# THE CONSTRUCTION OF EDUCATIONAL TASKS ON MATHEMATICS: MATHEMATICAL MODELS, ALGORITHMS, PROGRAMS

### Abstract

The problem of forming future and current teachers’ integrative knowledge as knowledge of a higher level in comparison with the knowledge of certain subjects (mathematics and informatics) is reduced to the solution of specific educational situations that require the simultaneous application of knowledge and skills from various subjects while solving them. These problems include the problem of computer-aided constructing of certain educational tasks with pre-defined properties. At the heart of solving this problem a creation of mathematical model of the required mathematical object, its study and solution are laid.

Mathematical model of a certain educational task contains a set of parameters, and selection of the values of these parameters determines the properties of the educational task. In turn, the properties of the educational task in the process of its formalizing turn into certain conditions, such as equations or inequalities. The formalization of a set of determined properties of the set of mathematical tasks leads to a system of equations and inequalities. So, our problem is reduced to the construction of a mathematical model as a system of equations and inequalities.

The first version of the mathematical model must be investigated for consistency, completeness, minimality of conditions. After adjustment (change, withdrawal or addition of certain conditions) the mathematical model is subjected to solving, namely finding the right settings. This process is called solving the mathematical model, a mean of solution created or found by the author of a mathematical model.

Mathematical models of educational tasks constructing in mathematics are created in such a way that the solutions of the mathematical models will be figures chosen beforehand, for example, integers from a certain period. Different vectors- solutions of mathematical model define specific examples of a particular type of the examples. In this article a continuous fractional rational function is constructed with exactly two extremes. Thus, certain scientific approaches and ways of solving the mathematical model are considered. In fact a method to find an acceptable solution of the mathematical model as a system of equations and inequalities is described.

Based on the method of solving the model, algorithms and programs written in Maple are established to automate the process of solving the model. Herewith the model solutions are generated beforehand by user’s choice. Different vectors- solutions of mathematical models define various mathematical problems of the same type.

This article describes a general approach developed by the author for creation and solving of a mathematical model constructing a specific function. However, given author's technology of constructing works equally well in constructing polynomials with a number of extremes, different types of irrational, logarithmic equations and inequalities, systems of linear equations, matrices with predetermined eigenvalues, fractional rational equations and inequalities, etc.

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