Point and direct are the main geometric figures on surface.
Ancient Greek scientist Euclid said: "Point" is something that does not have parts. " The word "point" translated from latin language means the result of instant touch, injection. The point is the basis for building any geometric shape.
A straight line or simply straight is a line, along which the distance between the two points is the shortest. The straight line is infinite, and it is impossible to port the entire straight and measure it.
The dots are indicated by the title Latin letters A, B, C, D, E, etc., and direct the same letters, but the linear a, b, c, d, e, etc., can be referred to as two letters corresponding to the points lying on her. For example, direct a can be labeled AB.
It can be said that the points of the AV lie on a direct A or belong to direct a. And we can say that straight and passes through points A and V.
The simplest geometric shapes on the plane are a segment, a beam, a broken line.
The segment is part of a straight line that consists of all points of this direct, limited two selected points. These points are the ends of the segment. The segment is indicated by an indication of its ends.
The beam or semi-straight is part of the straight line, which consists of all the points of this straight line, lying on one side of its point. This point is called the initial point of the semicircuit or the beginning of the beam. The beam has a start point, but does not end.
The semi-tray or rays are designated two line latin letters: the initial and any other letter corresponding to the point belonging to the semi-simplicate. At the same time, the initial point is made in the first place.
It turns out that direct is infinite: it does not have any beginning, no end; The beam has only the beginning, but there is no end, and the segment has the beginning and the end. Therefore, only the segment we can measure.
Several segments that are consistently interconnected in such a way that having one offshields (adjacent) are located not on one straight line, represent a broken line.
The broken line can be closed and unlocked. If the end of the last segment coincides with the beginning of the first, we have a closed broken line, if there is no - unlocked.
the site, with full or partial copying of the material reference to the original source is required.
We will look at each of themes, and at the end there will be tests on topics.
Point in mathematics
What is a point in mathematics? The mathematical point does not matter and denotes the title Latin letters: a, b, c, d, f, etc.
In the figure you can see the image of the points A, B, C, D, F, E, M, T, S.
Cut in mathematics
What is a segment in mathematics? In mathematics lessons, you can hear the following explanation: the mathematical segment has a length and ends. The segment in mathematics is a totality of all points lying on a straight line between the sections of the segment. Cuts segment - two border points.
In the figure, we see the following: segments ,,,, and, as well as two points B and S.
Straight in mathematics
What is a straight line in mathematics? Definition direct in mathematics: Straight does not ends and can continue in both sides to infinity. A straight line in mathematics is indicated by two any dots direct. To explain the concept of a direct student, we can say that direct is a segment that does not have two ends.
The figure shows two direct: CD and EF.
Bump in mathematics
What is the beam? Definition of the beam in mathematics: the beam is part of the straight, which has the beginning and has no end. In the title of the beam there are two letters, for example, DC. Moreover, the first letter always denotes the point of the start of the beam, so let the letters cannot be changed.
The figure shows the rays: DC, KC, EF, MT, MS. Rays KC and KD - one beam, because They have a common start.
Numerical straight in mathematics
The definition of a numerical straight in mathematics: straight, points of which marked numbers, called a numeric line.
The figure shows the numerical straight, as well as the beam OD and ED
To denote the geometric shapes and their projections, to display the relationship between them, as well as for brevity entries of geometric proposals, algorithms for solving problems and evidence by theorems in the course used geometric languageComposed of the designations and symbols adopted in the course of mathematics (in particular, in the new course of geometry in high school).
All varieties of designations and symbols, as well as connections between them can be divided into two groups:
group I - designations of geometric shapes and relations between them;
group II designations of logical operations that constitute the syntactic basis of the geometric language.
Below is given full list Mathematical symbols used in this course. Special attention It is paid to symbols that are used to designate projections of geometric shapes.
Group I.
Symbols denoting geometric shapes and relations between them
A. Designation of geometric shapes
1. The geometric figure is indicated - F.
2. Points are designated capital letters Latin alphabet or Arabic figures:
A, B, C, D, ..., L, M, N, ...
1,2,3,4,...,12,13,14,...
3. Lines arbitrarily located in relation to the planes of projections are denoted by the line letters of the Latin alphabet:
a, B, C, D, ..., L, M, N, ...
Line level indicate: H - horizontal; FRONTAL.
The following notation is also used for direct:
(AV) - straight, passing through points a A B;
[AV) - a beam with the beginning at point A;
[AV] - cut straight, limited to points A and V.
4. Surfaces are denoted by the line letters of the Greek alphabet:
α, β, γ, δ,...,ζ,η,ν,...
To emphasize the way of setting the surface, you should specify geometric elements with which it is determined, for example:
α (A || B) - the plane α is determined by parallel straight A and B;
β (D 1 d 2 Gα) - the surface β is determined by the guides d 1 and d 2, forming G and the plane of the parallelism α.
5. Corners are indicated:
∠abc is an angle with a vertex at point in, as well as ∠α °, ∠β °, ..., ∠φ °, ...
6. Corner: Value ( sigid measure) is indicated by the sign that is put on the angle:
The magnitude of the ABC angle;
The magnitude of the angle φ.
The straight angle is marked with a square with a point inside
7. The distances between the geometric figures are denoted by two vertical segments - ||.
For example:
| Av | - the distance between the points A and B (length of the CUT);
| Aa | - distance from point A to line A;
| Aα | - radiation from point A to the surface α;
| AB | - distance between lines A and B;
| αβ | The distance between the surfaces α and β.
8. For planes of projections, notation is taken: π 1 and π 2, where π 1 - horizontal plane projections;
π 2-Fulletal plane of projections.
When replacing the planes of projections or the introduction of new planes, the latter are indicated by π 3, π 4, etc.
9. The axes of projections are referred to: x, y, z, where X is the abscissa axis; y - axis ordinate; z - Applica axis.
The long straight line of Monge denote k.
10. Projections of points, lines, surfaces, any geometric shapes are denoted by the same letters (or numbers) as the original, with the addition of the upper index corresponding to the plane of the projection on which they are obtained:
A ", in", s ", d", ..., l ", m", n ", horizontal projection of points; a", in ", s", d ", ..., l", m " , N ", ... Front projections of points; a ", b", c ", d", ..., l ", m", n ", - horizontal projections of lines; a", b ", with", d ", ..., l", m ", N", ... Front projections of lines; α ", β", γ ", δ", ..., ζ ", η", ν ", ... horizontal surface projections; α", β ", γ", δ ", ..., ζ" , η ", ν", ... Frontal projections of surfaces.
11. Traces of planes (surfaces) are denoted by the same letters as the horizontal or front, with the addition of a substrate index 0α, emphasizing that these lines lie in the projection plane and belong to the plane (surface) α.
So: H 0α is a horizontal trace of the plane (surface) α;
f 0α - Frontal trace of the plane (surface) α.
12. Traces of direct (lines) are denoted by capital letters from which words begin with the name (in Latin transcription) of the projection plane, which the line crosses, with a substitution index indicating the belonging to the line.
For example: H a - horizontal trail line (line) A;
F a - frontal trail straight (line) a.
13. The sequence of points, lines (any figure) is marked with substrate indexes 1,2,3, ..., N:
A 1, a 2, a 3, ..., and n;
a 1, a 2, a 3, ..., a n;
α 1, α 2, α 3, ..., α n;
F 1, F 2, F 3, ..., F N, etc.
The auxiliary projection of the point obtained as a result of the transformation to obtain the actual magnitude of the geometric shape is indicated by the same letter with the substituum index 0:
A 0, B 0, C 0, D 0, ...
Axonometric projections
14. Axonometric projections of points, lines, surfaces are denoted by the same letters as the nature with the addition of the upper index 0:
A 0, 0, C 0, D 0, ...
1 0 , 2 0 , 3 0 , 4 0 , ...
a 0, B 0, C 0, D 0, ...
α 0, β 0, γ 0, δ 0, ...
15. Secondary projections are designated by adding an upper index 1:
A 1 0, in 1 0, C 1 0, D 1 0, ...
1 1 0 , 2 1 0 , 3 1 0 , 4 1 0 , ...
a 1 0, B 1 0, C 1 0, D 1 0, ...
α 1 0, β 1 0, γ 1 0, δ 1 0, ...
To facilitate the reading of the drawings in the textbook, several colors are used when designing an illustrative material, each of which has a certain semantic value: the line data (points) is indicated by the source data; green color used for lines of auxiliary graphic constructions; Red lines (points) show the results of constructions or those geometric elements to which special attention should be paid.
| № for the pores. | Designation | Content | An example of a symbolic record |
|---|---|---|---|
| 1 | ≡ | Match up | (AV) ≡ (CD) - direct, passing through points A and B, coincides with a straight line passing through points C and D |
| 2 | ≅ | Congrunny | ∠abc≅∠mnk - Angle Avs Congroiten Corner MNK |
| 3 | ∼ | Like | ΔAVS~Δmnk - ABC and MNK triangles are similar |
| 4 | || | Parallel | α || β - plane α parallel to the β plane |
| 5 | ⊥ | Perpendicular | a⊥b - direct a and b perpendicular |
| 6 | Crush | c d - straight C and D crossed | |
| 7 | Tangents | t L - Direct T is tangent to line L. βα - plane β tangent to surface α |
|
| 8 | → | Display | F 1 → F 2 - Figure Fig 1 is displayed in Figure F 2 |
| 9 | S. | Project project. If the project center is incompatible point, then his position is indicated by the arrow, Indicating the direction of projection | - |
| 10 | s. | Direction of projection | - |
| 11 | P. | Parallel projection | p S α parallel projection - parallel projection On the plane α in the direction s |
| № for the pores. | Designation | Content | An example of a symbolic record | An example of a symbolic record in geometry |
|---|---|---|---|---|
| 1 | M, N. | Set | - | - |
| 2 | A, B, C, ... | Elements of set | - | - |
| 3 | { ... } | Comprises... | F (a, b, c, ...) | F (a, b, c, ...) - Figure F consists of points A, B, C, ... |
| 4 | ∅ | Empty set | L - ∅ - the set L is empty (it does not contain elements) | - |
| 5 | ∈ | Belongs is an element | 2∈N (where n is a set natural numbers) - number 2 belongs to set n | A ∈ A - point A belongs to direct a (Point A lies on a direct a) |
| 6 | ⊂ | Includes contains | N⊂m - set n is part (subset) of the set M of all rational numbers | a⊂α - direct A belongs to the plane α (understood in the sense: many points of direct A is a subset of the points of the plane α) |
| 7 | ∪ | An association | C \u003d a U in - set with there is a set of sets A and B; (1, 2. 3, 4,5) \u003d (1,2,3) ∪ (4,5) | Abcd \u003d ∪ [Sun] ∪ - a broken line, ABCD is Combining segments [AV], [Sun], |
| 8 | ∩ | Intersection of many | M \u003d K∩L - the set M is the intersection of sets to and l (contains elements belonging to both the set to and the set L). M ∩ N \u003d ∅- The intersection of sets M and N is empty set (sets M and N do not have common elements) | a \u003d α ∩ β - direct, and there is an intersection Planes α and β a ∩ B \u003d ∅ - straight a and b do not intersect (do not have common points) |
| № for the pores. | Designation | Content | An example of a symbolic record |
|---|---|---|---|
| 1 | ∧ | Conjunction of proposals; corresponds to the Union "and". Proposal (R∧Q) is true then and only if R and Q are both true | α∩β \u003d (K: k∈α∧k∈β) the intersection of the surfaces α and β has a variety of points (line), consisting of all those and only those points to that belong both the surface α and the surface β |
| 2 | ∨ | Disjunction of proposals; Corresponds to the Union "or". Offer (P∨Q) true, when truly at least one of the proposals p or q (i.e. or p, or q, or both). | - |
| 3 | ⇒ | The implication is a logical investigation. Offer R⇒Q means: "If p, then and q" | (A || S∧B || C) ⇒a || b. If two straight parallel to the third, then they are parallel to each other |
| 4 | ⇔ | The proposal (Р⇔Q) is understood in the sense: "If p, then q; if q, then r" | A∈α⇔a∈L⊂α. The point belongs to the plane if it belongs to some line belonging to this plane. The reverse statement is also true: if the point belongs to some line, owned plane, then it belongs to the plane itself |
| 5 | ∀ | Quantitor community read: for any, for everyone, for any. Expression ∀ (x) p (x) means: "for any X: there is a property P (x)" | ∀ (ΔAVS) (\u003d 180 °) for all (for any) triangle Amount of its corners at the vertices is 180 ° |
| 6 | ∃ | Quantitor existence, read: exists. Expression ∃ (x) p (x) means: "There is x, having a property of P (x)" | (∀α) (∃a). For any plane α there is a straight A, which does not belong to the plane α and parallel plane α |
| 7 | ∃1 | Quantitor uniqueness of existence, read: There is only the only (s) ... expression ∃1 (x) (Px) means: "There is the only (only one) x, possessing the property of PC " | (∀ A, B) (A ≠ B) (∃1a) (A∋a, c) for any two different points A and B there is a single straight A, passing through these points. |
| 8 | (PX) | Rechange of statements P (x) | aB (∃α) (⊃⊃a, b). If straight a and b are cross, then there is no plane A, which contains them |
| 9 | \ | Knowledge denial | ≠-Opening [AV] is not equal to a segment. And? B - line and not parallel to the line B |
Abstract of the lesson in mathematics
Subject: "Straight. Straight designation »
Class: 1 "g"
Objectives lesson:
Educational: - know the concepts of direct and indirect line; be able to portray a straight line; be able to distinguish direct and indirect lines; be able to accept and maintain a learning task; be able to fulfill educational - informative actions in material and mental form; be able to work in a pair; ability to draw conclusions;
Developing: - develop observation, logical thinking, self-control skills; Thinking operations (analysis, synthesis, generalization); develop the skill of the right speech behavior;
Raising: value attitude to the subject, educate attentiveness, accuracy, perfection, adjacent; Positive attitude towards teaching; The desire to acquire new knowledge;
Type of lesson: Studying a new material
Technical support: Computer, Multimedia Projector, Screen, Interactive Board
Equipment:, Tutorial "Mathematics 1 class", workbook in mathematics
CMD: "Perspectic"
The date of the: 10/01/2016
Time spending: 45 minutes
Conductive: Boldueva Lyudmila Yuryevna
Organizing timeActualization of knowledge
Goaling
Familiarization with new material.
Fizkultminutka
Fixing
Fizkultminthork for eye
Fixing
Outcome
Reflection
10. Homework
Hello, sit down.
To begin with, we will spend an oral account.
On the board one by one, the maple leaves (or any other visibility) are attached at the expense of children.
Well done!
And now name the numbers in descending order.
Good, well done!
Guys, we fell into the country "geometry" and we meet the point. (The teacher attaches the first point on the board). Let's call her point A.
Now with the help of the line I will spend the line. Who knows how it is called?
What will be the topic of our lesson?
And what will we do today, what will we learn?
Good, well done!
View video.
So, how many direct can we spend through one point?
Open the tutorial on page 50 and look at the exercise 1. Here it is shown how the straight line is carried out through one point using the ruler.
Can I still spend direct through the point ah?
We continue to visit the girlfriend came to visit. This is a point B. (to the blackboard the teacher attaches the point b)
View video.
How much can you spend direct in two points?
Right!
Open workbooks on page 38, perform a task 1.
Land checking. Remind how to hold a pencil.
Two points A and B. We spend straight with a ruler. We celebrate the point of O. - - What are the directs from us?
How else can I designate the straight ab?
That's right, ba.
(all actions teacher performs on interactive blackboard)
Game on an interactive board (2)
But there are indirect lines, look at the second picture in the textbook. These are indirect lines. And on the board we have a straight and indirect line.
(on the board depicts a straight line and indirect line)
And who can tell with what we can learn a straight line or not?
That's right, using the ruler. If the ruler coincides with a straight line, then the line is straight, if not, then indirect.
Let's try (the teacher applies a ruler to 1 straight - the line coincided, then the line is straight; we apply to the second - does not coincide the line is indirect)
Game on an interactive board (1)
Return to K. working notebook, Number 2, we perform in pairs, and then check together. You need to spend direct de and MK, then spend even straight through points e, m, to. Cut. Think along with your neighbor in the desk and write down these straight lines.
Check, completed task. (The teacher depicts direct on an interactive board, discussing the correctness of execution with children)
On the computer (presentation)
We return to working notebooks and perform number 3.
(Teacher draws together with children on an interactive chalkboard)
Fingers.
Times, two, three, four, five, (compress and squeeze the cams.)
We went to the fishing rods.
This finger on the track, (flex your fingers starting from the big one.)
This path on the path,
This finger for mushrooms,
This finger for raspberries,
This finger is lost,
Very late returned.
My fingers with you scented and now perform number 4.
Landing rules.
Well, showed how we keep the handle? Good, well done!
AND last exercisethat we will perform on this lesson number 6.
Let's disassemble, we need to find out who from the artists will act next if it is not skating, not a clown and a bird.
Who is suitable for this description?
That's right, well done!
That came to the end of our lesson with you.
What new have we learned with you today?
What did you learn?
Today in the lesson, everyone worked actively, behaved well and so I now give you a sun on the sun.
Guys, raise your hands, those who understood everything in the lesson, easily coped with all the tasks.
And now those who had difficulty.
(And what exactly did you not understand what you did not work?)
At home, you can take number 7, in the textbook. Here, patterns and numbers need to redraw a notebook.
Hello, sit down.
Together with the teacher, leaflets are considered.
Straight and its designation
Let's learn to depict straight
Working with a textbook
Only one.
Come out in turn and perform the task
Conduct children to music
Working with workbooks
Work in parach
Exercise perform
Compress and squeeze cams
Fuck your fingers, I start with a big
Answers children
We learned what is direct, her name.
We learned to depict straight
Motivational basis learning activities (L);
Sense formation (l);
Formulation of cognitive goal (P);
Cognitive initiative (P);
Forecasting (P);
educational and informative interest (L);
Sense formation (l);
Volitional self-regulation (P);
Analysis, synthesis, comparison,
generalization, analogy (P);
Staging and formulation
problems (P);
Accounting different opinions,
coordination B.
cooperation
different positions (K);
Formulation and argument
of his opinion and position in