Planimetry is a branch of geometry in which figures on a plane are studied.

Figures studied by planimetry:

3. Parallelogram (special cases: square, rectangle, rhombus)

4. Trapeze

5. Circle

6. Triangle

7. Polygon

1) Point:

In geometry, topology and related branches of mathematics, a point is an abstract object in space that has neither volume, nor area, nor length, nor any other similar characteristics of large dimensions. Thus, a zero-dimensional object is called a point. The point is one of the fundamental concepts in mathematics.

Point in Euclidean geometry:

A point is one of the fundamental concepts of geometry, so "point" has no definition. Euclid defined a point as something that cannot be divided.

A straight line is one of the basic concepts of geometry.

A geometric straight line (straight line) is a non-closed on both sides, extended non-curving geometric object, the cross section of which tends to zero, and the longitudinal projection onto the plane gives a point.

In a systematic exposition of geometry, a straight line is usually taken as one of the initial concepts, which is only indirectly determined by the axioms of geometry.

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 along which the path is equal to the distance between two points.

3) Parallelogram:

A parallelogram is a quadrilateral whose opposite sides are pairwise parallel, that is, they lie on parallel lines. Special cases of a parallelogram are a rectangle, a square and a rhombus.

Special cases:

Square- a regular quadrilateral or rhombus, in which all angles are right, or a parallelogram, in which all sides and angles are equal.

A square can be defined as: a rectangle whose two adjacent sides are equal;

a rhombus with all right angles (any square is a rhombus, but not every rhombus is a square).

Rectangle is a parallelogram in which all angles are right (equal to 90 degrees).

Rhombus is a parallelogram with all sides equal. A rhombus with right angles is called a square.

4) Trapeze:

Trapeze A quadrilateral with exactly one pair of opposite sides parallel.

1. A trapezoid whose sides are not equal,

called versatile .

2. A trapezoid whose sides are equal is called isosceles.

3. A trapezoid, in which one side makes a right angle with the bases, is called rectangular .

The segment that joins the midpoints of the sides of a trapezoid is called middle line trapezoid (MN). middle line trapezium is parallel to the bases and equal to their half-sum.

A trapezoid can be called a truncated triangle, and therefore the names of trapeziums are similar to the names of triangles (triangles are versatile, isosceles, rectangular).

5) Circumference:

Circle is the locus of points in the plane equidistant from given point, called the center, by a given non-zero distance, called its radius.

6) Triangle:

Triangle- the simplest polygon having 3 vertices (corners) and 3 sides; a part of a plane bounded by three points and three line segments connecting these points in pairs.

7) Polygon:

Polygon is a geometric figure, defined as a closed polyline. There are three different definitions:

Flat closed broken lines;

Flat closed broken lines without self-intersections;

Parts of the plane bounded by broken lines.

The vertices of the polyline are called the vertices of the polygon, and the segments are called the sides of the polygon.

Basic properties of a line and a point:

1. Whatever the line, there are points belonging to this line and not belonging to it.

Through any two points you can draw a line, and only one.

2. Of the three points on a line, one and only one lies between the other two.

3. Each segment has a certain length greater than zero. The length of a segment is equal to the sum of the lengths of the parts into which it is divided by any of its points.

6. On any half-line from its starting point, a segment of a given length can be drawn, and only one.

7. From any half-line in a given half-plane, you can set aside an angle with a given degree measure less than 1800, and only one.

8. Whatever the triangle, there is an equal triangle at a given location with respect to a given half-line.

Triangle properties:

Relations between sides and angles of a triangle:

1) There is a larger angle opposite the larger side.

2) Opposite the larger angle lies the larger side.

3) Equal angles lie against equal sides, and conversely, equal sides lie against equal angles.

Relationship between interior and exterior angles of a triangle:

1) The sum of any two interior angles of a triangle is equal to the exterior angle of the triangle adjacent to the third angle.

2) The sides and angles of a triangle are also interconnected by relationships called the sine theorem and the cosine theorem.

The triangle is called obtuse, rectangular or acute , if its largest internal angle is respectively greater than, equal to, or less than 90∘.

middle line A triangle is a line segment that joins the midpoints of two sides of the triangle.

Properties of the middle line of a triangle:

1) The line containing the midline of the triangle is parallel to the line containing the third side of the triangle.

2) The middle line of the triangle is equal to half of the third side.

3) The midline of a triangle cuts off a similar triangle from a triangle.

Rectangle properties:

1) opposite sides are equal and parallel to each other;

2) the diagonals are equal and at the point of intersection they are divided in half;

3) the sum of the squares of the diagonals is equal to the sum of the squares of all (four) sides;

4) rectangles of the same size can completely tile the plane;

5) a rectangle can be divided into two equal rectangles in two ways;

6) the rectangle can be divided into two equal right triangles;

7) a circle can be described around the rectangle, the diameter of which is equal to the diagonal of the rectangle;

8) a circle cannot be inscribed in a rectangle (except for a square) so that it touches all its sides.

Parallelogram properties:

1) The middle of the diagonal of a parallelogram is its center of symmetry.

2) Opposite sides of a parallelogram are equal.

3) Opposite angles of a parallelogram are equal.

4) Each diagonal of a parallelogram divides it into two equal triangles.

5) The diagonals of the parallelogram are bisected by the point of intersection.

6) The sum of the squares of the diagonals of the parallelogram (d1 and d2) is equal to the sum of the squares of all its sides: d21+d22=2(a2+b2)

FROM square properties:

1) All corners of a square are right, all sides of a square are equal.

2) The diagonals of a square are equal and intersect at right angles.

3) The diagonals of a square bisect its corners.

Rhombus properties:

1. The diagonal of a rhombus divides it into two equal triangles.

2. The diagonals of the rhombus at the point of their intersection are divided in half.

3. Opposite sides of a rhombus are equal to each other, and its opposite angles are also equal.

In addition, the rhombus has the following properties:

a) the diagonals of the rhombus are mutually perpendicular;

b) the diagonal of a rhombus divides its angle in half.

Circle properties:

1) A straight line may not have common points with a circle; have one with the circle common point(tangent); have two common points with it (secant).

2) Through three points that do not lie on one straight line, it is possible to draw a circle, and moreover, only one.

3) The point of contact of two circles lies on the line connecting their centers.

Polygon properties:

1) The sum of the interior angles of a flat convex n-gon is equal to.

2) The number of diagonals of any n-gon is equal.

3). The product of the sides of the polygon by the sine of the angle between them is equal to the area of ​​the polygon.

This task is designed as a game in which the child has to change the properties of geometric shapes: shape, color or size. Such a developmental activity contributes to a more effective memorization of geometric shapes, since here the child not only remembers them visually, but also with the help of logical thinking changes their main properties by "processing" the figures in a magic factory.

In order to change the properties of geometric shapes in our magic factory, first read the instructions, download the task forms, print them out and prepare a simple pencil, an eraser and colored pencils in three colors - green, red and blue. Then the adult explains the rules of the game to the child.

"Now you and I are starting to work at the factory. There are special machines here that change the various characteristics of the figures: color, shape or size. Each figure that enters this machine is processed according to strict instructions and comes out already changed."

After that, the adult shows an example of how the machine works in this factory, changing the color of the figures:

Then the adult explains to the child the principle of operation of such a machine: "Any green figure that enters the car changes color to red (the arrow leads from a green circle with the letter "Z" to a red circle), any red figure changes to blue, and blue the shape changes to green.

There are other machines in the factory that change other properties of geometric shapes - not color (as in the example considered), but shape or size. Changes with figures occur according to a similar principle (we follow the arrows that show which figures the given ones should change to).

Also in some forms there are machines that change not one property of the figure, but two at once - for example, color and shape or shape and size.

Download tasks - Properties of geometric shapes - you can in attachments at the bottom of the page

In these tasks, you need to change only one property of the shapes - their color. Don't forget to color the shapes on the left before giving your child the task.

In the next task, you need to change another property of geometric shapes - their shape. The oval changes to a rectangle, the rectangle to a rhombus, the rhombus to an oval. Be careful! Ovals and rectangles in the task are different - horizontal and vertical. You need to change exactly the ones that are drawn in the car. Be sure to color in the shapes on the left before you get started.

In this task, the given figure first changes its shape (in the first car), and then its color (the second car).

In the next task, the machines change the size of the figures: large squares into small ones, small triangles into large ones.

On the following machines, we first change the shape of the figures, and then their size.

In this task, the figures change their color on the first machine, and their size on the second machine.

Well, the last task is the most difficult. Here the processing of the properties of the figures takes place on three machines. The first machine changes the color of the incoming geometric shapes, the second machine changes the size of some shapes, and the third machine completes the processing by changing their shape.

Groups of geometric shapes according to their characteristics

In this task, you will find groups of geometric shapes, each of which combines shapes according to some specific feature. For example, by color, shape or size. The child must determine on what basis the figures in each group are broken. Such activities develop the logical and mathematical abilities of children.

Download and print out the task forms, give the child and explain to him the rules for doing the exercise: “Look, geometric shapes are drawn here, which are divided into several groups. In each group, the shapes are united by one property or feature. figures of the same color (gray, white or black), the same shape (triangle, square or circle) or the same size (small, medium or large).

If it is difficult for a child to perform this exercise on his own, then help him with counter questions: "What geometric shapes do you see on the page? How do they differ from each other? What do they have in common?"

It is very important to conduct such classes systematically, using improvised materials. For example, you can use buttons of various shapes (square, round, oval, diamond-shaped and others), different colors, with different numbers of holes. The principle of completing the task is the same as in the submitted forms. An adult lays out buttons on the table, dividing them into groups according to a certain attribute. And the child must determine what is common in these groups. The lesson will be more effective if the child not only finds signs of groups, but also, at the request of an adult, will combine objects into different groups according to given characteristics.

You can download task forms - Groups of geometric shapes - in the attachments at the bottom of the page.

Properties of three-dimensional geometric shapes - Ladder of transformations

Here you will find an activity with which the child will learn to distinguish the properties of three-dimensional geometric shapes: color, shape and size. The lesson is presented in two versions of difficulty: easy (for children from 4 years old) and complicated (for children from 5-6 years old). An easy version of the task is in form No. 1, and a complicated one is in form No. 2. In forms No. 3 and No. 4 you can see the correct answers. Prepare colored pencils, printed forms with tasks and explain to the child the rules for doing the exercises:

"Look carefully at the picture. Here is a ladder of transformations of geometric shapes. Starting from the lowest step, each figure changes one of its properties with the transition to the next step: color (white, gray or black), shape (cube, cone or ball) or a value (large or small).For example, this large white ball (an adult shows an example of the transformation of a ball on form No. 1) on the second step changes its size and becomes small, on the third step it changes color from white to black, on the fourth - again becomes large, on the fifth step its shape changes and it turns into a cone.

Let the child analyze the transformations of the white ball in this example for some time in order to understand the logic of the transformations of the figures in the task. In the process of completing the task, the child must comment and justify his decisions and actions.

If the child liked the lesson, then you can invite him to independently draw another figure on the bottom step and draw the path of its transformations with a colored pencil. Similarly, you can draw another such staircase, and the child himself will draw the given figures on it and try to fill in all the steps with the figures, guided by the same rules as in the printed task.

You can download the task on the properties of three-dimensional figures in the attachments at the bottom of the page

Form number 1 - Easy option

Form number 2 - Complicated version

Form number 3 - Correct answers for an easy option

Form number 4 - Correct answers to the complicated option

Other materials on the study of geometric shapes will also be useful to you:

Fun and colorful tasks for children "Drawings from geometric shapes" are a very convenient learning material for preschool and younger children. school age on the study and memorization of basic geometric shapes.

Here you and your child can learn geometric shapes and their names with the help of fun picture tasks.

The tasks will introduce the child to the basic shapes of geometry - a circle, an oval, a square, a rectangle and a triangle. Only here is not a boring memorization of the names of the figures, but a kind of coloring game.

As a rule, they begin to study geometry by drawing flat geometric figures. The perception of the correct geometric shape is impossible without drawing it out with your own hands on a piece of paper.

This lesson will greatly amuse your young mathematicians. After all, now they will have to find familiar shapes of geometric shapes among many pictures.

Stacking shapes on top of each other is a geometry activity for preschoolers and junior schoolchildren. The meaning of the exercise is to solve addition examples. These are just unusual examples. Instead of numbers, here you need to add geometric shapes.

Here you can download tasks in pictures, which present the calculation of geometric shapes for math classes.

In this task, the child will get acquainted with such a concept as drawings of geometric bodies. In fact, this lesson is a mini-lesson on descriptive geometry.

Here we have prepared for you volumetric geometric shapes made of paper that need to be cut and glued. Cube, pyramids, rhombus, cone, cylinder, hexagon, print them on cardboard (or colored paper, and then stick on cardboard), and then give the child to remember.

Here we have prepared for you mental counting within 10 in the form of math tasks in pictures. These tasks form children's counting skills and contribute to more effective learning of simple mathematical operations.

And you can also play math games online from the fox Bibushi:

In this educational online game, the child will have to determine what is superfluous among 4 pictures. In this case, it is necessary to be guided by the signs of geometric shapes.

The sensory perception of the form of an object should be aimed not only at seeing, recognizing the forms, along with its other features, but also being able, by abstracting the form from the thing, to see it in other things as well. This perception of the shape of objects and its generalization is facilitated by the knowledge of standards by children - geometric shapes. Therefore, the task of sensory development is the formation in the child of the ability to recognize, in accordance with the standard (one or another geometric figure), the shape of various objects. Experimental data L.A. Wenger showed that children 3-4 months old have the ability to distinguish geometric shapes. Focusing on the new figure is evidence of this. Already in the second year of life, children freely choose a figure according to the model from such pairs: a square and a semicircle, a rectangle and a triangle. But children can distinguish between a rectangle and a square, a square and a triangle only after 2.5 years. The selection according to the model of figures of a more complex shape is available approximately at the turn of 4-5 years, and the reproduction of a complex figure is carried out by individual children of the fifth and sixth year of life.

At first, children perceive geometric shapes unknown to them as ordinary objects, calling them by the names of these objects:

cylinder - glass, column,

oval - testicle,

triangle - sail or roof,

rectangle - a window, etc.

Under the teaching influence of adults, the perception of geometric shapes is gradually being rebuilt. Children no longer identify them with objects, but only compare: a cylinder is like a glass, a triangle is like a roof, etc. And, finally, geometric figures are beginning to be perceived by children as standards, with the help of which knowledge of the structure of an object, its shape and size is carried out not only in the process of perceiving one or another form with vision, but also through active touch, feeling it under the control of vision and designation with a word.

The joint work of all analyzers contributes to a more accurate perception of the shape of objects. In order to better know an object, children tend to touch it with their hands, pick it up, turn it; moreover, viewing and feeling are different depending on the shape and design of the object being cognised. Therefore, the main role in the perception of an object and determining its form is played by an examination carried out simultaneously by visual and motor-tactile analyzers, followed by designation with a word. However, among preschoolers, there is a very low level of examination of the shape of objects; most often they are limited to cursory visual perception and therefore do not distinguish figures that are close in similarity (an oval and a circle, a rectangle and a square, different triangles). In the perceptual activity of children, tactile-motor and visual techniques gradually become the main way of recognizing the form. Examination of the figures not only provides a holistic perception of them, but also allows you to feel their features (character, directions of lines and their combinations, formed corners and peaks), the child learns to sensually distinguish the image as a whole and its parts in any figure. This makes it possible in the future to focus the child's attention on a meaningful analysis of the figure, consciously highlighting the structural elements in it (sides, corners, vertices). Children are already consciously beginning to understand such properties as stability, instability, etc., to understand how vertices, corners, etc. are formed. Comparing three-dimensional and flat figures, children already find a commonality between them (“A cube has squares”, “A bar has rectangles, a cylinder has circles”). Comparing a figure with the shape of an object helps children understand that different objects or parts of them can be compared with geometric shapes. So, gradually the geometric figure becomes the standard for determining the shape of objects.

Stages of learning:

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

The second stage of teaching children 5-6 years old should be devoted to the formation of systemic knowledge about geometric shapes and the development of their initial techniques and methods of "geometric thinking".

"Geometric thinking" is quite possible to develop even in preschool age. There are several different levels in the development of "geometric knowledge" in children.

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 the figures, perceives each of them separately. At the second level, the child already singles out the elements in the figure and establishes relationships both between them and between individual figures, but does not yet realize the commonality between the figures.

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

The place and nature of the use of visual (sample, display) and verbal (instructions, explanations, questions, etc.) teaching methods are determined by the level of assimilation of the material being studied by the children. When children get acquainted with new types of activity (counting, counting, comparing objects by size), a complete, detailed display and explanation of all methods of action, their nature and sequence, a detailed and consistent examination of the sample are necessary. Instructions encourage children to follow the actions of the teacher or the child called to his table, introduce them to the exact verbal designation of these actions. Explanations should be short and clear. The use of words and expressions incomprehensible to children is unacceptable.

In working with 5-year-old children, the role of verbal teaching methods increases. Instructions and explanations of the teacher direct and plan the activities of children. When giving instructions, he takes into account what children know and can do, and shows only new methods of work. The questions of the teacher during the explanation stimulate the manifestation of independence and ingenuity by children, prompting them to look for different ways to solve the same problem: “What else can be done? Verify? Say?"

As the ability to perform certain actions is accumulated, the child can be asked to first suggest what and how to do (build a number of objects, group them, etc.), and then perform a practical action. This is how children are taught to plan ways and order of completing a task.

The assimilation of the correct turns of speech is ensured by their repeated repetition in connection with the performance of different variants of tasks of the same type.

Knowledge of geometric shapes, their properties and relationships expands the horizons of children, allows them to more accurately and versatile perceive the form. surrounding objects, which positively affects their productive activities (drawing, modeling).

Stages of development of the ability to determine the shape of surrounding objects.

One of the properties of surrounding objects is their shape. The form of objects received a generalized reflection in geometric figures. Geometric figures are standards, using which a person determines the shape of objects and their parts.

The problem of introducing children to geometric shapes and their properties should be considered in two aspects:

in terms of sensory perception of the forms of geometric figures and their use as standards in the knowledge of the forms of surrounding objects, in the sense of knowing the features of their structure, properties, main connections and patterns in their construction, i.e. proper geometric material.

It is known that an infant recognizes by the shape of the bottle the one from which he drinks milk, and in the last months of the first year of life, a tendency is clearly revealed to separate some objects from others and to separate the figure from the background. The contour of an object is that common beginning, which is the starting point for both visual and tactile perception. However, the question of the role of the contour in the perception of form and the formation of a holistic image requires further development.

Primary mastery of the form of an object is carried out in actions with it. The form of an object, as such, is not perceived separately from the object, it is its integral feature. Specific visual reactions of tracing the contour of an object appear at the end of the second year of life and begin to precede practical actions.

The actions of children with objects at different stages are different. Toddlers tend, first of all, to grab an object with their hands and begin to manipulate it. Children 2.5 years old, before acting, in some detail visually and tactile-motorly get acquainted with objects. The importance of practical action remains paramount. From this follows the conclusion about the need to direct the development of perceptual actions of two-year-old children. Depending on the pedagogical guidance, the nature of children's perceptual actions gradually reaches a cognitive level. The child begins to be interested in various features of the object, including the shape. However, for a long time he cannot single out and generalize this or that feature, including the shape of various objects.

Teaching the ability to distinguish and name geometric shapes, compare and group them according to different criteria. Formation of generalizing concepts.

Second junior group

For the implementation of program tasks, as a didactic material in this group, models of the simplest flat geometric shapes (circle, square) of different colors and sizes are used.

Even before conducting systematic classes, the teacher organizes children's games with building materials, sets of geometric shapes, and geometric mosaics. During this period, it is important to enrich the perception of children, to accumulate them ideas about various geometric shapes, give their correct name. In the classroom, children are taught to distinguish and correctly name geometric shapes - a circle and a square. Each figure is known in comparison with the other. In the first lesson, the primary role is given to teaching children the methods of examining figures in a tactile-motor way under the control of vision and mastering their names. The teacher shows the figure, calls it, asks the children to pick up the same one. Then the teacher organizes the actions of children with these figures: roll a circle, put it, put a square, check whether it will roll. Children perform similar actions with figures of a different color and size. In conclusion, two or three exercises are carried out to recognize and label shapes with words (“What do I hold in my right hand, and what in my left?”; “Give the bear a circle, and parsley a square”; “Put one square on the top strip, and on the bottom many circles”, etc.).

In subsequent lessons, a system of exercises is organized in order to consolidate the children's ability to distinguish and correctly name geometric shapes: a) exercises to choose from according to the model: "Give (bring, show, put) the same." The use of the sample can be variable: only the shape of the figure is emphasized, no attention is paid to its color and size; figures of a certain color, a certain size and a figure of a certain color and size are considered; b) exercises to choose from according to the words: “Give (bring, show, put, collect) circles”, etc .; exercise options may contain instructions for choosing a figure of a certain color and size; c) exercises in the form of didactic and outdoor games: “What is this?”, “Wonderful bag”, “What is missing?”, “Find your house”, etc.

middle group

In children of the fifth year of life, it is necessary first of all to consolidate the ability to distinguish and correctly name a circle and a square, and then a triangle. For this purpose, game exercises are carried out in which children group figures of different colors and sizes. The color, size changes, but the signs of the form remain unchanged. This contributes to the formation of generalized knowledge about the figures. To clarify the ideas of children that geometric shapes come in different sizes, they. show (on a table, flannelgraph or typesetting canvas) known geometric figures. For each of them, children select a similar figure, both larger and smaller. By comparing the size of the figures (visually or by overlaying), the children establish that the figures are the same in shape, but different in size. In the next exercise, the children lay out three figures of different sizes in ascending or descending order. Then you can invite the children to consider the figures lying in individual envelopes, arrange them in rows of the same shape and offer to tell who has how much.

In the next lesson, children receive already unequal sets of figures. They, sorting out their sets, report who has what pieces and how many of them. At the same time, it is advisable to exercise children in comparing the number of figures: “Which figures do you have more and which are fewer? Do you have an equal number of squares and triangles? etc. Depending on how the geometric figures are assembled in individual envelopes, equality or inequality can be established between their number.

Performing this task, the child compares the number of figures, establishing a one-to-one correspondence between them. In this case, the techniques can be different: the figures in each group are arranged in rows, exactly one under the other, or are arranged in pairs, or overlap each other. One way or another, a correspondence is established between the elements of the figures of the two groups, and on this basis their equality or inequality is determined.

In a similar way, exercises are organized to group and compare figures by color, and then by color and size at the same time. Thus, constantly changing the visual material, we get the opportunity to exercise children in highlighting features that are essential and insignificant for a given object. Similar activities can be repeated as children learn new shapes.

Children are introduced to new geometric shapes by comparing with already known ones: a rectangle with a square, a ball with a circle, and then with a cube, a cube with a square, and then with a ball, a cylinder with a rectangle and a circle, and then with a ball and a cube. Examination and comparison of figures is carried out in a certain sequence: a) mutual overlap or application of figures; this technique allows you to more clearly perceive the features of figures, similarities and differences, highlight their elements; b) organization of examination of figures by tactile-motor way and selection of some elements and features of the figure; the effect of examining a figure largely depends on whether the teacher directs the children’s observations with his word, whether he indicates what to look at, what to learn (direction of lines, their connection, proportions of individual parts, the presence of corners, vertices, their number, color, size figures of the same shape, etc.); children must learn to verbally describe a particular figure. c) organization of various actions with figures (roll, put, put in different positions); acting with models, children reveal their stability or instability, characteristic properties. For example, children try to place a ball and a cylinder in different ways and discover that the cylinder can stand, it can lie down, it can roll, but the ball “always rolls”. Thus, the characteristic properties of geometric bodies and figures are discovered; d) organizing exercises for grouping figures in order of increasing and decreasing size (“Pick up by shape”, “Pick up by color”, “Put it in order”, etc.);

e) the organization of didactic games and game exercises to consolidate children's skills to distinguish and name figures (“What's gone?”, “What has changed?”, “Wonderful bag”, “Domino shapes”, “Shop”, “Find a pair”, etc. .).

Senior group

As already noted, the main task of teaching children 5-6 years old is the formation of a system of knowledge about geometric shapes. The initial link in this system is the idea of ​​some features of geometric shapes, the ability to generalize them on the basis of common features. Children are given figures known to them and are invited to examine the boundaries of a square and a circle, a rectangle and an oval with their hands and think about how these figures differ from each other and what is the same in them. They establish that the square and rectangle have corners, while the circle and oval do not. The teacher, tracing the figure with his finger, explains and shows the corners, vertices, sides of the figure on the rectangle and square. The vertex is the point where the sides of the figure meet. The sides and vertices form the border of the figure, and the border, together with its interior area, forms the figure itself.

On different figures, children show its inner area and its border - sides, vertices and corners as part of the figure's inner area1. You can invite the children to shade the inner area of ​​\u200b\u200bthe figure with a red pencil, and circle its border and sides with a blue pencil. Children not only show the individual elements of the figure, but also count the vertices, sides, angles of different figures. Comparing a square with a circle, they find out that a circle has no vertices and corners, there is only a circle border - a circle. In the future, children learn to distinguish between the inner area of ​​any figure and its border, count the number of sides, vertices, angles. Examining the triangle, they come to the conclusion that it has three vertices, three angles and three sides. Very often, children themselves say why this figure, unlike a rectangle and a square, is called a triangle.

To convince the children that the features they have identified are characteristic properties of the analyzed figures, the teacher offers the same figures, but of larger sizes. Examining them, children count the vertices, angles and sides of squares, rectangles, trapezoids, rhombuses and come to the general conclusion that all these figures, regardless of size, have four vertices, four corners and four sides, and all triangles have exactly three vertices, three corners and three sides. In such classes, it is important to put the children themselves in the position of those who are looking for an answer, and not be limited to communicating ready-made knowledge.

Angle (flat) - a geometric figure formed by two rays (sides) emerging from one point (vertex). It is necessary to teach the children to draw their own conclusions, to clarify and to generalize their answers. Such a presentation of knowledge puts children in front of questions to which it may not always be easy for them to find the right answer, but the questions make the children think and listen more carefully to the teacher. So, one should not rush to give the children ready-made tasks: it is necessary, first of all, to arouse interest in them, to provide the possibility of action. The task of the educator is to pedagogically correctly show the ways and methods of finding the answer.

The program of education and training in kindergarten It is planned to introduce older preschoolers to quadrangles. To do this, children are shown a lot of shapes with four corners and are asked to come up with a name for this group on their own. The children's proposals "four-sided", "quadrangular" must be approved and clarified that these figures are called quadrangles. This way of introducing children to the quadrilateral contributes to the formation of generalization. Grouping figures according to the number of corners, vertices, sides abstracts the thought of children from other, insignificant features. Children are led to the conclusion that one concept is included in another, more general one. This way of assimilation is most appropriate for the mental development of preschoolers.

In the future, the consolidation of children's ideas about quadrangles can go by organizing exercises on the classification of figures of different sizes and colors, sketching quadrangles different kind on lined paper, etc.

preschool group

Knowledge of geometric shapes preparatory group expand, deepen and systematize.

One of the tasks of the group preparatory to school is to introduce children to the polygonal and its features: vertices, sides, angles. Solving this problem will allow children to generalize: all figures that have three or more corners, vertices, sides belong to the group of polygons. Children are shown a model of a circle and a new figure - a pentagon. They offer to compare them and find out how these figures differ. The figure on the right differs from the circle in that it has angles, many angles. Children are invited to roll a circle and try to roll a polygon. It doesn't roll on the table. Angles interfere with this. They count angles, sides, vertices and establish why this figure is called a polygon. Then a poster is shown showing various polygons. Individual figures have their characteristic features. All figures have many sides, vertices, angles. How can you call all these figures in one word? And if the children do not guess, the teacher helps them.

To clarify knowledge about the polygon, tasks can be given to draw figures on paper in a cage. Then you can show different ways to transform the shapes: cut or bend the corners of the square and get an octagon. By superimposing two squares on top of each other, you can get an eight-pointed star. The exercises of children with geometric figures, as in the previous group, consist in identifying them by color, size in different spatial positions. Children count vertices, angles and sides, arrange shapes by their size, group by shape, color and size. They must not only distinguish, but also depict these figures, knowing their properties and features. For example, the teacher invites children to draw two squares on paper in a cage: one square should have a side length of four cells, and the other should have two cells more.

After sketching these figures, the children are invited to divide the squares in half, and in one square to connect two opposite sides with a segment, and in the other square to connect two opposite vertices; tell how many parts the square was divided into and what figures turned out, name each of them. In such a task, counting and measurement are simultaneously combined with conditional measurements (the length of the side of the cell), figures of different sizes are reproduced based on knowledge of their properties, figures are identified and named after dividing the square into parts (whole and parts).

According to the program, the preparatory group should continue to teach children how to transform figures. This work contributes, on the one hand, to the knowledge of figures and their features, and on the other hand, develops constructive and geometric thinking. The methods of this work are varied. Some of them are aimed at getting to know new figures when they are divided into parts, others - at creating new figures when they are combined.

Children are offered to fold the square in half in two ways: by combining opposite sides or opposite corners - and say which figures turned out after bending (two rectangles or two triangles). You can offer to find out what shapes turned out when the rectangle was divided into parts (Fig. 39), and how many shapes are now in total (one rectangle, and there are three triangles in it). Of particular interest to children are entertaining exercises for the transformation of figures. So, the analytical perception of geometric shapes develops in children the ability to more accurately perceive the shape of surrounding objects and reproduce objects when drawing, sculpting, and appliqué. By analyzing the different qualities of the structural elements of geometric figures, children learn the common thing that unites the figures. So, the guys will learn that some figures are in a subordinate relationship; the concept of a quadrilateral is a generalization of such concepts as "square", "rhombus", "rectangle", "trapezium", etc.; the concept of "polygon" includes all triangles, quadrangles, pentagons, hexagons, regardless of their size and type. Such interconnections and generalizations, quite accessible to children, raise their mental development to a new level. Children develop cognitive activity, new interests are formed, attention, observation, speech and thinking and its components (analysis, synthesis, generalization and concretization in their unity) develop. All this prepares children for the assimilation of scientific concepts in school.

The connection of quantitative representations with representations of geometric figures creates the basis for the general mathematical development of children.

Methods of dating children preschool age with the properties of geometric shapes.

Stages of introducing children to geometric shapes

Stage 1 (up to 3 years). We organize the performance of characteristic actions with objects of various shapes, we enter the name of geometric shapes in the passive dictionary of children. The kindergarten teacher uses common terms from the very beginning. Most often, young children use the name of a frequently occurring object for the name of the form. At the first stage, this is acceptable. However, one should not impose on a child a substitute word invented by an adult. The teacher can repeat his name after the child, but immediately pronounce the correct name in parallel.

At 3 years old, the name of geometric shapes is gradually translated into the active dictionary of children. To do this, children are asked questions: “What is this? What is the name of?"

Exercises are offered to find the figure by the model, and then by the name.

Stage 2 (3 - 6 years). We teach children to recognize the properties of geometric shapes based on the comparison of figures with each other. Enter the name of the shapes in the active dictionary. First, highly contrasting figures of the same volume are compared with each other, and then low-contrast figures of the same volume, and, finally, low-contrast figures of different volume (for example, a circle and a ball).

For children 3-4 years old, show and compare:

1. Circle and square (rolling - not rolling, no obstacles, there are obstacles);

2. Triangle and circle (rolling - not rolling, no obstacles, there are obstacles);

3. Square and triangle (they differ in the number of corners: one figure has 4 corners, the other has 3);

4. Ball and cube (rolling - not rolling, no obstacles - there are obstacles, you can build a turret - you can not build a turret);

1. Rectangle and square (not all sides are equal - all sides are equal);

2. Oval and circle (not all axes are equal - all axes are equal)

3. A cylinder with a ball and a cube (in one position, the cylinder has the properties of a ball, in another position, a cube);

4. A cone and a cylinder (a cone has a different thickness at the bottom and at the top, a cylinder has the same thickness, you can’t build a turret from cones; the cylinder rolls linearly, and the cone rolls in a circle);

1. Rhombus and square (a square has all angles equal, a rhombus has not all angles equal);

2. Trapezoid and rectangle (equality of angles, opposite sides; parallelism of opposite sides);

3. Pyramid and cone (different side surfaces, bases);

4. Ovaloid and ball (ovaloid rolls in one direction, and the ball rolls in different directions; the ball has the same thickness from bottom to top and from left to right, while the ovaloid has different thicknesses);

5. A quadrangular prism and a cube (a cube has equal edges, a prism has unequal ones);

6. Triangular prism and quadrangular (different base shapes; it is not always possible to build a turret from a triangular prism);

7. Ovaloid and cylinder (ovaloid is unstable in any position).

8. Comparison of flat and three-dimensional figures. We compare a circle with a ball, a square with a cube, an oval with an ovaloid, a rectangle with a prism, a rectangle with a cylinder, a triangle with a cone, a triangle with a pyramid, a triangle with a triangular prism.

Stage 3 (5-6 years). Tasks:

1. Teach children to generalize shapes in shape.

Children are given several models of the same figure, which differ in various ways (color, size, proportions of parts, location in space). It is proposed to examine all models and say what is common (characteristic features are indicated). Then the children have to name the figures in one word. Exercises are given for grouping figures (for various reasons)

2. Learn to determine the shape of surrounding objects.

Children are offered the most miscellaneous items, the question is raised: “what do these objects have in common?” Children must abstract from other properties and perceive the form as a property of the object.

Exercises:

Determine the shape of the shown object;

The host calls the form, and the children must find (name) an object of the same shape.

Games: "Geometric Lotto"; “Dapamazhy Oli” (cards are offered, divided into cells, a figure is depicted in the center, children select cards of the desired shape and fill in the windows); "Geometric Domino"; "Who will call correctly"; “Who will find it faster” (the host calls the form, the children are looking for objects of this form).

Notes:

It is very important to correctly reflect the shape of objects in speech. There are the following options:

1. For the name of the shape of the object, the name of the geometric figure is used.

The wardrobe (bedside table) has the shape of a quadrangular prism,

The surface of the table has the shape of a rectangle.

2. An adjective derived from the name of a geometric figure (rectangular) is used. Here it is necessary to indicate: volumetric form or planar (rectangular volumetric cabinet, table surface - rectangular flat).

The teacher must ensure that children do not use the name of flat geometric shapes to indicate the shape of three-dimensional objects in speech.

Methods for introducing children to the properties of geometric shapes

What is the name of?

Provocative (we show a new figure (oval) and ask: “Is this a circle?”)

How are they similar?

What is the difference?

Tactile motor examination. We examine flat figures with fingers, voluminous ones with a palm

Counting angles, sides; quantity comparison.

Comparison of sides, angles and axes in magnitude by superposition, by bending or by using a conventional measure. To compare angles in magnitude, a conditional measure equal to a right angle is used.

Figure rolling.

Superposition of one figure on another. When imposing, attention is drawn to the fact that the figures differ in the presence of extra pieces.

Building a turret (only for 3D items).

Hiding in the palms of figures (we check a flat or three-dimensional figure).

Creating the shape of an object: drawing, painting, cutting out flat figures, modeling and constructing three-dimensional figures.

Grouping exercises.

The figures differ only in form,

Figures of different colors, sizes, proportions.

Exercises to create a figure from parts.

Didactic games.

Finding a figure according to the model (“Find your house”, “Whose house will assemble faster”, “Cars and garages”).

Finding a figure by name (“Wonderful bag”, “Give me a named figure”).

Finding a figure according to the description (listing characteristic properties), "Guess".

Compilation of figures from parts (puzzle games: "Pythagoras", "Tangram", "Calumbo Egg", are actively used in the "Childhood" program).

Laying out figures from sticks. At the first stage, sticks of the same size are offered in the middle group, most often counting, matches cannot be used.

Task types

1. Construct a triangle, square, rectangle. After formulating the task, we analyze the figures and find out how many sides, angles, whether the sides are equal, how many sticks to take.

If children have difficulties, then an individual sample is given.

2. Provocative task: lay out a circle of sticks (it is impossible - the circle has no sides).

3. The task of an entertaining character for ingenuity: lay out two triangles of 5 sticks.

At the 2nd stage (senior group). In addition to sticks of the same length, we offer sticks of different lengths:

Build figures of different sizes;

Construct triangles with sides of different lengths;

Build a trapezoid, a rhombus.

Beforehand, children are asked questions (as in the first stage).

Tasks for ingenuity.

How to get a trapezoid from a rectangle. Offer one stick to make another figure.

You can offer to lay out a house, a boat, etc.

Methods for showing the difference between flat and three-dimensional figures:

We cover the figure on the table with a straight palm. If the palm touches the table, the figure is flat, if not, it is voluminous. Or: if the figure is hidden in the palms, then it is flat, if not, it is voluminous. Flat figures are “letters”, and voluminous “parcels” that do not fit into the mail slot.

Angle count is applied (for example, a square has 4, and a cube has 8).

Flat figures can be depicted on a sheet of paper in the process of drawing or application, and voluminous figures can be depicted in the process of modeling or constructing from paper or building parts. If it is necessary to draw a three-dimensional object, then we depict it in the form of a corresponding flat figure.

Notes on the rectangle.

1. First, the difference between a rectangle and a square is shown by overlaying. Pieces protrude from the square, which means the figures are different.

2. For a square, all sides are equal, but for a simple rectangle, adjacent sides are not equal. We check this with one of the following methods:

Bending the sheet until adjacent sides are aligned;

The use of a conditional measure.

It is important that children understand that a square is a rectangle. We can say that a square is a magic rectangle (all sides are equal). IN senior group a generalization of the concept of "rectangle" is carried out, the concept of "right angle" is preliminarily explained. First, let's define what an angle is.

We show and call that this piece of the plane is an angle (a part of the plane between sides that have a common point).

In order to give an idea of ​​the right angle, 2 pictures are considered:

1. The tree grows evenly, straight, which means there is a right angle between the tree and the ground.

2. The wind blew and the tree leaned over. The tree is not standing straight, so the angle is not straight.

Next, various figures are considered, their angles are compared and measured using a conditional measure. equal in size to a right angle. So that children do not confuse an angle with a triangle, the edge of the conditional measurement should not be a straight line.

Exercises are carried out on applying measurements to the corners of different figures. The origin of the word "rectangle" is explained: "straight" + "angle".

Exercise: measure the angles of objects in a group room using a conventional measure.

Notes on the oval. A more accurate way to show the difference between an oval and a circle is to measure the axes. Explanation of the concept of "axis": "A circle and an oval have no sides, we will draw a line inside the figures through the middle of the figure from one edge to the other. These lines are called "axes". Examples are given of rounded objects that have an axis, leading to the conclusion: a circle has all the axes equal to each other, while an oval has none. There are two ways to measure the axes: using a conditional measure or by bending along the axis.

Remarks on the rhombus. In senior age first, the similarity between a rhombus and a square is shown (4 corners; 4 sides, all sides are equal).

The difference is that not all angles of a rhombus are equal. This is shown using a conditional measure equal to the right angle. Acquaintance with a rhombus occurs in the process of application and drawing.

Notes on the trapezoid. In senior age when comparing a trapezoid with a rectangle, the following differences stand out:

1) A trapezoid does not have all right angles.

2) the parallel opposite sides of the trapezoid are not equal (checked by bending until the opposite sides are aligned, or by measuring with a conventional measure).

3) A trapezoid has 2 sides that are slanted (not parallel).

Parallelism is explained to children by showing that the distance between the sides of the rectangle is the same, but there is no distance between the sides of the trapezoid. We give examples of parallelism: electrical wires, rails, pieces of furniture.

Then the trapezoid is compared with a triangle (the roof can be of different shapes). Differences: a triangle has 3 angles and 3 sides, and a trapezoid has 4 angles and 4 sides.

Applique lessons show how to get a trapezoid first from a rectangle, and then from a triangle.

Cylinder notes. Average age a cylinder is compared to a sphere and a cube. First, it is shown how the cylinder is similar and how it differs from the ball, and then from the cube.

The cylinder for comparison with the ball is placed on its side and the similarities of the figures stand out:

1) the lateral surface of both figures has no obstacles.

2) the ball and the cylinder are rolling.

3) if you put a ball on a ball and a cylinder on a cylinder, then the turret does not work.

Then the cylinder is turned over on the base, so it does not look like a ball (there is an obstacle, it does not roll, a turret can be built from cylinders). It is noteworthy that in this position it looks like a cube. The conclusion is made: the cylinder is a cunning figure, if it lies on its side, it looks like a ball, if it stands on the base, then it looks like a cube.

At an older age, the cylinder is compared with the ovaloid in the process of modeling. First, it turns out how these figures are similar. Then the only difference is shown: if the cylinder is on the base, then it is stable, and the ovaloid is unstable in any position. There are also differences in sculpting techniques.

Cone notes. Differences between a cone and a cylinder:

1) a turret can be built from cylinders; but from cones - it is impossible;

2) the cylinder rolls forward - backward, the cone - in a circle;

3) the cylinder has both the floor and the ceiling in the shape of a circle;

4) the thickness of the cylinder at the bottom and top is the same, the cone is thick at the bottom, and thin at the top.

In senior asc. with a cone we compare a pyramid and a triangular prism.

The difference between a pyramid and a cone:

1) the pyramid has a ribbed side surface.

2) the base of the cone is a circle, the base of the pyramid is a polygon.

The difference between a cone and a triangular prism:

1) the surface of the prism is not smooth, ribbed,

2) the prism does not roll,

3) a triangular prism has 2 sharp peaks when lying on its side.

4) a triangular prism has a different base shape,

5) different number of vertices.

Similarity: both figures are used as a roof.

Prism notes. Acquaintance with a prism occurs at an older age on the basis of a comparison with a cube (similarly to how a rectangle was compared with a square).

Differences: all sides of a cube (edges) are equal, but for a general prism, adjacent sides are not equal (measured by a conventional measure).

By the end of Art. age, the differences between 4-angle and 3-angle prisms are shown:

The base of a 4-angled prism is a quadrilateral, and that of a triangular prism is a triangle. That is why they are called differently.

A 4-angled prism is stable (you can build a turret) if it lies on a side face, while a 3-angled prism is not. This figure is used as a roof in construction.

Notes on the ovaloid. The differences between the ovaloid and the ball are in the distinctive techniques in sculpting the figures: the ball is rolled out in a circular motion, the ovaloid is only forward and backward. It is shown that they have different thicknesses (usually on modeling). There are 2 ways:

With the help of a conditional measure - sticks. If you pierce the ball vertically and horizontally, then the thickness is the same. If you pierce an ovaloid, then the thickness is different.

With the help of a conditional measure - a thread - you can wrap the ball first vertically and then horizontally. For a ball, the length of the thread is the same. For an ovaloid, you need a thread of different lengths.

Stages of assimilation of space. Sensual and speech basis of spatial orientations.

Orientation in space requires the ability to use any reference system. In the period of early childhood, the child is oriented in space on the basis of the so-called sensory reference system, that is, along the sides of his own body. At preschool age, the child masters the verbal reference system in the main spatial directions: forward-backward, up-down, right-left. During the period of schooling, children master a new reference system - along the sides of the horizon: north, south, west, east. The development of each next frame of reference is based on a solid knowledge of the previous one. Thus, it has been established that the assimilation of the sides of the horizon by students of -V classes depends on the ability to differentiate the main spatial directions into geographical map . North, for example, is initially associated in children with the spatial direction above, south - below, west - with the direction to the left and east - with the location on the right. The differentiation of the main spatial directions is due to the level of orientation of the child “on himself”, the degree of mastery of the “scheme of his own body”, which, in essence, is the “sensory frame of reference”. Later, another reference system is superimposed on it - verbal. This happens as a result of assigning to the directions sensibly distinguished by the child the names relating to them: up, down, forward, backward, right, left. Thus, preschool age is the period of mastering the verbal frame of reference in the main spatial directions. How does the child master it? The child correlates the distinguished directions primarily with certain parts of his own body. This is how connections of the type are ordered: above - where is the head, and below - where are the legs, in front - where is the face, and behind - where is the back, to the right - where is the right hand, to the left - where is the left. Orientation on one's own body serves as a support in the development of spatial directions by the child. Of the three paired groups of main directions corresponding to the main axes of the human body (frontal, vertical and sagittal), the upper one stands out first, which is apparently due to the predominantly vertical position of the child's body. The isolation of the lower direction, both the opposite side of the vertical axis, and the differentiation of paired groups of directions characteristic of the horizontal plane (forward-backward, right-left), occurs later. Obviously, the accuracy of orientation on a horizontal plane in accordance with its characteristic groups of directions is a more difficult task for a preschooler than the differentiation of various planes (vertical and horizontal) of three-dimensional space. Having assimilated in the main groups of pairwise opposite directions, a small child is still mistaken in the accuracy of discrimination within each group. This is convincingly evidenced by the facts of mixing by children of the right with the left, the upper with the lower, the spatial direction forward with the opposite backward direction. Of particular difficulty for preschoolers is the distinction between right and left, which is based on the process of differentiation of the right and left sides of the body. Consequently, the child only gradually masters the understanding of the pairing of spatial directions, their adequate designation and practical distinction. In each of the pairs of spatial designations, first one is singled out, for example: under, on the right, above, behind, and on the basis of comparison with the first, the opposite ones are also recognized: above, on the left, below, in front. This should be taken into account in the teaching methodology, consistently forming interconnected spatial representations. How does a child acquire the ability to apply or use the system of reference he has mastered when orienting himself in the surrounding space? The first stage begins with "practical trying on", which is expressed in the actual correlation of surrounding objects with the starting point of reference. At the second stage, a visual assessment of the location of objects located at some distance from the starting point appears. In this case, the role of the motor analyzer is exceptionally great, the participation of which in spatial discrimination gradually changes. At first, the whole complex of spatial-motor connections is presented in a very detailed way. For example, a child leans back against an object and only after that says that this object is located behind; touches with his hand an object that is on the side, and only then says which side of it - on the right or on the left - this object is located, etc. sides of his own body. Direct movement to the object to establish contact with it is replaced later by the rotation of the body, and then by the pointing movement of the hand in the desired direction. Further, the broad pointing gesture is replaced by a less noticeable movement of the hand. The pointing gesture is replaced by a slight movement of the head and, finally, only a look turned towards the object being determined. So, from a practically effective method of spatial orientation, the child passes to another method, which is based on a visual assessment of the spatial placement of objects relative to each other and the subject that determines them. This perception of space is based on the experience of direct movement in it. With the acquisition of experience of spatial orientation in children, the intellectualization of outwardly expressed motor reactions occurs. The process of their gradual curtailment and transition to the plan of mental actions is a manifestation of a general trend in the development of mental action from a materialized, practical one. Peculiarities of children's orientation on the ground With the development of spatial orientation, the nature of the reflection of the perceived space also changes and improves. The perception of the external world is spatially dissected. Such dismemberment is "imposed" on our perception by the objective property of space - its three-dimensionality. Correlating objects located in space with different sides of his own body, a person, as it were, dismembers it in the main directions, i.e., perceives the surrounding space as a terrain, respectively divided into different zones: front (right-sided, left-sided) and rear (also right-sided and left-sided) . But how does a child come to such perception and understanding? What are the opportunities for preschoolers? At first, the child considers objects located in front, behind, to the right or to the left of himself only those that are directly adjacent to the corresponding sides of his body or as close as possible to them. Consequently, the area on which the child is oriented is at first extremely limited. The orientation itself is carried out in this case in contact proximity, that is, in the literal sense of the word, on oneself and away from oneself.

Features of mastering the ways of spatial orientation according to the scheme of one's own body, according to the arrangement of objects, according to the directions of space.

The younger preschooler orients himself on the basis of the so-called sensory frame of reference, i.e. on the sides of your body. Therefore, it is proposed to teach children distinguish between left and right hands, directions from oneself: forward (in front), backward (behind), above, below. Spatial representations develop in children of the fourth year of life, mainly during regime moments, in outdoor games, in all classes.

At the beginning of the school year, check if the kids know the names of their body parts, faces. Only after that you can teach them to determine the direction, focusing on themselves. For example, forward means facing me, behind means behind my back, etc.

Children should be introduced to the names of both hands (at the same time) and their various functions. For example, in drawing classes, a child is taught to hold a sheet of paper with his left hand so that it does not slide on the table, and to hold a pencil with his right hand. In applique classes, he learns to hold a brush with his right hand, spread what he sticks on, and hold it with his left hand, blot it with a cloth. In physical education, music classes, children are taught to orient themselves: “Let's go forward, turn back. Olya, stand in front. Seryozha, stand behind Olya.

Learn n Directions forward, backward, left, right help the game with the use of pointer arrows. On a walk, the teacher quietly hides the toy and tells the kids that an arrow will help to find it, the sharp end of which shows where to go.

Hanging ball games contribute to the assimilation of the concepts of up and down. In a ball consisting of two halves, a tape is clamped. It is hung on a crossbar above the height of the child. The teacher invites the children to swing the ball, then unnoticed by them raises the ball higher. Children reach out with their hands, but cannot reach. The teacher explains: “The ball is high - you can’t get it, but now I’ll lower it down so that I can swing it.” As soon as the children begin to swing the ball, the teacher picks it up again and asks: “Where is the ball, why don’t you play with it?” Then he clarifies: “The ball is at the top, and now it will be at the bottom again.”

To consolidate spatial directions, you can use another game - "Where does the bell ring?". Children become a semicircle, close their eyes. The teacher walks in a circle, stopping in turn at each child, and rings a bell to the left, then to the right of him, then above, then below. The child determines from which side the sound is heard. Opening his eyes, he can first show the direction with his hands, and then call it. In order not to disorientate children, the teacher must remember that in the classroom, where the special task of forming spatial representations is being solved, it is impossible to put or put the children against each other, in a circle, as this violates the uniformity of the perception of space.

The difference between children of the main directions from themselves in statics and during movements. Development of the ability to navigate in the surrounding space from oneself, from objects, determining the position of objects in relation to each other.

The tasks of orientation in space are becoming more complicated: children not only learn to identify direction away from you, but also move in that direction. Here you can use various game techniques and games like " Find the hidden toy”, “Where will you go and what will you find?”, "Travel" etc.

For example, in the game "Find the hidden toy", the child goes out the door, and everyone else hides the toy. In order to find it, the direction is indicated to the incoming person in one case verbally: “Go from the table to the carpet, turn right from the carpet, take three steps and look there!” Another time, the teacher marks the direction on the floor of the group room with arrows of different colors, and the child says: “First go where the red arrow points, then turn where the blue one points, then go three steps and look there.” When turning, the child must say where he turned: to the right or to the left.

Children also learn to identify and label the position of objects in relation to oneself. For example: “In front of me is a table, behind me is a closet, on the right is a door.” To consolidate skills, you can use didactic games such as “Where will we throw the ball?”, “What has changed?”, “Guess what is where” etc.

In the game "Where shall we throw the ball?" the children stand in a circle. The teacher gives tasks: "Throw the ball to the one who is standing in front of you", "Throw the ball to the one who is standing to your left." Game "What has changed?" can be held at the table. The leading child must say who is sitting in front of him, who is on the left, who is on the right. Then he closes his eyes and the children switch places. Opening his eyes, the driver determines what has changed. For example: “Masha was sitting in the back, and now she is sitting on the left. Vova was sitting on the left, and now in front of me.

Children also learn to navigate in space on a piece of paper. In the classroom, you often need to find the top and bottom strips of a score card, the right and left sides of a sheet, and place a certain number of objects in a certain place. Landmarks will help to assimilate the space of the sheet: the red line indicates the upper part of the sheet, the blue one - the bottom, the cross - the right side, the circle - the left. Such visual supports help to highlight the same parts of space in the paper and on your sheet and associate them with a specific name (above, above, below, below, right, left, middle).

Children of the sixth year of life continue to master spatial representations: left, right, above, below, in front, behind, far, near. A new task is to teach them to navigate in specially created spatial situations and determine their place according to a given condition. The child must be taught to complete tasks (such as: “Stand so that there is a closet to your right and a chair behind you. Sit so that Tanya is sitting in front of you and Kolya is behind you.”

In addition, children should learn to determine in a word the position of one or another object in relation to another: “To the right of the doll is a hare, to the left of the doll is a pyramid, in front of Tanya is a window, above Tanya’s head is a lamp.” The formation of spatial orientations is successful if the child constantly faces the need to operate with these concepts. The situations in which children are included should be entertaining for preschoolers.

In the development of spatial orientations, in addition to special games and tasks in mathematics classes, a special role is played by walks, outdoor games, physical exercises, music classes, visual activity classes, various regime moments (dressing, undressing, duty), bit orientation of children not only in their group room or on their site, but also in the premises of the entire kindergarten.

A technique for developing the ability to navigate in two-dimensional space.

Formation of the ability to navigate in two-dimensional space (3 - 6 years)

There are 6 directions in three-dimensional space: up, down, left, right, front, back. And in two-dimensional - only 4 directions (there are no directions: in front, behind).

Stage 1 (3 - 4 years). First, children are taught: where is the left (right) part of a sheet of paper. It is proposed to put your hands on a piece of paper: where the left hand is the left side of the piece of paper, and where the right hand is the right side.

Types of exercises: put 1 button on the left, a lot on the right, lay out objects from left to right.

Then they show what it means at the top, at the bottom of the sheet, then they explain: at the top - this is further from you, at the bottom - closer to you.

Task: put mushrooms at the top, Christmas trees at the bottom.

Stage 2 (4 - 5 years). Types of exercises:

Unfolding a certain number of items

right (left, top, bottom)

Creating a pattern on a plane. Options:

a) the teacher dictates which items to put in which place;

b) the children are given a ready-made card, and the children describe it;

c) children come up with a pattern and describe it.

When naming the location of an object on a plane, it is necessary to say: relative to what we place it (for example: above the triangle; below the entire plane)

Questions: What is at the top (bottom, left, right) on the sheet? Where is the triangle?

Games: - "Find your house" (children are looking for "houses" that match their pattern),

- “Paired pictures” (the same objects are drawn, but differently located in space; you need to find the same pictures).

You can create patterns on the application and drawing (postcard, house, apron).

Stage 3 (5 - 6 years). Children are offered exercises and games with complications. The patterns use large quantity objects, they are located in the corners. Children are explained such complex spatial directions as the “upper left corner” (lower right corner): if the object is both at the top and on the right, then we say that it is in the upper right corner. You can use a color: shade the top of the card with a strip of one color, the right side of the card with a strip of another color, at the intersection we get the upper right corner.

Exercise: "Creating a pattern on paper in a box." First, preparatory exercises are carried out:

Put a dot in the indicated place on the paper (for example, stepping back 3 cells from the top and 2 from the left),

Draw a line of a certain length in the indicated direction (for example, 3 cells from left to right).

Then the teacher dictates to the children a pre-thought-out pattern, it is desirable that it be symmetrical.

Stage 4 (5 - 6 years). They teach children to move from three-dimensional space to two-dimensional space and vice versa (transform), i.e. children are taught to make diagrams, a plan, and then find objects in three-dimensional space, focusing on the diagram.

Preparatory exercises: introduce children to conventional signs. Then the children are offered ready-made conventional signs, which they must lay out on a sheet of paper in accordance with the arrangement of objects in 3-dimensional space.

Basic exercises:

Draw on the diagram using conventional signs objects located in a room or on a plot,

According to the finished scheme, arrange objects.

Games: Furnish the doll with a room, Designer, Find the secret, Scouts, Find what is hidden. (The asterisk indicates the place where the secret is hidden, the arrows indicate the route along which you need to go. 2 teams can play: whoever finds it faster).

Features of the perception of time by children of early and preschool age.

Preschoolers form a clear idea for specific events about the past, present and future. Many teachers note this purely concrete character of preschool children's temporal representations. Children talk about days, months, hours as objects and even personify time: “Where did yesterday go?” To concretize temporal relations, the objectivity of which children cannot understand for a long time, they use any facts that in their experience turned out to be associated with certain indicators of time. For example: “Dad, why did you come! Is it already evening? Children 3-5 years old establish a connection between constantly repeating facts and the corresponding indicators of time: "Morning - when we get up, evening - when they take us home from the garden." As the experience of orientation in time accumulates, children establish more significant signs, how some objective phenomena begin to be used as indicators of time: “It’s already morning, it’s light, the sun is rising, and night is when it’s dark and everyone is sleeping.”

Younger preschoolers are already more clearly localizing in time events that have distinctive qualitative features, emotional attractiveness, which are well known to them: “Christmas tree - when it’s winter, we’ll go to the dacha, when it’s summer”, etc. How is the category of time practically reflected in the speech of preschool children? The most accessible, initial speech expressions of the category of time are undivided temporary relations. They are indicated by the words first, then, earlier, later, then the child begins to use words long ago and soon. Children 6-7 years old are already actively using temporary adverbs. But not all temporal categories are recognized by them and are correctly reflected in speech: adverbs denoting the speed and localization of events in time are better assimilated, worse are adverbs expressing duration and sequence. However, several training sessions that reveal the meaning of the most difficult tense adverbs for children clarify their understanding. From this follows the conclusion: the process of verbal expression of temporal concepts in children; 5-7 years is in the stage of continuous development, which proceeds especially intensively if this process is controlled. However, the fine differentiation of temporal relations at preschool age is still formed slowly and largely depends on the general mental and speech development of children. The nature of preschool children's ideas about time is associated with their understanding of the properties of time, mastering time concepts (at dawn, at dusk, at noon, at midnight, day, week, month, year), the ability to navigate the time of day according to natural phenomena, the idea of causal-temporal dependencies of rhythmic natural phenomena, the duration of a second, minute and hour and the ability to determine the time on the clock, evaluate time intervals. Teaching experience shows that in the process of organizing pedagogical influence in kindergarten and in the family, children learn only some of the listed temporal representations and the ability to navigate in time. The level of this knowledge is low. Temporal concepts of different meanings are often combined. For example, children do not recognize the difference between the words dawn and dusk, which denote the transitional periods from darkness to daylight. The meanings of the words midnight and noon are not perceived as designating moments of equal division of day and night. Children confuse the concepts of "day" and "day", they cannot name all parts of the day, they do not know that the day is part of the day. Most children do not notice differences in the color of the sky at different periods of the day, and cannot establish the sequence of parts of the day. In their view, the day ends at night and begins in the morning. Thus, some children have misconceptions about the isolation of each day and their discontinuity. Often preschoolers do not know the names of the days of the week, they cannot determine their sequence. In memorizing the days of the week, unevenness is observed; days that have a pronounced emotional coloring for the child are better remembered. This feature is also manifested in the memorization of the names of months by children. Even older preschoolers have insufficient knowledge of how to measure time (using a calendar, clock). The names of time intervals (minute, hour) remain purely verbal, abstract for children, since life experience of activity during these periods of time has not yet been accumulated. Can children evaluate the duration of small periods of time in the process of performing a variety of activities?

Experience shows that preschoolers are able to estimate the duration of one minute, but this assessment depends on the nature of the activity in a given period of time. Positive emotions in children that arise in the process of interesting activities cause a desire to prolong a pleasant moment. Therefore, when estimating time filled with events of interesting and rich content, the child allows an overestimation of the small time that passes imperceptibly and its duration seems to be shorter. Time filled with monotonous is little interesting activity seems to the child to be longer. The influence of these subjective factors can be significantly weakened as a result of the development of a "sense of time" in children, the accuracy of estimating various time intervals is improved under the influence of specially organized exercises. . So, the knowledge that children have about time is incomplete, isolated, not interconnected and static. This is due to the fact that episodic classes (conducted with preschoolers mainly by verbal methods), in which children are introduced to the signs of parts of the day, memorize the sequence days of the week, months, do not give them the necessary knowledge about time - about its fluidity and irreversibility, about rhythm, tempo and periodicity. The information received by children remains on the surface of consciousness, does not reveal temporal relationships.

Teaching children of different ages the difference between parts of the day, the ability to determine their sequence. The concept of "day". Assimilation of the words "yesterday", "today", "tomorrow".

The day is usually divided into four parts: morning, afternoon, evening, night. Such a division, on the one hand, is associated with objective changes occurring in the environment due to the different position of the sun, the illumination of the earth's surface, airspace, the appearance and disappearance of the moon, stars, and on the other hand, with a change in the types of human activities in different parts. day, alternating between work and rest. The duration of each part of the day is different, so their change is accepted conditionally. Familiarization of children with parts of the day according to the "Program of Education and Training in Kindergarten" begins with the second younger group. At this age, it is necessary to teach children to distinguish and designate in words all four parts of the day. The specific determinant of time for children is their own activity. Therefore, when teaching children, it is necessary to saturate parts of the day with specific, essential signs of children's activity, naming the appropriate time. What types of activities are recommended to be used as indicators of different parts of the day? Among the various activities that are repeated daily in the child’s day regimen, there are permanent ones that take place only once a day, at a certain time: this is coming to kindergarten, exercising, lunch, afternoon nap, etc. There are also variable activities, repeating several times during the day, in different parts of the day: playing, washing, dressing and undressing, walking, etc. They can also be used as indicators of parts of the day.

Familiarization with the parts of the day should begin with a conversation about the personal, concrete experience of the children. The teacher can ask such questions: “Children, you wake up at home, when mom says it's time to get up, it's already morning! What do you do at home in the morning? When do you come to kindergarten? What do you do in the morning in kindergarten? At the end of the conversation, the teacher summarizes: “In kindergarten, you do gymnastics every day, have breakfast. Then there is a lesson. All this happens in the morning. It's morning now and we're busy." Such conversations are held in mathematics classes, with special attention being paid to the exercise of children in the correct designation of parts of the day with words. In everyday life, it is important to exercise children in using the names of parts of the day, in correlating actions with a certain time of day.

Consolidation of the ability to determine parts of the day should be carried out in the classroom, showing children pictures depicting constant activities characteristic of each part of the day (you can use pictures of fabulous content), and discussing the question: “When does this happen?” In subsequent lessons, the task is complicated by offering to choose from several pictures those on which it is drawn what happens in any one of the periods of the day (morning, afternoon, evening or night).

To consolidate the knowledge of children, it is useful to read excerpts from stories, poems, which describe practical actions characteristic of each part of the day. You can also use the most simple word games to activate the dictionary through the names of the parts of the day. For example, in the game “Name the Missing Word”, the teacher skips the name of the part of the day in the sentence: “We have breakfast in the morning, but do we have lunch ...?” IN middle group it is necessary to consolidate in children the ability to name parts of the day, to deepen and expand their ideas about these periods of time, constantly paying attention to various phenomena characteristic of each part of the day. Here you can already show what is happening and what they do in the morning, afternoon, evening and night, not only the children themselves, but also adults. For this purpose, you can use pictures with a broader content: schoolchildren go to school in the morning, fireworks against the backdrop of the evening city, people leave the theater in the evening, etc. We also consider a series of pictures that depict everything that happens, for example, in the evening (children leave from kindergarten, play at home, watch the evening street from the balcony, grandmother reads a book to a child lying in bed). It is useful to invite the children themselves from the set to choose all the pictures that show what happens during the day.

The words “yesterday”, “today”, “tomorrow” are introduced into the passive dictionary at 3-5 years old. In active - at 5 - 6 years (according to the research of A.M. Leushina).

What did you do yesterday (today, tomorrow)? (in response - characteristic actions).

When did you go to the park (did the named activities)? (in response - yesterday, or today, or tomorrow).

Exercises on the rotation of 3 days: children are given 3 sets of cards of parts of the day and are invited to decompose these cards to make three days. It is explained: as soon as the night of the 1st day ends, the morning of the second day begins, those days that have passed are called “yesterday”, and those days that are coming are called “today”. After the night of today, there comes a day called “tomorrow”.

We have been talking for 3 days about some bright event. On the first day, we associate going to the theater with the word “tomorrow”: “Tomorrow we are going to the theater”, “When are we going to the theater?”, “Where are we going tomorrow?”. On the 2nd day, we associate a trip to the theater with the word “today”. On the 3rd day - with the word "yesterday".

Such conversations are repeated several times a year (about various bright events).

Exercises with three pictures, one of which depicts some event. The card with the event is placed in a certain place (“today” - in the middle, “tomorrow” - on the right, “yesterday” - on the left) and it turns out “When does this happen?” or the task is given “Put the card so that the event happens “tomorrow”.

A pair game can be organized: “When was that?”

After the children have mastered the sequence of days of the week well, a conversation is held daily: what day of the week is today, what was yesterday, what will be tomorrow.

Teaching children the ability to distinguish between temporary units and determine their sequence. The concepts of "week", "time of year", "month", "year".

mastering the sequence of days of the week. They are introduced to the fact that the day has its own names, that seven days make up a week. Each day of the week has its own name. In a week, the days follow each other in a certain order: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday. This sequence of days of the week is unchanged. The teacher tells the children that in the names of the days of the week one can guess which day of the week is on the account: Monday is the day after the week, that is, the first day after the end of the week, Tuesday is the second day of the week, Wednesday is the middle of the week.

You can involve children in determining the origin of the names: Thursday is the fourth day of the week, Friday is the fifth. In different classes, you can take 1-1.5 minutes to repeat the names of time periods and days of the week. To do this, children are asked questions: what day of the week is it today? What day of the week will be tomorrow? What day was yesterday? Consolidation and deepening of temporary representations occurs in various games that are used in the classroom. You can also use the game to learn the names and sequence of the days of the week.

Game "Live week". Seven children lined up at the blackboard and counted in order. The first child on the left takes a step forward and says, “I am Monday. What day is next? The second child comes out and says: “I am Tuesday. What day is next? The whole group gives the task to the “days of the week”, makes riddles. They can be very different: for example, name the day that is between Tuesday and Thursday, Friday and Sunday, after Thursday, before Monday, etc. Name all the weekend days of the week. Name the days of the week on which people work. The complication of the game is that the players can line up from any day of the week, for example, from Tuesday to Tuesday.

When the children learn the names and sequence of the days of the week, they willingly begin to solve such problems: “Two friends met on the street. “Come visit me,” Kolya said. “Thank you,” Petya answered. “Only on Monday my grandmother comes to me, and on Wednesday I leave to rest. But I will definitely come, ”On what day will Petya come to visit Kolya?” Another task: “Today is Wednesday, in one day there will be a holiday in kindergarten. What day will the holiday be? or "Name the day of the week between Thursday and Saturday."

The teacher can tell the children about how time was determined before. In the old days, to know how many days will pass, people used to use this method. They knew that a day passed from sunrise to the next sunrise. Therefore, every morning, that is, at sunrise, they strung a pebble with a hole (like a button) on a blade of grass. In this way, they determined how many or few days had passed before some event, for example, before the harvest.

Such a case is known. The ancient Persian king left the Greeks to guard the bridge. And he himself with his army went on a campaign against enemies. He handed over to the soldiers guarding the bridge a belt on which knots were tied. Every day, the soldiers had to untie the knot. When all the knots are untied, the warriors can return home. You can try with your children to use this old way of mastering time: bring a rope with several knots tied and agree that every day at the same time they will untie one knot; when all the knots are untied, there will be a holiday or an interesting math quiz.

As a rule, children do not experience difficulties in mastering temporal representations. However, the ability to navigate in temporary concepts is provided by everyday contact with them. Therefore, it is important not only in mathematics classes, but also in all others and in everyday life, to ask children questions: what day of the week is it today? What will tomorrow be like? What was yesterday? Children in this age group should also know what day of the week each activity takes place.

§ 4. Mathematical proof

26. Schemes of deductive reasoning.

§five. Text problem and the process of its solution

29. Structure of a text problem

30. Methods and methods for solving text problems

31. Stages of solving the problem and methods for their implementation

2. Search and drawing up a plan for solving the problem

3. Implementation of the plan for solving the problem

4. Verification of the problem solution

5. Modeling in the process of solving text problems

Exercises

32. Solving problems "in parts"

Exercises

33. Solving motion problems

Exercises

34. Main conclusions.

§6. Combinatorial problems and their solution

§ 7. Algorithms and their properties

Exercises

Exercises

Chapter II. Algebra elements

§ 8. Correspondences between two sets

41. The concept of compliance. Methods for specifying correspondences

2. Graph and correspondence graph. Correspondence inverse to this one. Types of matches.

3. One-to-One Correspondences

Exercises

42. One-to-one correspondences. The concept of a one-to-one mapping of a set x onto a set y

2. Equivalent sets. Ways to establish the equivalence of sets. Countable and uncountable sets.

Exercises

43. Main conclusions § 8

§ 9. Numerical functions

44. The concept of a function. Ways to set functions

2. Graph of the function. Monotonicity property of a function

Exercises

45. Direct and inverse proportionality

Exercises

46. ​​Main conclusions § 9

§10. Relations on the set

47. The concept of a relation on a set

Exercises

48. Relationship Properties

R is reflexive on x ↔ x r x for any x ∈ X.

R is symmetrical at x ↔ (x r y →yRx).

49. Equivalence and order relations

Exercises

50. Main conclusions § 10

§ 11. Algebraic operations on a set

51. The concept of an algebraic operation

Exercises

52. Properties of algebraic operations

Exercises

53. Main conclusions § 11

§ 12. Expressions. Equations. inequalities

54. Expressions and their identical transformations

Exercises

55. Numerical equalities and inequalities

Exercises

56. One Variable Equations

2. Equivalent equations. Equivalence theorems for equations

3. Solution of equations with one variable

Exercises

57. Inequalities with one variable

2. Equivalent inequalities. Equivalence theorems for inequalities

3. Solving inequalities with one variable

Exercises

58. Main conclusions § 12

Exercises

Chapter III. Natural numbers and zero

§ 13. From the history of the concept of a natural number

§ 14. Axiomatic construction of a system of natural numbers

59. On the axiomatic method of constructing a theory

Exercises

60. Basic concepts and axioms. Definition of a natural number

Exercises

61. Addition

62. Multiplication

63. Ordering of the set of natural numbers

Exercises

64. Subtraction

Exercises

65. Division

66. Set of non-negative integers

Exercises

67. Method of mathematical induction

Exercises

68. Quantitative natural numbers. Check

Exercises

69. Main conclusions § 14

70. Set-theoretic meaning of the natural number, zero and the relation "less than"

Exercises

Lecture 36

71. Set-theoretic meaning of the sum

Exercises

72. Set-Theoretic Meaning of the Difference

Exercises

73. The set-theoretic meaning of a product

Exercises

74. Set-Theoretic Meaning of Private Natural Numbers

Exercises

75. Main conclusions § 15

§16. Natural number as a measure of magnitude

76. The concept of a positive scalar quantity and its measurement

Exercises

77. The meaning of a natural number obtained as a result of measuring a quantity. Meaning of sum and difference

Exercises

78. The meaning of the product and quotient of natural numbers obtained as a result of measuring quantities

79. Main conclusions § 16

80. Positional and non-positional number systems

81. Writing a number in decimal notation

Exercises

82. Addition algorithm

Exercises

83. Subtraction algorithm

Exercises

84. Multiplication Algorithm

Exercises

85. Division Algorithm

86. Positional number systems other than decimal

87. Main conclusions § 17

§ 18. Divisibility of natural numbers

88. The ratio of divisibility and its properties

89. Signs of divisibility

90. Least Common Multiple and Greatest Common Divisor

2. Basic properties of the least common multiple and the greatest common divisor of numbers

3. Sign of divisibility by a composite number

Exercises

91. Prime Numbers

92. Ways to find the greatest common divisor and least common multiple of numbers

93. Main conclusions § 18

3. Distributivity:

§ 19. On the extension of the set of natural numbers

94. The concept of a fraction

Exercises

95. Positive rational numbers

96. The set of positive rational numbers as an extension

97. Writing positive rational numbers as decimals

98. Real numbers

99. Main conclusions § 19

Chapter IV. Geometric figures and quantities

§ 20. From the history of the emergence and development of geometry

1. The essence of the axiomatic method in the construction of a theory

2. The emergence of geometry. Geometry of Euclid and Geometry of Lobachevsky

3. The system of geometric concepts studied at school. The main properties of the belonging of points and lines, the relative position of points on a plane and a line.

§ 21. Properties of geometric figures on the plane

§ 22. Construction of geometric figures

1. Elementary tasks for construction

2. Stages of solving the construction problem

Exercises

3. Methods for solving construction problems: transformations of geometric figures on a plane: central, axial symmetry, homothety, movement.

Main conclusions

§24. Image of spatial figures on a plane

1. Properties of parallel design

2. Polyhedra and their representation

Tetrahedron Cube Octahedron

Exercises

3. Ball, cylinder, cone and their image

Main conclusions

§ 25. Geometric quantities

1. The length of the segment and its measurement

1) Equal segments have equal lengths;

2) If a segment consists of two segments, then its length is equal to the sum of the lengths of its parts.

Exercises

2. The magnitude of an angle and its measurement Every angle has a magnitude. Special name for her

1) Equal angles have equal magnitudes;

2) If an angle consists of two angles, then its value is equal to the sum of the values ​​of its parts.

Exercises

1) Equal figures have equal areas;

2) If a figure consists of two parts, then its area is equal to the sum of the areas of these parts.

4. Polygon area

5. The area of ​​an arbitrary flat figure and its measurement

Exercises

Main conclusions

1. The concept of a positive scalar quantity and its measurement

1) The mass is the same for bodies balancing each other on the scales;

2) Mass is added when bodies are joined together: the mass of several bodies taken together is equal to the sum of their masses.

Conclusion

Bibliography

§ 21. Properties of geometric figures on the plane

Lecture 53

1. Geometric figures on the plane and their properties

2. Angles, parallel and perpendicular lines

3. Parallel and perpendicular lines

A geometric figure is defined as any set of points. Segment, straight line, circle, ball - geometric shapes.

If all points of a geometric figure belong to the same plane, it is called flat. For example, a segment, a rectangle are flat figures. There are figures that are not flat. This is, for example, a cube, a ball, a pyramid.

Since the concept of a geometric figure is defined through the concept of a set, we can say that one figure is included in another (or is contained in another), we can consider the union, intersection and difference of figures.

For example, the union of two rays AB and MK is the straight line KB, and their intersection is the segment AM.

There are convex and non-convex figures. A figure is called convex if, together with any two of its points, it also contains a segment connecting them.

Figure F₁ is convex, and figure F₂ is non-convex.

Convex figures are a plane, a straight line, a ray, a segment, a point, a circle.

For polygons, another definition is known: a polygon is called convex if it lies on one side of each line containing its side. Since the equivalence of this definition and the one given above for a polygon has been proved, both can be used.

Consider some of the concepts studied in the school course of geometry, their definitions and properties, accepting them without proof.

corners

Injection- This is a geometric figure that consists of a point and two rays emanating from this point. The rays are called the sides of the angle, and their common beginning is its vertex.

An angle is designated differently: either its vertex, or its sides, or three points are indicated: the vertex and points on the sides of the angle: A,  (k, l), ABC.

Angle is called deployed if its sides lie on the same straight line.

An angle that is half a straight angle is called direct. An angle smaller than a right angle is called sharp. An angle greater than a right angle but less than a straight angle is called stupid.

flat corner- this is a part of the plane bounded by two different rays emanating from the same point.

There are two plane angles formed by two rays with a common origin. They're called additional.

ABOUT

The angles that are considered in planimetry do not exceed the developed angle.

The two corners are called related, if they have one side in common, and the other sides of these angles are complementary half-lines.

The sum of adjacent angles is 180º. The validity of this property follows from their definition of adjacent angles.

The two corners are called vertical, if the sides of one angle are the complementary half-lines of the sides of the other.

Vertical angles are equal.

Parallel and perpendicular lines

Two lines in a plane are called parallel if they do not intersect

If line a is parallel to line b, then write a║b.

Let's consider some properties of parallel lines, and first of all signs of parallelism.

Signs are called theorems in which the presence of any property of an object in a certain situation is established. In particular, the need to consider the signs of parallelism of lines is due to the fact that in practice it is often necessary to resolve the issue of the relative position of two lines, but at the same time it is impossible to directly use the definition.

Consider the following signs of parallel lines:

1. Two lines parallel to a third are parallel to each other.

2. If the internal cross-lying angles are equal or the sum of internal one-sided angles is 180º, then the lines are parallel.

A true statement reverse the second sign of parallelism of lines: if two parallel lines are intersected by a third, then the internal cross-lying angles are equal, and the sum of one-sided angles is 180º.

An important property of parallel lines is revealed in theorem, bearing the name of the ancient Greek mathematician Thales: if parallel lines intersecting the sides of an angle cut off equal segments on one of its sides, then they cut off equal segments on its other side.

The two lines are called perpendicular if they intersect at right angles.

If line a is perpendicular to line b, then write ab.

The main properties of perpendicular lines are reflected in two theorems:

1. Through each point of a straight line it is possible to draw a straight line perpendicular to it, and only one.

2. From any point not lying on a given line, one can drop a perpendicular to this line, and only one.

A perpendicular to a given line is a line segment perpendicular to a given line that has an intersection point at its end. The end of this segment is called the base of the perpendicular.

The length of the perpendicular from a given point to the line is called distance from point to line.

Distance between parallel lines is the distance from any point on one line to another.

Lecture 54

4. Triangles, quadrangles, polygons. Formulas for the area of ​​a triangle, rectangle, parallelogram, trapezoid.

5. Circle, circle.

triangles

The triangle is one of the simplest geometric shapes. But its study gave rise to a whole science - trigonometry, which arose from practical needs in measuring land, compiling maps of the area, designing various mechanisms.

triangle a geometric figure is called, which consists of three points that do not lie on one straight line, and three pairwise segments connecting them.

Any triangle divides the plane into two parts: internal and external. A figure consisting of a triangle and its interior is also called a triangle (or flat triangle).

In any triangle, the following elements are distinguished: sides, angles, heights, bisectors, medians, midlines.

The angle of the triangle ABC at the vertex A is the angle formed by the half-lines AB and AC.

Height of a triangle dropped from a given vertex is called a perpendicular drawn from this vertex to the line containing the opposite side.

bisector triangle is the segment of the bisector of the angle of the triangle, connecting the vertex with a point on the opposite side.

median a triangle drawn from a given vertex is called a line segment that connects this vertex to the midpoint of the opposite side.

middle line A triangle is a line segment that joins the midpoints of two of its sides.

Triangles are said to be congruent if their corresponding sides and corresponding angles are equal. In this case, the corresponding angles must lie against the corresponding sides.

In practice and in theoretical constructions, signs of the equality of triangles are often used, which provide a faster solution to the question of the relationship between them. There are three such signs:

1. If two sides and the angle between them of one triangle are equal, respectively, to two sides and the angle between them of another triangle, then such triangles are equal.

2. If the side and angles adjacent to it of one triangle are equal, respectively, to the side and angles adjacent to it of another triangle, then such triangles are equal.

3. If three sides of one triangle are equal, respectively, to three sides of another triangle, then such triangles are congruent.

The triangle is called isosceles if its two sides are equal. These equal sides are called the sides, and the third side is called the base of the triangle.

Isosceles triangles have a number of properties, for example:

In an isosceles triangle, the median drawn to the base is the bisector and height.

We note several properties of triangles.

1. The sum of the angles of a triangle is 180º.

This property implies that any triangle has at least two acute angles.

2. The middle line of the triangle, connecting the midpoints of the two sides, is parallel to the third side and equal to half of it.

3. In any triangle, each side is less than the sum of the other two sides.

For a right triangle, the Pythagorean theorem is true: the square of the hypotenuse is equal to the sum of the squares of the legs.

Quadrangles

quadrilateral A figure is called a figure that consists of four points and four segments connecting them in series, and no three of these points should lie on one straight line, and the segments connecting them should not intersect. These points are called the vertices of the quadrilateral, and the segments connecting them are called its sides.

Any quadrilateral divides the plane into two parts: internal and external. A figure consisting of a quadrilateral and its interior is also called a quadrilateral (or flat quadrilateral).

The vertices of a quadrilateral are called adjacent if they are the ends of one of its sides. Vertices that are not adjacent are called opposite. Line segments connecting opposite vertices of a quadrilateral are called diagonals.

Sides of a quadrilateral that originate from the same vertex are called adjacent sides. Sides that do not have a common end are called opposite sides. Quadrilateral ABCD has opposite vertices A and B, sides AB and BC are adjacent, BC and AD are opposite; segments AC and BD are the diagonals of the given quadrilateral.

There are convex and non-convex quadrilaterals. Thus, the quadrilateral ABCD is convex, and the quadrilateral KRMT is non-convex. Among convex quadrilaterals, parallelograms and trapezoids are distinguished.

A parallelogram is a quadrilateral whose opposite sides are parallel.

Let ABCD be a parallelogram. From the vertex B to the line AD, let's take the perpendicular BE. Then the segment BE is called the height of the parallelogram corresponding to the sides BC and AD. Section

M

CM is the height of the parallelogram corresponding to the sides CD and AB.

To simplify the recognition of parallelograms, consider the following feature: if the diagonals of a quadrangle intersect and the intersection point is divided in half, then this quadrilateral is a parallelogram.

A number of parallelogram properties that are not contained in its definition are formulated in the form of theorems and proved. Among them:

1. The diagonals of the parallelogram intersect and the intersection point is divided in half.

2. A parallelogram has opposite sides and opposite angles.

Consider now the definition of a trapezoid and its main property.

Trapeze A quadrilateral is called if only two opposite sides are parallel.

These parallel sides are called the bases of the trapezium. The other two sides are called lateral.

The segment connecting the midpoints of the sides is called the midline of the trapezoid.

The midline of a trapezoid has the following property: it is parallel to the bases and equal to half their sum.

Of the many parallelograms, rectangles and rhombuses are distinguished.

Rectangle A parallelogram is called if all angles are right angles.

Based on this definition, it can be proved that the diagonals of a rectangle are equal.

Rhombus A parallelogram is called if all sides are equal.

Using this definition, one can prove that the diagonals of a rhombus intersect at right angles and are the bisectors of its angles.

From the set of rectangles, squares are selected.

A square is a rectangle in which all sides are equal.

Since the sides of a square are equal, it is also a rhombus. Therefore, a square has the properties of a rectangle and a rhombus.

Polygons

A generalization of the concept of a triangle and a quadrilateral is the concept of a polygon. It is defined through the concept of a broken line.

A polyline A₁A₂A₃…An is a figure that consists of points A₁, A₂, A₃, …, An and segments A₁A₂, A₂A₃, …, An-₁An connecting them. Points А₁, А₂, А₃, …, Аn are called polyline vertices, and segments А₁А₂, А₂А₃, …, Аn-₁Аn are its links.

If a broken line has no self-intersections, then it is called simple. If its ends coincide, then it is called closed. About the broken lines shown in the figure, we can say: a) - simple; b) - simple closed; c) is a closed broken line that is not simple.

a B C)

The length of a broken line is the sum of the lengths of its links.

It is known that the length of a broken line is not less than the length of the segment connecting its ends.

polygon is called a simple closed broken line if its adjacent links do not lie on the same straight line.

The vertices of the polyline are called the vertices of the polygon, and its links are called its sides. Segments connecting non-neighboring vertices are called diagonals.

Any polygon divides the plane into two parts, one of which is called the internal, and the other - the external region of the polygon (or flat polygon).

There are convex and non-convex polygons.

A convex polygon is called regular if all sides and all angles are equal.

An equilateral triangle is a regular triangle, a square is a regular quadrilateral.

The angle of a convex polygon at a given vertex is the angle formed by its sides converging at that vertex.

It is known that the sum of the angles of a convex n-gon is 180º (n– 2).

In geometry, in addition to convex and non-convex polygons, polygonal figures are also considered.

A polygonal figure is a union of a finite set of polygons.

a B C)

The polygons that make up a polygonal figure may not have common interior points, they may have common interior points.

A polygonal figure F is said to consist of polygonal figures if it is their union, and the figures themselves do not have common interior points. For example, the polygonal figures shown in figures a) and c) can be said to consist of two polygonal figures, or that they are divided into two polygonal figures.

Circle and circle

circumference called a figure that consists of all points of the plane equidistant from a given point, called center.

Any line segment connecting a point on a circle with its center is called the radius of the circle. Radius also called the distance from any point on the circle to its center.

A line segment that joins two points on a circle is called chord. The chord passing through the center is called diameter.

A circle is a figure that consists of all points of the plane that are at a distance not greater than a given distance from a given point. This point is called the center of the circle, and this distance is called the radius of the circle.

The boundary of a circle is a circle with the same center and radius.

Recall some properties of a circle and a circle.

A line and a circle are said to touch if they have a single common point. Such a line is called a tangent, and the common point of the line and the circle is called the tangent point. It is proved that if a line touches a circle, then it is perpendicular to the radius drawn to the point of contact. The converse statement is also true (Fig. a).

A central angle in a circle is a flat angle with a vertex at its center. The part of the circle located inside the flat angle is called the arc of the circle corresponding to this central angle (Fig. b).

An angle whose vertex lies on a circle, and the sides intersect it, is called inscribed in this circle (Fig. c).

An angle inscribed in a circle has the following property: it is equal to half of the corresponding central angle. In particular, the angles based on the diameter are right angles.

A circle is said to be circumscribed near a triangle if it passes through all its vertices.

To describe a circle around a triangle, you need to find its center. The rule for finding it is justified by the following theorem:

The center of the circle circumscribed about the triangle is the point of intersection of the perpendiculars to its sides, drawn through the midpoints of these sides (Fig. a).

A circle is said to be inscribed in a triangle if it touches all of its sides.

The rule for finding the center of such a circle is justified by the theorem:

The center of a circle inscribed in a triangle is the point of intersection of its bisectors (fig.b)

Thus, the perpendicular bisectors and bisectors intersect at one point, respectively. It has been proven in geometry that the medians of a triangle intersect at one point. This point is called the center of gravity of the triangle, and the point of intersection of the altitudes is called the orthocenter.

Thus, in any triangle there are four remarkable points: the center of gravity, the centers of the inscribed and circumscribed circles, and the orthocenter.

A circle can be circumscribed about any regular polygon, and a circle can be inscribed in any regular polygon, and the centers of the circumscribed and inscribed circles coincide.

Summary of the lesson of mathematics

Topic "Signs of geometric shapes»

Grade 2

(EMC "Primary School of the 21st Century")

Tatarinova Natalya Vasilievna

teacher primary school

MBOU "Komsomolskaya secondary school"

THE PURPOSE OF THE LESSON: Introduce the essential features of a rectangle and a square.LESSON OBJECTIVES: - educational: to clarify the concepts of a rectangle and a square, to form the ability to recognize them on the basis of essential properties, show the difference and similarity of a rectangle and a square, form the skill of determining shapes by sides and corners, introduce the term "geometry", improve computational skills. - developing: develop spatial skills, counting skills, thinking, attention, memory. - educational: cultivate love for the subject, a sense of cooperation, accuracy.Equipment for the lesson: interactive whiteboard, laptops,individual assistant cards, figure templates, handouts. Teaching method : active, practical, visualEquipmentfor the teacher :

textbook, interactive whiteboard, document Camera,

for students:

The card is an assistant pen, simple pencil, ruler, model right angle, glue, sheet of white cardboard geometric figures

During the classes:

Org. moment. Psychological mood.

– Smile to each other, to our guests, because to me “Big success begins with a little luck!” Children speak in unison: We are smart! We are friendly! We are attentive! We are diligent! We are great learners! Everything will work out for us!

Updating of basic knowledge

Work in pairs

- Task in the assistant card No. 1 EVALUATION- Today you yourself will evaluate your work with the help of symbols, which are located on the margins of the card. Pay attention to what these flowers mean: Flower with five petals - Excellent!Flower with four petals - Keep it up!Flower with three petals - Can be better. Evaluate how you coped with the first task and color one of the flowers.

Repetition of geometric concepts

Hush, hush .... Working with geometric shapes. (I put it on the board)- What has changed? - What figure is superfluous? Why? - Reading riddles. All my angles are straight, There are four sides, But not all of them are equal. I am a quadrilateral What? ...(rectangle) . (open on the board) I am a figure - no matter where, Always very even, All angles in me are equal And four sides. The cube is my favorite brother, Because I .... (square). (open on the board) What figures are we talking about? (Rectangle, square) What do you think we will talk about in the lesson? Name the topic of our lesson. What do you want to learn in the lesson?

Message about the topic and purpose of the lesson.

"Discovery" of new knowledge

1.Introduction of the terms "top", "width", "length". - Task in the assistant card No. 2 (textbook p. 111 No. 1)Highlight the corners in each shape.Highlight the sides in each shape.Select the vertices in each shape.CHECK according to the standard on the board. EVALUATION - What do these figures have in common? - What is the difference between these figures? Tell me, please, what is a rectangle? Square? - Is it possible to say that a square is a rectangle?2. Working with a rule - Task in the assistant card No. 3 (textbook p. 111 No. 1)- Read the rule on the card and fill in the missing words.- CHECK in the textbook (Compare the rule that we have derived with the rule in the textbook) EVALUATION Fizminutka for the eyes (musical)

Write down the numbers of the figures.

Polygons - Quadrangles - Rectangles - Squares - CHECK EVALUATION

Incorporating new content into the knowledge system

Finding the area of ​​a rectangle and a square.

- What can be determined from these geometric shapes? (Area) - How to determine the area of ​​\u200b\u200ba figure? (To find the area, you need to multiply the length by the width.) - What do you need to know in order to correctly determine the area of ​​\u200b\u200bthe figure? (Multiplication table)

Do-It-Yourself Solution examples on the multiplication table Task number 4

CHECK WITH DOCUMENT CAMERA SCORE A LAPTOP ( the first work on a laptop)

Determination of the area of ​​​​figures (DISC)

CHECK4 cm 2 3cm 2 7cm 2 8cm 2 16cm 2 (on the board) EVALUATION

Game "Molecules"

Do you know what molecules are? These are particles that move freely. (Music sounds, children dance, the music is over; children, at the signal of the teacher, unite in groups of 3,4, 5 people)

Practical work in groups. (Don't forget the rules of friendly work)

Make an application from geometric shapes 1 group - house 2 group - lifting edge 3 group - car 4 group - elephant 5 group - robot Place your work on the board. What geometric shapes did we use? What a wonderful courtyard we got. - Who can use it? - What needs to be done to make our site safe? (enclose with a fence) 6. Solving the problem The length of the playground is 9 meters, the width is 7 meters. What is the length of the entire fence? INDEPENDENT SOLUTION Examination- What did you find? -What is the perimeter

Summary of the lesson.

Now let's sum up What went to the guys for the future. I have 5 questions, Answer - ka, friends. We will knead each finger. We will conduct a reflection. M(LITTY FINGER) - What discovery did I make in the lesson? B(UNNAMED) - What have I learned? FROM(MEDIUM) - What professions would benefit from knowledge of geometric shapes? At(INDICATIONAL) - Which of the classmates worked perfectly today? B(BIG) - What's my mood like? (Show) Flowers Throughout the lesson, you assessed yourself. Evaluate your work throughout the lesson. Take a flower, decorate our courtyard with flowers. THANK YOU FOR YOUR WORK Additional tasks. How many squares are in the picture ... (p. 112 v. 4) Do you know ... (what is geometry) Geometry is the science of the properties of geometric shapes. The word "geometry" is Greek, translated into Russian means "land surveying". This name was given to this science because in ancient time the main purpose of geometry was to measure distances and areas on the earth's surface. Geometry is often used in practice. Workers, engineers, architects, and artists need to know it too. In a word, everyone should know geometry.