How many rays can be drawn from one point. Ray: starting point, ray symbol

Technology: developmental education L. V. Zankova.

Lesson objectives:

• create conditions for the formation of a primary idea of ​​a ray, teach to distinguish between a straight line, a segment, a ray, and check the degree to which children have previously assimilated this information;
• develop memory, attention, thinking, the ability to observe, compare, classify, analyze and generalize, develop the intellectual and practical skills of children;
• develop an active personality.

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 work. And today we are setting off 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.

(Figure No. 1)

2. Updating basic knowledge.

Teacher: You are already familiar with many residents of the city and can easily recognize them.

Game: “Get to know me.”

(On each child’s desk there is a set of geometric shapes.)

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

(Students select a triangle from the handout and show it to the teacher. The teacher attaches a blue triangle to the board.)

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

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

(picture No. 2)

Teacher: How are all the figures similar?

Children: They are all the same color.

Teacher: How are they different?

Children: They have different shapes.

Children: They are different sizes.

Teacher: Which figure is the odd one out?

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

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

Children: I think it’s an extra circle. It is round and has no straight lines.

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

Fizminutka.

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

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

(Figure No. 3)

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

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

Children: Circle.

Teacher: Who is the saddest?

Children: Square.

Teacher: Now let’s continue our journey. Together with our guide Karandash 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: The segment still lives there.

Children: Straight and curved lines live there.

Teacher: Well done. And now I will tell the story that happened to Pencil. And you will help me. Agreed? But before listening to the story about Pencil, I suggest you rest a little.

Fizminutk A.

(Exercises that correct posture.)

Exit on the topic of the lesson.

Teacher: This is what happened to Pencil.

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

How much longer do I have to go? Will I make it to the end? - he asks Straight.

What will Direct Line answer him?

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

Teacher: Correct.

“Oh, I have no end,” answered Straight.

Then I’ll go the other way,” said Pencil.

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

Teacher: That's right. And Straight even sang a song to him.

Without end and edge straight line,
Walk along it for at least a hundred years,
You won't find the end of the road.

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

Pencil was upset.

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

What advice do you guys have for Pencil?

(Children give various advice.)

Teacher: Then mark 2 points on me, - Direct advised him. That's what Pencil did.

(Students put two points on a straight line.)

Hooray! - Pencil shouted. – Two ends appeared. Now I can walk from one end to the other. But then I started thinking.

And what happened on Direct?

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. Studying new material.

Teacher: And one day the Pencil decided to take away the Straight segment. He took scissors with him and slowly cut out the segment. I connected the remaining ends and tied it off. He just doesn’t understand what happened.

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

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

Teacher: What happened was that there were 2 rays on a straight line coming out from one point. The ray has a beginning, but no end.

5. Practical part.

Work according to the textbook. ( I. Arginskaya, mathematics, part 1, p. 52, No. 100)

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

(Figure No. 4)

Children: We knew a straight line, a segment.

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

Children: This line is called a ray.

Teacher: Find the beam and trace it with a red pencil.

Think and explain how a ray differs from a straight line? From the segment?

Draw two rays.

Teacher: Ray has prepared a riddle for you.

Among the blue field -
The bright shine of a large fire.
The fire moves slowly here,
It goes around Mother Earth,
There is a cheerful light shining in the window.
Well, of course it is…….

Children: The sun.

Physical exercise.

(Exercises for hands.)

Teacher: Why did Ray tell you a riddle about the sun?

D: Because the sun also has rays.

Teacher: Draw a sun in your notebooks.

Teacher: How many rays does your sun have?

(Children say how many rays they drew from the sun. The number of rays varies.)

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

(Children express their opinions.)

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

Work according to the textbook. (page 54 no. 105)

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

(Figure No. 5)

Teacher: In your notebook, draw 3 segments and 2 rays.

6. Lesson summary.

Teacher: Our imaginary journey has ended. 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 ray has a beginning and no end.

Teacher: I hope our journey was exciting and interesting. Let's smile goodbye to all the inhabitants of the magical land 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.

Ray- is a part of a straight line located on one side of any point lying on this straight line. The beam is also called semidirect.

Any ray has a beginning and a direction. Beam start, starting point or beam apex is the point from which the ray emanates. Thus, the ray has a beginning, but no end.

Let's consider three rays with a common origin:

All 3 rays have a common starting point O, but in different directions. About each of them we can say: the ray comes from a point O or a ray emanating from a point O .

Additional rays

Any point lying on a straight line divides this straight line into two half-lines, that is, into two parts. Each of these parts will be called an additional ray relative to the second ray:

Additional rays- these are rays that have a common origin, opposite directions and lie on the same straight line. We can also say that rays that complement each other to a straight line are called complementary.

Ray designation

The beam is denoted by one lowercase Latin letter:

Ray h.

The ray can also be designated by two points lying on it:

When designating a ray with two points, the first place is marked with a letter indicating the beginning of the ray, and the second place with a letter indicating some other point: ray B.C..

Let's look at next example:

Beam with origin at point A can be denoted as AB or A.C..

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

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

point A, point B, point C

point 1, point 2, point 3

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

A A A

A line is a set of points. Only the length is measured. It has no width or thickness Indicated by lowercase (small)

with Latin letters

a b c

1. The line may be
2. closed if its beginning and end are at the same point,

open if its beginning and end are not connected

closed lines

1. You left the apartment, bought bread at the store and returned back to the apartment. What line did you get? That's right, closed. You are back to your starting point. You left the apartment, bought bread at the store, went into the entrance and started talking with your neighbor. What line did you get? Open. You haven't returned to your starting point. You left the apartment and bought bread at the store. What line did you get? Open. You haven't returned to your starting point.
2. self-intersecting

without self-intersections

self-intersecting lines

1. lines without self-intersections
2. straight
3. broken

crooked

straight lines

broken lines

curved lines

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

Indicated by a lowercase (small) Latin letter. Or two capital (capital) Latin letters - points lying on a straight line

straight line a

straight line AB

Direct may be

1. intersecting if they have a common point. Two lines can intersect only at one point.
   • perpendicular if they intersect at right angles (90°).
2. Parallel, if they do not intersect, 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 continued indefinitely in only one direction

The ray of light in the picture has its starting point as the sun.

Sun

A point divides a straight line into two parts - two rays A A

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

ray a

beam AB

The rays coincide if

1. located on the same straight line
2. start at one point
3. directed in one direction

rays AB and AC coincide

rays CB and CA coincide

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

Through one point you can draw any number of lines, including straight lines

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

curved lines passing through two points

straight line AB

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 (capital) Latin letters, where the first is the point at which the segment begins, and the second is the point at which the segment ends

segment AB

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

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

A long segment was “broken” into several short ones

The links of a broken line (similar to the links of a chain) are the segments that make up the broken line. 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.

A broken line is designated 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

broken link AB, broken link BC, broken link CD, broken link DE

link AB and link BC are adjacent

link BC and link CD are adjacent

link CD and link DE are adjacent

The length of a broken line 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 more peaks ? The first line has all the links of the same length, namely 13 cm. The second line has all links of the same length, namely 49 cm. The third line has all links of the same length, namely 41 cm.

A polygon is a closed polygonal line

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

The vertices of a polygon are the vertices of a broken line. Adjacent vertices are the 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

CD side and DE side are adjacent

side DE and side EF are adjacent

side EF and side FA are adjacent

The perimeter of a polygon is the length of the broken line: 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, etc.

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

Equipment: three-dimensional toy, album, pencils, notebook, pen, projector, flashlight

Annotation: During the work, children answer the 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 straight line) -1 size ., surface – 2 sizes, space – 3 sizes. Draw a fish in your album. Color it. Make the same one from plasticine. Place it in a transparent jar. How are the images of fish different? 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 its properties: straightness and the existence of a beginning. We will consider the light source to be the beginning of the beam; straightness is determined by the presence of a shadow (the beam cannot go around an obstacle). Using the example of the sun's rays, one more of their properties can be shown - infinity. To do this, a flashlight is used as a small sun, sending a beam of light towards a field or along the road; it is impossible to say where it ends. Analyze what is considered a ray and what is a segment. Let's agree that a ray has a beginning and a direction, and a segment has a beginning and an end. What to do with the sun's rays? Is this a segment or a ray? (some of them fall on the Earth, some are scattered in space, if a physical object is encountered on the path of the beam, then it is no longer a beam, but a segment). Give your own 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 a beam you need to cut it at any point in the area that lies on the desk. We get two threads (beams), 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 point 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. Reasoning task: can rays that have a common origin intersect somewhere else at another point? Explain your answer. A task to expand your horizons: a splashing fish knocks down its prey with a stream of water at a distance of 1.5 m. The length of the fish is 10 cm. Determine how much longer the length of the stream is than the length of the fish’s body.

4. Project 1-2 grades “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 ​​angle, teach to recognize and designate it.

Annotation: This topic is associated with the negative experiences of children, so the teacher should pay attention to the subject being studied, and not record the child’s memories. Consider different examples: hands on a clock (they have a beginning and a direction - that's why they are rays). The arrows are spaced at different distances, that part of the plane that shows up. between them called angle. Complete various tasks on this topic that show that angles can be compared with each other (find such tasks 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 to create a right angle. Show how you can use a triangle to construct different angles. What time does the clock show if the hands form a right angle and the minute hand is at 12? Find a picture where students can count the angles shown. Draw 4 clock faces in your notebook with images of right and indirect angles.