Bertiscanova К.Т., Berdalieva А.А.

Karaganda State University named after E.A.Buketov,

Kazakhstan

About interrelation of teaching materials of algebra and geometry at training of concept of relation

Knowledge and abilities that pupils are getting during the study of the course of mathematics, should not be the simple mechanical sum of them. The solution of many mathematical problems is based on application of system of knowledge not only by one discipline. The quality of pupils´ studying depends on application of various components of scientific knowledge by algebra and geometry in their interrelation. Therefore, in the article is considered the question of interrelation of teaching materials of algebra and geometry while training the pupils to the school mathematics on example of concept of the relation.

The set-theoretical approach is taken as a principal in the school course of mathematics [1]. It allows us to enter some set-theoretical concepts during some question studying. One of these general concepts of mathematics is the concept of relation which is not defined and is given by means of the description. The relation can be considered both between elements of one set and between elements of two sets. Thus speaking about the relation between sets we deal with set of ordered pairs, pairs that are in the given relation. The relation between two sets can be defined with the set of points of the plane with co-ordinates x and y, i.e. graphically. In other words, some set of points of a co-ordinate plane is the graph of relation between abscissas and ordinates of points.

The special kind of the relation is the concept of mapping which can be considered as: maps of number sets on number sets (number functions); maps of number sets on dot sets (a method of co-ordinates); maps of sets of geometrical figures on number sets (measurement of geometrical values); maps of dot sets on dot sets (geometrical transformations) [2]. Such peculiarity of concept of mapping, extending on many questions of a school course of mathematics, gives the chance during some concepts studying to pass from algebraic language on geometrical and vice versa. It allows to establish interrelation in studying of algebraic and geometrical materials, to lead integration at studying of related questions of mathematics [3].

The concept of function of a school course of mathematics is entered on the basis of the general concept of the relation. Such concepts as domain, range of reflection, graph of reflection, etc. are entered into mathematics with concept of reflection.

It is known, that function is the relation special case, namely, any function is the relation, but not any relation can be a function. Really, there are graphs of some sets of points of a co-ordinate plane on two pictures (see pictures 1 and 2), in other words, the graphs of relations between abscissas and ordinates of these points.

Рic. 1 Рic. 2

In the first case (pic. 1) we have the graph of relation which is a function, in the second case (pic. 2) – there is some relation which is not the graph of a function. If some relation between elements of two sets is a function it means that to each element of one set there corresponds a unique element of another set. Here the first set is the domain of the given relation (in our case, of function), and the second is the range of this relation. Thus, the concept of function is treated as correspondence and the last concept is initial concept. Authors of modern school textbooks consider the concept of function as some correspondence because in such representation the concept of function becomes intuitively possible and accessible to pupils.

There is no a distinction between concepts of relation and correspondence in the school textbooks. The definition of function accepted by authors of modern school textbooks, gives the chance to consider only concept of relation because the general theory of relations and correspondences does not play a special role in the course of mathematics for pupils [2]. The relation between sets is defined by the law of correspondence which is established by means of formula or by description.

The concept of function, as some subset of the relation as its specific difference, is one of fundamental concepts of mathematics. For the first time the term "function" has been entered into a science at the end of 17 century and only in 1918 it began to be applied in a secondary school course.

The concept of function has big theoretical and practical value. In the school course of mathematics acquaintance with concept of function, with the first graph of function (linear) begins in 6 form and comes to the end in 11 form with studying of exponential, logarithmic and power functions. Therefore studying of a theme "Function" represents one of the important methodical lines of a school course of mathematics.

Studying the concept of function, pupils get acquainted with variables, which are more than constants. It is known, that there are not many constants in nature, and in mathematics. As an example of a constant in mathematics, it is possible to talk about two formulas (the relation of the circle´s length to its diameter and the sum of interior angles in a triangle) which do not depend on the chosen values. One type of variables changes arbitrary, and others - in certain dependence, last values represent a subject of special studying in mathematics.

In the school course of mathematics the graph of relation is not a subject of special studying. But this concept is the main base for the course of mathematics in the 6-8 th forms. So, the concept of relation and its graph can be considered more deeply in the high school to generalize and systematize the pupils’ knowledge by the theme "Function".

And now, let's talk about the graph of relation. For this purpose let's examine various variants of the same relation between the elements of sets. This is relation with an absolute value. Besides, we will stop at the link between the given algebraic problem and concept of symmetry from geometry course which is carried out by the set-theoretical approach.

Pupils get acquainted with concept of symmetry in 6 form, and its regular studying begins in 9 form in part «Transformation of figures».This concept plays the important role at studying geometry and has a wide application not only in geometry, but also in reality. Ideally beautiful forms of buildings and other constructions, patterns on fabrics, room wall-papers, leaves of plants, wings of butterfly and others often contains elements of symmetry, more precisely, are symmetric concerning some axis. Therefore it is necessary to pay an attention for studying the theme «Axial symmetry».There are not enough time in the school course of geometry to study the symmetry question. This question´s studying has fragmentary character. So pupils have incomplete, superficial idea about this theme. Meanwhile returning to idea of symmetry at lessons not only in a geometry course, but also in algebra would allow to develop pupil´s space imagination and to acquire properties of figures more strongly, for example, while constructing the graphs of functions with an absolute value. It is possible to connect closely the question about the relations´ graphs with axial symmetry. Let´s consider how the symmetry concerning the axis can be represented in analytical form, and vice versa. It becomes possible due to the property of relation´s symmetry. Pupils should have representations about properties of relations to consider this question. Such approach of teaching the properties that carry out the connection between materials of courses of algebra and geometry will allow to systematize pupils´ knowledge, to make them more realized and strong.

There are the main types of the graph of the function with an absolute value: ; ; . In the school course of mathematics one of the simplest types of graph of the given functions is . From these main types by various combinations of the module sign it is possible to receive the following types of relations:

1. . 13. .

2. . 14. .

3. . 15. .

4. . 16. .

5. . 17. .

6. . 18. .

7. . 19..

8. . 20. .

9. . 21. .

10.. 22. .

11. . 23. .

12. .

As it was said earlier, we will consider not, but some set of points of co-ordinate plane (relation), represented on pic. 3 and we will plot various graphs of the given relation.

Pic.3

While plotting the graphs we should use not only the concept of symmetry but also the concept of an absolute value of a number that is not resulted here.

We will consider the graphs of relations , , as elementary and won’t pay attention to their plotting. We will note only that graphs of relations and - are symmetric in y-axis and graphs of relations and - are symmetric in x-axis.

Let's consider more detail the plotting of some graphs of relations from the offered system of tasks.

1. The graph of relation , is symmetric in the y-axis. Therefore at first we plot the part of graph at non-negative values of х, and then we will take the symmetry in the y-axis (pic. 4). If to bend a sheet of paper on the y-axis the left and right parts of the relation´s graph will coincide.

Pic.4 Pic.5

3. For plotting the graph of relation , proceeding from the definition of the absolute value of a real number and symmetry of the graph in the y-axis, we will consider two graphs of relations: a) , at x ≥ 0; and , at x < 0. The graph of the given relation is presented in at the picture 5.

4. We obtain the graph of relation from the graph that is represented at the picture 5, by taking the symmetry with respect to the x-axis (pic. 6).

Pic.6 Pic.7

5. The graph of relation set by formula , will be situated in the top half plane, i.e. above the x-axis , including the axis itself because the expression is inside an absolute value. Therefore we keep the part of the graph above the x-axis, and symmetrically reflect in the x-axis the part of the graph which is below the x-axis (pic. 7).

6. The graphs of relations and (pic.7) are symmetric in the y-axis. So we can easily plot the graph of (pic.8).

Pic.8 Pic.9

7. We obtain the graph of relation from the graph of relation (pic.7) by symmetric reflection in the x-axis (pic.9).

9. The graph of relation is symmetric in y-axis, because the variable is inside of absolute value and is situated above the x-axis, because the whole expression is inside of absolute value. The given graph is turned out from the graph of relation (pic.4) by symmetric reflection in the x-axis. The graph received in this way is presented at picture 10.

Pic.10 Pic.11

11. As in the previous task (pic.10) the graph is symmetric in the y-axis and will be situated above the x-axis, including this axis. As the initial graph the graph of relation (pic. 5) is taken. The given graph is presented at picture 11.

13. The relation set by the formula has a property of symmetry in the x-axis. The given relation has the condition , so let´s plot its graph, keeping the top part that is above the x-axis without changing, and reflect it symmetrically in the x-axis (pic.12). If to bend a sheet of paper on the x-axis the top and bottom parts of the given relation’s graph will coincide.

Pic.12 Pic.13

15. The graph of relation can be easily obtained from the graph of relation (pic.12), if we take the symmetry in the y-axis. By another way, we plot the graph of relation , keep its top part and reflect symmetrically in the x-axis (pic.13).

17. To plot the graph of relation let´s use the graph of relation (pic.7), and then take the symmetry of it with respect to the x-axis (pic.14).

Pic.14 Pic.15

18. To obtain the graph of relation we take the symmetry of the graph of relation with respect to the y-axis. By another way, we can use the graph of the relation y (pic.8) and reflect it symmetrically in the x-axis (pic.15).

19. Let’s plot the graph of relation . Here we have the doubled symmetry: as in the x-axis and in y-axis as well. If to bend a sheet of paper on the co-ordinate axes the parts of the given relation’s graph will coincide (pic.16). The graph of the given relation can be also obtained from the graph of relations that are drawn on the pictures 4 and 12. In the first case we keep the top part of the graph , that is above the x-axis and symmetrically reflect it in the x-axis. In the second case we take the symmetry twice of the part of the relation’s graph that is situated in the first quadrant with respect to the both axes (pic.16).

Pic.16 Pic.17

20. To plot the graph of relation we can use the graphs of relations: (pic.5), we keep the part of the graph that is in the first quadrant and take the symmetry twice with respect to the co-ordinates axes; or we can use the graph (pic.13), here we keep without change the part of these graph at x ≥ 0 and reflect this part symmetrically in the y-axis. In the both cases we obtain the graph of relation represented at the picture 17.

22. In this task we deal with three absolute values and doubled symmetry: with respect to the both axes of co-ordinates. We can use the graph of relation that is at the picture 10, by reflecting it symmetrically in the x-axis (pic.18). Or we can use the graph of relation (pic.7) at x ≥ 0 and take the symmetry twice with respect to the axes of co-ordinates. Also we can use the graph of relation (pic.14) at x ≥ 0 and symmetrically reflect it in the y-axis.

Pic.18 Pic.19

23. To construct the graph of relation let´s use the graph of relation (pic.5). At first we take the symmetry of the bottom part of the graph with respect to the x-axis, keeping the part of the graph that is in the first quadrant we take the symmetry two times in the axes of co-ordinates (pic.19). Or we can obtain the given graph from the graph of relation (pic.11) by reflecting it symmetrically in the x-axis.

The rest numbers: 2, 8, 10, 12, 14, 16, 21 we suggest to do by yourselves.

Carrying out this system of tasks above, pupils study to analyze, abstract the particular objects, situations. Designing various images, they develop the space imagination; start to learn the mathematical objects in movement, and also the connections between algebra and geometry. Besides, the research skills are formulated because such tasks have no certain stereotype of solution.

Teachers can use the materials of this article, system of the developed tasks as additional lessons by mathematics, at planning and performance of scientific projects, and also at lessons of the elementary mathematics for students.