concrete activities of paper folding,
which I found in Harold Jackob’s book Mathematics the Human Endeavor. Together we
proceeded to design a consistent and rigorous unit encompassing all of the conic curves.
Working as a team was very valuable. My understanding greatly benefited from Ming’s logical
and analytical skills– he was the only person in our group with a formal degree in mathematics,
and I quickly discovered that a way the trained mathematician approaches a problem differs from
my own thinking. Moreover Ming was the only member of our group proficient in using the
Geometer’s Sketchpad, so when we decided to use the technology, our investigation revealed
very interesting and surprising results.
We were rather pleased with the results of our analysis1. We felt that activities and their
mathematical analysis presented in our project formed a consistent and rigorous framework for a
unit that can be taught in Algebra II or (with some modifications) in the Mathematical Analysis
course. The unit would extend the traditional approach of deriving standard equations of conic
sections using analytic geometry. Moreover, after completion students would gain a complete
understanding of physical (reflective) and analytical properties of conics. Our project also
included the geometric proofs why the lines we folded formed the various conic sections.
We knew that our unit will motivate practice of mathematical techniques found in most algebra
textbooks specific to conic sections (e.g. completing the square) and problems including practical
applications.
Somehow I lost contact with John but Ming and I do continue to work together. On Oct. 20th,
2001 we gave a presentation for the Bay Area Mathematics Project in Albany. Right now we are
focussed on investigating the mathematics of conic curves and their properties and search for a
unifying approach that would allow us to look for similarities between them and predict common
characteristics. Ming’s studies are moving in purely analytical direction while mine are centered
on the practice of teaching.
This school year, I decided to teach conic sections in Algebra II course using concrete activities
and proofs researched earlier with my colleagues. I would incorporate concrete activities:
drawing (using various techniques) and paper folding as well as geometric proofs of properties,
and applications of reflective properties of conics. I would also teach the mathematical
techniques needed for transforming and graphing these curves, as indicated by California
Content Standards 16.0, and 17.0. Moreover, I decided to postpone teaching getting standard
equations of conics using coordinate proofs till next course (Mathematical Analysis).
I designated eight teaching days for the activities: Oct. 16–24 and Nov. 20-21. (El Cerrito HS
has a block schedule, so we teach mathematics every day for 1.5 hrs for half of a year). Because
the important skill of completing the square is taught in the chapter on Quadratic Functions
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See Conic Sections: From Concrete activities to Abstract Understanding by Ming Ho, John W. Holmes and Jolanta
Walukiewicz.
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Jolanta Walukiewicz – Analysis of the Process of Teaching
Conic Sections
Fall 2001/2002
(Chapter 5) - I decided to teach Chapter 9 - Quadratic Relations and Systems immediately
following the Chapter 5.
Below is the brief summary of the lesson plans for Chapter 9: Conic Sections:
These lesson plans aim at creating lessons to teach for understanding for all students by
providing concrete experience with mathematical curves and motivating mathematical inquiry
about their properties. (For detailed proofs and abstract analysis check your handout or e-mail
the presenters).

Day One – Circles
Concrete activity: Drawing using compass. Draw a circle with a center (h, k) and a radius r on
the coordinate plane.
Abstraction: The picture will be used to introduce the geometrical definition of the circle, discuss
the properties of the shape, as well as motivate the equation of the circle in standard form
obtained using the distance formula
(x – h) 2 + (y – k) 2 = r 2 .
Practice ( in class and at home) will include graphing circles with given (h, k) and a radius r ,
transforming quadratic relations in a form
Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0
to standard form, and analyzing the effects of constants A, B, C, D, E, and F on the graph of the
circle. (Many textbooks supply lots of good examples for practice problems.)

Day two – Ellipses
Concrete activity #1: Constructing ellipses using two fixed points and a piece of string
(described on page 473). Choose string of length 10 units and position F 1 and F 2 at (-4,0), (4,0).
This results in a familiar 3, 4, 5 right triangle, which will be used to relate the focal radius c and
the length of semi-major a and semi-minor b axis; c 2 + b 2 = a 2 .
The resulting drawing of an ellipse will be used to introduce the geometrical definition of an
ellipse - a set of points, such that the sum of its distances from two fixed points F 1 and F 2 is
constant and equals 2a. The activity also motivates the discussion of the properties of the shape,
its symmetries, and the relationship a 2 + b 2 = c 2 .
Concrete activity #2: Graphing ellipses. Problem Introduction to Ellipses page 467( see below)
allows students to use the previously learned skills (point plotting and solving equations) to
obtain graph of an ellipse with a = 5, b = 3, c = 4 which matches the ellipse they constructed
using two pins and a string.
After students carefully graph the quadratic relation 9 x 2 + 25 y 2 = 225 on a graph paper we
can show how the characteristics of the ellipse match the coefficients in the equation in the
standard form of an ellipse centered at (0, 0). Dividing both sides of
9 x 2 + 25 y 2 = 225 by 225
renders:
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Jolanta Walukiewicz – Analysis of the Process of Teaching
Conic Sections
Fall 2001/2002
( x  0) 2 ( y  0) 2

1.
52
32
Connection to Prior Knowledge: Using the observations from the analysis of both activities and
knowledge about transformations of functions students can write the standard equation
( x  h) 2 ( y  k ) 2

 1,
a2
b2
which describes ellipse with vertices at (a, 0) and (0, b) and foci at (c, 0) where
c  a2  b2 .
Practice: Students graph ellipses given equations in standard and general form and find equations
of ellipses given their graphs.
Introduction to Ellipses (page 467)
In this problem you will find out what the graph looks like if the coefficients of x and y are not
equal, but have the same sign. Do the following things for the relation whose equation is
9 x 2 + 25 y 2 = 225
a) Solve the equation for y in terms of x.
b) Explain why there are two values of y for each value of x between –5 and 5.
c) Explain why there are no real values of y for x>5 and for x<-5.
d) Calculate values of y for each integer value of x from –5 to 5, inclusive. Use a calculator to
approximate any radicals , which do not come out integers. Round off to one decimal place.
e) Plot the graph. If you are successful, you should have a closed figure called an ellipse.

Day 3 – Ellipses. Reflective properties.
Concrete activity. Paper folding.
Materials: Regular copier paper, tracing paper or patty paper works well. Compass and ruler.
Each student should use a compass to draw a circle with a radius of about 10 cm on a sheet of
paper. Mark the center of the circle as B, then cut out the circle.
Now draw one point inside the circle and label it A. I like the students to choose the point ~6 cm
away from the center of the circle.( This generates the uniform ellipse and makes analysis much
easier.)
Fold the circle so that its edge falls on the point marked A and make a sharp crease.
Then unfold the paper and fold it again in a different direction so that the edge again falls on the
point. Repeat this many times so that points all around the edge have been folded onto the point
A. After many folds you should be able to identify the boundary of an ellipse and identify the foci
of the curve, as well as find the lines of symmetry of this shape.
While teaching the activity I like to demonstrate folding using a circle (congruent to students’
circles) cut from clear overhead transparency. The folds are clearly visible on the white board. We
can also see the shape of an ellipse. At that point I like to ask a students to compare the shapes of
our ellipses. Students can superimpose my transparent circle over their paper circles and verify
that ellipses are indeed congruent. Then we can trace the shape of the ellipse on the white board
and use it for the mathematical analysis.
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Jolanta Walukiewicz – Analysis of the Process of Teaching
Conic Sections
Fall 2001/2002
Abstraction: To describe the analytical properties of elliptical curve choose a point P on the
circumference of the circle. Then fold the paper so that P falls on A. Notice that the fold touches
the curve at one point. Label it Q.
Open the paper flat again, pick another point on the circumference and repeat the directions in the
preceding paragraph using the new point as P.
It is important to motivate and made conjecture about the proof. I like to point out to my students
that the sum of distances from Q to points A and B equals to the radius of the circle by showing
how they can fold the major axis of the ellipse. This motivates the proof.
After analyzing the experiment with folding paper students should be able to prove that the
resulting shape is an ellipse 1 . Students can also locate foci, vertices, and the center of the ellipse.
The measurement should yield the satisfactory values for the length of the minor and major axis.
Using observations and measurements, students can even derive the standard equation
x2 y 2

1
a 2 b2
which describes ellipse with vertices at (a, 0) and (0, b) and foci (c, 0)
where c  a 2  b 2 .
The folded ellipse has a = 5 cm, b = 4 cm, c = 3 cm. (If you would like students to obtain ellipse
matching the one studied the day earlier – you’ll need to put point A 8 cm from the center B.)
Teacher can then lead the students to the proof of the reflective property of the ellipse. That is, a
light beam travelling from one focus will reflect from the surface of an ellipse to arrive at the
other.
Practice: Students will fold two more ellipses by choosing point A at different distances from
c
point B. The analysis of their investigations motivates the concept of eccentricity as e = .
a

Day 4- Hyperbola
Concrete activity #1: Introduction to hyperbolas, page 477 (see below). Similar to Introduction to
ellipses this problem guides students graphing the quadratic relation 9 x 2 – 16 y 2 = 144 to
obtain the graph of hyperbola a non-closed curve with two branches. The problem also clearly
illustrates why the hyperbola exists only from x = -4 to x = -8, and from x = 4 to x = 8. After the
experience with graphing ellipses students can easily obtain equation of the hyperbola in standard
x2 y 2
form 2  2  1 .
a b
Concrete activity #2: Paper folding
Materials: Regular copier paper, tracing paper or patty paper works well. Compass and ruler.
Each student should use a compass to draw a circle with a radius of about 6 cm on a sheet of
paper. Mark the center of the circle and mark it as B.
Next each student should draw one point outside the circle and label it A. Students should pick a
point approx. 10 cm away from the center of the circle.
Now the students should fold the paper so that points on the circumference of the circle fall on the
point marked A and make a sharp crease.
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Jolanta Walukiewicz – Analysis of the Process of Teaching
Conic Sections
Fall 2001/2002
Then unfold the paper and fold it again in a different direction so that the point on the
circumference again falls on the point A. Repeat this many times so that points all around the
edge have been folded onto the point.
Again, I like to demonstrate folding using the overhead transparency. On the white board I can
then outline the shape of a hyperbola and use it for the mathematical analysis.
After many folds students should be able to identify the boundary of a hyperbola and identify the
foci and the symmetries of the curve.
Abstraction: To describe the analytical properties of hyperbolical curve students should answer
the following question.
? - How is the difference of the distances from the point on the curve to points A and B related to
the radius of the circle? It is good to advise students to pick a few random points on the hyperbola
and measure the distances from points A and B and then calculate the difference. It is not a very
rigorous approach, but it motivated proof for my students.
After completing the experiment with folding paper students should be able to prove that the
resulting shape is a hyperbola. Also they should be able to locate the foci, vertices, and the center
of the hyperbola. The analysis of the relationship between the length of the transverse axis and the
radius should yield the satisfactory values for the length of the minor and major axis, which can
be verified by the measurements. And analysis of the properties of the curve as well as the
process of folding (concentrate on the folds tangent to the circle) should allow students to find
equations of asymptotic lines. Using the observations and measurements students can derive the
equation
x2 y 2

1
a 2 b2
which describes hyperbola with vertices at (a, 0) and foci at (c, 0) where c  a 2  b 2 .
Students can also show the reflective property of the hyperbola. That is, a light beam traveling
from one focus will reflect to move away from the other.
Practice: Textbook offers good set of problems providing practice with graphing and finding
equations of hyperbolas.
Introduction to hyperbolas (page 477)
The ellipses you have plotted in this section have equations in which the x and y terms have the
same sign. In this problem you will find out what the graph looks like if they have opposite signs.
Do the following things for the relation whose equation is
9 x 2 – 16 y 2 = 144,
a) Solve the equation for y in terms of x.
b) Explain why there will be no real values for y for –4<x<4.
c) Explain why there will be two values for y for each value of x when x>4 and x<-4.
d) Make a table of values of y for each integer value of x from 4 to 8 inclusive, finding decimal
approximations by calculator if necessary.
e) Explain why you can use the same table of values of y for values of x between –4 and –8.
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Jolanta Walukiewicz – Analysis of the Process of Teaching
Conic Sections
Fall 2001/2002
f) Plot the graph of the relation from x=-4 to x=-8, and from x = 4 to x = 8.
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3
g) On the same Cartesian coordinate system, plot graphs of the lines y = x and y =  x .
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4
If your graph in part f is correct, then it should have these two lines as diagonal asymptotes.
The graph is called a hyperbola, and is a non-closed curve with two branches.

Day 5 – Applications
It is good to take a break and celebrate the progress! We were lucky since the Odyssey space craft
launched on April the 7th, just entered the orbit around Mars and all the major newspapers printed
articles detailing the voyage completed with drawings of beautiful elliptical orbits that Odyssey
traveled on its journey.
We also talked about lithotripters used in medicine to break the kidney stones, as well as
brainstormed all the ways the properties of conic section can render them useful.
Practice: Two good problems in the book: Stadium Problem (page 476) and “Loran” (page 511) –
see below.
I announced that the students are expected to do a project that utilizes the properties of conic
sections, and explains the mathematics involved. Students can work individually or in groups of
two, and they have three weeks to complete their projects.
Stadium Problem
Suppose that you are a Chief Mathematician for a construction company, which has a contract to
build a football stadium in the form of two concentric ellipses, with the field inside the inner
ellipse, and the seats between the two ellipses. The seats are in the intersection of the graphs of
x 2  y 2  100 and 25 x 2  36 y 2  3600 ,
where each unit represents 10 meters.
a) Draw a graph of the sitting area.
f) From the handbook, you found that the area of an elliptical region is ab , where a and b are
the semi axis, and   3.14158. The Engineering Department estimates that each seat
occupies 0.8 square meters. What is the seating capacity of the stadium?
Loran Problem
Suppose that you are abroad a space ship. The Earth is at the origin of a Cartesian coordinate
system, and the path of your space ship is the graph of
9 x 2  25 y 2  72 x  81.
a) Which conic section is this path, and how do you tell?
b) Sketch the graph of this path.
c) Show that the Earth is at the focus of this conic section.
d) To determine your position at a particular time, you tune in to the LORAN station and find out
that you are also on the graph of
9 x 2  15 y 2  9.
Which conic section is this graph, and how can you tell?
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Jolanta Walukiewicz – Analysis of the Process of Teaching
Conic Sections
Fall 2001/2002
e) By solving the system formed by the equations of the two conic sections above, find the three
possible points at which your spaceship could be located.
f) Draw the graph of the conic section in part d) on the same Cartesian coordinate system as in
part b), and thus show that your answer to part e) is correct.
g) To ascertain at which one of the three points you are located, you find that you are also on the
graph of 3x – 5y = 27.
h) At which of the points are you located?

Day 6 - Parabolas
Concrete Activity: Paper folding.
Materials: Sheet of clear binder paper, ruler, pencil.
Measure and mark one edge of the paper in one cm increments. On the other side of the paper
have the students mark a point A. Fold the paper so the first mark matches up with point A.
Then fold the paper so the second mark matches up with point A. Proceed with this pattern until
all the marks have been matches up with point A. Trace the curve with your pencil.
Analysis: The given point and the given line l resemble the focus-diretrix definition of the
parabola. Perhaps the folds are tangent to the parabola with A and l as focus and diretrix. To
show that the folded line is tangent to a parabola, we need to show that a unique point on that
line is equidistant to points A and P (point on the diretrix line). The details of the proof are
presented in 1 .
Similar to the other curves we can prove the reflective properties of parabolas.
It is important to reinforce the connection between two equations for the parabolic curve
1 2
y = ax 2 and y =
x .
4p
Practice: Book contains good selection of problems about graphing parabolas by finding the
vertex and intercepts, or focus and diretrix, and finding equations for parabolas given geometric
descriptions.
Day 7 & 8 – The Research Project: Beyond the Grade Point Average - The True Value of
Alternative Assessment
I hesitated before assigning the independent research project to students. We had a quiz, an
individual test, I also checked student’s notebooks for complete descriptions of activities and
completed homework. I felt that student’s learning was documented quite well. Now we were
about to plunge into the exponential and logarithmic functions. From experience, I know that this
particular chapter although not very complicated, can be very tricky for kids who need to quickly
recall various properties of exponents and logs. And I didn’t want to divert their attention as they
worked with the compound interest or exponential decay type of problems.
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Jolanta Walukiewicz – Analysis of the Process of Teaching
Conic Sections
Fall 2001/2002
Nevertheless, I was curious how well the class understood conic sections and how many
applications were they willing to uncover for each other. So I gave them about three weeks to
work on the Conic Sections project. I stated the objective quite generally: to build/investigate a
concrete object representing conic section, describe any mathematical equations involved, and
prepare a presentation to the class and a small write-up summarizing the mathematical properties
of an object. Before the presentation I handed students a general rubric which specified the
elements they were to be graded on.
The presentations happened two days before Thanksgiving break, and the tense energy of the
classes was perfectly captured by a frenzy of presentations. We were all captivated by the
originality, amount of student-hours put in by the groups, and the creativity involved.
The most popular were the investigations of reflective properties of ellipses (elliptical pools) and
parabolas (solar cookers and models that demonstrated reflection of light). Students build several
elliptical pools; from the big, coffee table size and the model of an elliptical golf course through
several medium sized, to the tiny –destined to became a holiday tree ornament.
It was interesting to read how students recall struggling with the engineering of their elliptical
pools. As Jose & Andrew put it…When this project was assigned, we thought that it would be
the easiest thing ever…As it was, the project turned out to be harder than we thought. We had to
measure everything, which took time. We had to cut wood, which was not pretty, because we kept
cutting into our ellipse. We also had problems with our first wall (plastic one) which didn’t
bounce the ball perfectly either…
Aemad, Suzanne & Catherine….Making the whole thing was frustrating because even if the
wood was level, the ground often wasn’t, so it had to be just right. If you combine the hours of
each group member the total was around eight or nine hours per student. In the end, when we
finally saw that it worked, it made the effort worth it…
Alice and Lauren’s investigation of the properties of curves obtained while rolling eggs
captivated the audience. Who would have guessed that rolling eggs will leave trace wobbly
ellipses! Girls dipped the eggs in paint and estimated the mathematical parameters of those
ellipses. At that point we all felt like born scientists yearning to get eggs of various species,
compare generated ellipses, gather and compare the results into some exciting hypothesis…
The LORAN project (the global positioning system used by the NAVY) -here the mathematics
was a bit harder and more understanding of physics was required but the work done by Jay, Kayo
and Robert truly captivated the audience. Just imagine being able to explain that GPS works
thanks to a system of intersecting hyperbolas.
Mike and Robert captivated the class with the demonstration and explanation of the “Mirage”
toy. Mike got the “Mirage” as a gift many years ago and played with it to creating illusions of
real objects appearing in space. Now the team proudly supplied the mathematical explanation
how reflective properties of parabolas can be used to explain the mirages.
Other popular project topics were the circular orbits, whispering galleries, parabolic and hanging
bridges and the analysis of architectural shapes involving conics.
Maria investigated the architecture of Saint Mary’s Cathedral in San Francisco. ….There are no
curves in the design. What appear to be curves is the result of the juncture of straight lines. This
concept of design is called a hyperbolic paraboloid - Pietro Bellaschi….
When Maria cited the Italian engineering consultant from MIT the class gasped - that’s exactly
what the paper folding activities demonstrated - curves as the result of intersection of straight
lines !!!
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Jolanta Walukiewicz – Analysis of the Process of Teaching
Conic Sections
Fall 2001/2002
I am proud of the work students did in the unit on quadratic relations. I am confident that we did
justice to NCTM Standards 16.0 and 17.0, and that concrete activities described here allowed all
students to better understand and appreciate the mathematics involved.
I think that kids share my opinion. A few days ago some of them offered to scan all the projects
and pictures I took while doing the activity and assemble a class portfolio. They plan to put it in
the Power Point format and burn a CD. (Later their presentations were put on the school Web
site.)
PS. The case of a parabolic heater – For most of the last week (Dec. 3-Dec. 7) central heating
in our school didn’t function at all. Some teachers survived by dressing in many warm layers,
and by advising students to do the same. Others survived by using the alternate sources of heat.
As my students observed - the most effective was a parabolic heater in Ms. Berman’s room.
They all agreed that superior reflective properties of parabolic mirrors made it a sure winner….
And so it goes….
Kids learn to observe and analyze and this knowledge helps them to better understand and make
sense of the real world around.
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Jolanta Walukiewicz – Analysis of the Process of Teaching
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