Abstract:

Apollonius's problems have always occupied a particularly important place among planimetric problems not only because its solution requires knowledge of various mathematical relationships, but also because during its solution many new knowledge have been discovered.

After fifteen years of teaching mathematics at university in the preparation of prospective teachers for secondary education, we always encounter the situation that the students have not yet encountered Apollonius' problem. However, some specific easier Apollonius’s problems are already solved by pupils of lower secondary education, e.g. using Fixed Point Sets, without knowing that this is a type of Apollonius’s problem. Other complex ones require deeper knowledge of geometry, such as homothety, power of point theorem, circular inversion, dilatation or analytical solution based on other known geometric knowledge. In upper secondary education, these are mainly problems using homothety.

In general, Apollonius's problems are not standard problems in mathematics teaching in secondary education. Within the project "Non-standard problems in mathematics education", we want to point out the possibilities of integrating non-standard problems into mathematics education, which are more difficult to solve, but with the help of ICT can be solved by pupils of secondary education.

The use of ICT brings new possibilities for pupils to solve problems. Since lower secondary education, pupils have been using digital technology (PC, interactive whiteboard, tablets) and related software in the educational process. On the one hand, it is a trend and an inherent nature to use didactic aids that are currently available and have an impact on increasing the effectiveness of teaching. On the other hand, there is the effort of the state through the respective national institutions, strategic plans and curricular documents, which form the basis for education, to develop pupils' digital literacy.

To solve geometric problems are currently mainly used dynamic geometry tools - GeoGebra. This tool has the potential to enable pupils of lower secondary education to solve geometric problems more quickly, more precisely, more clearly and also solve more complex problems. In upper secondary education, these technologies and GeoGebra will enable pupils to solve more complex Apollonius's problems and these tools will also allow them to use methods of solution that would be unrealistic without ICT such as e.g. for solving Apollonius's problems use such Fixed Point Sets like conic sections, with which pupils of upper secondary education are acquainted in teaching.

Finally yet importantly, it is possible to use these specific problems in teaching, including historical notes in the context of interdisciplinary relations, to acquaint pupils with the history of assigned problems, with historical publications describing these tasks for the first time and with the life and work of famous mathematicians who dealt with these problems.

Keywords:

Apollonius’s problem, secondary education, ICT, non-standard problems.