Applications of Symmetry to Problem Solving

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:[email protected]

(Received 4 January 1999)

Symmetry is an important mathematical concept which plays an extremelyimportant role as a problem-solving technique. Nevertheless, symmetry israrely used in secondary school in solving mathematical problems. Severalinvestigations demonstrate that secondary school mathematics teachers are notaware enough of the importance of this elegant problem-solving tool. In thispaper we present examples of problems from several branches of mathematicsthat can be solved using diVerent types of symmetry. Teachers ’ attitudes andbeliefs regarding the use of symmetry in the solutions of these problems arediscussed.

1. IntroductionResearch on mathematical problem solving emphasizes the potential of

problem solving in: (a) developing students’ mathematical ability, their intuitionsand reasoning; (b) enhancing students’ motivation and enthusiasm with respect tomathematics [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 theproblem 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 usedin secondary school mathematics as a problem-solving technique.

Teachers ’ perceptions of a particular mathematical concept and their beliefsabout its role in problem solving strongly in� uence the kinds of learningexperiences they design for their students, and consequently the processes oflearning mathematics in their classrooms. Dreyfus and Eisenberg [20] claim thatteachers need to develop awareness of the importance of symmetry for secondaryschool mathematical problem solving. They support this claim by their � ndingthat high school mathematics teachers who participated in an investigationconducted by Dreyfus and Eisenberg [20] did not recognize symmetry as a toolthat can simplify the solutions.

This paper is based on a study, which focused on symmetry as a tool forprofessional 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 TechnologyISSN 0020–739X print/ISSN 1464–5211 online # 2000 Taylor & Francis Ltd

http://www.tandf.co.uk/journals

* Author for correspondence.

study a careful analysis of the Israeli secondary mathematics curriculum has beenmade in order to identify problems related to diVerent mathematical topics thatcould be solved using symmetry. The problems were applied in a series ofworkshops 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 wereinvolved.

Symmetry was de� ned in the study as follows:

De� nition. Symmetry is a triplet consisting of an object …S†, a speci� c 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 totransformation M for any object S having this property.

This de� nition covers diVerent types of symmetry such as geometric symmetry(see, for example, Problems 1 and 2 for line symmetry, Problem 3 for centralsymmetry 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 ofmathematical problems, which can be solved using symmetry. These problemswere 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 onteachers ’ attitudes and beliefs regarding application of symmetry.

2. ExamplesIn 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 solvedusing 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 aretaken from Israeli textbooks and tests in chapters that do not make any connectionsto the concept of symmetry.

Problems 1 and 2 are optimization problems that usually are associated withuse 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 theangle’ s sides, such that perimeter of the triangle ABP is minimal (� gure 1).

Solution. Let P 0 and P 00 be two points that are symmetrical to P with respectto the sides m and n respectively. Then point A on m and point B on n ful� l theconditions AP 0 ˆ AP and BP 00 ˆ BP and the perimeter of the triangle ABP…PA ‡ AB ‡ BP† is equal to the sum of segments P 0A ‡ AB ‡ BP 00 (� gure 2).

800 R. Leikin et al.

Let A¤ and B¤ be the intersection points of the segment P 0P 00 with lines m and nrespectively, then for every A on m and B on m such that A and B are diVerentfrom A¤ and B¤, P 0A¤ ‡ A¤B¤ ‡ B¤P 00 < P 0A ‡ AB ‡ BP 00. Thus, the triangleA¤B¤P has minimal perimeter.

In the following problem we use role symmetry in proof in addition to geometricsymmetry.

Problem 2. Given an acute angled triangle ABC. Find a triangle PQRinscribed in the triangle ABC having minimal perimeter (� gure 3).

Solution. Let P1 be a point in the side BC of the given triangle and points P 01

and P 001 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 minimalperimeter if Q1 and R1 are the intersection points of the segment P 0

1P 001 with sides

Applications of symmetry 801

Figure 1.

Figure 2.

AB and AC of the triangle (� gure 4). The perimeter of the triangle P1Q1R1 is equalto the length of the segment P 0

1P 001 .

For any point Pi in BC the triangle P 0i AP 00

i is an isosceles triangle (AP 0i ˆ

APi ˆ AP 00i as symmetrical segments). For any i, the angle P 0

i AP 00i ˆ 2BAC and

therefore constant (� gure 5). Of all the isosceles triangles with equal anglesopposite to the bases, the triangle having the shortest lateral side has the shortestbase. Hence, P 0

i P00i 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 P 0

i P00i

and its length is minimal when APi is minimal. Hence P¤Q¤R¤ is of the minimalperimeter if and only if P¤A is the altitude of the triangle (� gure 6).

The points P, Q and R play symmetrical roles in the problem. Thus all threepoints P¤, Q¤, and R¤ are heels of altitudes of the given triangle (� gure 7). Hence,


Figure 3.

Figure 4.

of all the triangles inscribed in the acute angled triangle ABC the triangle withvertices 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 rathercomplicated calculus manipulations.

Problem 3. Given the quadratic function y ˆ x2 ¡ 3 and a family of linearfunctions y ˆ ax ‡ 3, for which value of the parameter a is the area of the regionbounded by the parabola and by the line minimal (� gure 8)?

Solution. Consider the line y ˆ 3. The � gure bounded by the parabola and bythis 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 � gure (� gure 9).


Figure 5.

Figure 6.

Let the line y ˆ 3 intersect the parabola at points P and M. Let line ` intersectthe parabola at the points L and S and the y-axis at Q. Let PN and MK be parallelto the y axis where points N and K are on ` . Triangles QPN and QMK aresymmetrical 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 familyof functions y ˆ ax ‡ 3 is minimal when a ˆ 0.

In Problem 4 we use geometric rotational symmetry in the theory of complexnumbers.

Problem 4. Prove that the sum of all complex roots of the equation zn ˆ 1 isequal to 0 for all natural numbers n.


Figure 7.

Figure 8.

Solution. All the roots are points on the unit circle (� gure 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 iszero.

Solution of the system of equation in Problem 5 presents use of algebraicsymmetry 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 unknownsand 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


Figure 9.

Figure 10.

also solutions of this system. It is easy to check that the system has only onesolution, 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 symmetryand geometric symmetry.

Problem 6. Prove that the straight line y ˆ x intersects the ellipsex2 ‡ xy ‡ y2 ˆ 12 at right angles.

Solution. The equation of the ellipse is algebraically symmetrical. In otherwords, 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 tangentlines to the curve is also symmetrical with respect to the line y ˆ x. Thus, thetangent line to the curve at the point of intersection of the curve with the line y ˆ xis symmetrical with respect to this line, if it exists. An ellipse is convex and has atangent line at each of its points, therefore the tangent line to the curve at the pointof intersection of the curve with the line y ˆ x is perpendicular to this line.

3. The interventionThe 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 preferenceswith respect to the use of symmetry. The teachers’ responses to the problems theysolved were recorded and analysed. The problems were presented to the teachersin diVerent ways. For example, for Problems 1, 2, 4 and 5 the teachers were askedto suggest as many solutions as possible before any discussion of solutions tookplace. On the other hand, for Problem 6, two diVerent solutions were presented tothe teachers by the researcher, one of which was based on symmetry and the otherwas based on more conventional ways typically used in secondary school. Follow-ing the presentation of the two diVerent solutions, the teachers were asked tochoose one of these suggested ways when solving a similar problem. Succeedingtheir 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 theuse of symmetry in problem solving. Problem 3 was brought by a group of teacherswho knew how to solve the problem using calculus but, having gained appreciationof symmetry, asked us to help them to use symmetry in solving the problem.

4. DiscussionIn 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 usesymmetry in problem solving. The teachers ’ tendencies to use symmetry dependedon their familiarity with symmetry and on their beliefs about mathematics. As aresult of the intervention, teachers ’ inclinations to use symmetry in solvingmathematical problems were enhanced. For example, at the beginning of the


study teachers tended not to use role symmetry at all, later on they began to ‘feelsymmetry’ , and by the end of the intervention they used symmetry in solving mostof the problems.

As noted above, at the beginning of the study few teachers used symmetrywhen solving the problems. Moreover, when these teachers presented theirsolutions based on symmetry, they were not able to convince the other teachersthat the use of symmetry in their solution was suYcient.

For example, when Problem 1 was solved using geometric symmetry theteachers tended to try to ‘complete’ a solution by conventional tools such asderivative and congruency. One of the teachers claimed that ‘we need to use bothsymmetry and congruence of segments in order to prove that these segments areequal ’ .

Role symmetry in proofs was the most diYcult type of symmetry for teachersto accept. When the teachers were asked to explain why the vertices Q¤, and R¤ ofthe inscribed triangle are ‘heels ’ of the altitudes of the given triangle (see solutionto Problem 2 above) several teachers justi� ed it by using similarity and congru-ence. These teachers seemed to reject the use of role symmetry. Other teachersonly 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 sideBC …P¤† was obtained as a vertex of the inscribed triangle P¤Q¤R¤ having theminimal perimeter. Then, in the same way, the heels of altitudes to the sidesAC and AB of the given triangle (points Q¤¤ and R¤ 0) could be obtained asvertices of two other triangles: Q¤¤R¤¤P¤¤ and R¤ 0P¤ 0Q¤ 0. Thus, the peri-meters of these triangles must be compared and the smallest one has to bechosen.

Only one teacher could explain how ‘thinking symmetry’ helped in proving thestatement. He solved the problem as follows:

For any point P in BC which diVers from P¤ the perimeter of the inscribedtriangle with vertex P is greater than the perimeter of the obtained triangleP¤Q¤R¤. The same consideration holds with respect to any point Q whichdiVers from Q¤ and to any point R which diVers from R¤. Hence, these threepoints are ‘heels ’ of the altitudes of the given triangle.

As mentioned above, teachers ’ beliefs about the nature of mathematical proof aswell as their familiarity with symmetry in� uenced their preferences regarding useof symmetry in problem solving. For example, when the teachers were presentedwith two diVerent solutions to Problem 6 all of them agreed that the solution basedon symmetry was more elegant and beautiful. Nevertheless, most of the teachersthought that the solution using calculus was more ‘trustworthy ’ . They felt more‘safe ’ to teach their students to solve problems using calculus. In addition, whenthe teachers were asked to choose one of the presented solutions of Problem 6 inorder to solve a similar problem, only two teachers chose to use symmetry in theirsolutions but were not sure that the solution based on symmetry was ‘goodenough’ . In order to be on the ‘safe ’ side, these teachers also solved the problemin a standard way. During the discussion of the problem one of the teachers saidthat the presented solution is ‘too sophisticated and philosophical’ .


It is interesting to note that while the intervention enhanced the teachers ’

knowledge of symmetry it did not equally in� uence their beliefs regarding whatconstitutes a valid mathematical proof. Although, at the end of the study most ofthe teachers were able to use diVerent types of symmetry in problem solving, andeven solved problems using symmetry, still, many were not convinced that theproofs were ‘mathematical enough’ . For example one of the teachers, who solvedthe problem using role symmetry, presented this solution to the other teacher andthen concluded: ‘but this is not mathematics, this is . . . ’ . Only by the end of theintervention were most of the teachers able and willing to use symmetry inproblem solving. Teachers ’ perception of the solutions based on symmetry as‘more elegant’ and conveying the beauty of mathematics served as a springboardfor these changes. Almost 50% of the teachers who preferred not to use symmetryat 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 tosolve Problem 8 using symmetry. All the teachers agreed that this solution waseasier, more interesting, and more elegant.

The way in which Problem 3 was raised by the teachers themselves, is also anindication of changes in teachers ’ tendencies to use symmetry when solving theproblems. The teachers solved this problem with their twelfth-grade studentsusing 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 theteachers 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 ourinvestigation the teachers seemed to be convinced that:

‘Special cases of symmetric objects are often prime candidates for ex-amination’

([5], p. 81)

AcknowledgementsThe 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 theTechnion. The � rst draft of this paper was prepared while the � rst author was apost-doctoral fellow with Ed Silver at the Learning Research and DevelopmentCenter, University of Pittsburgh.

References[1] Brown, S. I., and Walter, M. I. (eds), 1993, Problem Posing: Re� ection and Application

(New Jersey: Lawrence Erlbaum Associates).[2] Polya, G., 1963, Amer. Math. Monthly, 70, 605–619.[3] Polya, G., 1973, How To Solve It. A New Aspect of Mathematical Method (Princeton,

NJ: Princeton University Press).[4] Schoenfeld, A. H., 1983, Problem Solving in the Mathematics Curriculum: A Report,

Recommendations and An Annotated Bibliography (Reston, VA: The MathematicalAssociation of America).

[5] Schoenfeld, A. H., 1985, Mathematical Problem Solving (Academic Press).


[6] Silver, E. A., 1985, in Teaching and Learning Mathematical Problem Solving: MultipleResearch Perspectives, edited by E. A. Silver (London: Lawrence ErlbaumAssociates) pp. 247–266.

[7] Polya, G., 1981, Mathematical Discovery (New York: John Wiley & Sons).[8] Allendoefer, C. B., 1969, Math. Teacher, 62, 165–169.[9] Eccles, F. M., 1972, Math. Teacher, 65, 165–169.

[10] Ellis-DAVIES, A., 1986, Math. in School, 15(3), 27–30.[11] Leikin, R., 1997, Symmetry as a Way of Thought—A Tool for Professional

Development of Mathematics Teachers. Unpublished Doctoral Dissertation(Technion. Israel).

[12] Leikin, R., Berman, A., and Zaslavsky, O., 1995, Culture & Science, 6, 332–335.[13] Ram, B., 1975, Math. Teacher, 68, 298–300.[14] Rosen, J., 1989, in Symmetry 2, Unifying Human Understanding, edited by I. Hargittai

(Oxford: Pergamon Press), pp. 13–15.[15] Rosen, J., 1995, Symmetry in Science. An Introduction to the General Theory (New

York: Springer-Verlag).[16] Sonin, A. S., 1987, Comprehension of Perfection (Moscow: Znanie) (in Russian).[17] Weyl, H., 1952, Symmetry (Princeton, NJ: Princeton University Press).[18] Witman, N., 1991, Math. Teacher, 84, 296–302.[19] Yaglom, I. M., 1962, Geometric Transformations. Vol. 1. Displacements and Symmetry

(New York: Random House).[20] Dreyfus, T., and Eisenberg, T., 1990, ZDM–Int. Rev. Math. Educ., 2, 53–59.[21] Arbel, B., 1991, Problem-solving Strategies (Tel-Aviv: Open University).


Applications of Symmetry to Problem Solving

Documents

Transcript of Applications of Symmetry to Problem Solving