The main geometric shapes are straight cut ray. The main elements in the figure. What geometric shapes meet

19.03.2019

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Four-year-old baby knows and distinguishes such geometric figureslike a circle, square and triangle. Difficulties arise in distinguishing a circle and oval, square and rectangle. When comparison of the items, the child takes into account several properties: length, width, height. The above games and tasks will help you teach the baby to distinguish between geometric shapes and compare items by different signs. Older children offer tasks with volumetric figures.

Geometric lotto.

1 . Take a sheet of paper and divide it on 6 squares or rectangles. Make as many such cards. Draw on them geometric shapes. If your child knows how to read, instead of the image of the shape on paper, write the name of this figure. Let the cards be with the pattern. Baby's task to read the name of the shape and put the card with the image of this figure.

2 . Another version of the geometric lotto - you call the cell to which the child should put a specific figure.
For example: "Put the circle into the upper left corner, or place a triangle to the bottom right corner." If you have a multi-colored geometric shapes, then specify the color of the shape you want to see in the cell. So you fasten the concept on the right / left, on top / bottom and color name. Fill your card with the child. When all cells are filled, compare your cards.

Comparison of objects

The essence of the task is to come down to the fact that the child is proposed to compare a picture with geometric shapes.
To do this, it is necessary to find (or draw a picture of the items, which will resemble a geometric shape. For example: Circle - Buttons, Ball, Watermelon. Oval - melon, cucumber. Rectangle - door, table, etc.

Find the subject

Nabumage draws geometric shapes. The child's task is to draw objects similar to the shapes depicted on paper or find the items of this form in the room.

"Magic bag"

Figures are folded into the bag, and at your request, the child to the touch pulls the item you need. The baby can feel the objects both through the fabric, and dropping his hands into the bag. The main condition is not to look into the bag with figures.

Form and size

1. Prepare paper geometric shapes different sizes. Now ask the child all the circles to build in a row ascending (from a small circle to large), then all the triangles - descending (from a large triangle to small). In each row there should be no more than 5 items.

2. Take the box different in size, but the same in shape. Invite the child to put in the boxes of toys and close them suitable on the size of the lid. First help the baby, show how to close the box.
When he learns to distinguish the dimensions of one form, complicate the task: together with the boxes, give the child more and the cans of different values \u200b\u200bwith the lids. Now the baby needs not only to distinguish between "big / small", but also - "round / square".

Size and color

Work with the child's concept "Size", "Form" and "Color" of the subject can be as follows: Take the sheet of Watman and the color scotch \u003d "Circle") the contours of your geometric figures (it can be the details of the designer or homemade models). Now the child, taking one figure, fills all the fields on Watman, given the form of the item, as well as its size and color.
To complicate tasks, use one-color tape. In this case, the color will not act as a tip.

Exercise simulator

Before you start playing, consider the table with the child. Pay attention that the table has rows and columns (columns). List the shapes and colors. Make sure the child distinguishes the figures in size. Now proceed to the exercises:

1. Calculate!
- How much is the table depicted small circles?
- How many small red circles?
- How many big green squares?
- How many blue figures? etc.

2. Who lives?
The child needs to call the location of the specified figure. For example, you specify a big oval. The child should answer that the big oval is in the first column, in the second row.
You can play and vice versa: You call the "address" of the figure (for example, the fifth row, the fifth column), and the child finds the figure you said and calls it (Big Blue Square).

3. Right / left, top / bottom
For this simulator, you can teach (repeat) directions of the parties. For example, what figure is to the left of a big red rectangle? (Big Blue Circle) And what is on top of a large blue circle? (Big Blue Square), etc.

Speak shape

Offer the child from pre-prepared details to fold the circle (square, etc.). First, offer to fold the figure of two parts (two identical semicircles for a circle), then out of 3, etc. At first, the details for each figure are kept in separate envelopes. Later, the details from different geometric figures can be mixed. To facilitate the task, cut out each figure in a separate color (the circle is red, square - blue, etc.).

Classification of objects in form

The child needs to decompose the pictures into envelopes or bugs in accordance with the image form, thus creating several groups. At first, offer to sort pictures into two groups: round items in one envelope, quadriginal - in another. At this stage, it is important that the child distinguishes round items from the corners with corners - four-deeds, so square items will fall into the second group (for example, wall clocks), and rectangular (for example, book). Then add a group with triangular objects.

Later you can complicate the task, adding similar images, for example, round and oval, square and rectangular, triangular and trapezoids. Most sophisticated view Tasks - Surride all the pictures at once.

House for figurines

Show the child image of the housing (slag, needle, multi-storey building). Ask what geometric shapes they resemble the baby. Now he needs to be found for geometric shapes (triangle, circle, square) suitable in the form of a house.

Draw and Guess

Adult and child in turn draw in the air and guess various geometric shapes. You can draw the figures with your finger on the back.

Count the geometric figures

Ask a child to look at the picture. Consistently name the geometric shapes. Then ask him to count, call and designate the number of depicted squares, rectangles, triangles, rhombuses, trapezoids, circles and ovals.

Circuit figure

Cut from dense cardboard geometric shapes (circle, square, rectangle, triangle, rhombus, trapezium, oval). Ask a child to circulate on the contour of the figure. Let the child, rubbing the figure, considers her parties.

The main elements in the figure

Offer a child:

show parties at a square (rectangle, triangle, trapezoids, circle, oval). Show how you need to lead your finger on the side of the figure;
calculate the vertices of the square (rectangle, triangle, trapezoids) or mark the vertices with points in the image of a color pencil;
show square angles (rectangle, triangle, trapezium). Teach the child to show an angle with two fingers: big and index;
cut the border of the depicted figure of the depicted figure;
sharp the color pencil internal area of \u200b\u200bthe depicted figure;
find similarities of geometric shapes (for example, a rectangle, a square and a trapezium have 4 sides, 4 vertices and 4 corners);
similar geometric shapes (square, rectangle, trapezium, rhombus - quadricles; triangle, four-broth, five-hexagon - polygons).

Volumetric figures

1. Talking by Ob. volumetric figures, Try to achieve a child of understanding the difference between flat and volumetric geometric shapes (square - cube, circle - sphere (ball), etc.). Compare them, try to make cardboard or plasticine.

2. Consider parties Volume figures. Please note that they can be different from one figure. For example, in cone 2 sides: one is a circle at the base, and the second is the entire side surface of the cone.

3. Ask a child to compare coneand Pyramid.
Tell us that at the base of the pyramid there may be a triangle, a quadricle or polygon. And the side faces of the pyramid will be triangles converging in one vertex. If there is a circle at the base, then the cone will be.

4. Ask a child call or draw Items resembling volumetric geometric shapes.

Geometric - page number 1/1

MKOU "Svetlovskaya Sosh Zavyalovsky district"

Geometric

Names in the names

Performed:

Merzlova Tanya,

class 5 student

Leader:

mathematic teacher

Zhukova G.V.

from. Light 2012 year

Introduction ................................................................................. 3.

Chapter 1. What is geometry? .......................................... ...........................four

What is a geometric figure? ....................... ........................ ...four

The main geometric shapes .................................... 4

Mathematics and its role in human life .............................. 5

What geometric shapes meet ........................ .6

Chapter 2. Research ................................................ 7

Conclusion .................................................................................... ..8.

Literature .............................................................................. .9

Appendix ........................................................................... 10.

Introduction:

Is it possible to imagine the world without geometry? Remember that we do every day with you: Without numbers, you will not do a purchase, do not know the time, do not have the phone number, but without the drawings there would be no houses, different buildings, bridges, monuments and many other things. And space ships, lasers and all other achievements! They would simply be impossible if it were not for the science of geometry.

Geometry allows you to express the results of the account or measure something. People so often enjoy geometry, which is difficult to even imagine that they existed not at all times, but were invented by a person. In mathematics lessons, we got acquainted with some geometric figures and imagine what point, straight, cut, beam, angle, etc. In the seventh grade, we have to expand and deepen our knowledge of geometric figures. We learn a lot of important and interesting properties that are in geometry. I noticed that many names associated with mathematics are found in the names of people. I became curious, and I decided to study this material in more detail.
Relevance: To be inquisitive this is the way to expand knowledge.

Subject: Geometrical names in the names

Purpose: Create a bibliographic list of surnames in which the names of geometric shapes are found.

Tasks:

Explore literature on mathematicians and mathematics
Collect and process information on geometric figures

3. To find out when and in connection with what the human needs appeared science geometry

4. Media surnames in the telephone directory

Hypothesis:If the Russian language is great and mighty, then the surnames can be infinitely a lot

Subject of study: phonebook

Object of study: Families of the villagers of the village Light
Scientific novelty of research: If you explore surnames by syllables, you can find a lot interesting words And then you can look at the world with great curiosity. The execution of this type of work is expanding the horizons.
Methods: Search, learning, analysis, generalization, modeling, classification.
Practical significance: The knowledge gained teach to be attentive. This material can be used on cooling hours.

Chapter 1. What is geometry?

Geometry is one of the most ancient sciences, it originated for a very long time, even before our era. Translated from the Greek word "geometry" means "SEREMERIA" ("Geo" - in Greek, and "Mettero" - to measure). This name is explained by the fact that the origin of the geometry was associated with various measuring work, which had to be carried out with the markup of land, carrying out roads, construction of buildings and other structures. As a result of this activity, appeared and gradually accumulated various rulesassociated with geometric measurements and construction. Thus, the geometry arose on the basis of practical activities of people and at the beginning of its development served mainly practical purposes. In the future, geometry was formed as an independent science, engaged in the study of geometric shapes.

X | Week became a century of geometry, new sections appear, designed geometry develop.

What is a geometric figure?

We are familiar with some geometric figures and imagine what point, straight, cut, ray, triangle, rectangle, circle, circle, etc.

For example, Pythagoras, ancient Greek mathematician

and the philosopher idealist, skillfully used the knowledge gained in light travel. He is credited with a number of important discoveries at the time, for example, the theorem about the sum of the inner corners of the triangle. He built "cosmic" figures, i.e. five right polyhedra.

And what is a geometric figure?

The figure in geometry is the term applied to a variety of multiple points; Typically, the figure is called such sets that can be represented by consisting of a finite number of points of lines or surfaces.

Another great thinker This is Euclide. He brought the order in the accumulated knowledge of geometry, lived in the 3rd century BC in Alexandria. He processed and in a new one comprehended already famous results. For centuries, mathematicians seemed that 13-languid labor, which was called "Beginning", cannot be improved. In it, the entire geometry known by the time was set out.

Basic geometric shapes

TO the main geometric figure On the plane are related point and straight. Section, ray, loan line - the simplest geometric shapes on the plane.

Point - This is the most small geometric figurewhich is the basis of all other constructions in any image or drawing.

Straight line , or straight, can be imagined as countless pointswhich are located on one line that does not have any beginning, no end. On a sheet of paper, we see only part of the straight line, as it is infinite.

Part straight linelimited from two sides points, called a straight line, or cut .

Ray - This is a directed semi-bypass that has point Beginning and has no end.

Loan line - these are several segmentsinterconnected in such a way that the end of the first segment is the beginning of the second segment, and the end of the second segment - the beginning of the third segment, etc., with the neighboring (having one common point), Segments are located not on one straight line. If the end of the last segment does not coincide with the beginning of the first, then such a broken line is called unlucky.

If the end of the last segment of the broken coincides with the beginning of the first segment, then such a broken line is called closed.

Plane , as well as direct, is a primary concept that has no definition. At the plane, as in direct, it is impossible to see neither beginning or end. We consider only part of the plane, which is limited by a closed broken line.

Example plane It is the surface of the desktop, a notebook sheet, any smooth surface.

The study of geometry begins with planimetry.

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

There are figures that are not flat. This is, for example, a cube, ball, pyramid.

Mathematics and its role in human life

Mathematics - the basis of all sciences. Without it, just do not do. Yes, and science is interesting, ancient; With its help, still ancient scientists began to know the world.

Without knowledge of mathematics all modern life It would be impossible. We would not have good homes. Our clothes would be very rude. There would be no railwaysnor ships, no aircraft, no large industry. There would be no radio, television, cinema, telephone and thousands of other things that make up part of our civilization. Using mathematics, measurement "How much?", "How long?" are a vital part of the world in which we live.

Mathematics inexhaustible and multifaceted. The great German mathematician Gauss called Mathematics Queen of Science. It is a close connection of mathematics with other sciences, its role in creating new disciplines and theories at the junction of science gives it beauty and value.

A.S. Pushkin said: "Inspiration is needed in poetry, as in geometry"

In life, do not meet a single person who would not engage in mathematics. Each of us knows how to count, knows the multiplication table, knows how to build geometric shapes. With these figures, we often meet in the surrounding life.

Thanks to mathematics, computational counting machines appeared. Computing equipment has passed the way from simple accounts, arithmometers, logarithmic lines to microcalculators and computers. Now computing machines are used in all sectors of the national economy: in statistics, trade, automated management of factories and factories. Machines not only believe they can make translations from one language to another, can compose music, play chess.

What geometric shapes meet

straight	square	parallelepiped
point	rectangle	polyhedron
ray	triangle	prism
angle	polygon	pyramid
section	rhombus	ball
curve	parallelogram	cone
a circle	trapeze	cylinder
oval		cubic

Chapter 2. Research Part

Surname Translated from Latin is a family. Hereditary generic name of a person indicating the origin of a person from a certain kind. The centuries-old history stores many surnames. The origin of this or that surname is associated with the professions of our ancestors, regions where there were ancestors, their life, customs, nicknames, characters and appears.
We will conduct research and study the frequency of use geometric titles In the names of residents with. Light. Base for research - telephone directory of 2011

	Surname, name	name of geometric shape
1.	Kovalev	oval
2.	Sipovov	oval
3.	Victor	tOR
4.	Victoria	tOR
AR
1.	Arkhipova	aR
2.	Zakharov	aR
3.	Nazarov	aR
4.	Natarov	aR
5.	Taravatov	aR
6.	Tarasova	aR
7.	Harchenko	aR
8.	Arina	aR
9.	Daria	aR
10.	Illarion	aR
It was interesting to meet in addition to the names related to geometry more different words. Names
1.	Buchnev	Eve
2.	Persiavet	Eve
3.	Sobolev	Eve
4.	Tukoyev	Eve
5.	Danilchuk	Chuk

6.	Anastasia	Stas.
7.	Zhanna	Anna
8.	Tatyana	Yana.
9.	Yaroslav	Glory
Different words
1.	Vladimir	peace
2.	Miroshnik	peace
3.	Mironenko	peace

1.	Catherine	boat
2.	Ivan.	willow
3.	Sivakov	willow
4.	Danilchuk	il.

5.	Nikita	whale
6.	Olesia	forest
7.	Denisov	owl

Output:

148 surnames from the telephone directory and all surnames and the names of students are considered. In the two names and in two names there are names of geometric shapes. In 7, the names and 3 names met the concept of ar (weaving) - the square of the square with a side of 10 m. So, 1 A \u003d 100m².

In addition, found many different words.

I was interested to work on this topic.

There are a number of professions that mathematics are needed in "pure" form. This is an accountant, builder, teacher, cook and mechanic. They need to calculate the ability to use various formulas etc. And there are a number of professions, which, at first glance, mathematics is not needed at all. For example, actors, makeup artists, journalists, art historians, mannequins, singers, etc. But the people of these professions it will come in handy in everyday life: get a salary, make repairs, pay for utilities, pay in store for shopping, buy thing on credit, etc.

So, mathematics simply must be studied by a person of any profession.

Literature:

1) Geometry, 7-9: studies. For general education. institutions / A.V.Pogorelov, - 5th ed. - M.: Enlightenment, 2004.

2) Geometry, 10-11: studies. For general education. institutions / A.V.Pogorelov, - 7th ed. - M.: Education, 2007

3) telephone directory of 2011

4) encyclopedic Dictionary Young mathematics / Sost. A.P.Savin.-M.: Pedagogy, 1985.-352c., Il.

With spatial geometric shapes (cube, ball, parallelepiped, etc.) children get acquainted in practical activities (when designing, during the game) much earlier than with flat figures. Features of thinking of younger preschoolers determine the choice of visual material. At this age it is important that the object being studied is large, bright, so that they can be performed (play). The survey is on a touch basis, so with models bulk figures Children get to know easier. Cubes, balls, bars, etc. Participated in the game of children at the same time as the first toys. Typically, mathematical names are not given to them, but it is familiar with various volumetric forms, and only some terms are introduced into speech.

The main figures in space are considered: point, straight, plane. Each plane performs all planimetry approvals. In stereometry, as well as in planimetry, a number of axioms are introduced, which are studied in school course Geometry.

Volumetric geometric shapes called geometric bodies. The space is distinguished by polyhedra (prism, pyramid, etc.) and rotation bodies (ball, cone, cylinder, etc.).

Polyhedra

Polyhedron- This is a limited body, the surface of which consists of a finite number of polygons. These polygons are called edges, their parties are ribs, and the vertices are the vertices of the polyhedron.

The polyhedron is called convex if it is located one way from the plane of any of its face (Fig. 70).

Convex polyhedron Nonyochny polyhedron

Task 43.

Show peaks, ribs and edges of the polyhedra shown in Figure 70. What geometric figures they are!

The correct convex polyhedron has a face - the right equal polygons, and in each of its top it converges the same number of ribs.

Task 44.

1. Remember what figures are called equal.

2. What are the correct convex polyhedra known to you.

In total, there are 5 correct polyhedra, unlike the correct polygons, which are infinitely a lot. This is due to two reasons:

Rectangular parallelepiped - This is a parallelepiped, all the face of which rectangles (Fig. 75).

Cubic - This is a rectangular parallelepiped with equal edges (or all the face of which are squares) (Fig. 76).

Preschoolers, studying the cube, may noted that its surface consists of six squares, which has 8 vertices. The properties of the cube are mastered by them, for example, when performing such a task: "Locking a cube with colored paper. What is needed for this? " (Cut 6 identical squares).

Rectangular parallelepiped B. children's garden It is often called "brick" or "bar", which is permissible in pre-group preparation. These words are the pre-metallon names of geometric shapes, as well as the "cube", "roof" (triangular prism), "Column" (cylinder), etc.

Junior schoolchildren You can offer task: "Cut out the pattern for the box. What form does each part have? " Thus, children find out that rectangles are clearly rectangles, without formulating this clearly.

Pyramid - a polyhedron consisting of a flat polygon (base), a point that is not lying in the plane of the base (vertex), and all segments connecting the vertex of the pyramid with the points of the base (Fig. 77). Segments connecting the vertex of the pyramid with the tops of the base are called side ribs. The surface of the pyramid consists of the base and side faces. All side grains are triangles.

Depending on the number of angles of the polygon, which is the base, the pyramids are: triangular (Fig. 77a), quadrangular (Fig. 776), pentagonal and other pyramids.

Any faith triangular Pyramid It can serve as its basis. The same name is also the correct polyhedron, tetrahedron, whose faces are equal equilateral triangles.

The shape of the tetrahedron has a milk package (old packaging), and the Egyptian pyramids have the shape of a quadrangular correct pyramid. Preschoolers call the "pyramid" a completely different model - a toy from rings of different magnitude, which has a cone form. This situation may cause difficulties in memorization and proper use Geometric terms in children. This problem It is overcome with the timely competent explanation and separation of the names of toys from the names of their form, the standards for the definition of which geometric shapes serve.

Task 47.

1. Draw a pentagonal pyramid. Show its base, side surface, side faces and ribs. What geometric figures are they?

2. Allow the height of the pyramid and proper Pyramid.

Body of rotation

Studying the form of surrounding objects, preschoolers face with the bodies of rotation (Fig. 78).

These figures are called rotation bodies, as they can be obtained by rotating some flat geometric shapes.

Cylinder - This is the body of rotation that can be obtained by rotating the rectangle around one of its sides, like axis (Fig. 79).

Cone - this is the body of rotation, which can be obtained by rotating the rectangular triangle around one of its cathets, like axis (Fig. 80),

Ball - This is the body of rotation, which can be obtained by rotating half the circle around its diameter, like axis (Fig. 81).

Definitions of these figures from the course of geometry high School:

Cylinder - body that consists of two circles (bases) combined parallel

transfer and all segments connecting

corresponding points of these circles.

Cone - The body that consists of a circle (base), points (vertices), not lying in the plane of this circle, and all segments connecting the vertex of the cone with points of the base.

Ball- The body that consists of all points of space located at a distance of no more specified (radius) from this point (center).

Task 48.

Give definitions:

- spheres;

- cylinder heights and direct cylinder;

- forming cone, cone height and direct cone.

Preschoolers do not get acquainted with these wording, but can distinguish and recognize volumetric bodies, And if you spend a special job, and correctly call them. Children assume the properties of these figures in comparison with others. For example, during the game "rolls - it does not roll", they find out that: "The cylinder standing on the ground is stable, like a cube, but if it puts it - rolls like a ball."

The surface survey gives knowledge that the base of the cylinder and the cone is a circle. Drawing volume objects of different shapes On the plane teaches children to compare, to carry out an analogy, simulate, transform space on the plane. For example, in the process of discussing such questions: "What is the ball? What kind of figure should I draw to portray the ball? "

Acquaintance with bulk figures is expanding the knowledge of children about the world around, lays the foundation for studying geometry at school, enriches their speech, forms surveillance skills, develops thinking.

Questions for self-controlling topic number 3

1. What does geometry study?

2. What does the planimetry study?

3. What is studied stereometry?

4. What is called a geometric figure?

5. Name the rules for constructing geometry.

6. Name the main figures on the plane and in space.

7. What figures are called flat?

8. What are the figures called convex?

9. Give the definition of a segment.

10. Give the definition of the beam.

11. Give the definition of the angle.

12. What line is called broken?

13. What is the broken is called simple?

14. Give the definition of a polygon.

15. What kind of polygon is called convex?

16. What kind of polygon is called correct?

17. Give the definition of a triangle.

18. What triangle is called equilateral, which is an equilibried, what a versatile?

Topic 4.

Values \u200b\u200band their measurement

The concept of magnitude

Value - One of the main mathematical concepts that arose in antiquity and in the process of long-term development has been subjected to a number of generalizations. Length, area, volume, weight, speed are values.

Task 49.

Give examples of various greatness studied at school in the lessons of mathematics, physics, chemistry. Remember the methods for measuring them and the units of these values.

The values \u200b\u200bare the special properties of real objects or phenomena, which are manifested when comparing them for this property, and each value is associated with a certain comparison method. For example, the length of the segments can be compared with the method of overlay, and the mass of objects - weighing. Values \u200b\u200bcan be estimated quantitatively based on comparison.

The amount is considered as a generalization of the properties of some objects and how individual characteristic Properties of a specific object. For example, the property of items "have a length" is called "long".

Uniform values - The values \u200b\u200bthat express the same property of some class objects. For example, length, width, perimeter - homogeneous values.

Droinen values Express the various properties of objects (one item may have a mass, volume, etc.).

Formation of the concept of a geometric shape

Historically, the concept of geometric shape, as well as the concept of natural number, was one of the initial concepts of mathematics. Like natural numbers, the concept of a geometric shape was formed using the abstraction of identification, which is based on some equivalence ratio. In this case, such an attitude is "similarity", "similarity" of objects in their form, with which many objects are divided into equivalence classes so that any two objects of the same class have the same form, And any two subjects of various classes are various forms. Abstraining from other properties of objects (colors, values, material from which they are made, appointments, etc.), we get an independent concept of a geometric shape.

In mathematics, they are also coming: the class of similar objects is determined by any subject belonging to it and is called the form.

In connection with the consideration of equivalence relationship (Chapter IV, § 4), an example of the classification of blocks in their form was given. Solving this task, children receive classes of square, round, triangular and rectangular blocks, then each of these classes, as well as their individual representatives, are called a square, circle, triangle, rectangle, respectively. The discharge of these concepts is the equivalence attitude

"Have the same form."

In the study of geometry, and in particular geometric figures,

there are several levels of thinking.

The first, the most simplest level is characterized by the fact that geometric shapes are treated as integers and differ only in their form. If you show the preschooler Circle, square, rectangle and tell it the corresponding names, 1 then after some time he will be able to unmistakably recognize } these shapes are solely in their form (and not yet aiaiahio-bath), not distinguished square from the rectangle. At this level, the square is opposed to a rectangle.

In the following, second, the level analysis of perceived forms is carried out, as a result of which their properties are detected. Geometric figures appear already as carriers of their properties and are recognized by these properties, the properties of the figures are logically not yet ordered, they are set empirically. The figures themselves are also not ordered, as they are only described, but are not defined. This level of thinking in the geometry region does not yet include the structure of the logical retention.

What is the geometric figure?

Any geometric shape is meant consisting of points, i.e., every geometric shape is a variety of points, kommersantincluding one point is also considered to be

geometric figure.

Therefore, operations on sets and relationships between sets, considered in Chapter W, can be transferred to geometric shapes as on many points.

For example, in Figure 11, various relationships are depicted in which the square and the circle can be:

/ - The circle is in a square;

Square is in the circle;

Square and circle intersect;

Square and circle do not intersect.

Offering children to place a square and a circle of all sorts of ways or draw them and paint them with a common part (intersection) in a certain color, thereby help them identify the features of each of the relationships shown in the figure and:

a) all the points of the circle are points - a query;

Fig. eleven.

b) all square points are also circle points;

c) square and circle have common and non-dots;

d) Square and Circle have no common points.

At the pre-group level, children get acquainted with the simplest, but most. Common geometric shapes: various lines, shapes of blocks - square, circle, triangle, and a pentagon, hexagon. Strict definitions, of course, is not given at this level.

§ 2. Types of geometric shapes

All geometric shapes are divided into flat and spatial. So, for example, a square, a circle - flat figures; Cube, ball - spatial. Let's start with the consideration of the lines. Under the line we will keep in mind the flat line -Thine, all points of which lie on some plane, and the line itself is a subset of the points of the plane.

Obviously, such clarifications, as "length without width" or "surface boundary", cannot be taken for accurate definitions, since we do not know the exact meaning of the terms "length", "width", "border", "surface" and t . P. Essentially in the elementary geometry, the concept of the line is considered intuitively clear and their study is reduced to consideration different examples: straight, broken, curve, closed line, impelled line, segment, etc.

Direct line, or simply direct, can be allocated among other lines using its characteristic properties, i.e. such properties that only direct and no other lines have.

Figure 12, several paths are laid between the tree and the house. On the geometric language it means: in two points D.and FROMmultiple lines passes. The direct stands out among them by the fact that this is the line of the shortest distance.

Fig. 13.

Another characteristic property direct: after two points D.and with you can spend many different lines, directly - only one, i.e., in two points, one and only one passes

The lines are closed and unlucky. For example, a straight line is an open line, a circle is closed.

In relation to the direct two points may be "one way" from it or "on different directions". For example, a house and tree can be on one side of the river and then you can walk from the house to the tree or back, without passing through the bridge. If they are located on different sides of the river, then reach the garden or back, without passing through the bridge, it is impossible.

On the geometric language, this situation is described by the following

Sf; way. Two points And to B.are one way from

straight / if the segment connecting these points does not cross

straight / (Fig. 13).

Two points L and C (Fig. 13) are located on different sides from straight / if the segment l with connecting these points crosses

straight I.

Essentially straight I.cars the set of all the points that do not belong to it in two classes (two subsets), called P about l conclusted I m and with the boundary. This partition is generated by the equivalence ratio introduced into a plurality of all non-plane-belonging / points as follows: two points are in this respect, if the segment connecting them does not cross the direct /, and are not in this respect, if this segment crosses the straight /.

Children are quite early assimilated, which means "inside" and "outside" some closed line. An example of this is a children's game in classes. To successfully move from class to class, you need, jumping and throwing a bit, to accurately get inside a certain class (square). The first ideas about "inside" and "outside" are fixed in wrap games (Chapter III), when children meet with all the complicating situations

defining blocks inside and outside one hoop, inside one and outside of another hoop, inside all three hoops, inside two hoops and outside the third, etc. Therefore, before solving tasks related to the classification of blocks, or figures in games with hoops it is necessary to find out whether children recognize the inner and external domain in relation to each hoop.

We now translate these situations into the language of the geometry. It is intuitive that every circumference breaks the set of all the points that do not belong to it in two areas (Fig. 14). If two points l and in or D.and E.lie in the same area, then the segment connecting them does not cross the line /; If two points, such as C and D,belong to different areas, the interpretation of their segment crosses the line / (at the point TO)-

One of these areas is called the inner, the other is external. What geometric property can characterize the inner or outer region?

The area that is intuitively accepted for external has the following property: two points can be found in this area, for example D.and E,such a straight line passing through them is entirely in this area. The second area, which is intuitively accepted for the inner, does not have this property or is characterized by a property representing the dedication of the characteristic properties of the external domain, i.e. it is impossible to find such two points in it so that the direct passing through them lay in this area ( Or, otherwise, straight, passing through any two points of this area, be sure to cross the line /).

Above we used the concept of "segment" and connected it unchanged with two points: "Cut Av ","Cut connecting points l and in" etc. What is a segment? Sometimes they say "part straight". This can be understood as a subset of points direct. But what is a subset?

Sometimes use the relationship between "applicable to three

Fig. fourteen.

points. This attitude corresponds to a visual representation of a point lying on a straight line between two other points: if the point FROMles between points BUTand IN,it is impossible to "walk" in a straight line from L to C, without passing through the point S. These visual views are prompted and some properties of the relation "between": if the point C lies between BUTand IN,that FROMlies between B and L; of three points only one lies between two others, i.e. if FROMlies between L and B, then already BUT

does not lie between C and B as well INdoes not lie between BUTand S.

There are two various interpretations Concepts of cut (Shthere are two different concepts.) For one of them, the segment AUbelong to the points themselves l and in (secking ends) and all points direct ABlying between BUTand V. on a different interpretation of the point And I B.not considered to be segment belonging ABalthough still called its ends (i.e. the sections of the segment do not belong

We will adhere to the first interpretation, didactically

more appropriate.

Since in two points l and in the only straight line ABthen these two points are determined by the only segment with

ends of L and V.

Knowing what segment is, you can clarify the concept of broken

If l2, Ah,... , BUT"_!, A P.- points, no successive three of which do not lie on one straight line, then a line consisting of segments L1l2, L2L3, ..., L P - | L ", is called a broken line, these segments are called loloral links, and points A, AG,..., Ap- \\,L "- her vertices, points BUT\\and A P.cameras are also called broken ends.

If the ends of the broken coincide, then the broken is called the name, otherwise, "E is closed to that (strict definitions of a closed and unlocked line curve in the elementary

geometry are not given).

Figure 15, 1 depicted a closed broken line, on

figure 15.2 - unlocked.

Like any closed line, the closed broken line breaks the set of the points that do not belong to it into two areas - inner and external.

Among the broken lines are distinguished simple (without self-integration) broken lines, i.e., those that do not intersect themselves.

Pictures depicted in Figure 15 simple. On the

P "s. fifteen

figure 16, /, 2 impropy depicted, intersecting lines themselves.

We now turn to the consideration of polygons. There are two main approaches that are essentially determining various concepts: according to one of them, under the polygon, they understand the simple closed broken line, according to the second, a simple closed broken broody together with its internal area or the union of a simple closed broken and its inner field.

According to the first interpretation, the polygon model, for example, can be made of wire, on the second - cut out of the paper. What of the two interpretations is more appropriate from a didactic point of view? (From a logical point of view, both interpretations are correct and have the right to exist.) For small children, it is more natural to call a square, a triangle, etc. It is the figure that they painted and cut out, that is, the broken along with its internal area . Therefore, it seems that for school, the second interpretation is more appropriate.

Polygons are classified by the number of parties or angles: triangles, quadrangles, pentagons, hexagons, etc. Watching various polygons, you can detect the presence or absence of a property called convexity.

Figure 17 shows polygons possessing (in cases /, 2, 4, 6) and not possessing (in cases 3, 5, 7) this property.

Fig. sixteen.

How to geometrically describe this intuitive property? Any of the polygons in cases / 2, 4, 6 (Fig. 17) is located one way from a straight spent through each direction, i.e., if you continue any side, the resulting direct will cross the polygon (for this purpose, in the figure, the side of these polygons are continued by dotted). In each of the polygons in cases 3, 5, 7 there is at least one such party, the continuation of which crosses the polygon. The first is called convex, the second - non-depth. "

Fig. 17.

Triangle, square, rectangle - convex quadrangles. Five-pointed asterisk - a non-depth decidagon.

Parties, including vertices, polygon, i.e., closed broken, form the boundary of the polygon. This is an intuitive concept. For example, an intuitive idea of \u200b\u200bthe border of the figure is preparing children to geographical concept borders.

What is the difference between the boundary point, i.e. the point belonging to the border, from the inner point of the polygon (and in general figures)? How is this difference to describe geometrically?

For this purpose, we introduce the concept of the neighborhood of the point. Under the neighborhood of a point BUTwe will understand the circle of any radius with the center at the point BUT.Now, using this, a very visual concept, describing the difference between the inner and boundary points of the polygon.

For any inner point BUT,no matter how close it is to the border, you can always find the surroundings, all the points of which are internal (Fig. 18, 1,2).

For a boundary point INthere is no such neighborhood, i.e. whatever the neighborhood INthey took, inside it will be found both internal and external points. The same properties have internal and boundary points on geographic map, representing some geometric shape.

internal dots you can find the surroundings inside which all points belong to the territory of the USSR. For any point on the border of the USSR there is no such neighborhood, i.e., in any neighborhood of such a point, there are both points belonging to the USSR and the points belonging to the neighboring state.

Among the forms of the blocks used by us (or figures), except for a triangle, square, rectangle, and a circle. In addition, many items that children meet (plates, saucers, bicycle wheels, etc.), have a round shape. We consider it inappropriate for preschoolers to introduce the term "circle".

In the elementary geometry, the circle is defined as a set (or geometric place) of all points of the plane removed from some point, called the center, not exceeding the distance R.(/? -Redius circle); The circle is defined similarly as a set of all points of the plane removed from the point, called the center, at the same distance R.

Fig. nineteen.

Note that if in these wording the word "plane" to replace the word "space", then we obtain the determination of the ball and the sphere, respectively, the spatial analogues of the circle and circle. Circle, circle, ball and sphere can be determined and genetically, that is, the description of the formation of these figures. This process is easy to simulate: if the segment is fixed at one end and rotate it near this end, then it will describe the circle, and the second end is a circle. If the semicircle rotate around the diameter, then it will describe the ball, and the limiting semi-laboriousness - the sphere.

Preschoolers also get acquainted with one of the simplest polyhedra, what is the cube.

Cube is a spatial analogue of the square. It is limited six squares. It can be constructed (glue) from a flat pattern shape shown in Figure 19.

The familiarization of children with the above-described geometric figures described above is a propaedeutic basis for further formation and development of geometric, including spatial, representations.