A point is an abstract object that has no measuring characteristics: no height, no length, no radius. Within the framework of the task, only its location is important

The point is indicated by a number or a capital (large) Latin letter. Several dots - different numbers or different letters so that they can be distinguished

point A, point B, point C

A B C

point 1, point 2, point 3

1 2 3

You can draw three "A" points on a piece of paper and invite the child to draw a line through the two "A" points. But how to understand through which? A A A

A line is a set of points. She only measures length. It has no width or thickness.

Indicated by lowercase (small) Latin letters

line a, line b, line c

a b c

The line could be

1. closed if its beginning and end are at the same point,
2. open if its beginning and end are not connected

closed lines

open lines

You left the apartment, bought bread in the store and returned back to the apartment. What line did you get? That's right, closed. You have returned to the starting point. You left the apartment, bought bread in the store, went into the entrance and talked to your neighbor. What line did you get? Open. You have not returned to the starting point. You left the apartment, bought bread in the store. What line did you get? Open. You have not returned to the starting point.

1. self-intersecting
2. without self-intersections

self-intersecting lines

lines without self-intersections

1. straight
2. broken line
3. crooked

straight lines

broken lines

curved lines

A straight line is a line that does not curve, has neither beginning nor end, it can be extended indefinitely in both directions

Even when a small section of a straight line is visible, it is assumed that it continues indefinitely in both directions.

It is denoted by a lowercase (small) Latin letter. Or two capital (large) Latin letters - points lying on a straight line

straight line a

a

straight line AB

B A

straight lines can be

1. intersecting if they have a common point. Two lines can only intersect at one point.
   • perpendicular if they intersect at a right angle (90°).
2. parallel, if they do not intersect, they do not have a common point.

parallel lines

intersecting lines

perpendicular lines

A ray is a part of a straight line that has a beginning but no end, it can be extended indefinitely in only one direction

The starting point for the beam of light in the picture is the sun.

sun

The point divides the line into two parts - two rays A A

The beam is indicated by a lowercase (small) Latin letter. Or two capital (large) Latin letters, where the first is the point from which the ray begins, and the second is the point lying on the ray

beam a

a

beam AB

B A

The beams match if

1. located on the same straight line
2. start at one point
3. directed to one side

rays AB and AC coincide

rays CB and CA coincide

C B A

A segment is a part of a straight line that is bounded by two points, that is, it has both a beginning and an end, which means that its length can be measured. The length of a segment is the distance between its start and end points.

Any number of lines can be drawn through one point, including straight lines.

Through two points - unlimited number of curves, but only one straight line

curved lines passing through two points

B A

straight line AB

B A

A piece was “cut off” from the straight line and a segment remained. From the example above, you can see that its length is the shortest distance between two points. ✂ B A ✂

A segment is denoted by two capital (large) Latin letters, where the first is the point from which the segment begins, and the second is the point from which the segment ends

segment AB

B A

Task: where is the line, ray, segment, curve?

A broken line is a line consisting of successively connected segments not at an angle of 180°

A long segment was “broken” into several short ones.

The links of a polyline (similar to the links of a chain) are the segments that make up the polyline. Adjacent links are links in which the end of one link is the beginning of another. Adjacent links should not lie on the same straight line.

The tops of the polyline (similar to the tops of mountains) are the point from which the polyline begins, the points at which the segments forming the polyline are connected, the point where the polyline ends.

A polyline is denoted by listing all its vertices.

broken line ABCDE

vertex of polyline A, vertex of polyline B, vertex of polyline C, vertex of polyline D, vertex of polyline E

link of broken line AB, link of broken line BC, link of broken line CD, link of broken line DE

link AB and link BC are adjacent

link BC and link CD are adjacent

link CD and link DE are adjacent

A B C D E 64 62 127 52

The length of a polyline is the sum of the lengths of its links: ABCDE = AB + BC + CD + DE = 64 + 62 + 127 + 52 = 305

Task: which broken line is longer, a which one has more peaks? At the first line, all the links are of the same length, namely 13 cm. The second line has all the links of the same length, namely 49 cm. The third line has all the links of the same length, namely 41 cm.

A polygon is a closed polyline

The sides of the polygon (they will help you remember the expressions: "go to all four sides", "run towards the house", "which side of the table will you sit on?") are the links of the broken line. Adjacent sides of a polygon are adjacent links of a broken line.

The vertices of the polygon are the vertices of the polyline. Neighboring vertices are endpoints of one side of the polygon.

A polygon is denoted by listing all its vertices.

closed polyline without self-intersection, ABCDEF

polygon ABCDEF

polygon vertex A, polygon vertex B, polygon vertex C, polygon vertex D, polygon vertex E, polygon vertex F

vertex A and vertex B are adjacent

vertex B and vertex C are adjacent

vertex C and vertex D are adjacent

vertex D and vertex E are adjacent

vertex E and vertex F are adjacent

vertex F and vertex A are adjacent

polygon side AB, polygon side BC, polygon side CD, polygon side DE, polygon side EF

side AB and side BC are adjacent

side BC and side CD are adjacent

side CD and side DE are adjacent

side DE and side EF are adjacent

side EF and side FA are adjacent

A B C D E F 120 60 58 122 98 141

The perimeter of a polygon is the length of the polyline: P = AB + BC + CD + DE + EF + FA = 120 + 60 + 58 + 122 + 98 + 141 = 599

A polygon with three vertices is called a triangle, with four - a quadrilateral, with five - a pentagon, and so on.

Target: conduct a research experiment using the tactile method of comparison to identify the differences between the plane and space in terms of dimension

Equipment: 3D toy, album, pencils, notebook, pen, projector, flashlight

Annotation: in the course of work, children answer questions: how to get a flat figure and how to get a three-dimensional figure. Take a three-dimensional toy, draw it in an album and compare the toy itself and its image on paper. Analyze the difference between a plane and space using the example of children's games (table hockey (1 control lever), a car on a plane (2 control levers), an airplane (3 control levers)): line (including a straight line) -1 size ., surface - 2 sizes, space - 3 sizes. Draw a fish in the album. Color her. Sculpt the same from plasticine. Plant it in a transparent jar. What is the difference between the images of fish. You can even make an aquarium with fish and analyze this model as well. The concept of a ray can be considered using the example of a ray of light as an abstract concept that has St. you: straightness and the existence of a beginning. We will consider the light source as the beginning of the beam, the straightness is determined by the presence of a shadow (the beam cannot go around the obstacle). Using the example of the sun's rays, one more property of them can be shown - infinity. To do this, a flashlight is used as a small sun, sending a beam of light towards the field or along the road, one cannot tell where it ends. Analyze what is considered a ray and what is a segment. We agree that a ray has a beginning and a direction, and a segment has a beginning and an end. What about sun rays? Is it a line segment or a ray? (some of them hit the Earth, some are scattered in space, if a physical object is encountered on the path of the beam, then this is no longer a beam, but a segment). Give your examples of rays and segments, for example, is a projector a ray or a segment? Complete a practical task: take a rope longer than the desktop, position it so that one end hangs from the table, to get the beam you need to cut it at any point, in the area that lies on the desk. We get two threads (rays), the beginning of which lies on the desk. The place of the cut is the beginning of the rays and there are two directions to the left and to the right. Complete the task: draw a straight line in the album and divide it with a dot into two rays. How are they located relative to each other? How many different rays can be drawn from one point A? Draw 5 such rays emanating from point A. Assignment-reasoning: can rays that have a common origin intersect somewhere else? Explain your answer. A task for broadening one's horizons: a splasher fish knocks down its prey with a jet of water at a distance of 1.5 m. The length of the fish is 10 cm. Determine how much the jet is longer than the length of the body of the fish.

4. Project 1-2 class "Flat and volumetric: corner"

This topic is a continuation of the previous one. The definition of the angle follows from the definition. beam.

Target: form an idea of ​​\u200b\u200bangle, teach to recognize and designate it.

Annotation: This topic is related to the negative experiences of children, so the teacher should pay attention to the subject being studied, and not fix the child's memories. Consider different examples: the hands on a clock (they have a beginning and a direction - that's why they are rays). The arrows are separated at different distances, that part of the plane that nah. between them called angle. Complete various tasks on this topic that show that angles can be compared with each other (find such problems yourself). You can compare like this: draw two corners, transfer one of the corners to translucent paper and compare the images, the image on the other corner. Fold a sheet of paper twice - you get a right angle. Show how a triangle can be used to construct different angles. What time does the clock show if the hands form a right angle, and the minute hand is at 12? Choose a picture in which students count the angles shown there. Draw in a notebook 4 clock faces with images of right and indirect angles.

Technology: developmental education L. V. Zankova.

Lesson Objectives:

• create conditions for the formation of a primary idea of ​​the beam, teach to distinguish between a straight line, a segment, a beam, check the degree of assimilation of previously given information by children;
• develop memory, attention, thinking, the ability to observe, compare, classify, analyze and generalize, develop the intellectual and practical skills of children;
• educate an active person.

During the classes

1. Organizational moment.

Teacher: Hello guys. I am very glad to see your kind, cheerful eyes. I see that you are ready to go. And today we are going on another journey through the Great Country of Mathematics and will visit the city of Geometry already known to us. Our guide will be Pencil.

(picture No. 1)

2. Updating basic knowledge.

Teacher: You already know many of the inhabitants of the city and you can easily recognize them.

Game: Get to know me.

(Each child has a set of geometric shapes on their desks.)

I am a polygon with 3 sides. What is my name?

(Students choose a triangle from the handout and show it to the teacher. The teacher puts a blue triangle on the board.)

I am a polygon, I have 4 equal sides . (square)

But I'm not a polygon at all. But I can find it in a watch, in a car, in a cup, even the sun looks like me from afar. Who am I? (a circle)

(picture No. 2)

Teacher: How are all the shapes alike?

Children: They are all the same color.

Teacher: How are they different?

Children: They have different shapes.

Children: They are different sizes.

Account: Which figure is missing?

Children: The extra figure is a triangle, because it is the smallest.

Children: I agree that the triangle is an extra figure, because the square and the circle have a slightly similar shape. If you cut off the corners of a square, it will look like a circle.

Children: And I think it's an extra circle. It is round and does not have straight lines.

Children: And the circle has no corners. I also think that the circle is superfluous.

Fizminutka.

(Gymnastics for the eyes according to the method of G. A. Shichko.)

Teacher: And now draw these figures, following the requests of the letters.

(picture No. 3)

(F. - shape, C. - color, R. - size. Children draw geometric shapes, changing the shape, color and size according to this task.)

Teacher: Well done. Everyone completed the task. And also, guys, these figures had a different character. The circle was more fun than the triangle, and the triangle was more fun than the square. Who was the funniest?

Children: Circle.

Teacher: Who is the saddest?

Children: Square.

Teacher: Now let's continue our journey. Together with our guide Pencil we will go to Lineiny Avenue. Our cheerful and kind friends live here.

Who do you think they are?

Children: Straight lines live in these houses.

Children: A segment still lives there.

Children: Straight and curved lines live there.

Teacher: Well done. And now I will tell the story that happened to the Pencil. And you will help me. Deal? But before listening to the fairy tale about the Pencil, I suggest you take a break.

Fizminutk a.

(Posture Correction Exercises

Output on the topic of the lesson.

Teacher: This is the story that happened to the Pencil.

One day Pencil decided to take a walk along the Straight Line. He goes, he goes, he is tired, but the end of the line is still not visible.

How long do I have to go? Will I make it to the end? he asks Direct.

What will the Direct Line answer him?

Children: The pencil will not reach the end of the line, because the line has no end.

Teacher: Right.

Oh, you, I have no end, - answered Direct.

Then I'll go the other way, - said the Pencil.

Children: And in the other direction, the Pencil will not reach the end of the line, because the line has no beginning and end.

Teacher: Right. And Direct, even sang a song to him.

Without end and edge, the line is straight,
At least a hundred years go along it,
You won't find the end of the road.

Teacher: Let's draw a straight line in the notebook.

Upset Pencil.

What should I do? I don't want to walk the line. I'm tired.

What do you guys advise Pencil?

(Children give different advice.)

Teacher: Then mark 2 points on me, Direct advised him. So Pencil did.

(Students put two points on a straight line.)

Hooray! shouted Pencil. - There are two ends. Now I can walk from one end to the other. But then I thought about it.

And what is it on the Direct happened?

Guys, help Pencil.

Children: This is a segment.

Teacher: What do you know about the segment?

Children: A segment is a part of a straight line. It has a beginning and an end.

4. Learning new material.

Account: And one day the Pencil decided to take away the straight line. He took scissors with him and slowly cut out a segment. Connected the remaining ends and tied. He just doesn't understand what happened.

Do you guys know? Could this be a new cut?

Children: No, it can't. One line has no beginning and has an end, and the other has a beginning but no end.

Teacher: And it turned out on a straight line 2 rays coming out of one point. The beam has a beginning, but there is no end.

5. Practical part.

Textbook work. ( I. Arginskaya, mathematics, part 1, p. 52, No. 100)

Teacher: Compare the lines. How are they similar? What is the difference? What lines are you already familiar with?

(picture no. 4)

Children: We knew a straight line, a segment.

Teacher: Trace a straight line with a blue pencil, a line segment with green. What is the name of the line you met today?

Children: This line is called a beam.

Account: Find a beam and circle it with a red pencil.

Think and explain how a ray differs from a straight line?. From a cut?

Draw two rays.

Teacher: The Ray has a riddle for you.

Among the field of blue -
The bright brilliance of a great fire.
Slowly the fire is walking here,
Bypasses mother earth
Shines cheerfully in the window.
Well, of course it is…….

Children: Sun.

Fizminutka.

(Exercises for the hands.)

Teacher: And why did Ray give you a riddle about the sun?

D: Because the sun also has rays.

Teacher: Draw the sun in your notebooks.

Teacher: How many rays does your sun have?

(Children say how many rays they drew in the sun. The number of rays is different.)

Teacher: How many rays can be drawn from one point?

(Children express their opinion.)

Account: Well done. Indeed, from one point we can draw any number of rays.

Textbook work. (p. 54 no. 105)

Under each picture in the left cell, write how many lines are on it, and in the right cell, how many rays.

(picture No. 5)

Account: In a notebook, draw 3 segments and 2 rays.

6. The result of the lesson.

Teacher: This is the end of our imaginary journey. We say goodbye to the city of Geometry, its beautiful inhabitants - geometric figures. Let's remember once again what we know about a straight line, a segment and a ray.

Children: A straight line has no beginning and no end.

Children: A segment has a beginning and an end.

Children: And the beam has a beginning and no end.

Account: I hope our trip was exciting and interesting. Let's smile goodbye to all the inhabitants of the magical country of Mathematics, to each other and rejoice at our successes. But this is only a small part of what can be learned in mathematics lessons. There are many more journeys ahead of you in the Great Country, whose name is Mathematics.