Abstract:

The study of axial symmetry in a Dynamic Geometry System in combination with suitably formed open learning activities could create favorable conditions, on the one hand for its understanding and on the other hand for overcoming students’ misconceptions, regarding the symmetricalness (bilateral symmetry) of a shape and the finding, subsequently, the number of its axes of symmetry.
This work, that no similar one has been presented so far, puts a double target and tries, mainly via Cabri Geometry, to acquire learning benefits during the study of symmetry in regular polygons and the circle. Moreover, this paper, through this axial symmetry, attempts to investigate the types of infinity.
Cantor proved the existence of different types of infinity, namely that there are infinite sets of different sizes. The infinity of the natural, integers or prime numbers on the one side and the infinity of the points of a line or the infinity of the real and irrational numbers, on the other. The infinity of the set of integers is the smallest infinity. This set can be placed in one-to-one correspondence with the set of natural numbers. Each such set is called denumerable (countably infinite) and its size is the first infinite cardinal.
All the infinite sets do not have the same cardinality. The sets of real or irrational numbers are “more numerous» than the set of natural numbers and have a cardinal called continuum (C). Even the number of points of any segment is C! These sets as above are called uncountable or nondenumerable infinite sets.
The first references about the particularity and the “problematic nature” of infinity are presented in the age of ancient Greeks. Zeno's Paradoxes are well-known, while Aristotle and the Pythagoreans had important contributions.
A lot of researches have been carried out regarding the comprehension of infinity. Most of them showed that students’ perceptions are erroneous and contradictory and more research is required on teaching and learning infinity.
The goal of this instructive intervention is the recognition of regular polygons and circles as symmetric shapes and flowingly the finding of the number of their axes of symmetry. As parallel objectives are determined, firstly, the “discovery” that the number of sides of a regular polygon is equal to its axes of symmetry. Secondly, as far as the circle is concerned, the number of its axes (that are its diameters) is uncountable.
The proposed interactive activities allow students to experiment with the concept of bilateral symmetry. Cabri automatically constructs polygons with up to 30 sides. Students making “virtual folding” using a lot of potential axes, that go through the centre of the polygon each time, can observe the matching or not of 2 of its parties.
Students via the dynamic movement of the line of fold may understand that this line is rendered axis of symmetry in the case of linking two opposite vertices or the midpoints of two opposite vertices of a polygon with an even number of sides, as well as when connecting a vertex with the midpoint of the opposite side, in the case of polygons with an odd number of sides. Similar method is followed, during the study of a circle.
Any regular polygon is inscribed in a circle. By increasing the sides, the polygon becomes circle in the limit that has uncountable number of axes.
Thus the countable axes become uncountable ones in the limit, result that constitutes the last learning target of this work.

Keywords:

Axial symmetry, types of infinity, regular polygons, circle, Cabri Geometry.