© FORMATEX 2006 The concept of similarity in triangles within the context of tools of Cabri-Geometry II A. Mastrogiannis *,1 and M. Kordaki 2 1 DEPT OF MATHEMATICS, Patras University, 26500, Rion Patras, Greece 2 DEPT OF COMPUTER ENGINEERING AND INFORMATICS, Patras University, 26500, Rion Patras, Greece This paper presents the role of tools provided by the well known educational software Cabri-Geometry II [1] or the construction of the concept of similarity in triangles by the students. An a priori analysis showed that a variety of solution strategies could be invented by the students to tackle this concept in the context of the tools of Cabri-Geometry II. In fact, at least seventeen solution strategies can be performed by the students. For the construction of these strategies a variety of geometrical concepts could be integrated with the concept of similarity in triangles such as: the specific circles of a triangle (circumscribed circle, inscribed circle, and escribed circle), regular polygons, different forms of triangles (isosceles triangles, right-angled triangles), specific linear elements of a triangle (altitude of a triangle, bisector of an angle; medians of a triangle, parallel segments joining the mid points of two sides of a triangle), parallel and perpendicular lines, angles, as well as specific elements of a circle such as tangents of a circle and chords intersected outside and inside of it. Keywords Similarity in triangles; Cabri-Geometry II; multiple representations; learning activities; multi- ple solution strategies 1. Introduction The concept of similarity is essential for the students to grasp as it consists part of both mathematics and of their every-day life. Students have difficulty in understanding the concept of similarity as they confuse it with equality. Similarity constitutes a relationship between shapes/figures. A shape F1 is similar to a shape F2 if there is a transformations such as s(F1)=F2. In fact, the concept of similarity has a dynamic character as it implies an understanding of the fact that the size of the angles of a rectilinear figure could be conserved although the length of its sides and its area are altered according some specific ratio [2]. In addition, the concept of similarity is not an isolated geometrical concept but it is also interconnected to a variety of geometrical concepts. In fact, similar shapes can be produced as results of specific geometric transformations as well as results of specific geometrical constructions. To help students acquire a dy- namic perspective regarding similarity as well as to integrate this concept in a wide context of geometri- cal concepts the role of Dynamic Geometry Systems (DGS) is crucial. Cabri Geometry II [1] is a widely known DGS that provides students with potential opportunities in terms of: a) Means of construction, providing a rich set of tools to perform a variety of geometrical con- structions referring to a variety of concepts concerning Euclidean Geometry. b) Tools to construct a variety of representations, both numerical and visual, such as geometrical figures, tables, equations, graphs and calculations. c) Linking representations, by exploiting the interconnection of the different representation modes provided. d) Dynamic, direct manipulation of geometrical constructions by using the ‘drag mode’ operation. In fact, Cabri-Geometry II provides possibilities for dynamic transformations of the geometrical constructions presented on the screen of the computer by using the ‘drag mode’ opera- tion. These types of geometrical constructions retain their geometrical properties through dragging. By experimenting with this type of dynamic transformations, students can form hypotheses, generalizations and abstractions regarding the learning of the concepts in focus [3]. e)The possibility of collecting large amounts of numerical data. f) Interactivity and feedback; intrinsic visual feedback and extrinsic numeri- * Corresponding author: e-mail: kordaki@cti.gr, Phone: +30-2610- 997-751 Current Developments in Technology-Assisted Education (2006) 641