The concept of geometric transformation is instrumental in Geometry, since the properties of geometrical shapes which remain invariant, under the transformations, classify Geometry into several forms such as Euclidean, Affine, Projective etc. Generally, a form of Geometry is characterized by the group of its transformations, under which its theorems remain true.
Euclidean Geometry studies the properties of shapes that remain constant after shifts. Euclidean transformations are the most common ones, where the form of objects is not altered, since lengths and measures of angles remain the same. Only the place and the orientation of objects change. Euclidean transformations are considered the following: a) translation, b) axial symmetry and c) rotation.
The above transformations are isometries, namely transformations where distances of points are preserved.
The natural ways of shifting an object through these three Euclidean transformations can be easily defined in simple mathematic terms. As a result, this fact encourages students to study geometric transformations.
The Piagetian theory declares that students of medium school age are able to execute Euclidean transformations, by means of them, their cognitive processes can be revealed. Also, according to this theory, students can comprehend the concept of conservation of length.
Nevertheless, data research does not confirm the above considerations. Students 9 -13 years old are getting confused and, generally, fail in executing simple transformations, as well as in composing them. Students have difficulties in understanding the conservation of distances as well.
This work suggests that Euclidean transformations can be considered as cognitive vehicles, so that they may become helpers in assessing the area of basic shapes of plane geometry and proving simple identities and, thus, this study has two goals. Through the proposed activities, students of the upper classes of elementary and the lower ones of middle school can study ways of finding types of areas, using transformations, for all the basic shapes (parallelograms, triangles, trapeziums, circles, polygons), but they can also comprehend Euclidean transformations. As ally in this attempt, there is the well known educational software Cabri-Geometry II, which contributes decisively.
This software, provides a lot of technical and training advantages, such as high interactivity, continuous optical feedback, dynamic handling of objects. It is an open training environment, with a plethora of tools, giving opportunities for students to learn many geometrical subjects and concepts. The entire menu is dedicated to the implementation of transformations.
Activities of measurement and assessment of area are considered as main mathematic material and due to significant learning obstacles there must be paid particular attention in their study. As a compensatory factor, resorting often to splitting parts of an object and recomposing them, offers instructive benefits.
For example, via this work, students can discover the type of area of a trapezium ABCD by cutting and rotating the triangle DGE (GE=EB) around point E, by 180o.
Afterwards, they can recompose it, forming a triangle with the same height and a base which is equal to the sum of the trapezium’s bases.
Also, students using transformations and measuring areas through splitting parts and recomposing them can prove basic identities such as (a^2-b^2) = (a+b)(a-b).