The introduction of new standards of mathematical education requires to stop understanding of the learning process as the transfer of ready-made knowledge and experience. Educational activity built on the principle of self-construction of knowledge by schoolchildren is highly demanded in new environment. Tasks with parameters have high learning, development, research and diagnostic potential. It allows to identify and in the process of teaching mathematics to prepare students who possess subject knowledge at the highest level, corresponding to the trends of the time. The urgency of the problem under study is determined by the need for students to form the ability to solve problems with parameters in order to achieve high results in mathematical, intellectual and personal development. The aim of the research is to develop a methodology for teaching students how to solve problems with parameters as an effective means of high-quality mathematical studies. The authors have identified main methods for solving problems with parameters and approaches to their study, and proved the theoretical basis for the application of these methods in the learning process. Therefore, they have shown the role of the propaedeutic stage of teaching graphic methods for solving problems with parameters, its goals, objectives and content. The authors suggest a methodology for designing a system of tasks that contributes to achieving high learning outcomes, which has passed multi-stage approbation. Moreover, they prove the need to use the Live Mathematics software as an effective visualization tool for studying graphic methods for solving problems with parameters. The methodology described in the article can be used by teachers at school and extracurricular mathematics classes, by the authors of textbooks for students and teachers, and it can also be the basis for a special course for students of pedagogical universities.
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