Keyword: «methods of teaching mathematics in high school»

The relevance of this work is determined by the need to implement the principle of continuity in education, not only in the transition school-University, but also in the study of individual mathematical disciplines. This ensures the establishment of strong inter-disciplines relations and is one of the most important components in the formation of the future University graduate competences, as it contributes to the wide application of knowledge in various fields. This issue is examined on the example of studying the questions of second-order curves orthogonality within the discipline "Ordinary differential equations and partial differential equations". Students first broach the topic “Second-order curves” starting from grade 7, and they are constantly referring to it throughout the school mathematics course. Then students expand their knowledge about the properties of these curves and study their canonical equations within the University program. Meanwhile, students comprehend the full picture only after they study differential geometry and apply mathematical analysis to solve some tasks related to finding tangents and normals to the line on the plane, as well as finding the distance from the point to the curve. The purpose of this article is to reveal the potentials of the discipline "Ordinary differential equations and partial differential equations" for generalization of students ' knowledge on the topic "Second-order curves" on the example of the concept "orthogonality of plane curves". The main argument is that it is possible to combine both geometrical and functional approaches within this discipline, and also to use tools of the mathematical analysis. The article provides recommendations in the framework of methods of this discipline teaching to achieve these purposes. Namely, a series of tasks on the topic "Orthogonal trajectories" is given, which allows us to return to the second-order curves once again, to systematize and expand the knowledge previously available to students. At the same time, the solution of these tasks makes it possible to work out the necessary skills within the discipline itself on the available material, which will easily illustrate the results obtained and show their consistency with the previously known facts from the school and University course of mathematics. One of the main approaches to the study of this problem is an integrative approach, which helps to ensure the integrity and continuity of the educational program content during the transition from school mathematics to higher mathematics. It also helps to establish strong interdisciplinary connections between different sections of higher mathematics. The article studies the potentials of the discipline "Ordinary differential equations and partial differential equations" to establish interdisciplinary connections within mathematical disciplines, such as geometry, analytical geometry, differential geometry and mathematical analysis on the example of the topic "Second-order curves". The practical significance of the article lies in the fact that it provides an indicative list of practical tasks with solutions that can be used in the process of teaching this discipline to achieve the above mentioned purposes.