This document discusses using symmetry as a problem solving tool in mathematics education. It presents six mathematical problems that can be solved using different types of symmetry, such as geometric, algebraic, and role symmetry. The problems were used in workshops for secondary mathematics teachers to develop their understanding and preference for using symmetry. Initially, teachers did not often use symmetry in their own problem solving approaches. Through discussion of alternative symmetric solutions, teachers' inclination to utilize symmetry gradually increased over the course of the intervention.

1. Applications of symmetry to problem solving
ROZA LEIKIN,*{ ABRAHAM BERMAN{ and ORIT ZASLAVSKY{
{ Faculty of Education, University of Haifa, Haifa, 31905, Israel
{ Technion, Israel Institute of Technology, Haifa, 32000, Israel; e-mail:
rozal@construct.haifa.ac.il
(Received 4 January 1999)
Symmetry is an important mathematical concept which plays an extremely
important role as a problem-solving technique. Nevertheless, symmetry is
rarely used in secondary school in solving mathematical problems. Several
investigations demonstrate that secondary school mathematics teachers are not
aware enough of the importance of this elegant problem-solving tool. In this
paper we present examples of problems from several branches of mathematics
that can be solved using diVerent types of symmetry. Teachers’ attitudes and
beliefs regarding the use of symmetry in the solutions of these problems are
discussed.
1. Introduction
Research on mathematical problem solving emphasizes the potential of
problem solving in: (a) developing students’ mathematical ability, their intuitions
and reasoning; (b) enhancing students’ motivation and enthusiasm with respect to
mathematics [1–6]. The concept of symmetry has a special role in problem solving.
The importance of the role of symmetry in problem solving is expressed by Polya
[7] as follows:
We expect that any symmetry found in the data and conditions of the
problem will be mirrored by the solution ([7], p. 161)
Symmetry connects between various branches of mathematics such as algebra,
geometry, probability, and calculus [8–19]. Nevertheless, symmetry is rarely used
in secondary school mathematics as a problem-solving technique.
Teachers’ perceptions of a particular mathematical concept and their beliefs
about its role in problem solving strongly inuence the kinds of learning
experiences they design for their students, and consequently the processes of
learning mathematics in their classrooms. Dreyfus and Eisenberg [20] claim that
teachers need to develop awareness of the importance of symmetry for secondary
school mathematical problem solving. They support this claim by their Žnding
that high school mathematics teachers who participated in an investigation
conducted by Dreyfus and Eisenberg [20] did not recognize symmetry as a tool
that can simplify the solutions.
This paper is based on a study, which focused on symmetry as a tool for
professional development of mathematics teachers [11]. In the framework of this
int. j. math. educ. sci. technol., 2000, vol. 31, no. 6, 799–809
International Journal of Mathematical Education in Science and Technology
ISSN 0020–739X print/ISSN 1464–5211 online # 2000 Taylor & Francis Ltd
http://www.tandf.co.uk/journals
* Author for correspondence.

2. study a careful analysis of the Israeli secondary mathematics curriculum has been
made in order to identify problems related to diVerent mathematical topics that
could be solved using symmetry. The problems were applied in a series of
workshops for mathematics teachers on the use of symmetry in problem solving.
In order to follow their preferences and tendencies regarding the use of symmetry,
the workshops were designed to document processes in which the teachers were
involved.
Symmetry was deŽned in the study as follows:
DeŽnition. Symmetry is a triplet consisting of an object …S†, a speciŽc property
…Y† of the object, and a transformation …M† satisfying the following two conditions:
(i) The object belongs to the domain of the transformation.
(ii) Application of the transformation to the object does not change the
property of the object.
Symmetry …S; Y; M† is called a trivial symmetry if property Y is immune to
transformation M for any object S having this property.
This deŽnition covers diVerent types of symmetry such as geometric symmetry
(see, for example, Problems 1 and 2 for line symmetry, Problem 3 for central
symmetry and Problem 4 for rotational symmetry), algebraic role symmetry (see,
for example, Problems 5 and 6), and role symmetry in proofs (see, for example,
Problem 2).
The purpose of this paper is to present symmetry as a powerful problem-
solving tool in secondary mathematical curriculum, and to discuss teachers’
attitudes and preferences towards this tool. In section 2 we present examples of
mathematical problems, which can be solved using symmetry. These problems
were introduced to teachers in diVerent ways, which are described in section 3.
We conclude in section 4 with a discussion of the inuence of this intervention on
teachers’ attitudes and beliefs regarding application of symmetry.
2. Examples
In this section we present six problems and demonstrate how to solve them
using symmetry. Problem 1 is a classical example of a question that can be solved
using symmetry. Problems 2 and 5 are taken from Arbel [21] and Polya [7],
respectively, in which they are solved using symmetry. Problems 3, 4 and 6 are
taken from Israeli textbooks and tests in chapters that do not make any connections
to the concept of symmetry.
Problems 1 and 2 are optimization problems that usually are associated with
use of calculus. The solutions presented below are based on geometric symmetry,
i.e. line symmetry.
Problem 1. Given a point P in an acute angle. Find points A and B on the
angle’s sides, such that perimeter of the triangle ABP is minimal (Žgure 1).
Solution. Let P0
and P00
be two points that are symmetrical to P with respect
to the sides m and n respectively. Then point A on m and point B on n fulŽl the
conditions AP0
ˆ AP and BP00
ˆ BP and the perimeter of the triangle ABP
…PA ‡ AB ‡ BP† is equal to the sum of segments P0
A ‡ AB ‡ BP00
(Žgure 2).
800 R. Leikin et al.

3. Let A¤
and B¤
be the intersection points of the segment P0
P00
with lines m and n
respectively, then for every A on m and B on m such that A and B are diVerent
from A¤
and B¤
, P0
A¤
‡ A¤
B¤
‡ B¤
P00
< P0
A ‡ AB ‡ BP00
. Thus, the triangle
A¤
B¤
P has minimal perimeter.
In the following problem we use role symmetry in proof in addition to geometric
symmetry.
Problem 2. Given an acute angled triangle ABC. Find a triangle PQR
inscribed in the triangle ABC having minimal perimeter (Žgure 3).
Solution. Let P1 be a point in the side BC of the given triangle and points P0
1
and P00
1 symmetrical to P1 with respect to the lines AB and AC. According to
Problem 2, for this particular point P1 the triangle P1Q1R1 is of the minimal
perimeter if Q1 and R1 are the intersection points of the segment P0
1P00
1 with sides
Applications of symmetry 801
Figure 1.
Figure 2.

4. AB and AC of the triangle (Žgure 4). The perimeter of the triangle P1Q1R1 is equal
to the length of the segment P0
1P00
1 .
For any point Pi in BC the triangle P0
i AP 00
i is an isosceles triangle (AP0
i ˆ
APi ˆ AP00
i as symmetrical segments). For any i, the angle P0
i AP 00
i ˆ 2BAC and
therefore constant (Žgure 5). Of all the isosceles triangles with equal angles
opposite to the bases, the triangle having the shortest lateral side has the shortest
base. Hence, P0
i P00
i is minimal when APi is minimal, in other words, when APi is an
altitude of the triangle ABC.
The perimeter of the triangle PiQiRi is equal to the length of the segment P0
i P00
i
and its length is minimal when APi is minimal. Hence P¤
Q¤
R¤
is of the minimal
perimeter if and only if P¤
A is the altitude of the triangle (Žgure 6).
The points P, Q and R play symmetrical roles in the problem. Thus all three
points P¤
, Q¤
, and R¤
are heels of altitudes of the given triangle (Žgure 7). Hence,
802 R. Leikin et al.
Figure 3.
Figure 4.

5. of all the triangles inscribed in the acute angled triangle ABC the triangle with
vertices in the heels of the altitudes of the given triangle has minimal perimeter.
The solution to Problem 3 is based on central symmetry instead of rather
complicated calculus manipulations.
Problem 3. Given the quadratic function y ˆ x2
¡ 3 and a family of linear
functions y ˆ ax ‡ 3, for which value of the parameter a is the area of the region
bounded by the parabola and by the line minimal (Žgure 8)?
Solution. Consider the line y ˆ 3. The Žgure bounded by the parabola and by
this line is symmetrical with respect to the y-axis. Each line …`: y ˆ ax ‡ 3†
diVerent from y ˆ 3 ‘breaks’ the symmetry of this Žgure (Žgure 9).
Applications of symmetry 803
Figure 5.
Figure 6.

6. Let the line y ˆ 3 intersect the parabola at points P and M. Let line ` intersect
the parabola at the points L and S and the y-axis at Q. Let PN and MK be parallel
to the y axis where points N and K are on ` . Triangles QPN and QMK are
symmetrical with respect to the point Q, then SLMQ > SKMQ ˆ SNPQ > SSPQ.
Hence the region bounded by the parabola y ˆ x2
¡ 3 and by a line from the family
of functions y ˆ ax ‡ 3 is minimal when a ˆ 0.
In Problem 4 we use geometric rotational symmetry in the theory of complex
numbers.
Problem 4. Prove that the sum of all complex roots of the equation zn
ˆ 1 is
equal to 0 for all natural numbers n.
804 R. Leikin et al.
Figure 7.
Figure 8.

7. Solution. All the roots are points on the unit circle (Žgure 10).
The rotation of the circle does not change the location of the set of points.
Hence the sum of the solution vectors for z does not change. Hence this sum is
zero.
Solution of the system of equation in Problem 5 presents use of algebraic
symmetry of role.
Problem 5. Solve the following system of equations:
3x ‡ 2y ‡ z ˆ 30
x ‡ 3y ‡ 2y ˆ 30
2x ‡ y ‡ 3z ˆ 30
8
>
<
>
:
Solution. The given system consists of three equations with three unknowns
and is invariant under cyclic permutation of the unknowns. Thus, if a triplet
…t1; t2; t3† is a solution of the system, then the triplets …t2; t3; t1† and …t3; t1; t2† are
Applications of symmetry 805
Figure 9.
Figure 10.

8. also solutions of this system. It is easy to check that the system has only one
solution, hence t1 ˆ t2 ˆ t3. Consequently: x ˆ y ˆ z, 6x ˆ 30, x ˆ y ˆ z ˆ 5.
Note. The uniqueness of the solution does not follow from symmetry.
Finally in the solution of the following problem we use both algebraic symmetry
and geometric symmetry.
Problem 6. Prove that the straight line y ˆ x intersects the ellipse
x2
‡ xy ‡ y2
ˆ 12 at right angles.
Solution. The equation of the ellipse is algebraically symmetrical. In other
words, permutation of the variables x and y does not change the equation. Hence,
if a point …a; b† belongs to the curve, then the point …b; a† also belongs to the curve.
Therefore, the curve is symmetrical with respect to line y ˆ x. The set of tangent
lines to the curve is also symmetrical with respect to the line y ˆ x. Thus, the
tangent line to the curve at the point of intersection of the curve with the line y ˆ x
is symmetrical with respect to this line, if it exists. An ellipse is convex and has a
tangent line at each of its points, therefore the tangent line to the curve at the point
of intersection of the curve with the line y ˆ x is perpendicular to this line.
3. The intervention
The problems presented above are examples of mathematical problems that
were used in a secondary mathematics teachers’ professional development pro-
gramme in order to identify their problem-solving strategies and their preferences
with respect to the use of symmetry. The teachers’ responses to the problems they
solved were recorded and analysed. The problems were presented to the teachers
in diVerent ways. For example, for Problems 1, 2, 4 and 5 the teachers were asked
to suggest as many solutions as possible before any discussion of solutions took
place. On the other hand, for Problem 6, two diVerent solutions were presented to
the teachers by the researcher, one of which was based on symmetry and the other
was based on more conventional ways typically used in secondary school. Follow-
ing the presentation of the two diVerent solutions, the teachers were asked to
choose one of these suggested ways when solving a similar problem. Succeeding
their engagement with the problems, teachers were involved in authentic discus-
sions focused on advantages and disadvantages of the diVerent kinds of solutions.
Additionally, the teachers were asked about their own preferences regarding the
use of symmetry in problem solving. Problem 3 was brought by a group of teachers
who knew how to solve the problem using calculus but, having gained appreciation
of symmetry, asked us to help them to use symmetry in solving the problem.
4. Discussion
In this section we discuss the main factors that inuenced teachers’ problem-
solving performance and their preferences to use symmetry in problem solving.
Our investigation demonstrated that mathematics teachers usually did not use
symmetry in problem solving. The teachers’ tendencies to use symmetry depended
on their familiarity with symmetry and on their beliefs about mathematics. As a
result of the intervention, teachers’ inclinations to use symmetry in solving
mathematical problems were enhanced. For example, at the beginning of the
806 R. Leikin et al.

9. study teachers tended not to use role symmetry at all, later on they began to ‘feel
symmetry’, and by the end of the intervention they used symmetry in solving most
of the problems.
As noted above, at the beginning of the study few teachers used symmetry
when solving the problems. Moreover, when these teachers presented their
solutions based on symmetry, they were not able to convince the other teachers
that the use of symmetry in their solution was suYcient.
For example, when Problem 1 was solved using geometric symmetry the
teachers tended to try to ‘complete’ a solution by conventional tools such as
derivative and congruency. One of the teachers claimed that ‘we need to use both
symmetry and congruence of segments in order to prove that these segments are
equal’.
Role symmetry in proofs was the most diYcult type of symmetry for teachers
to accept. When the teachers were asked to explain why the vertices Q¤
, and R¤
of
the inscribed triangle are ‘heels’ of the altitudes of the given triangle (see solution
to Problem 2 above) several teachers justiŽed it by using similarity and congru-
ence. These teachers seemed to reject the use of role symmetry. Other teachers
only partly accepted the idea of role symmetry. For these teachers it was ‘clear’
that:
If the construction started in the side BC, the ‘heel’ of the altitude to the side
BC …P¤
† was obtained as a vertex of the inscribed triangle P¤
Q¤
R¤
having the
minimal perimeter. Then, in the same way, the heels of altitudes to the sides
AC and AB of the given triangle (points Q¤¤
and R¤ 0
) could be obtained as
vertices of two other triangles: Q¤¤
R¤¤
P¤¤
and R¤ 0
P¤ 0
Q¤ 0
. Thus, the peri-
meters of these triangles must be compared and the smallest one has to be
chosen.
Only one teacher could explain how ‘thinking symmetry’ helped in proving the
statement. He solved the problem as follows:
For any point P in BC which diVers from P¤
the perimeter of the inscribed
triangle with vertex P is greater than the perimeter of the obtained triangle
P¤
Q¤
R¤
. The same consideration holds with respect to any point Q which
diVers from Q¤
and to any point R which diVers from R¤
. Hence, these three
points are ‘heels’ of the altitudes of the given triangle.
As mentioned above, teachers’ beliefs about the nature of mathematical proof as
well as their familiarity with symmetry inuenced their preferences regarding use
of symmetry in problem solving. For example, when the teachers were presented
with two diVerent solutions to Problem 6 all of them agreed that the solution based
on symmetry was more elegant and beautiful. Nevertheless, most of the teachers
thought that the solution using calculus was more ‘trustworthy’. They felt more
‘safe’ to teach their students to solve problems using calculus. In addition, when
the teachers were asked to choose one of the presented solutions of Problem 6 in
order to solve a similar problem, only two teachers chose to use symmetry in their
solutions but were not sure that the solution based on symmetry was ‘good
enough’. In order to be on the ‘safe’ side, these teachers also solved the problem
in a standard way. During the discussion of the problem one of the teachers said
that the presented solution is ‘too sophisticated and philosophical’.
Applications of symmetry 807

10. It is interesting to note that while the intervention enhanced the teachers’
knowledge of symmetry it did not equally inuence their beliefs regarding what
constitutes a valid mathematical proof. Although, at the end of the study most of
the teachers were able to use diVerent types of symmetry in problem solving, and
even solved problems using symmetry, still, many were not convinced that the
proofs were ‘mathematical enough’. For example one of the teachers, who solved
the problem using role symmetry, presented this solution to the other teacher and
then concluded: ‘but this is not mathematics, this is . . . ’. Only by the end of the
intervention were most of the teachers able and willing to use symmetry in
problem solving. Teachers’ perception of the solutions based on symmetry as
‘more elegant’ and conveying the beauty of mathematics served as a springboard
for these changes. Almost 50% of the teachers who preferred not to use symmetry
at the beginning of the study were convinced by their experience in our investiga-
tion that it is worthwhile. At the end of our study most of the teachers decided to
solve Problem 8 using symmetry. All the teachers agreed that this solution was
easier, more interesting, and more elegant.
The way in which Problem 3 was raised by the teachers themselves, is also an
indication of changes in teachers’ tendencies to use symmetry when solving the
problems. The teachers solved this problem with their twelfth-grade students
using calculus. In their opinion this solution was too complicated and they felt that
‘this problem can be solved with the help of symmetry’. When in our workshop the
teachers solved this problem using symmetry they enjoyed the fact that they
‘found such a good example for the use of symmetry’. Moreover, as a result of our
investigation the teachers seemed to be convinced that:
‘Special cases of symmetric objects are often prime candidates for ex-
amination’
([5], p. 81)
Acknowledgements
The study was supported by the Technion Project for the Enhancement of
Secondary Mathematics Education directed by O. Zaslavsky. The work of A.
Berman was supported by the Fund for the Promotion of Research at the
Technion. The Žrst draft of this paper was prepared while the Žrst author was a
post-doctoral fellow with Ed Silver at the Learning Research and Development
Center, University of Pittsburgh.
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Applications of symmetry 809