ABOUT TEACHING METHODS FOR SOLVING TEXT TASKS IN THE COURSE OF ALGEBRA

The article is devoted to the solution of text tasks in the course of algebra. Recently, it has been observed that the solution of text tasks causes difficulties not only for schoolchildren, but also there are cases when the solution of such problems causes problems for practicing mathematics teachers. Therefore, in the article, we have considered several types of tasks for which solutions have been proposed, which we consider useful material for preparing existing teachers for algebra lessons when studying the topics of solving text tasks using equations and their systems. Also, reading this article is recommended for students-future teachers of mathematics before or during their teaching practice. The article does not contain solutions to equations and their systems, since the purpose of this article is to teach how to make equations and systems of equations for solving text tasks, and not to solve them. For the design of the task conditions, both drawings, tables, and just writing short conditions are used.

APPLICATION OF INVARIANCE PROPERTIES IN PROBLEMS DEPENDING ON THE PARAMETER

In this paper, we comprehensively consider the problems that depend on the parameter. The necessary and sufficient conditions for the existence of a unique solution of equations and systems of equations that depend on the parameter are investigated. Finding solutions to equations and systems of equations that depend on parameters is one of the most difficult areas of elementary mathematics. To solve them, you need to find ways that require special logical thinking, which must satisfy some additional conditions, for example, to determine a single solution, a solution in a set of all possible values of unknowns, cases when all the solutions of one system are solutions of another, and so on. One of these cases is the use of invariant properties. The article uses various examples of using invariant properties of transformations to consider methods and techniques for solving equations and systems of equations that depend on a single parameter. As a result, due to the presence of invariant properties of transformations, we see that the desired solution is found quickly and easily.

Embeddings of the 1-3-1 lattice

This chapter studies embeddings of the 1–3–1 lattice, proving Theorem 1.6. One of the central and longstanding areas of classical computability theory concerns the structure of the degrees of unsolvability, and particularly the computably enumerable degrees. In the same way that studying symmetries in nature and solutions to equations leads to group theory, studies of the computational content of mathematics lead naturally to the structure of sets of integers under reducibilities. Understanding these structures should lead to insights into relative computability. Notable in these studies is the question of embeddability into the c.e. degrees. There are nondistributive lattices that can be embedded. Ultimately, the embedding of the 1–3–1 lattice was an amazing result, and introduced the “continuous tracing” technique into computability theory.