Let's project a point BUT orthogonally on both projection planes:

AA 1 _|_ P 1 ;AA 1 ^P 1 =A 1 ;

AA 2 _|_ P 2; AA 2 ^P 2 \u003d A 2;

Projection beams AA 1 and AA 2 mutually perpendicular and create a projecting plane in space AA 1 AA 2, perpendicular to both sides of the projections. This plane intersects the projection planes along the lines passing through the projections of the point BUT.

To get a flat drawing, we match the horizontal projection plane P 1 with the frontal plane P 2 rotation around the axis P 2 / P 1 (Fig. 61, a). Then both projections of the point will be on the same line perpendicular to the axis P 2 /P 1. Straight A 1 A 2, connecting the horizontal A 1 and frontal A 2 point projection is called vertical line of communication.

The resulting flat drawing is called complex drawing. It is an image of an object on several combined planes. A complex drawing consisting of two orthogonal projections connected to each other is called a two-projection one. In this drawing, the horizontal and frontal projections of the point always lie on the same vertical connection line.

Two interconnected orthogonal projections of a point uniquely determine its position relative to the projection planes. If we determine the position of the point a relative to these planes (Fig. 61, b) its height h (AA 1 =h) and depth f(AA 2 =f ), then these the values ​​in the multidrawing exist as segments of the vertical connection line. This circumstance makes it easy to reconstruct the drawing, i.e., to determine the position of the point relative to the projection planes from the drawing. To do this, it is enough at point A 2 of the drawing to restore the perpendicular to the plane of the drawing (considering it to be frontal) with a length equal to the depth f. The end of this perpendicular will determine the position of the point BUT relative to the plane of the drawing.

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7. Self-test questions

SELF-CHECK QUESTIONS

4. What is the name of the distance that determines the position of a point relative to the plane of projections P 1, P 2?

7. How to build an additional projection of a point on a plane P 4 _|_ P 2 , P 4 _|_ P 1 , P 5 _|_ P 4 ?

9. How can I build a complex drawing of a point by its coordinates?

33. Elements of a three-projection complex drawing of a point

§ 33. Elements of a three-projection complex drawing of a point

To determine the position of a geometric body in space and obtain additional information on their images, it may be necessary to build a third projection. Then the third projection plane is placed to the right of the observer perpendicular to the simultaneously horizontal projection plane P 1 and the frontal plane of projections P 2 (Fig. 62, a). As a result of the intersection of the frontal P 2 and profile P 3 projection planes we get a new axis P 2 / P 3 , which is located on the complex drawing parallel to the vertical communication line A 1 A 2(Fig. 62, b). Third point projection BUT- profile - turns out to be connected with the frontal projection A 2 a new line of communication, which is called horizontal

Rice. 62

Noah. The frontal and profile projections of a point always lie on the same horizontal line of communication. And A 1 A 2 _|_ A 2 A 1 and A 2 A 3 , _| _ P 2 / P 3.

The position of a point in space in this case is characterized by its latitude- the distance from it to the profile plane of the projections P 3, which we denote by the letter R.

The resulting complex drawing of a point is called three-projection.

In a three-projection drawing, the point depth AA 2 is projected without distortion on the plane P 1 and P 2 (Fig. 62, a). This circumstance allows us to construct the third - frontal projection of the point BUT along its horizontal A 1 and frontal A 2 projections (Fig. 62, in). To do this, through the frontal projection of the point, you need to draw a horizontal line of communication A 2 A 3 _|_A 2 A 1 . Then, anywhere on the drawing, draw an axis of projections П 2 / П 3 _|_ A 2 A 3, measure the depth f of a point on a horizontal projection field and set aside it along the horizontal line of communication from the axis of projections P 2 /P 3 . Get a profile projection A 3 points BUT.

Thus, in a complex drawing consisting of three orthogonal projections of a point, two projections are on the same line of communication; communication lines are perpendicular to the corresponding projection axes; two projections of a point completely determine the position of its third projection.

It should be noted that in complex drawings, as a rule, the projection planes are not limited and their position is set by the axes (Fig. 62, c). In cases where the conditions of the problem do not require this

It turns out that projections of points can be given without depicting axes (Fig. 63, a, b). Such a system is called baseless. Communication lines can also be drawn with a gap (Fig. 63, b).

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34. The position of a point in the space of a three-dimensional angle

§ 34. The position of a point in the space of a three-dimensional angle

The location of the projections of points in the complex drawing depends on the position of the point in the space of a three-dimensional angle. Let's consider some cases:

• the point is located in space (see Fig. 62). In this case, it has depth, height, and breadth;
• the point is located on the projection plane P 1- it has no height, P 2 - no depth, Pz - no breadth;
• the point is located on the axis of projections, P 2 / P 1 has no depth and height, P 2 / P 3 - has no depth and latitude and P 1 / P 3 has no height and latitude.

35. Competing points

§ 35. Competing points

Two points in space can be located in different ways. In a particular case, they can be located so that their projections on some projection plane coincide. Such points are called competing. On fig. 64, a a complex drawing of points is given BUT and AT. They are located so that their projections coincide on the plane P 1 [A 1 \u003d= B 1]. Such points are called horizontally competing. If the projections of the points A and B coincide on the plane

P 2(Fig. 64, b) they're called frontally competitive. And if the projections of the points BUT and AT coincide on the plane P 3 [A 3 \u003d= B 3] (Fig. 64, c), they are called profile competitive.

Competing points determine the visibility in the drawing. Horizontally competing points will see the one with a greater height, frontally competing ones - the one with more depth, and profile competing ones - the one with more latitude.

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36. Replacing projection planes

§ 36. Replacement of projection planes

The properties of a three-projection drawing of a point allow, using its horizontal and frontal projections, to build a third onto other projection planes introduced instead of the given ones.

On fig. 65 a showing dot BUT and its projections - horizontal A 1 and frontal A 2 . According to the conditions of the problem, it is necessary to replace the planes П 2 . Let's designate the new projection plane P 4 and place it perpendicularly P 1. At the intersection of planes P 1 and P 4 we get a new axis P 1 / P 4 . New point projection A 4 will be located on communication line passing through a point A 1 and perpendicular to the axis P 1 / P 4 .

Since the new plane P 4 replaces the frontal projection plane P 2 , point height BUT depicted equally in full size and on the plane P 2 and on the plane P 4 .

This circumstance allows us to determine the position of the projection A 4 , in the system of planes P 1 _|_ P 4(Fig. 65, b) on the complex drawing. To do this, it is enough to measure the height of the point on the replaced plane

sti projection P 2, put it on a new line of communication from the new axis of projections - and a new projection of the point A 4 will be built.

If a new projection plane is introduced instead of the horizontal projection plane, i.e. P 4 _ | _ P 2 (Fig. 66, a), then in the new system of planes the new projection of the point will be on the same line of communication with the frontal projection, and A 2 A 4 _|_. In this case, the depth of the point is the same on the plane P 1, and on the plane P 4 . On this basis they build A 4(Fig. 66, b) on the line of communication A 2 A 4 at such a distance from the new axis P 1 / P 4 at what A 1 is located from the axis P 2 /P 1.

As already noted, the construction of new additional projections is always associated with specific tasks. In the future, a number of metric and positional problems solved using the method of replacing projection planes will be considered. In tasks where the introduction of one additional plane will not give the desired result, another additional plane is introduced, which is denoted by P 5 . It is placed perpendicular to the already introduced plane P 4 (Fig. 67, a), i.e. P 5 P 4 and produce a construction similar to those previously considered. Now the distances are measured on the replaced second of the main projection planes (in Fig. 67, b on surface P 1) and deposit them on a new line of communication A 4 A 5, from the new projection axis P 5 /P 4 . In the new system of planes P 4 P 5, a new two-projection drawing is obtained, consisting of orthogonal projections A 4 and A 5 , connected by a communication line

When constructing a point according to the given coordinates, it must be remembered that, in accordance with the drawing rules, the scale along the axis Oh decreases in 2 times compared to the scale along the axes OU and Oz.

1.Build points: A(2; 1; 3) x A = 2; y A = 1; z A = 3

a) usually, first of all, they build the projection of a point onto a plane Ohu. Mark points x A =2 and y A=1 and draw straight lines through them parallel to the axes Oh and OU. The point of their intersection has coordinates (2;1; 0) point built A 1 (2;1; 0.)

A(2; 1; 3)

0 y A=1

x A =2 at

A 1 (2; 1; 0) 0 y A=1at

X x A \u003d 2 A 1 (2; 1; 0)

X

b) further from the point A 1 (2; 1; 0) restore perpendicular to the plane Ohu (draw a line parallel to the axis Oz ) and lay a segment equal to three on it: z A = 3.

2.Build points: B(3; - 2; 1) x B = 3; y B = -2; Z B = 1

z

y B = - 2

B(3; -2; 1) O at

B 1 (3;-2) x B \u003d 3

X

3. Build a point C(-2; 1; 3 ) z C (-2; 1; 3)

X A \u003d -2; Y A = 1; Z A = 3

x C \u003d - 2 C 1 (-2; 1; 0)

y A =1 y

4.Dan cube. A ... D 1, whose edge is 1 . The origin is the same as the point AT, ribs VA, Sun and BB 1 coincide with the positive rays of the coordinate axes. Name the coordinates of all other vertices of the cube. Calculate the diagonal of a cube.

z

AB = BC = BB 1 BD 1 = =

B 1 (0; 0; 1) C 1 (0; 1; 1) = =

A 1 (1; 0; 1) D 1 (1; 1; 1)

В(0;0;0) С(0;1;0)

A(1;0;0) D(1;1;0)

5.Plot points A(1;1;-1) and B(1; -1; 1). Does the segment intersect the coordinate axis? coordinate plane? Does the line segment pass through the origin? Find the coordinates of the intersection points, if any. z The points lie in a plane perpendicular to the axis Oh.

The segment intersects the axis Oh and plane hoy at the point

B(1; -1; 1)

0(0;0;0)

С(1;0;0)

A(1;1;-1)

6.Find the distance between two points: A(1;2;3) and B(-1;1;1).

a)AB = = = =3

b)С(3;4;0) and D(3;-1;2).

CD = = =

In space, to determine the coordinates of the middle of the segment, a third coordinate is introduced.

B (x B; y B; z B)

With( ; ; )

A(x A; y A; z A)

7.Find coordinates With midpoints of segments: a)AB, if A(3; - 2; - 7), B(11; - 8; 5),

x M = = 7; y M = = - 5; z M = = - 1; C(7; - 5; - 1)

8. Point coordinates A(x; y; z). Write the coordinates of points that are symmetrical to the given one with respect to:

a) coordinate planes

b) coordinate lines

in) origin

a) If point A 1 symmetrical to the given one with respect to the coordinate plane ho, then the difference in
coordinates of points will only be in the sign of the coordinate z: A 1 (x; y; -z).

dot A 2 Ohz, then A 2 (x; -y; z).

dot A 3 symmetrical to the given one with respect to the plane Ouz, then A 2 (-x; y; z).

b) If point A 4 symmetrical to the given one with respect to the coordinate line Oh, then the difference in
coordinates of points will be only in signs of coordinates at and z: A 4 (x; -y; -z).

dot A 5 OU, then A 5 (-x; y; -z).

dot A 6 symmetrical to a given one with respect to a straight line Oz, then A 6 (-x; -y; z).

in) If point A 7 is symmetrical to the given one with respect to the origin, then A 6 (-x; -y; -z).

COORDINATE CONVERSION

The transition from one coordinate system to another is called coordinate system transformation.

We will consider two conversion cases coordinate systems, and derive formulas for the dependence between the coordinates of an arbitrary point of the plane in different coordinate systems. (The technique of transforming the coordinate system is similar to transforming graphs).

1.Parallel transfer. In this case, the position of the origin of coordinates changes, while the direction of the axes and the scale remain unchanged.

If the origin of coordinates goes to the point 0 1 with coordinates 0 1 (x 0; y 0), then for the point M(x; y) relationship between system coordinates x0y and x 0 0y 0 expressed by the formulas:

x \u003d x 0 + x "

y = y 0 + y"

The resulting formulas allow us to find old coordinates from known new ones. X" and at" and vice versa.

y M(x; y) M(x"; y")

0 1 (x 0; y 0), x "

x 0 x"

2.Rotation of coordinate axes. In this case, both axes are rotated by the same angle, while the origin and scale remain unchanged.

M(x; y)

y 1 x 1

Point coordinates M in the old system M(x; y) and M(x"; y") - in the new one. Then the polar radius in both systems is the same, and the polar angles are respectively equal + and , where - polar angle in the new coordinate system.

According to the formulas for the transition from polar to rectangular coordinates, we have:

x = rcos( + ) x = rcos cos - rsin sin

y = rsin( + ) y = rcos sin + rsin cos

But rcos = x" and rsin = y", That's why

x \u003d x " cos - y "sin

y \u003d x "sin + y" cos

Answer the following questions in writing:

1. What is a rectangular coordinate system in a plane? in space?
2. What is the applicate axis? Ordinate? Abscissa?
3. What is the notation for unit vectors on the coordinate axes?
4. What is an ort?
5. How is the length of a segment given by the coordinates of its ends calculated in a rectangular coordinate system?
6. How are the coordinates of the middle of a segment given by the coordinates of its ends calculated?
7. What is a polar coordinate system?
8. What is the relationship between the coordinates of a point in rectangular and polar coordinate systems?

Complete the tasks:

1. How far from the coordinate planes is the point A(1; -2; 3)

2. How far is the point A(1; -2; 3) from coordinate lines a)OU; b) OU; in)Oz;

3. What condition is satisfied by the coordinates of points in space that are equally distant:

a) from two coordinate planes Ohu and Оуz; AB

b) from all three coordinate planes

4. Find the coordinates of a point M middle of the segment AB, A(-2; -4; 1); B(0; -1; 2) and name the point symmetrical to the point M, relatively a) axes Oh

b) axes OU

in) axes Oz.

5. Given a point B(4; - 3; - 4). Find the coordinates of the bases of the perpendiculars dropped from a point on the coordinate axes and coordinate planes.

6.On axle OU find a point equidistant from two points A(1; 2; - 1) and B(-2; 3; 1).

7. Flat Ohz find the point equidistant from three points A(2; 1; 0); B(-1; 2; 3) and C(0;3;1).

8. Find the lengths of the sides of the triangle ABC and its area , if the vertex coordinates : A (-2; 0; 1), B (8; - 4; 9), C (-1; 2; 3).

9. Find the coordinates of the projections of points A(2; -3; 5); In (3;-5; ); WITH(- ; - ; - ).

10. Points are given A(1; -1; 0) and B(-3; - 1; 2). Calculate the distance from the origin to the given points.

VECTORS IN SPACE. BASIC CONCEPTS

All quantities that are dealt with in physics, technology, everyday life are divided into two groups. The former are fully characterized by their numerical value: temperature, length, mass, area, work. Such quantities are called scalar.

Other quantities such as force, speed, displacement, acceleration, etc. determined not only by their numerical value, but also by direction. These quantities are called vector, or vectors. A vector quantity is geometrically represented as a vector.

Vector-this is a directed straight line segment, i.e. segment that has
specified length and direction.

A point is one of the basic concepts of geometry. In modern mathematics, points are called elements of a different nature that make up spaces, for example, in Euclidean space, a point is an ordered set of n numbers.

In descriptive geometry, the position of a point in space can be determined by its coordinates. A remarkable feature is that the coordinate characterizing the distance of a point from the projection plane is of the same name with the axis, which is not present in the formation of this projection plane. So, the removal of a point from P 2 is measured by the y coordinate, and the frontal plane of the projections P 2 itself is formed by the intersection of the axes OX and OZ.

Thus, each of the three projections of a point is characterized by two coordinates, their name corresponds to the names of the axes that form the corresponding projection plane: horizontal - A 1 (X A; Y A); frontal - A 2 (X A; Z A); profile - A 3 (Y A; Z A).

Translation of coordinates between projections is carried out using communication lines. Thus, in the system of projection planes П 1 П 2 the coordinate x, common for frontal and horizontal projections, is broadcast by the vertical communication line А 2 А 1 perpendicular to the OX axis.

According to these two projections, you can build projections of a point either using coordinates or graphically. Graphically, a profile projection is built by translating the Z parameter with a horizontal connection line drawn from the frontal projection, and the Y parameter is transferred from the horizontal projection using the constant straight line of the drawing k - the bisector of the split axis angle: Y 1 OY 3 , on which the horizontal connection line drawn from the horizontal projection perpendicular to OY 1 is refracted at a right angle. At the same time, a square with a side equal to the Y coordinate of the original is formed at the origin of coordinates, which ensures the transfer of the Y coordinate between the horizontal and profile projections. In table. Figures 3.1 and 3.2 present general algorithms for constructing point A by coordinates in a spatial model of a system of three projection planes P 1 P 2 P 3 and on a complex drawing.

Table 3.1

Algorithm for constructing a visual image of a point by coordinates
Word form	Graphic form
1. Set aside the corresponding coordinates of point A on the X, Y, Ζ axes. Get the points A x , A y , A z
2. Horizontal projection A 1 is located at the intersection of communication lines from points A x and A y drawn parallel to the X and Y axes
3. Frontal projection A 2 is located at the intersection of communication lines from points A x and A z, drawn parallel to the axes X and z
4. Profile projection A 3 is located at the intersection of communication lines from points A y and A z drawn parallel to the axes Y and z
5. Point A is located at the intersection of communication lines drawn from points A 1, A 2 and A 3

Instruction

Build three coordinate planes to have a reference point at point O. In the drawing, the projection planes are in the form of three axes - ox, oy and oz, with the oz axis pointing up, the oy axis to the right. To build the last x-axis, divide the angle between the y- and z-axes in half (if you are drawing on a sheet of paper in a cage, just draw this axis).

Please note that if the coordinates of point A are written as three in brackets (a, b, c), then the first number a is from the x plane, the second b is from y, the third c is from z. First take the first coordinate a and mark it on the x-axis, to the left and down if a is positive, to the right and up if it is negative. Name the resulting letter B.

Then plot the last number c up on the z-axis if it's positive, and down the z-axis if it's negative. Mark received point letter D.

From the obtained points, draw the projections of the desired point on the planes. That is, at point B draw two straight lines that will be parallel to the axes oy and oz, at point C draw straight lines parallel to the axes ox and oz, at point D - straight lines parallel to ox and oy.

If one of the coordinates of the point is equal to zero, the point lies in one of the projection planes. In this case, just mark the known coordinates on the plane and find point intersections of their projections. Be careful when plotting points with coordinates(a, 0, c) and (a, b, 0), do not forget that the projection onto the x-axis is at an angle of 45⁰.

Related videos

Sources:

• build by coordinates

Tip 2: How to check that the points do not lie on the same line

Based on the axiom describing the properties straight: whatever the line is, there is points belonging and not belonging to it. Therefore, it is quite logical that not all points will lie on one straight lines.

You will need

• - pencil;
• - ruler;
• - pen;
• - notebook;
• - calculator.

Instruction

In the event that (x - x1) * (y2 - y1) - (x2 - x1) * (y - y1) is less than zero, the point K is located above or to the left of the line. In other words, only if an equation like (x - x1) * (y2 - y1) - (x2 - x1) * (y - y1) = 0 is true, points A, B and K will be located on the same straight.

In other cases, only two points(A and B), which, according to the assignment, lie on straight, will belong to it: the line will not pass through the third point (point K).

Consider the second membership option points note: this time you need to check if the point C(x,y) belongs to the segment with endpoints B(x1,y1) and A(x2,y2), which is part of straight z.

Describe the points of the segment under consideration by the equation pOB+(1-p)OА=z, provided that 0≤p≤1. OB and OA are vectors. If there is a number p that is greater than or equal to 0, but less than or equal to 1, then pOB + (1-p) OA \u003d C, and, the point C will lie on the segment AB. Otherwise, the given point will not belong to this segment.

Write down the equality pOB+(1-p)OА=С coordinatewise: px1+(1-p)x2=x and py1+(1-p)y2=y.

Find the number p from the first and substitute its value into the second equality. If the equality will correspond to the conditions 0≤p≤1, then the point C belongs to the segment AB.

note

Make sure your calculations are correct!

Helpful advice

To find k - the slope of a straight line, you need (y2 - y1) / (x2 - x1).

Sources:

• Algorithm for checking whether a point belongs to a polygon. Ray tracing method in 2019

Three-dimensional space consists of three basic concepts that you gradually learn in the school curriculum: point, line, plane. In the course of working with some mathematical quantities, you may need to combine these elements, for example, to build a plane in space from a point and a line.

Instruction

To understand the algorithm for constructing planes in space, pay attention to some axioms that describe the properties of a plane or planes. First: through three points that do not lie on one straight line, a plane passes, and only one. Therefore, to construct a plane, you need only three points that satisfy the axiom in position.

Second: a straight line passes through any two points, and only one. Accordingly, it is possible to construct a plane through a straight line and a point that does not lie on it. If from the opposite: any line contains at least two points through which it passes, if one more point is known, not on this line, a line can be built through these three points, as in the first paragraph. Each point of this line will belong to the plane.

Third: a plane passes through two intersecting straight lines, and only one. Intersecting lines can form only one common point. If in space, they will have an infinite number of common points, and therefore form one straight line. When you know two lines that have a point of intersection, you can construct at most one plane passing through these lines.

Fourth: through two parallel lines it is possible to draw a plane, and only one. Accordingly, if you know that the lines are parallel, you can draw a plane through them.

Fifth, an infinite number of planes can be drawn through a straight line. All these planes can be considered as the rotation of one plane around a given straight line, or as an infinite number of planes having one line of intersection.

So, you can build a plane if all the elements that determine its position in space are found: three points that do not lie on a line, a line and a point that does not belong to a line, two intersecting or two parallel lines.

Related videos

Did you know that the human body is a mini power plant? Each of us generates a small amount of electricity. This happens both in motion and at rest - then the generation of electricity occurs in the internal organs, one of which is the heart.

One of the medical studies that can determine the condition of the heart is an ECG. The cardiologist takes an electrocardiogram to find out where it is located in the chest, how the atria, valves and ventricles work, their shape and whether there are any functional changes. One of the most important ECG indicators is the direction of the electrical axis of the heart.

What is the axis of the heart and how to find it?

The heart axis (as well as the earth axis) cannot be seen or touched. It is determined only with the help of an electrocardiograph, because it captures the electrical activity of the heart. When the cells of the heart muscle tense and relax, in obedience to impulses coming from the nervous system, they form an electric field, the center of which is the EOS (the electrical axis of the heart).

But if you look at the anatomical atlas, you can draw a vertical line that will divide the heart into two equal parts - this is approximately how the axis of the heart is located. From this we can conclude that the EOS coincides with the so-called anatomical axis. Of course, each person is individual, therefore, the electric axis can be located differently for different people (for example, if we start from the heart-statistical value, then the EOS is located vertically in a thin person, and horizontally in an obese person).

When does the cardiac axis change position?

After taking an ECG and finding out how the EOS is located, the cardiologist can tell you how in the chest, whether the myocardium is healthy ( cardiac), how nerve impulses pass to different parts of the heart.

If the electrocardiogram shows that the electrical axis is to the right or to the left, this will indicate to the doctor any pathological process. Deviation to the right can lead to suspicions about the incorrect position of the heart (its displacement may be congenital or occur due to aortic expansion, the occurrence of neoplasms and other pathologies). In addition, EOS deviation is a sign of life-threatening conditions: dextrocardia, blockade of the His bundle, myocardial infarction (its anterior wall).

If the EOS is significantly deviated to the left side, this may be a sign of cardiomyopathy, hypertrophy of some parts of the heart, apical infarction, or congenital malformation.

A number of heart diseases can be asymptomatic for the time being. Therefore, it is so important to periodically undergo a medical examination, one of the components of which is an ECG. After all, the disease is easier to prevent. And heart disease is a must, because they are a direct threat to life.

Duration: 1 lesson (45 minutes).
Class: 6th grade
Technology:

• multimedia presentation Microsoft Office PowerPoint, Notebook;
• use of an interactive whiteboard;
• student handout created with Microsoft Office Word and Microsoft Office Excel.

annotation:
On the topic "Coordinates" in the thematic planning, 6 hours are allotted. This is the fourth lesson on the topic "Coordinates". At the time of the lesson, students have already become familiar with the concept of "coordinate plane" and the rules for constructing a point. Updating knowledge is carried out in the form of a frontal survey. In repetition lessons, all students are included in various activities. In this case, all channels of perception and reproduction of the material are used.
The assimilation of the theory is also checked during oral work (the task is to solve the crossword puzzle, in which quarter the point is located). For strong students, additional tasks are provided.
The lesson uses multimedia equipment and an interactive whiteboard to demonstrate presentations and tasks in Microsoft Office PowerPoint and Notebook. To create test tasks and handouts were used: Microsoft Office Excel, Microsoft Office Word.
The use of an interactive whiteboard expands the possibilities of presenting material. In the Notebook program, students can independently move objects to the right place. In the Microsoft Office PowerPoint program, it is possible to set the movement of objects, so a physical minute for the eyes is provided.

The lesson uses:

• checking homework;
• front work;
• individual work of students;
• presentation of the student's report;
• performing oral and written exercises;
• the work of students with an interactive whiteboard;
• independent work.

Lesson outline.

Target: consolidate the skills of finding the coordinates of the marked points and build points according to the given coordinates.
Lesson objectives:
educational:

• generalization of knowledge and skills of students on the topic "Coordinate plane";
• intermediate control of knowledge and skills of students;

developing:

• development of communicative competence of students;
• development of students' computing skills;
• development of logical thinking;
• development of students' interest in the subject through non-traditional forms of teaching;
• development of mathematically literate speech, horizons of students;
• developing the ability to work independently with a textbook and additional literature;
• development of aesthetic feelings of students;

educational:

• education of discipline in the organization of work in the classroom;
• education of cognitive activity, sense of responsibility, culture of communication;
• education of accuracy in the execution of constructions.

During the classes.

• Organizing time.

Greeting students. Message of the topic and purpose of the lesson. Checking the readiness of the class for the lesson. The task is set: to repeat, generalize, systematize knowledge on the announced topic.

2. Actualization of knowledge.

Verbal counting.
1) Individual work: several people do work on cards.

2) Work with the class: calculate examples and make a word. Table on the interactive whiteboard screen, letters are entered into the table with an electronic marker from the interactive whiteboard.

Students take turns going to the blackboard and writing letters. It turns out the word "Prometheus". One of the students, who has prepared a report in advance, tells what this word means. (The ancient Greek astronomer Claudius Ptolemy, who used latitude and longitude as coordinates already in the 2nd century.)

Front work.

The task “Solve the crossword puzzle” will help you remember the basic concepts on the topic “Coordinate plane”.
The teacher shows a crossword puzzle on the screen of the interactive whiteboard and invites students to solve it. Students use electronic markers to write words in a crossword puzzle.
1. Two coordinate lines form a coordinate ....
2. Coordinate lines are coordinate ....
3. What angle is formed at the intersection of the coordinate lines?
4. What is the name of a pair of numbers that determine the position of a point on a plane?
5. What is the name of the first number?
6. What is the name of the second number?
7. What is the name of the segment from 0 to 1?
8. How many parts is the coordinate plane divided by coordinate lines?

3. Consolidation of skills and abilities to build a geometric figure according to the given coordinates of its vertices.

Construction of geometric figures. Work with the textbook in notebooks.

• No. 1054a “Construct a triangle if the coordinates of its vertices are known: A (0; -3), B (6: 2), C (5: 2). Specify the coordinates of the points where the sides of the triangle intersect the x-axis.
• Construct a quadrilateral ABCD if A(-3;1), B(1;1), C(1;-2),D(-3;-2). Determine the type of quadrilateral. Find the coordinates of the intersection of the diagonals.

4. Physical exercise for the eyes.

On the slide, students should follow the movements of the object with their eyes. At the end of the physical minute, a question is asked about the geometric shapes obtained as a result of the movement of the eyes.

5. Control over the ability to build points on the coordinate plane according to the given coordinates.

Independent work. Artists competition.
The coordinates of the points are written on the slide. Cards are also printed for each student. If you correctly mark the points on the coordinate plane and sequentially connect them, then you get a picture. Each student completes the task independently. After completing the work, the correct drawing opens on the screen. Each student receives an assessment for independent work.

6. Homework.

• No. 1054b, No. 1057a.
• Creative task: draw a drawing by points on the coordinate plane and write down the coordinates of these points.

7. Summing up the lesson.

Questions for students:

• What is a coordinate plane?
• What are the names of the coordinate axes OX and OY?
• What angle is formed when the coordinate lines intersect?
• What is the name of a pair of numbers that determine the position of a point on a plane?
• What is the name of the first number?
• What is the name of the second number?

Literature and resources:

• G.V. Dorofeev, S.B. Suvorova, I.F. Sharygin “Mathematics. 6cl”
• Mathematics. Grade 6: Lesson plans (according to the textbook by G.V. Dorofeev and others)
• http://www.pereplet.ru/nauka/almagest/alm-cat/Ptolemy.htm