The goal of this master thesis is to study the additional mathematical topics for secondary education that interweave the curriculum prescribed content. A thesis is divided into six chapters. In the first chapter we will investigate divisibility and congruence properties, as well as their applications on to more complex tasks. In the second chapter we will get acquainted with irrational equations and inequalities with a real parameter, that, in practice, proved to be the most demanding of equations and inequalities within elementary algebra. In the third chapter we will state axioms of constructive geometry and elementary constructions that will be applied when solving more complex constructive problems. In the fourth chapter we will study synthetic-constructive approach to the second-order curves as opposed to standard analytical approach. In the fifth chapter we will describe the methods and general principles of counting used to solve chosen combinatorial problems. In the sixth chapter will be singled out interesting and current probabilistic problems with geometric interpretation defined on \(\mathbb{ℝ}\) and \(\mathbb{ℝ}_2\).