EQUI-AFFINE TRANSFORMATIONS IN MATHEMATICAL EDUCATION OF SCHOOLCHILDREN AND FUTURE MATHEMATICS TEACHERS - Pratsiovytyi, Mykola; Pravitska, Natalia; Ratushniak, Sofiia (Physical and Mathematical Education, 2025)
The work is devoted to a specific class of affine transformations of the plane (bijections of the plane onto itself that preserve the collinearity of points), namely transformations whose primary invariant is the areas of quadrilateral figures. These are referred to as equi-affine transformations and are considered metric transformations. They hold significance both in mathematics and its practical applications.Formulation of the problem. Affine transformations, particularly equi-affine transformations of the plane, are not studied by students in the school geometry curriculum. However, they are included in the university course in Analytical Geometry for future mathematics teachers. Equi-affine transformations represent a distinct segment within the broader topic of "Affine Transformations of the Plane" (they form a subgroup of the group of affine transformations under the operation of "a composition of transformations"). Notable examples of this class of transformations are hyperbolic and elliptical rotations. Theoretical Exposition. Equi-affine transformations can be presented as a fully autonomous subject. Yet, it is difficult to find a filtered exposition of the "Equi-affine Transformations of the Plane" in instructional and methodological literature (in fact, it simply does not exist). This challenge served as the primary motivation for working on this research. A motivated mathematics teacher can use the proposed material as an introduction to the theory of equi-affine transformations of the plane.Materials and methods. Theoretical methods of scientific and pedagogical search were used. The theoretical analysis of the sources of educational literature shows the practical absence of information concerning the equiaxial transformations of the plane, and they form an important subgroup of the group of all affine transformations and are important for applications. Moreover, students' acquaintance with affine transformations should begin with equiafine transformations.Results. This article provides a basic presentation of theoretical educational material on the topic of "Equi-affine Transformations of the Plane." It is supplemented with commentary, application examples, solved problems, and tasks for independent problem-solving. In particular, the work derives formulas for calculating the area of a triangle constructed on two vectors as sides, as well as the area of a triangle determined by the coordinates of its vertices in a rectangular Cartesian coordinate system. These are auxiliary results used to substantiate the criterion of equi-affinity for a transformation. Special attention is given to two "progenitor equi-affine transformations: the hyperbolic and elliptical rotations of the plane. Additionally, the work proves a criterion for isometry within the family of affine transformations of the plane.Conclusions. The presented educational material can be utilized by mathematics teachers in extracurricular activities at schools or by instructors of analytical geometry for future mathematics teachers. The article discusses the feasibility and significance of studying the topic, the alternative approaches to proving certain facts, and addresses issues related to the methodology of teaching the topic, particularly the balance of problem sets.
Keywords: Geometric transformation of the plane; affine transformation; equi-affine transformation; a criterion of equi-affinity; hyperbolic rotation; elliptical rotation; school geometry curriculum; extracurricular activities in schools; future mathematics teacher