Geometric shapes and their names. Geometric shapes, or where geometry begins

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 at the same time 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 Performances already as carriers of their properties and are recognized on these properties, the properties of the shapes 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.

© Wrone-eyed two levels are quite accessible to children 4-6 years old, and this circumstance should be taken into account when drawing up the program "Definition and developing a technique.

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: after 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.

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

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 (Shcreature 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,… ,

Many people mistakenly believe that for the first time they meet geometric shapes in high School. There they are studying their names, properties and formulas. But in fact, from childhood, any item that sees the child feels, feels his smell or interacts with it in any other way, represents the geometric shape. The couch on which the woman has just been lied is a rectangle, a lamp, which gives obstetrices the light is a round figure, the windows in the window - squares. This list can be continued infinitely.

Geometric shapes, directly as an element of science, first meets schoolchildren in middle classes. You can even say that geometry begins with them. However, as mentioned above, the first interactions with them occur long before that. Take, for example, point. It is the smallest figure in geometry. In addition, it is considered to be the basis of all others (as atoms in chemistry). All triangles, squares and other figures on any drawing consist of a variety of points. They have certain properties, each of which is inherent only in one figure (no other may be endowed with them).

It can be assumed that all geometric shapes consist directly from the lines, but what is she? This is a variety of points located in a row. They can continue infinitely because the straight line does not end. If it is limited from two sides, it is customary to be called a segment. If there is only one limit, then beam before you. Consequently, all flat figures in geometry consist of segments, since the constituents have the end and the beginning. It is worth noting that the straight line, which was divided by the point, is two rays directed in opposite parties to each other.

Not only from flat elements is geometry, there are also volumetric geometric shapes. They are started to study in school later, closer to the end of study, but it faces people, again, much earlier. For example, when a child takes a cube, keeps the cube in the palms. Or, if he looks at the chest, then in front of him a rectangular parallelepiped. All volumetric figures consist of planes (that is, it is an uncertain primary concept, as well as straight). The same parallelepiped consists of six such elements. You can clearly familiarize yourself with the plane, looking at the surface of any table. But it will only be part of it, because there are limitations. Directly plane is the same infinite as the straight line.

Thus, there is no sphere where geometric shapes would not meet. Their names are different, they define properties and features. For example, the formula is not suitable for a rectangle or square.

It is advisable to attach a child to geometric pieces in preschool age. You can make them with your own hands, and then lay out various drawings on paper (if it is flat elements). However, you should not refuse from the bulk figures. On the Internet you can find a variety of related. But you can not postpone the acquaintance with them, because everything we see is geometric shapes. Even a person consists of them!

"Geometric shapes are flat and voluminous"

Objectives lesson:

1. Cognitive: Create conditions for familiarization with the concepts flat and volumetric geometric shapes,expand the idea of \u200b\u200bthe types of volume figures, teach to determine the look of the shape, compare the figures.

2. Communicative : Create conditions for the formation of the ability to work in pairs, groups; upbringing a benevolent relation to each other; Relieve mutual assistance in students, mutual execution.

3. Regulatory: Create conditions for forming to plan a learning task, build a sequence of necessary operations, adjust your activities.

4. Personal: Create conditions for the development of computing skills, logical thinking, interest in mathematics, formation cognitive interests, intellectual abilities Pupils, independence in the acquisition of new knowledge and practical skills.

Planned results:

- personal:

the formation of cognitive interests, intellectual abilities of students; the formation of value relations to each other;

independence in the acquisition of new knowledge and practical skills;

the formation of skills to perceive, recycle the information obtained, allocate the main content.

- metaPered:

mastering the skills of independent acquisition of new knowledge;

the development of theoretical thinking based on the formation of skills to establish facts.

- subject:

assimate the concepts of flat and volumetric figures, learn to compare figures, find flat and volumetric figures in the surrounding reality, learn how to work with the spread.

Wood general scientific:

search and selection of necessary information;

application of information search methods, conscious and arbitrary construction of speech statement orally.

Wood Personal:

evaluate their own and other people's actions;

manifestation of confidence, care, goodwill,

ability to work in a pair

express a positive attitude to the process of knowledge.

Equipment: tutorial, interactive board, emoticons, models of figures, expanding figures, traffic lights customized, rectangles feedback, Dictionary.

Type of lesson: Studying a new material.

Methods: verbal, research, visual, practical.

Work forms: Frontal, group, steam room, individual.

1. Organization of the beginning of the lesson.

In the morning, the sun rose.

New day brought us.

Strong and kind

We meet a new day.

Here are my hands, I reveal

They will meet the sun.

Here are my legs, they firmly

Stand on earth and lead

I'm loyal.

Here is my soul, I reveal

Her towards people.

Wood, new day!

Hello, new day!

2. Actualization of knowledge.

1. Create good mood. Smile to me and each other, sit down!

To walk to the goal, you must first go.
Before you say, read. What does this statement mean?

(To achieve something, you need to do something)

And indeed, the guys falling into the goal can be only the one who sets himself to the collaboration and the organization of its actions. And so I hope that we will achieve your goal in the lesson.

Let's start our way to achieve the purpose of today's lesson.

3. Preparatory work.

Look at the screen. What do you see? (Geometric figures)

Name these figures.

What a task, can you offer your classmates? (divide the figures into groups)

You have cards with these figures. Perform this task in pairs.

What kind of sign did you share these figures?

· Flat and bulk figures

· Based on volume figures

What figures did we already worked with? What did you study find them? What figures do we occur on geometry for the first time?

What theme of our lesson? (Teacher adds words on the board: Volumenny, on the board appears the topic of the lesson: volumetric geometric shapes.)

What should we learn in the lesson?

V. "Opening" of new knowledge in practical research work.

(The teacher shows the cube and square.)

What are they like?

Is it possible to say that this is the same?

What is the difference between the cube from the square?

Let's do experience. (Pupils receive individual shapes - cube and square.)

Let's try to attach a square to the flat surface of the ports. What do you see? Is it all (entirely) Loe to the surface of the party? Close?

! How do you call the figure, which can be completely located on one flat surface?

(Flat figure.)

Is it possible a cube completely (all) to press to the party? Check.

Is it possible to call a cube flat figure? Why? Is there any space between hand and desk?

! So what can we say about Cuba? (Occupies a certain space, is a volume figure.)

Conclusions: What do flat and volumetric figures differ? (Teacher hangs out the conclusions on the board.)

Flat volumetric

You can entirely place a certain space,

on one flat surface. Tower over a flat surface.

Volumetric figures: pyramid, cube, cylinder, cone, ball, parallelepiped.

4. Recreation of new knowledge.

1. Name the shapes depicted in the picture.

What form are the founding of these figures?

What other forms can be seen on the surface of the cube and prism?

2. Figures and lines on the surface of the volume figures have their names.

Offer your names.

Side sides forming flat figure Clauses are called. And the side lines - Rybra. Corners of polygons - vertices. These are elements of bulk figures.

Guys, how do you think, what are such volumetric figures, who have many faces? Polyhedra.

Working with notebooks: reading new material

Correlation of real objects and volumetric bodies.

And now pick up for each subject volume figureon which it looks like.

Box - parallelepiped.

Apple - ball.

Pyramid - Pyramid.

Bank - cylinder.

Flower pot - cone.

Cap - cone.

Vase - cylinder.

Ball - ball.

5. Fizminutka.

6. 1. Imagine a big ball, intend it from all sides. He is big, smooth.

(Pupils "clasp" with hands and stroke imaginary ball.)

Now imagine a cone, touch it top. The cone grows up, so it is already above you. Take care before it is top.

Imagine that you are inside the cylinder, praise on its top base, sweep along the bottom, and now your hands on the side surface.

The cylinder has become a small gift box. Imagine that you are a surprise that is in this box. I press the button and ... Surprise pops out of the box!

7. Group work:

(Each group receives one of the figures: cube, pyramid, parallelepiped. Children's figure are learned, the conclusions are recorded in the card prepared by the teacher.)

Group 1. (To explore the parallelepiped)

Group 2. (For studying the pyramid)

Group 3. (To explore Cuba)

8. Crossword decision

9. The outcome of the lesson. Reflection.

Crossword solution in the presentation

What have you been opened for yourself today?

All geometric shapes can be divided into volumetric and flat.

And I learned the names of the volume figures