Introducing Conics in 9
th
Grade: An Experimental Teaching
Borislav Yordanov Lazarov
1
and Dimitar Georgiev Dimitrov
2
1
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Blok 8 Georgi Bonchev Str., Sofia, Bulgaria
2
125
th
Secondary School, Sofia, Bulgaria
Keywords: Conics, Loci, Envelopes, Integrated Approach, Synthetic Competence.
Abstract: Under consideration is an experimental teaching focused on presenting the conics in a dual manner: as loci
and envelopes. A bunch of computer technologies is drawn to explore and investigate this duality of the
conics. An example of how it is done for a particular conic is given. The target group includes secondary
school students who are advanced in math and information technologies. The theoretical base is an original
didactical model for designing individual educational trajectories that is adapted for the team-working mode.
The educational goal includes developing synthetic competence of an entire team. The individual
characteristics of the team members complement one another for resolving complex problems from the local
behavioral environment, which were specifically formed for the purposes of the experimental teaching.
1 INTRODUCTION
The new Bulgarian educational legislation allows a
type of schools (so-called innovative schools) to vary
the syllabus including integrated subjects into the
school plan. Such status quo provides the opportunity
for creative teaching. However, there are no
textbooks neither educational plans nor guidelines for
the new subjects. Therefore, any good practice is
welcome in this slippery twilight zone. Below we are
going to share our experience in teaching conics in 9
th
grade, hoping it is a kind of good practice.
The approach we applied integrates areas of
several types.
1) Organizational: it is an example of integrating
school and academic staff.
2) Technological: it integrates classical geometry
with computer technologies.
3) Didactical: we combine the deductive with
algorithmic method in math.
4) Mathematical: we synthesized the concepts of
locus and envelope by a common algorithm for any
of the conics.
The experimental teaching took part in 2018/2019
scholastic year with a team of three ninthgraders, who
are advanced both in math and ICT. Further we refer
to this team as the Team. The small numbers of the
students in the Team allowed to apply individual
approach on one hand, and to take advantages of the
team work on the other hand. The didactical model
we applied is a modification of DMT (Lazarov, 2013)
for indiviual educational trajectory of small team.
Some details of the model are given in (Lazarov,
2019).
2 CHOOSING THE TOPIC
We carefully selected the topic for our experimental
teaching to satisfy two controversial requirements:
the matter to be in the zone of proximal development
and to be beyond the curriculum. Our previous
practice in teaching conics is presented in (Lazarov
and Todorova, 2014), where parabola was introduced
in an inquiry based mode of classroom activity. The
modest results of this experiment was related with the
attempt to organize Socratic style teaching in a large
group of students. Nevertheless, some of them made
considerable progress, which gave us reasons to
reconsider the topic, this time applying individual
approach.
The concept of locus used to be a part of the 8
th
grade Bulgarian curriculum until 2016. Nowadays it
remains just a trace of this concept in the 7
th
grade
curriculum. The loci are included implicitly in the
properties of the segment bisector and the angle
bisector. Fortunately, these two loci was just we need
as a base for developing the topic. The concept of
envelope of one parametric family of lines was
436
Lazarov, B. and Dimitrov, D.
Introducing Conics in 9th Grade: An Experimental Teaching.
DOI: 10.5220/0009430204360441
In Proceedings of the 12th International Conference on Computer Supported Education (CSEDU 2020) - Volume 1, pages 436-441
ISBN: 978-989-758-417-6
Copyright
c
2020 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
completely new matter for the Team (it is rather far
from the secondary school mathematics). The modern
dynamic geometry software (DGS) provides some
opportunities to elaborate the envelope in pure
geometrical mode, which is acceptable for the
advanced school students (Lazarov, 2011). However,
some concepts, which are routine for the calculus,
should be reconsidered from the secondary school
viewpoint.
3 ABOUT THE DEFINITIONS
The proper selection of definitions is another
challenge we met. A variety of didactical approaches
could be used for introducing a math concept: by
examples and counterexamples, constructively,
deductively etc. Our key reason was to stay as close
to the curriculum as possible.
3.1 Basic Concepts
In our opinion, the closest to the 9
th
grade math
curriculum are the following definitions:
Parabola is the locus of points in the plane that are
equidistant from a given point (focus) and a given
line (directrix).
Ellipse is the locus of points for which the sum of
the distances to two given points (foci) is constant.
Hyperbola is the locus of points for which the
absolute value of the difference of the distances to
two given points (foci) is constant.
The ‘is constant’ in the last two bullets means ‘equals
a given segment’. These definitions also match the
GeoGebra operators for drawing the conics
(GeoGebra, 2019). In fact, the GeoGebra operators
require a third point for constructing the ellipse and
hyperbola (as say the operator icons). The ‘given
segment appears as a sum or difference of the
distances from this third point to the foci.
3.2 Calculus-circumventing
The calculus definition of tangent line to a curve is
inapplicable in 9
th
grade. We defined the tangency in
a specific manner for any particular conic. A line is
tangent to:
parabola, iff it has exactly one common point with
it and is not perpendicular to its directrix;
ellipse, iff it has exactly one common point with
it;
hyperbola, iff it has exactly one common point
with it and is not parallel to any of its asymptotes.
These definitions extended the familiar concept of
tangent line to a circle.
Further, we need to elaborate into geometry the
concept of parameter that is common for algebra. The
parameter was introduced as:
point at a straight line when consider parabola;
point at a circle when consider ellipse;
point at two circle arcs when consider hyperbola.
We considered only one-parameter families of
straight lines, which allowed us to restrict the notion
of envelope of a one-parametric family of lines. It is
a conic, which touches each line from the family in a
single point and any line from the family is tangent to
this conic.
So, we managed to avoid the concepts from the
calculus and to stay inside the secondary school
mathematics. Let us note that a similar approach is
adopted in (Lazarov, 2011) (but due to the larger
generality of considerations in that work, a need to
introduce the concept of tangent curves appears).
4 THE CONSTRUCTIONS
Three algorithms for constructing conics were the
core of the experimental teaching. Geometrical
constructions are included in 7
th
and 8
th
grade of the
Bulgarian curriculum. Among the conics, only
parabola is studied but just as the graph of quadratic
function and with no properties of the curve itself.
We decide to develop a constructive method to
unify in one algorithm the dual nature of a conic: as
locus on one hand and as envelope on the other hand.
Here it is the algorithm for the hyperbola H having
foci 𝐹
,
and passing through a given point 𝑇:
1) Denote 𝑐|𝐹
𝑇𝐹
𝑇| and draw the circle
𝑘 with radius 𝑐 centred at 𝐹
.
2) Take an arbitrary point 𝐾∈𝑘.
3) Draw the segment bisector 𝑠

of 𝐹
𝐾 (Figure 1).
Figure 1: Thе hyperbola construction algorithm.
Lemma 1. When it exists, the intersection point 𝑃
of 𝐹
𝐾 and 𝑠

is at H.
Introducing Conics in 9th Grade: An Experimental Teaching
437
(The proofs of this lemma and the statements that
follow are given in the Appendix.)
Lemma 2. If 𝑃 exists then 𝑠

is the tangent to H
at 𝑃.
Parametrizing the algorithm by 𝐾, we can obtain
both objects: points at the hyperbola H (according to
Lemma 1) and tangents to it (according to Lemma 2).
Let us note that 𝑃 exists iff 𝐹
𝐾 is not a tangent to 𝑘.
The tangency of 𝐹
𝐾 and 𝑘 happens in two points
𝐾
,
that split 𝑘 into two arcs 𝑘
,
. So, we can take
these two arcs as the domain of 𝐾. In the case, when
𝐾 runs along 𝑘
∪𝑘
, we obtain a one-parametric-
family of lines 𝓗
𝑠

:𝐾 𝑘
∪𝑘
. (The lines
𝑠

,
are the asymptotes of H.)
Theorem 1. H is the envelope of 𝓗
Figure 2: The hyperbola as the envelope of 𝓗 .
Thus, the hyperbola H occurs in the same time
the locus of 𝑃 and the envelope of 𝑠

. Similar
constructions were done for parabola and ellipse. In
fact, the construction algorithm for the parabola
(Lazarov, 2011) serves as a template for the ellipse
and hyperbola algorithms, which were done by the
Team. It is clear that the algorithms are device-
independent: any one could be realized either as a
DGS applet or as a traditional ruler-and-compass
construction.
5 DIVIDENDS
The unified algorithmic introduction of the conics
allows an easy explanation of the reflective properties
of these curves. Let us note that according to the laws
of the geometric optics a smooth curve reflects an
incoming ray as it does the tangent line at the
reflection point. Having a conic as locus of points,
one operate with a ‘real object’. The envelope
configures the conic as a phantom, but this phantom
performs the reflection of a beam: any ray is reflected
by the corresponding tangent line. Combining the
dual nature of the conic, one gets the big picture of
the reflection. We will illustrate this again with the
hyperbola.
Theorem 2. A ray coming from inside of one of
the branches of H and directed to the focus inside the
other branch, after reflection by H, passes through the
focus inside the first branch (Figure 3).
Figure 3: Thе reflective property of the hyperbola.
Scholium. The reflective property of the hyperbola
could work in the reverse manner of the one that is
described in Theorem 3: in Figure 3 the sound coming
from 𝐹
, after reflection by 𝐻 , is uniformly
distributed inside the branch, which contains 𝐹
. In
our opinion, Netrebko and Garanca (2007) use this
property when they sing facing the wall. It is also
applied to improve the acoustics of the cathedral La
Sagrada Familia in Barcelona (Burry et al., 2011).
Connecting mathematical results with high
achievements in arts significantly lifts the students’
attitude to math (Lazarov, 2019).
6 TEAM EDUCATIONAL
TRAJECTORY
We followed the general structure of the model DMT
(Lazarov, 2013, 2019) for the experimental teaching,
which consists of iterative steps (climbing floors).
However, the design of the individual educational
trajectory (IET) needed slight modification taking
into account the students’ interaction inside the Team.
In our general educational plan, the proximal
educational goals were stated to the Team but any
student was in charge for some details. Nevertheless,
the elaboration of the details happened in
collaboration of the team members and as result a
team product appears when the particular educational
goal was reached.
CSEDU 2020 - 12th International Conference on Computer Supported Education
438
6.1 On the First Floor of the IET
We started the experimental teaching during the
summer holyday. The students were proposed to
examine closely GeoGebra by themselves and to get
an idea about TeX.
During the first term of the scholastic year, we had
lectures once weekly. The introductory part of the
teaching we devoted to the basic facts about the conic
sections. Initially their properties were examined
experimentally but then were rigorously proven.
Meanwhile, students mastered their skills in
GeoGebra and TeX.
6.2 On the Second Floor of the IET
The proximal educational goal after introductory part
was to apply the knowledge about conics and the
DGS skills for modifying the parabola module to
similar modules about ellipse and hyperbola. These
activities were directed to prepare a paper for the
Bulgarian national math journal for school students
Matematika.
The Team was motivated and the students did
their best to polish all details. E.g., 39 GeoGebra
applets were made and there were done several
interim variants for the most of them. The paper also
needed several redactions. These efforts paid
themselves: the paper was published (Dimitrov et al.,
2019). Let us note that students’ papers were not
published in the journal for a rather long time before.
This paper resurrected the column Students’ works (in
Bulgarian „Ученическо творчество“ – Figure 4).
Figure 4: Thе first page of the Team’s paper.
6.3 On the Third Floor of the IET
The final educational goal of the experimental
teaching was to foster a kind of synthetic competence.
This means to examine the students’ knowledge-
skills-attitude (KSA) package for multifunktionality
and transferability. The didactical innovation was in
the team-competence we developed. The stimulus of
the Team to continue studying the conics was the
presentation of the project at a school symposium and
then at an international conference for school
students.
6.3.1 Implementation of Old-tech
The pre-computer technology for visualization of
envelopes require ingenuity and deep understanding
of the matter. The Team prepared a short movie about
paper-and-pencil drawing of a parabolic envelope
(Figure 5).
Figure 5: An old-tech method performed and filmed by the
Team.
The theoretical base for this drawing relates to
another parametrization of the tangents to a parabola:
the right-angled vertex of the wooden triangle moves
along the line, which is located in the middle between
the focus and the directrix. The Team proved the
corresponding theorem relying on the parabola
construction algorithm.
6.3.2 Explanation of the Terminology
The definitions for conics we adopted do not explain
the terminology ‘conic sections’, but we did not need
other definitions for the project. However, it was
good manner for the Team to show their classmates
the origin of the concept.
Figure 6: Thе Team’s illustration of conic sections.
For us, it was another opportunity to check the
transferability and multifunktionality of the Team’s
Introducing Conics in 9th Grade: An Experimental Teaching
439
KSA-package. Therefore, we urge the Team to make
the images on Figure 6 and to argue that the three
curves, which appears on the wall, are sections of
conic shape by plane.
6.3.3 Searching for Applications
Our role on the final stage of the experimental
teaching was to state some guidelines for informal
learning in accordance with the findings of Petrovic
(2018). The didactical goal we stated to the Team was
to find appropriate implementations of the reflective
properties of the conics. We adopted a kind of
tutoring style to force the multifunktionality of the
Team’s KSA-package beyond the context of its
development. The testing area for the transferability
of the students’ knowledge and skills was real-life
applications of conic surfaces.
We already discussed some potential applications
of hyperbolic surfaces in the Scholium above. The
implementation of the reflective property of parabolic
surface in modern communications could be observed
just walking around the city (Figure 7, left).
Figure 7: Applications of conic sections surfaces.
More sophisticated was the case with the ellipse.
To find some proper applications, the Team was
urged to investigate different areas including
professional medical sources. The image in the Figure
7 (right) illustrates how an elliptic surface focuses
shockwaves for pulverizing renal and ureteric stones
(Pearl, 2012). The Team prepared an explanation of
the theoretical background of how the lithotripter
works.
7 CONCLUSIONS
On the top floor of the IET, the Team demonstrates a
large arsenal of multifunctional knowledge and skills
cemented by a positive attitude toward mathematics
and ICT. The collaboration between the students
allowed reaching the stated educational goals via a
synergetic effect, which was a kind of revelation for
us.
The team synthetic competence developed on this
final stage of the experimental teaching encouraged
us to continue our individual experimental teaching
with the same students for the upcoming school term.
The complex way we introduced conics created a
solid fundament for consideration more advanced
mathematical topics. We plan to enlarge students’
synthetic competence entering higher mathematics
via computer algebra system.
ACKNOWLEDGEMENTS
The authors are grateful to Mrs. S. Petrova who is the
vice-principle of 125
th
Secondary School, Sofia, for
upholding the experimental teaching. The study is
partially supported by the Chernorizec Hrabar
Educational and Research Program of the Institute of
Mathematics and Informatics, Sofia. All figures in the
text are done by the Team.
Authors’ contributions: D. Dimitrov
management and didactical support; B. Lazarov
conceptual frame, tutoring and the paper text.
REFERENCES
Burry, J., Davis, D., Peters, B., Ayres, P., Klein, J., Pena de
Leon, A., & Burry, M. (2011). Modelling Hyperboloid
Sound Scattering: The Challenge of Simulating,
Fabricating and Measuring. In C. Gengnagel, A. Kilian,
N. Palz, & F. Scheurer (Eds.), Computational Design
Modelling: Proceedings of the Design Modelling
Symposium Berlin 2011 (pp. 89-96). Berlin: Springer
Dimitrov, M., Peeva, G., and Stoyanov, B. (2019) Conics
as loci and envelopes. Matematika, Vol. LIX, No 1, pp
46-57 (in Bulgarian)
GeoGebra (2019). https://www.geogebra.org (last visited in
Nov 2019)
Lazarov, B., (2011). Teaching envelopes in secondary
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Information Models and Analyses. Vol. 2, No1, 2013,
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Lazarov, B. (2019). Improving the Attitude towards
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th
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th
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15th International BASOPED Conference, Lumina
Publishing, Bucharest, pp 504-510
Netrebko, A., Garanca, E. (2007). Duo des fleurs by Leo
Delibe. Baden-Baden Opera Gala https://
www.youtube.com/watch?v=Vf42IP__ipw (6:00-6:25)
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XKL9Bnk8 (last visited in Nov 2019). This link refers
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367:50-57,
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APPENDIX
We follow the notations from section 4.
Proof of Lemma 1. Consider the configuration
on Figure 8. We have 𝑃∈𝑠

𝐹
𝑃𝐾𝑃. Now
𝐹
𝑃 𝐹
𝑃
𝐹
𝐾𝐾𝑃
𝐹
𝑃
𝐹
𝐾𝐾𝑃
𝐾𝑃 𝐹
𝐾  𝑐 ⟹ 𝑃 H.
Figure 8: Configuration of Lemma 1.
Proof of Lemma 2. Suppose the contrary: 𝑠

has a second common point 𝑄 with H, i.e. there exists
△𝑄𝐹
𝐾. Let 𝑍 𝐹
𝑄∩𝑘 (Figure 9). Now:
𝑄∈ H 𝑄𝐹
𝑄𝐹
𝑐
𝑄𝑍  𝑍𝐹
𝑄𝐹
𝑐
𝑄𝑍  𝑐
𝑄𝐹
𝑐⟹𝑄𝑍𝑄𝐹
.
𝑄∈𝑠

𝑄𝐹
𝑄𝐾⟹𝑄𝑍𝑄𝐾.
Applying the triangle inequality for △𝑄𝐹
𝐾, we
get
𝑄𝐾  𝐾𝐹
𝑄𝐹
𝑄𝑍𝑍𝐹
⟹𝐾𝐹
𝑍𝐹
,
which is a contradiction.
Proof of Theorem 1. According to Lemma 2,
𝑠

is tangent to H for every 𝐾∈𝑘
∪𝑘
. Now let
𝑃 be an arbitrary point at H such that 𝑃𝐹
𝑃𝐹
𝑐.
Let 𝐾 𝐹
𝑃∩𝑘. Hence 𝑃𝐾 𝑃𝐹
𝑐𝑃𝐹
, i.e.
𝑃∈𝑠

, which is tangent to H.
Figure 9: Configuration of Lemma 2.
Proof of Theorem 2. Consider the configuration
in Figure 10 where 𝐴 is an arbitrary point inside the
hyperbola branch containing 𝐹
. Let 𝐴𝐹
meets this
branch at 𝑃 and 𝐴𝐹
∩𝑘𝐾. Denote by 𝑀 the
midpoint of 𝐾𝐹
. We have to prove that the rays 𝐴𝑃
and 𝑃𝐹
conclude equal angles with the tangent to
H at 𝑃. According to Lemma 2, this tangent is 𝑠

.
Following the notations in Figure 10, we have to
prove that 𝛼𝛽.
Figure 10: Configuration of Theorem 2.
Since 𝑃∈𝑠

, then 𝐾𝑃 𝑃𝐹
. In the isosceles
triangle 𝐾𝑃𝐹
, the segment bisector of 𝐾𝐹
is angle
bisector of ∠𝐾𝑃𝐹
. Hence, 𝛽𝐾𝑃𝑀 .
Furthermore, 𝛼𝐾𝑃𝑀 as vertical angles. Thus
𝛼𝛽.
Introducing Conics in 9th Grade: An Experimental Teaching
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