Математика/4.Прикладная математика

Ismagul R.

Kostanay State University named after A.Baitursynov,  Kazakhstan.

Decision of non-standard tasks in the process of educating

Problem solving in mathematics is of great educational and educational value. Search nonstandard tasks develops initiative, perseverance and ingenuity. The main problem solving Olympiad level - logical basis, so the decision should be logically complete reasoning that does not contain errors.

The task is considered non-standard, if its solution requires a non-standard methods of solution. These kinds of problems are mainly pre d seems to solve a mathematical Olympiads, which are often conducted among students and among students. The main objective Olympiad movement - promotion of mathematical knowledge among young people, preparing for future research activities, sharpening the intellect. Students addicted to solving mathematical problems, they must be interested in, and do it quite easily. But if the problem drew a-negative, the interest in it, regardless of the wishes of the student, mobilizes its UMS tons vennuyu energy facilitates memorization.

Problem solving is one of the means of mastering the system of scientific knowledge on a particular academic subject. Particularly important the solution of problems in mastering the system of concepts. Invaluable role in the process of problem solving skills and mastery of cognitive skills and practical character.Development of creative thinking, self-reliance, preparing students for creative work is only possible if a systematic problem solving.

Are important tasks as a diagnostic tool of general mental development of special abilities of students. In the process of solving problems, students master the methods of study of various phenomena of nature, which are based on fundamental laws. Through the knowledge of the students research methods in the process of assimilation of methods and ways of solving problems in specific disciplines.

Solving problems is also a means of control over the knowledge, skills and abilities of students. Solving problems is one of the important conditions for the prevention of formalism in the students' knowledge and develop their ability to apply knowledge in practice.

Vector concept is one of the fundamental concepts of modern mathematics. From on power vectors can be solved meaningful geometrical problems and their solutions vector often much easier and more efficient solutions by means of elementary geo meters. In solving problems find application information known from school geometry course: effect of addition of vectors and its laws, subtraction of vectors, the effect of multiplying the vector by a number and its laws, the concept of collinear of vectors, vector decomposition in this basis, the only of decomposition. Through these actions and their properties, we can but to solve the problem on parallel lines, belonging three points one straight line, the calculation of the segments of parallel lines and some others. However, the problem for the calculation of the races distances and angles with these actions can not be resolved. To solve the problems associated with the lengths and angles(called metric) applies scalar multiplication of vectors and its properties. Ability to use vector tional methods requires certain skills. First, you need a good knowledge of theory. We must learn to translate the relation between g eometricheskie figures on a vector language, as well as, on the contrary, receivedvector ratio interpret geometrically. It is useful to remember some common for solving vector correlation and arr atit attention to their greater generality.

Basic formulas and ratios used in solving problems.

1) Rule of vector addition: : .

2) Rule subtracting vectors: , where O - arbitrary point.

3) conditions for membership three points A, B and C one line:

 a); ; b), where k - Some number

4) Condition parallel lines AB and CD: .

5) The formula dividing the segment in this respect:

If , so, where O - arbitrary point.

6) Formula midpoint: if C - the midpoint of AB, then.

In solving problems often is used the following property vectors:

7) The uniqueness of the expansion of the vector in two non-collinear vectors: if the vectors and are collinear, then the equality it follows that х = х₁ у= у₁

8) If M and N - midpoints of the sides AB and CD quadrangular  ka ABCD, then  .

9) Quadrilateral ABCD is a parallelogram if and only if one of the relations:

а) ; b) ; c)

10) If ABCD - a trapezoid with base AB, AB and BC side which while continuing to intersect at a point P, and Diago  Nali - then at the point O, is, and, , where    [1].

Here is an example of using vector method for the solution of the problem that can be solved by other methods.

References:

1. Cherkasov, A.Yakushev. Mathematics Intensive preparation for the exam (Basic methods for solving problems) Iris - Press, 2003.