**On Symmetry in School Mathematics**

**Tommy Dreyfus ^{1)}**

**Center for Technological Education, ^{1)} **

tommy.dreyfus@weizmann.ac.il

eisen@black.bgu.ac.il

Everywhere we turn we can see symmetrical relationships. They are both
visual and audio, and they are so pervasive in our daily lives that one
is led naturally to wonder if the notion of symmetry is innate in human
beings. It is almost as though the notion of symmetry is built into us
as a standard against which we measure aesthetic appeal to assess both
mental and physical constructs. Hargittai and Hargittai
in their text Algebra, geometry, trigonometry and calculus are four main domains
of school and collegiate mathematics. And in each of these domains, students are introduced
to generalizations on the notions of symmetry; e.g., in algebra they are introduced to
symmetric functions, symmetric determinants, symmetric
groups, symmetric systems of equations and to symmetric forms, as in the symmetric
form of the equatation of a line; in geometry
they meet the notions of point and line symmetry, and
It is well known that there seems to be a small set of real numbers
which appeal to our psyche more than other numbers. E.g., more than a hundred
years ago the psychologist Gustav Fechner made literally thousands of measurements
of rectangles commonly seen in everyday life; playing cards, window frames,
writing papers, book covers, etc., and he noticed that the ratio of the
length to the width seemed to approach the golden ratio
The golden ratio is a number closely tied to symmetry. The ratio is
obtained as follows. Given the line segment
The Golden Ratio
Constructing the points The aesthetic appeal of the golden ratio and its ties to the Fibonacci sequence, as well as its far reaching connections to nature and science are well documented in the literature (Huntley, 1970; Herz-Fischler, 1998; Dunlap, 1998). But the connection of this number to the human psyche, in the spirit of Fechner's work with investigating our attractions toward it, is an open question. Nevertheless, there are many aspects of symmetry which are embedded in the golden ratio and which are instructive for students to study. (Space limitation allow us to only mention a few.) The two points
Perhaps the ratio is not connected to our subconsciousness, but it and its reciprocal, ((Ö5 - 1)/2), are somehow connected to our aesthetic attraction towards simplicity and minimalism. Hidden symmetries in the golden ratio abound. For example, We now have the simplest continued fraction and root expansion known
to man; each of which can be continued on
Whether or not we have a subconscious gravitation towards symmetry,
and special numbers such as the golden ratio, is admittedly a hazy area
and perhaps best left to psychologists to investigate. But like learning
to appreciate art and music, where one learns what to look for with respect
to a painting, and what to listen for with respect to a piece of music
of a particular period, one must be taught how to look for symmetrical
relationships; gravitation towards symmetry might happen naturally, but
At the most basic level it helps to look at texts like the one written by the Hargittais, where the ubiquity of point and line symmetries are vividly pointed out and awake our sensitivity to geometrical symmetry. But the notion of symmetry enters many domains of school mathematics other than geometry. One of these domains is problem solving, where symmetry must be seen or imposed on a problem to effect its solution. Another domain is in concept formation, where it is often advantageous to think of basic mathematical notions in terms of symmetrical properties which surround them. Figure 3 lists three problems whose solutions depend on symmetry. Do you see how to solve them and to generalize the problem? It has been our experience that most students cannot solve these problems, because they do not use symmetry as a heuristic tool. (Partial answers are presented at the end of this paper.) (c)
But symmetry need not be limited to geometry and algebra. Let us consider the notion
of Magic Squares.
There are many problems associated with magic squares. E.g.,
Students seem to love problems with magic squares, but constructing
the square itself, even a normal magic square, is not easy. There are two
main classes of normal magic squares of order The following algorithm can be extended to all magic squares of odd order; we exemplify it in Figure 4 for the 3×3 case. Start with 1 in the top middle cell. Now move along the diagonal, always moving upward and to the right. If in so moving, we find ourselves outside of the square's frame at the top, we continue filling in the cells of the square at the bottom of that column (Fig 4a). If we find ourselves outside the frame on the right, we move to the first cell on the left of that row (Fig.4b). If a cell is blocked within the frame itself because we have already filled it in, we drop to the cell immediately under the one from which we came, fill it in (Fig.4c) and continue filling in the square moving upward and to the right.
There are several ways to see that the square is really magic. One way is to actually check the sums in each row, column and diagonal. Another is a little easier; e.g., to check to see if a 5×5 square is indeed magic, write each number in each cell in base 5. Now from the number in each cell subtract one from its base 5 representation. What remains is all combinations of two digit numbers in base five. It can be seen that each row and column and one of the diagonals are some permutation of the same digits. In other words, one need not do the actual computations, but simply check to make sure that all digits are there. Students love to play with magic squares, but they are often unable to construct ones which are not normal. Here is where symmetry can enter because all magic squares of order 3 are of a particular form as shown in Figure 5.
Students are often amazed at how simple this is for them
and they often try to generalize the method of building symmetry
along the diagonals of higher ordered squares. This exercise opens up many
doors for discussion, building in symmetry to effect a solution and degrees
of freedom are just two avenues for deeper work.
Its point of inflection is at
b/3a, f(-b/3a)),
So, using its graph to guide us we wish to show that:
f((-b/3a)-x)-f(-b/3a)
= f(-b/3a)-f((-b/3a)+x)
The algebra is not easy but straightforward. Another way to do the problem is to play on its symmetry from the start.
First we look at the graphs of We then return to the given equation and ask if there is some way to
transform it into one that will result in a cubic of
the form
In this paper we have tried to show that symmetry is a concept which can be exploited and used as a red thread connecting different branches and skills in school mathematics. But the main message we have tried to argue is that symmetry must be taught. It is too useful and important of a topic to let it develop casually, if at all, as one passes through the school curriculum.
We would like to thank Slavik Jablan for the
wonderful animation of the static diagrams we submitted to him, and for
the suggestions he made for improving the original draft of this
paper as well as for the encouragement he gave us along the way.
a) Let b) This is a symmetric polynomial. As such, we should focus
on its middle term, namely
= tx+1/x and noting that
t^{2}-2 = x^{2}+1/x^{2} transforms
the equation into a quadratic
which gives a surprising solution.
c) Systematically consider special cases.
A conjecture for the general case will soon be apparent. An elegant
way to prove the conjecture is to use symmetry, by repeatedly
reflecting a table of given dimensions about its edges. The path of the ball
becomes a straight line.
: 15.01.2000
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