Geometry is an exact mathematical science that deals with the study of spatial and other similar relationships and shapes. But it is often called "dry" because it is unable to describe the shape of many natural objects, because clouds are not spheres, mountains are not cones, and lightning does not propagate in straight lines. Many objects in nature are complex in shape compared to standard geometry.

However, there are a number of amazing figures that are not usually studied in school lessons geometry, but they are the ones that surround a person in the real world: in nature and architecture, puzzles, computer games, etc.

The main property of this complex geometric figure is self-similarity, that is, it consists of several parts, each of which is similar to a whole object. It is this property that distinguishes fractals from objects of classical (or, as they say, Euclidean) geometry.

At the same time, the term "fractal" itself is not mathematical and does not have an unambiguous definition, therefore it can be applied to objects that are self-similar or approximately self-similar. It was invented in 1975 by Benoit Mandelbrot, borrowing latin word"Fractus" (broken, crushed).

Fractal forms are the best fit to describe the real world and are often found among natural objects: snowflakes, plant leaves, the blood vessel system of humans and animals.

This is one of the most extraordinary 3D shapes in geometry and is easy to make at home. To do this, it is enough to take a paper strip, the width of which is 5-6 times less than its length, and, twisting one of the ends by 180 °, glue them together.

If everything is done correctly, then you can check its amazing properties yourself:

• The presence of only one side (without dividing into internal and external). This is easy to check if you try to paint over one of its sides with a pencil. Regardless of where and in what direction you start painting, as a result the entire ribbon will be painted over with the same color.
• Continuity: If you drag a line along the entire surface, the end of the line connects to the start point without crossing the surface boundaries.
• Two-dimensionality (connectivity): when cutting the Mobius strip along, it remains solid, just new shapes are obtained (for example, when cutting in two, one larger ring will turn out).
• Lack of orientation. Travel along such a Mobius strip will always be endless, it will lead to the starting point of the path, only in a mirror image.

The Mobius strip is widely used in industry and science (in belt conveyors, dot matrix printers, sharpening mechanisms, etc.). In addition, there is a scientific hypothesis according to which the Universe itself is also a Mobius strip of incredible dimensions.

Polyomino

These are flat geometric shapes that are formed by connecting several squares of equal size on their sides.

Polyomino names depend on the number of squares from which they are formed:

• monomino - 1;
• dominoes - 2;
• trimino - 3;
• tetrimino - 4, etc.

At the same time, for each variety there is a different number of types of figures: dominoes have 1 type, trimino has 3 types, hexamino (from 6 squares) has 35 types. The number of different variations depends on the number of squares used, but at the same time, none of the scientists have yet managed to find an amazing formula that will express this dependence. From the details of the polyomino, you can lay out both geometric shapes and images of people, animals, objects. Despite the fact that these will be sketchy silhouettes, the main features and shapes of objects make them quite recognizable.

Polyamond

Along with polyomino, there is another amazing geometric figure used to compose other figures - polyamond. It is a polygon formed from several equilateral triangles of equal size.

The name was invented by the mathematician T. O'Bairn based on one of the names of the rhombus in English language- diamond, which can be composed of 2 equilateral triangles. By analogy, O'Beirn called a figure from 3 equilateral triangles triamond, from 4 - tetriamond, etc.

The main question of their existence is the question of the possible number of polyamides that can be made up of a certain number of triangles. The use of polyamond in real life also similar to using polyominoes. These can be all sorts of puzzles and logic tasks.

Reuleaux triangle

Surprisingly it sounds, but with a drill you can drill a square hole, and the Reuleaux triangle helps in this. It is an area formed by the intersection of 3 equal circles, the centers of which are the vertices of a regular triangle, and the radii are equal to its side.

The Reuleaux triangle itself is named after the German scientist-engineer, who was the first to investigate its features in the most detail and used it for his mechanisms at the turn of the 19th and 20th centuries. century, although its amazing properties were known by Leonardo da Vinci. Whoever was its discoverer, in modern world this figure has found wide application in the form:

• the Watts drill, which allows you to drill holes of an almost perfect square shape, only with slightly rounded edges;
• a pick required for playing musical plucked instruments;
• cam mechanisms used to create zigzag seams in sewing machines, as well as German watches;
• pointed arches, characteristic of the Gothic style in architecture.

Impossible figures

Special attention should be paid to the so-called impossible figures - amazing optical illusions, which at first glance seem to be a projection of a three-dimensional object, but upon closer inspection, unusual combinations of elements become noticeable. The most popular of these are:

Tribar, created by father and son Lionel and Roger Penrose, which is an equilateral triangle, but has strange patterns. The sides that form the top of the triangle appear perpendicular, but the right and left edges at the bottom also appear perpendicular. If we consider each part of this triangle separately, it is still possible to recognize their existence, but in reality such a figure cannot exist, since the correct elements were incorrectly connected during its creation.

The endless staircase, which also belongs to the father and son of the Penrose, is why it is often called by their name - the "Penrose staircase", as well as the "Eternal staircase". At first glance, it looks like an ordinary staircase leading up or down, but at the same time a person walking along it will continuously rise (counterclockwise) or descend (clockwise). If you visually travel along such a staircase, then at the end of the "journey" the gaze stops at the starting point of the path. If such a staircase existed in reality, it would have to climb and descend an infinite number of times, which can be compared with an endless Sisyphean labor.

The Impossible Trident is an amazing object, looking at which it is impossible to determine where the middle prong begins. It is also based on the principle of irregular connections, which can only exist in two-dimensional, not three-dimensional space. Considering the parts of the trident separately, 3 round teeth are visible on one side, and 2 rectangular teeth on the other.

Thus, the parts of the figure enter into a kind of conflict: firstly, there is a change in the foreground and background, and secondly, the round teeth in the lower part are transformed into flat ones in the upper part.

There are an infinite number of forms. The form is called the external outline of an object.

The study of forms can begin from early childhood, drawing the attention of your child to the world around us, which consists of figures (the plate is round, the TV is rectangular).

From the age of two, the baby should know three simple shapes - a circle, a square, a triangle. At first, he should just show them when you ask for it. And at the age of three already call them yourself and distinguish a circle from an oval, a square from a rectangle.

The more exercises for consolidation of forms the child performs, the more new figures he will remember.

The future first grader should know all simple geometric shapes and be able to make applications from them.

What do we call a geometric figure?

A geometric figure is a standard with which you can determine the shape of an object or its parts.

The figures are divided into two groups: flat figures, volumetric figures.

We will call flat figures those figures that are located in the same plane. These include a circle, an oval, a triangle, a quadrilateral (rectangle, square, trapezoid, rhombus, parallelogram) and all kinds of polygons.

Three-dimensional figures include: a sphere, a cube, a cylinder, a cone, a pyramid. These are the shapes that have height, width, and depth.

Follow two simple tips when explaining geometric shapes:

1. Patience. What we, adults, seem to be simple and logical to a child, will seem simply incomprehensible.
2. Try drawing shapes with your child.
3. A game. Start learning shapes in game form. Good exercise for fixing and studying flat forms - applications of geometric shapes. For volumetric - you can use ready-made purchased games, as well as choose applications where you can cut and glue a volumetric shape.

The text of the work is placed without images and formulas.
Full version work is available in the tab "Files of work" in PDF format

Introduction

Geometry is one of the most important components of mathematical education, necessary for the acquisition of specific knowledge about space and practically significant skills, the formation of a language for describing objects of the surrounding world, for the development of spatial imagination and intuition, mathematical culture, as well as for aesthetic education. The study of geometry contributes to the development logical thinking, building skills of proof.

In the 7th grade geometry course, knowledge about the simplest geometric shapes and their properties is systematized; the concept of equality of figures is introduced; the ability to prove the equality of triangles using the studied signs is developed; a class of construction problems is introduced using a compass and a ruler; one of the most important concepts is introduced - the concept of parallel lines; new interesting and important properties of triangles are considered; one of the most important theorems in geometry is considered - the theorem on the sum of the angles of a triangle, which allows one to give a classification of triangles by angles (acute-angled, rectangular, obtuse).

During classes, especially when moving from one part of the class to another, changing activities, the question arises of maintaining interest in classes. Thus, relevant the question of the use of tasks in geometry classes in which there is a condition of a problem situation and elements of creativity becomes. Thus, aim This study is the systematization of tasks of geometric content with elements of creativity and problem situations.

Object of study: Problems in geometry with elements of creativity, entertainment and problem situations.

Research objectives: Analyze existing problems in geometry aimed at developing logic, imagination and creative thinking... Show how interesting techniques can be used to develop interest in a subject.

Theoretical and practical significance research is that the collected material can be used in the process additional classes in geometry, namely at Olympiads and competitions in geometry.

The scope and structure of the study:

The research consists of an introduction, two chapters, a conclusion, a bibliographic list, contains 14 pages of the main typewritten text, 1 table, 10 figures.

Chapter 1. PLANE GEOMETRIC FIGURES. BASIC CONCEPTS AND DEFINITIONS

1.1. Basic geometric shapes in the architecture of buildings and structures

In the world around us, there are many material objects of different shapes and sizes: residential buildings, car parts, books, jewelry, toys, etc.

In geometry, instead of the word object, they say a geometric figure, while dividing geometric figures into flat and spatial ones. This paper will consider one of the most interesting sections of geometry - planimetry, in which only flat figures are considered. Planimetry(from Lat. planum - "plane", Old Greek. μετρεω - "I measure") - a section of Euclidean geometry that studies two-dimensional (one-plane) figures, that is, figures that can be located within one plane. A plane geometrical figure is called such, all points of which lie on the same plane. An idea of ​​such a figure is given by any drawing made on a sheet of paper.

But before considering flat figures, it is necessary to get acquainted with simple, but very important figures, without which flat figures simply cannot exist.

The simplest geometric shape is dot. This is one of the main figures in geometry. She is very small, but she is always used to build different forms on surface. The point is the main figure for absolutely all constructions, even of the highest complexity. From the point of view of mathematics, a point is an abstract spatial object that does not have such characteristics as area, volume, but at the same time remains a fundamental concept in geometry.

Straight- one of the fundamental concepts of geometry. In a systematic presentation of geometry, a straight line is usually taken as one of the original concepts, which is only indirectly determined by the axioms of geometry (Euclidean). If the basis for constructing geometry is the concept of the distance between two points in space, then a straight line can be defined as a line, the path along which is equal to the distance between two points.

Straight lines in space can occupy various provisions, consider some of them and give examples that are found in the architectural appearance of buildings and structures (Table 1):

Table 1

Parallel lines	Parallel Line Properties
	If the straight lines are parallel, then their projections of the same name are parallel:	Essentuki, building of mud baths (author's photo)
Intersecting straight lines	Intersecting line properties	Examples in the architecture of buildings and structures
	Intersecting straight lines have a common point, that is, the points of intersection of their projections of the same name lie on a common communication line:	Mountain buildings in Taiwan https://www.sro-ps.ru/novosti_otrasli/2015_11_11_pervoe_zdanie_iz_grandioznogo_proekta_big_v_tayvane
Crossed straight lines	Properties of intersecting straight lines	Examples in the architecture of buildings and structures
	Lines that do not lie in the same plane and are not parallel to each other are intersecting. None is a common communication line. If the intersecting and parallel lines lie in the same plane, then the intersecting lines lie in two parallel planes.	Robert, Hubert - Villa Madama near Rome https://gallerix.ru/album/Hermitage-10/pic/glrx-172894287

1.2. Flat geometric shapes. Properties and definitions

Observing the forms of plants and animals, mountains and meanders of rivers, the features of the landscape and distant planets, man borrowed from nature its correct forms, sizes and properties. Material needs prompted a person to build dwellings, make tools of labor and hunting, sculpt utensils from clay, and so on. All this gradually contributed to the fact that a person came to an understanding of the basic geometric concepts.

Quadrangles:

Parallelogram(Old Greek παραλληλόγραμμον from παράλληλος - parallel and γραμμή - line, line) is a quadrangle, in which opposite sides are pairwise parallel, that is, lie on parallel lines.

Signs of a parallelogram:

A quadrilateral is a parallelogram if one of the following conditions is met: 1. If in a quadrilateral opposite sides are pairwise equal, then a quadrilateral is a parallelogram. 2. If in a quadrilateral diagonals intersect and the intersection point is divided in half, then this quadrilateral is a parallelogram. 3. If in a quadrilateral two sides are equal and parallel, then this quadrilateral is a parallelogram.

A parallelogram in which all angles are straight is called rectangle.

A parallelogram in which all sides are equal is called rhombus.

Trapezium— it is a quadrilateral in which two sides are parallel and the other two are not parallel. Also, a trapezoid is called a quadrilateral, in which one pair of opposite sides is parallel, and the sides are not equal to each other.

Triangle- This is the simplest geometric figure formed by three segments that connect three points that do not lie on one straight line. These three points are called vertices. triangle, and the segments - by the sides triangle. It is because of its simplicity that the triangle was the basis of many dimensions. Surveyors in their calculations of land areas and astronomers when finding distances to planets and stars use the properties of triangles. This is how the science of trigonometry arose - the science of measuring triangles, expressing the sides through its angles. Through the area of ​​a triangle, the area of ​​any polygon is expressed: it is enough to break this polygon into triangles, calculate their areas and add the results. True, it was not immediately possible to find the correct formula for the area of ​​a triangle.

The properties of the triangle were especially actively studied in XV-XVI centuries... Here is one of the most beautiful theorems of the time, due to Leonard Euler:

A huge amount of work on the geometry of the triangle, carried out in the XY-XIX centuries, created the impression that everything was already known about the triangle.

Polygon - it is a geometric shape, usually defined as a closed polyline.

A circle- the locus of points on the plane, the distance from which to a given point, called the center of the circle, does not exceed a given non-negative number, called the radius of this circle. If the radius is zero, then the circle degenerates into a point.

There are a large number of geometric shapes, they all differ in parameters and properties, sometimes surprising with their shapes.

In order to better remember and distinguish flat figures by their properties and characteristics, I came up with a geometric fairy tale, which I would like to present to your attention in the next paragraph.

Chapter 2. PUZZLES FROM PLANE GEOMETRIC FIGURES

2.1 Construction puzzles complex figure from a set of flat geometric elements.

Having studied the flat figures, I wondered if there are some interesting problems with flat figures that can be used as tasks-games or tasks-puzzles. And the first problem I found was the Tangram puzzle.

This is a Chinese puzzle. In China it is called chi tao tu, a seven-piece mental puzzle. In Europe, the name "Tangram" most likely arose from the word "tan", which means "Chinese" and the root "gram" (Greek - "letter").

First, you need to draw a 10 x 10 square and divide it into seven parts: five triangles 1-5 , square 6 and parallelogram 7 ... The essence of the puzzle is to put together the figures shown in Fig. 3 using all seven parts.

Fig. 3. Elements of the game "Tangram" and geometric shapes

Fig. 4. Tangram quests

It is especially interesting to make “shaped” polygons from flat figures, knowing only the outlines of objects (Fig. 4). I came up with several such tasks-outlines myself and showed these tasks to my classmates, who happily began to solve the tasks and made many interesting polyhedron figures, similar to the outlines of objects in the world around us.

For the development of imagination, you can also use such forms of entertaining puzzles as tasks for cutting and reproducing given figures.

Example 2. Tasks for cutting (parquetting) may seem, at first glance, very diverse. However, most of them use only a few basic types of cuts (as a rule, those with the help of which one can get another from one parallelogram).

Let's consider some cutting techniques. In this case, the cut figures will be called polygons.

Rice. 5. Techniques for cutting

Figure 5 shows geometric shapes from which you can assemble various ornamental compositions and make an ornament with your own hands.

Example 3. One more interesting task, which you can independently invent and exchange with other students, while whoever collects the cut figures more, is declared the winner. There can be a lot of tasks of this type. For coding, you can take all existing geometric shapes, which are cut into three or four parts.

Fig. 6 Examples of cutting tasks:

------ - recreated square; - cut with scissors;

Main figure

2.2 Equal and equal-sized figures

Let's consider another interesting technique for cutting flat figures, where the main "heroes" of cutting will be polygons. When calculating the areas of polygons, a simple technique is used called the partitioning method.

In general, polygons are called scissor-congruent if, after cutting the polygon in a certain way F into a finite number of parts, it is possible, by arranging these parts in a different way, to compose from them the polygon H.

This implies the following. theorem: equal polygons have the same area, so they will be considered equal.

Using the example of scissor-congruent polygons, one can also consider such an interesting dissection as the transformation of a "Greek cross" into a square (Fig. 7).

Fig. 7. Conversion of the "Greek cross"

In the case of a mosaic (parquet) made up of Greek crosses, the parallelogram of the periods is a square. We can solve the problem by superimposing a mosaic made up of squares onto a mosaic formed using crosses, so that the congruent points of one mosaic coincide with the congruent points of the other (Fig. 8).

In the figure, the congruent points of the mosaic of the crosses, namely the centers of the crosses, coincide with the congruent points of the "square" mosaic - the vertices of the squares. Moving the square mosaic in parallel, we always get a solution to the problem. Moreover, the task has several options for solutions, if color is used when drawing up the parquet ornament.

Fig. 8. Parquet assembled from the Greek cross

Another example of scaly shapes can be seen on the example of a parallelogram. For example, a parallelogram is equal to a rectangle (Fig. 9).

This example illustrates the method of partitioning, which consists in the fact that to calculate the area of ​​a polygon, they try to divide it into a finite number of parts so that these parts can be used to compose a simpler polygon, the area of ​​which we already know.

For example, a triangle is equidistant with a parallelogram having the same base and half the height. From this position, the formula for the area of ​​a triangle is easily derived.

Note that the above theorem also holds true. converse theorem: if two polygons are of the same size, then they are scissor-congruent.

This theorem, proved in the first half of the 19th century. Hungarian mathematician F. Boyai and German officer and lover of mathematics P. Gervin, can be presented in this form: if there is a cake in the shape of a polygon and a polygonal box, completely different in shape, but of the same area, then you can cut the cake into a finite number of pieces (without turning them upside down) so that they can be put in this box.

Conclusion

In conclusion, I note that problems for flat figures are sufficiently represented in various sources, but those on the basis of which I had to come up with my own puzzle problems were of interest to me.

After solving such problems, one can not only accumulate life experience, but also acquire new knowledge and skills.

In puzzles, when constructing actions-moves using turns, shifts, transfers on planes or their compositions, I got independently created new images, for example, polyhedron figures from the game "Tangram".

It is known that the main criterion for the mobility of a person's thinking is the ability, by means of recreational and creative imagination, to perform certain actions in a set period of time, and in our case, the moves of figures on a plane. Therefore, studying mathematics and, in particular, geometry at school will give me even more knowledge in order to further apply it in my future professional activities.

Bibliographic list

1. Pavlova, L.V. Unconventional approaches to teaching drawing: tutorial/ L.V. Pavlova. - Nizhny Novgorod: Publishing house of NSTU, 2002 .-- 73 p.

2. Encyclopedic Dictionary of a Young Mathematician / Comp. A.P. Savin. - M .: Pedagogy, 1985 .-- 352 p.

3.https: //www.srops.ru/novosti_otrasli/2015_11_11_pervoe_zdanie_iz_grandioznogo_proekta_big_v_tayvane

4.https://www.votpusk.ru/country/dostoprim_info.asp?ID=16053

Annex 1

Questionnaire for classmates

1. Do you know what the Tangram puzzle is?

2. What is a "Greek cross"?

3. Would you be interested to know what "Tangram" is?

4. Would you be interested to know what a "Greek cross" is?

22 pupils of the 8th grade were interviewed. Results: 22 pupils do not know what "Tangram" and "Greek cross" are. 20 students would be interested to learn how to use the Tangram puzzle, which consists of seven flat shapes, to get a more complex shape.

Appendix 2

Elements of the game "Tangram" and geometric shapes

Conversion of the "Greek cross"

Geometric figures are a complex of points, lines, bodies or surfaces. These elements can be located both on a plane and in space, forming a finite number of straight lines.

The term "shape" refers to multiple sets of points. They should be located on one or more planes and at the same time be limited to a specific number of completed lines.

The main geometric shapes are a point and a straight line. They are located on a plane. Besides them, among simple figures select a ray, a broken line and a segment.

Dot

This is one of the main figures in geometry. It is very small, but it is always used to construct various shapes on a plane. The point is the main figure for absolutely all constructions, even of the highest complexity. In geometry, it is customary to denote it with a letter of the Latin alphabet, for example, A, B, K, L.

From the point of view of mathematics, a point is an abstract spatial object that does not have such characteristics as area, volume, but at the same time remains a fundamental concept in geometry. This zero-dimensional object simply has no definition.

Straight

This figure fits completely in one plane. The straight line does not have a specific mathematical definition, since it consists of a huge number of points located on one endless line, which has no limit and boundaries.

There is also a segment. This is also a straight line, but it starts and ends with a point, which means it has geometric constraints.

Also, the line can turn into a directional beam. This happens when a straight line starts from a point, but does not have a clear end. If you put a point in the middle of the line, then it will split into two rays (additional), and oppositely directed to each other.

Several segments that are sequentially connected to each other by their ends at a common point and are not located on one straight line are usually called a broken line.

Injection

Geometric shapes, the names of which we discussed above, are considered key elements used in building more complex models.

An angle is a structure consisting of a vertex and two rays that extend from it. That is, the sides of this figure are connected at one point.

Plane

Let's consider one more primary concept. A plane is a figure that has neither end nor beginning, as well as a straight line and a point. When considering this geometric element, only its part is taken into account, bounded by the contours of a polyline closed line.

Any smooth bounded surface can be considered a plane. This could be an ironing board, a piece of paper, or even a door.

Quadrangles

A parallelogram is a geometric figure whose opposite sides are parallel to each other in pairs. Among the private types of this design, rhombus, rectangle and square are distinguished.

A rectangle is a parallelogram in which all sides touch at right angles.

A square is a rectangle with equal sides and angles.

A rhombus is a shape in which all faces are equal. In this case, the angles can be completely different, but in pairs. Each square counts as a rhombus. But this rule does not always work in the opposite direction. Not every rhombus is a square.

Trapezoid

Geometric shapes can be completely different and whimsical. Each of them has a unique shape and properties.

A trapezoid is a shape that is somewhat similar to a quadrangle. It has two parallel opposite sides and is considered curved.

A circle

This geometric figure implies the location on the same plane of points equidistant from its center. In this case, a given nonzero segment is usually called the radius.

Triangle

It is a simple geometric figure that is very often seen and studied.

A triangle is considered a subspecies of a polygon located on one plane and bounded by three faces and three points of contact. These elements are connected in pairs.

Polygon

The vertices of polygons are the points that connect the line segments. And the latter, in turn, are considered to be parties.

Volumetric geometric shapes

• prism;
• sphere;
• cone;
• cylinder;
• pyramid;

These bodies have something in common. All of them are limited to a closed surface, inside which there are many points.

Three-dimensional bodies are studied not only in geometry, but also in crystallography.

Curious facts

Surely you will be interested in reading the information provided below.

• Geometry was formed as a science in ancient times. It is customary to associate this phenomenon with the development of art and various crafts. And the names of geometric shapes indicate the use of the principles of determining similarity and similarity.
• Translated from ancient Greek, the term "trapezium" means a table for a meal.
• If you take different shapes, the perimeter of which will be the same, then the circle is guaranteed to have the largest area.
• Translated from Greek, the term "cone" means a pine cone.
• There is a famous painting by Kazemir Malevich, which has attracted the views of many painters since the last century. The work "Black Square" has always been mystical and mysterious. The geometric figure on a white canvas delights and amazes at the same time.

There are many geometric shapes. They all differ in parameters, and sometimes even surprise with their shapes.

Chukur Lyudmila Vasilievna
Geometric figures. Features of children's perception of the shape of objects and geometric shapes

« GEOMETRIC FIGURE.

CHILDREN'S PERCEPTIONS

Prepared: st. educator Chukur L... IN.

1. Concept « geometric figure» . Features of the development of ideas about the shape of objects in preschool children

One of the properties of others objects is their shape. Shape of objects was summarized in geometric shapes.

Figure is a latin word means "image", "view", "Mark"; it is a part of a plane bounded by a closed line, or a part of a space bounded by a closed surface. This term came into general use in the XII century. Before that, another Latin word was used more often - « form» , also meaning "External view", "Outer outline subject» .

Watching over objects of the surrounding world, people have noticed that there is some common property that allows you to combine items in one group... This property has been named geometric figure. Geometric figure is a standard for determining the shape of an object., any non-empty set of points; generalized abstract concept.

Itself the definition of the concept of a geometric figure was given by the ancient Greeks... They are identified, what geometric figure is the inner area bounded by a closed line on the plane. Euclid actively used this concept in his work. The ancient Greeks classified everything geometric shapes and gave them names.

Mention of the first geometric shapes found among the ancient Egyptians and ancient Sumerians. Scientists-archaeologists have found a papyrus scroll with geometric problems which mentioned geometric figures... And each of them was called some a certain word.

Thus, understanding of geometry and studied by this science figures had people from ancient times, but the name, « geometric figure» and names for everyone geometric shapes given by ancient Greek scientists.

Nowadays, acquaintance with geometric shapes begins in early childhood and continues throughout the learning path. Preschoolers learning the world face diversity shapes of objects, learn to name and distinguish them, and then get acquainted with the properties geometric shapes.

Form- this is the outer outline subject... A bunch of forms endlessly.

Representations of the shape of objects occur in children early enough. In the studies of L.A. Wenger, it is clarified whether it is possible to distinguish shapes of objects by children who have not yet an act of grabbing has formed... As an indicator, he used the tentative reaction of a child at the age of 3-4 months.

For children were presented two volumetric bodies of the same steel color and size (a prism and a ball, one of them was suspended above the arena in order to extinguish the approximate reaction; then a pair was again suspended figures... One of them (prism) the reaction is extinguished, the other (ball)- new. The little ones looked to the new figure and fixed it with a gaze for a longer time than the old one.

L.A. Wenger also noted that on geometric figure with a change in spatial orientation, the same visual concentration arises as on a new geometric figure.

Research by M. Denisova and N. Figurin showed that the baby is on shape to the touch defines the bottle, pacifier, mother's breast. Children begin to distinguish visually shape of objects from 5 months... In this case, the indicator of discrimination is the movements of the hands, the body towards the experimental object and grasping it (with food reinforcement).

Other studies have found that if items differ in color, then a child of 3 years old selects them form only if, if item familiar to the child from practical experience (experience of manipulation, action).

This is proved by the fact that the child equally recognizes straight and inverted images (he can view and understand familiar pictures while holding a book "upside down", subjects painted in unusual colors (a black apple, but a square rotated at an angle, that is, in the form of a rhombus, does not recognize, since the immediate similarity disappears object shape, which is not in experience.

2. Features of perception by children preschool age shapes of objects and geometric shapes

One of the leading cognitive processes preschool children is perception. Perception helps to distinguish one subject from another, highlight some subjects or phenomena from others similar to him.

Primary mastery the shape of the object Item shape as such is not subject precede practical actions. Actions of children with subjects at different stages are different.

Research by the psychologist S.N. Shabalin shows that the geometric figure is perceived preschoolers in a peculiar way. If an adult perceives bucket or glass like subjects having a cylindrical shape then into it perception turns on knowledge geometric shapes ... In a preschooler, the opposite occurs.

Children 3-4 years old objectify geometric shapes since they are in their experience presented inseparably with objects are not abstracted. The geometric figure is perceived by children as a picture. like some item: a square is a handkerchief, a pocket; a triangle - a roof, a circle - a wheel, a ball, two circles next to it - glasses, several circles next to it - beads, etc.

At 4 years old objectification of a geometric figure occurs only when a child collides with an unfamiliar figure: a cylinder is a bucket, a glass.

At 4-5 years old, the child begins to compare geometric figure with subject: says about the square "It's like a handkerchief".

As a result of organized learning, children begin to distinguish familiar geometric shapes, compare object with a figure(the glass is like a cylinder, the roof is like a triangle, learns to give the correct name geometric shape and shape of the object, words appear in their speech "square", "a circle", "square", "round" etc.

The problem of acquaintance of children with geometric shapesand their properties should be considered in two aspects:

In terms of touch perception of shapes of geometric shapes and using them as standards in cognition shapes of surrounding objects;

In the sense of knowledge features of their structure, properties, basic connections and patterns in their construction, that is, actually geometric material.

Circuit subject is a common beginning, which is the source for both visual and tactile perception... However, the question of the role of the circuit in perception of shape and formation a holistic image requires further development.

Primary mastery the shape of the object carried out in actions with him. Item shape as such is not perceived separately from the subject, she is his essential feature. Specific visual contour-tracking reactions subject appear at the end of the second year of life and begin precede practical actions.

Actions of children with subjects at different stages are different. The little ones tend, above all, to capture item hands and start manipulating him. Children 2.5 years old, before acting, in some detail visually and tactilely - motorly get acquainted with subjects... The importance of practical action remains the main one. Hence, it follows that it is necessary to guide the development of perceptual actions in two-year-old children. Depending on the pedagogical leadership, the nature of children's perceptual actions gradually reaches cognitive level... The child begins to be interested in various signs. subject including form... However, for a long time he cannot isolate and generalize this or that sign, including shape of different objects.

Sensory perception of the shape of an object should be aimed not only at see, learn shape, along with its other features, but be able to abstract shape from thing, see her in other things too... Such the perception of the shape of objects and its generalization and contributes to the knowledge of standards by children - geometric shapes... Therefore, the task sensory development is an shaping the child's ability to recognize in accordance with the standard (one or another geometric figure) shape of different objects.

The experimental data of L.A. Wenger showed that the ability to distinguish geometric figures are possessed by children 3-4 months. Focusing on the new figure- evidence of this.

Already in the second year of life, children freely choose figuremodeled on such pairs: square and semicircle, rectangle and triangle. But children can distinguish between a rectangle and a square, a square and a triangle only after 2.5 years. Selection by sample more complex shapes available around the turn of 4-5 years, and reproduction of a complex figure carried out by children of the fifth and sixth year of life.

Under the teaching influence of adults perception of geometric shapes is gradually being rebuilt. Children begin to perceive geometric shapes as standards., with the help of which the knowledge of the structure subject, his shape and size is carried out not only in the process perception of one or another form of vision, but also by active touch, feeling it under the control of vision and designation with a word.

Collaboration of all analyzers contributes to a more accurate perception of the shape of objects... To get to know better item, children tend to touch it with their hand, pick it up, turn it; moreover, viewing and feeling are different depending on shape and the construction of the cognizable object. Therefore, the main role in the perception of the object and the determination of its shape has an examination, carried out simultaneously by visual and motor-tactile analyzers, followed by designation with a word. However, in preschoolers, there is a very low level surveys shapes of objects; most often they are limited to fluent visual perception and therefore do not distinguish between close ones by similarity figures(oval and circle, rectangle and square, different triangles).

In the perceptual activity of children, tactile-motor and visual techniques gradually become the main way of recognizing the form... Survey figures not only ensures their holistic perception but also allows you to feel them peculiarities(the character, directions of the lines and their combinations, the resulting corners and peaks, the child learns to sensually highlight in any figure the image as a whole and its parts. This makes it possible to further focus the child's attention on meaningful analysis. figures deliberately highlighting in it structural elements (sides, corners, tops)... Children are already consciously beginning to understand properties such as stability, instability, etc., to understand how tops, angles, etc. are formed. figures, children already find commonality between them ( "The cube has squares", "The bar has rectangles, the cylinder has circles" etc.).

Comparison figures with the shape of an object helps children understand that with geometric shapes you can compare different objects or parts thereof... So, gradually geometric figure becomes a benchmark determining the shape of objects.

3. Peculiarities examinations and stages of learning examinations children preschool age shapes of objects and geometric shapes

It is known that cognition is always based on sensory examination, mediated by thinking and speech. In the studies of L. Wenger with children 2-3 years as an indicator of visual discrimination the forms of objects served the child's object actions.

According to research by S. Yakobson, V. Zinchenko, A. Ruzskoy, children of 2-4 years old learned better objects in shape, when it was suggested to first touch the object, and then find the same one. Lower results were observed when the subject was perceived visually.

T. Ginevskaya's research reveals peculiarities hand movements during examination objects in shape... Children were blindfolded and offered to get acquainted with the subject by touch.

At 3-4 years old - executive movements (roll, knock, carry)... Movement is sparse, inside figures, sometimes (once) along the centerline, many wrong answers, a mixture of different figures... At 4-5 years old - setting movements (gripped in hand)... The number of movements doubles; judging by the trajectory, focused on size and area; large, sweeping, groups of closely spaced fixations are found that are among the most characteristic features figures; give better results. At 5-6 years old - survey movements (contour tracking, elasticity check)... Movements appear that follow the contour, but they cover the most characteristic part of the contour, other parts are unexplored; movements within the contour, the amount is the same, high results; As in previous period, there is a mixture of close figures... At 6-7 years old - movement along the contour, crossing the field figures, and the movements focus on the most informative signs, excellent results are observed not only in recognition, but also in reproduction.

Thus, in order for the child to highlight the essential features geometric shapes, their visual and motor examination is necessary. Hand movements organize eye movements and this needs to be taught to children.

Inspection Learning Steps

The task of the first stage of teaching children 3-4 years old is sensory perception of the shape of objects and geometric shapes.

The second stage of education for children 5-6 years old should be devoted to formation of systemic knowledge about geometric shapes and their development initial receptions and ways« geometric thinking» .

« Geometric thinking» it is quite possible to develop back in preschool age... In development « geometric knowledge» children have several different levels.

The first level is characterized by the fact that the figure is perceived by children as a whole, the child still does not know how to distinguish individual elements in it, does not notice the similarities and differences between figures, each of them perceives apart.

At the second level, the child already selects elements in figure and establishes relationships both between them and between individual figures, however, is not yet aware of the commonality between figures.

At the third level, the child is able to establish connections between properties and structure. figures, the relationship between the properties themselves. The transition from one level to another is not spontaneous, parallel to the biological development of a person and dependent on age. It takes place under the influence of purposeful learning, which helps to accelerate the transition to a higher level. Lack of training hinders development. Therefore, training should be organized in such a way that, in connection with the assimilation of knowledge about geometric shapes children also developed elementary geometric thinking.

Cognition geometric shapes, their properties and relationships expands the horizons of children, allows them more accurately and versatile perceive the shape of surrounding objects, which has a positive effect on their productive activities (e.g. drawing, sculpting).

Great value in development geometric thinking and spatial views have conversion actions figures(make a square from two triangles or add two triangles from five sticks).

All of these types of exercises develop spatial representations and beginnings of geometric thinking in children, form they have the ability to observe, analyze, generalize, highlight the main, essential and at the same time educate such personality traits as purposefulness, perseverance.

So, at preschool age, the mastery of perceptual and intellectual systematization occurs. shapes of geometric shapes... Perceptual activity in cognition figures outstrips the development of intellectual systematization.

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