Geometry is an important part of mathematics, which they begin to study in schools from grade 7 as a separate subject. What is geometry? What is she studying? What useful conclusions can be drawn from it? All these issues are discussed in detail in the article.
Concept of geometry
This science is understood as a branch of mathematics that studies the properties of different figures on the plane and in space. The word "geometry" from the ancient Greek language means "measurement of the earth", that is, any real or imaginary objects that have a finite length along at least one of the three coordinate axes (our space is three-dimensional), are studied by the science under consideration. We can say that geometry is the mathematics of space and plane.
In the course of its development, geometry acquired a set of concepts with which it operates in order to solve various problems. Such concepts include a point, a line, a plane, a surface, a segment, a circle, a curve, an angle, and others. The basis of this science are axioms, that is, concepts that connect geometric concepts within the framework of statements that are accepted as true. Based on the axioms, theorems are constructed and proved.
When did this science appear
What is geometry in terms of history? It should be said here that it is a very ancient teaching. So, it was used by the ancient Babylonians in determining the perimeters and areas of simple shapes (rectangles, trapezoids, etc.). It was developed in ancient Egypt. It is enough to recall the famous pyramids, the construction of which would be impossible without knowledge of the properties of three-dimensional figures, as well as without the ability to navigate the terrain. Note that the famous number "pi" (its approximate value), without which it is impossible to determine the parameters of the circle, was known to the Egyptian priests.
Scattered knowledge about the properties of flat and three-dimensional bodies was collected into a single science only in the time of Ancient Greece thanks to the activities of its philosophers. The most important work on which modern geometric teachings are based is the "Elements" of Euclid, which he composed about 300 BC. For about 2000 years, this treatise was the basis for every scientist who studied the spatial properties of bodies.
In the 18th century, the French mathematician and philosopher Rene Descartes laid the foundations of the so-called analytical science of geometry, which described using numerical functions any spatial element (line, plane, and so on). Since that time, many branches in geometry begin to appear, the reason for the existence of which is the fifth postulate in the "Elements" of Euclid.
Euclidean geometry
What is Euclidean geometry? This is a fairly coherent doctrine of the spatial properties of ideal objects (points, lines, planes, etc.), which is based on 5 postulates or axioms set forth in the work entitled "Elements". Axioms are given below:
- If two points are given, then you can draw only one straight line that connects them.
- Each segment can be continued indefinitely from any of its ends.
- Any point in space allows you to draw a circle of arbitrary radius so that the point itself is in the center.
- All right angles are similar or congruent.
- Through any point that does not belong to a given line, you can draw only one line parallel to it.
Euclidean geometry forms the basis of any modern school course in this science. Moreover, it is precisely humanity that uses it in the course of its life in the design of buildings and structures and in compiling topographic maps. It is important to note here that the set of postulates in the Elements is not complete. It was expanded by the German mathematician David Hilbert at the beginning of the 20th century.
Types of Euclidean geometry
We figured out what geometry is. Consider what its types are. In the framework of classical teaching, it is customary to distinguish two types of this mathematical science:
- Planimetry. She studies the property of flat objects. For example, calculating the area of a triangle or finding its unknown angles, determining the perimeter of a trapezoid or the circumference are tasks of planimetry.
- Stereometry. The objects of study of this branch of geometry are spatial figures (all points that form them lie in different planes, and not in one). So, determining the volume of a pyramid or cylinder, studying the symmetry properties of a cube and a cone are examples of stereometry problems.
Non-euclidean geometries
What is geometry in its broadest sense? In addition to the usual science of the spatial properties of bodies, there are also non-Euclidean geometries in which the fifth postulate in the Elements is violated. These include elliptical and hyperbolic geometries, which were created in the 19th century by the German mathematician Georg Riemann and the Russian scientist Nikolai Lobachevsky.
Initially, it was believed that non-Euclidean geometries have a narrow scope (for example, in astronomy in the study of the celestial sphere), and the physical space itself is Euclidean. The fallacy of the last statement was shown by Albert Einstein at the beginning of the 20th century, developing his theory of relativity, in which he generalized the concepts of space and time.
Geometry at school
As mentioned above, the study at the school of geometry begins with grade 7. At the same time, students are shown the basics of planimetry. Class 9 geometry already includes the study of three-dimensional bodies, that is, stereometry.
The main objective of the school course is to develop students' abstract thinking and imagination, as well as to teach them to think logically.
Many studies have shown that when studying this science, students have problems with abstract thinking. When a geometric problem is formulated for them, they often do not understand its essence. In high school students, difficulties in understanding mathematical formulas are added to the problem with imagination to determine the volume and surface area of the spread of spatial figures. Often, high school students in studying geometry of grade 9 do not know which formula should be used in a particular case.
School books
There are a large number of teaching aids for teaching schoolchildren this science. Some of them provide only basic knowledge, for example, the textbooks of L. S. Atanasyan or A. V. Pogorelov. Others are pursuing an in-depth study of science. Here you can highlight the textbook of A. D. Alexandrov or the complete geometry course of Bevza G. P.
Since in recent years a single standard of the Unified State Examination has been introduced to pass all exams at school, textbooks and resolvers have become necessary, which allow the student to quickly figure out the necessary topic on their own. A good example of such aids is the geometry of A. Ershova, V. V. Goloborodko
Any of the textbooks mentioned above has both positive and negative reviews from teachers, so teaching in the school of geometry is often carried out using several textbooks.