Straight ray segment, broken segment length. The simplest geometric figures: point, straight line, segment, ray, broken line

Along with such concepts as point, segment, line, there is one more concept in geometry. It is called ray. A ray is a part of a straight line, limited on one side by a point, and on the other side - infinite, i.e. not limited by anything.

An analogy can be drawn with nature. For example, a beam of light that we can direct from earth into space. On the one hand it is limited, but on the other hand it is not. Each ray has one extreme point at which it begins. It is called the beginning of the ray.

If we take an arbitrary straight line a, and mark some point on it ABOUT, then this point will split our line into two parts. Each of which will be a ray. Point O will belong to each of these rays. Point O will be at in this case the beginning of these two rays.

The beam is usually designated by one Latin letter. The figure below shows ray k.

You can also denote the beam with two capital Latin letters. In this case, the first of them is the point at which the beginning of the beam lies. The second is the point that belongs to the ray, or in other words, through which the ray passes.

The figure shows the OS beam.

Another way to designate a ray is to indicate the starting point of the ray and the line to which this ray belongs. For example, the figure below shows the ray Ok.

Sometimes they say that the ray comes from point O. This means that point O is the beginning of the ray. Rays are also sometimes called semi-straight.

Task:

Draw a straight line and mark points A B on it and mark point C on segment AB. Among the rays AB, BC, CA, AC and BA, find pairs of coinciding rays.

The rays coincide if they lie on the same straight line and have general beginning and none of them is a continuation of another ray.
The figure shows that these conditions are met by rays AB and AC, as well as rays BC and BA. Therefore, they are coincident.

Straight line - one of the fundamental concepts of geometry.

Clearly straight line can demonstrate a taut cord, the edge of a table, the edge of a sheet of paper, a place, the junction of two walls of a room, a beam of light. When drawing straight lines, a ruler is used in practice.

Straight line have such characteristic peculiarities:

1.U straight line there is no beginning or end, that is, it is endless . It is possible to draw only part of it.

2.In two arbitrary points can be carried out straight line, and only one at that.

3. Through n arbitrary point You can draw an unlimited number of straight lines on a plane.

4.Two mismatched straight lines on a plane or intersect at the only point, or they parallel.

To indicate straight line use either one small letter Latin alphabet, or two capital letters, written in two different places on this line.

If you indicate on a straight line point, then as a result we get two beam:

Beam call part straight line, limited on one side. To designate a beam, either one small letter of the Latin alphabet or two large letters are used, one of which is designated at the beginning of the beam.

The part of a straight line limited on both sides is called segment. A segment, like straight line, is designated either by one letter or two. In the latter case, these letters indicate the ends of the segment.

A line formed by several segments that do not lie on the same straight line is usually called broken line. When the ends of the broken line coincide, then broken line is called closed.

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 dots - 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

A straight line is a line that is not curved, has neither beginning nor end, it can be continued endlessly in both directions

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 vertices of a broken line (similar to the tops of mountains) are the point from which the broken line begins, the points at which the segments that form the broken line are connected, and the point at which the broken line ends.

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 has more vertices? The first line has all the links 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 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.

Math lesson notes

in 1st grade.

Subject: Point. Curved line. Straight line. Line segment. Ray.

Compiled and conducted

Buvailova Elena Ivanovna

Subject: Point. Curved line. Straight line. Line segment. Ray

Target: in progress practical tasks and observations to teach to distinguish different types lines.

Planned results: Students will learn to distinguish and name a straight line, curve, segment, ray, broken line; use a ruler for drawing; correlate real objects and their elements with studied geometric lines and figures; perform mental operations of analysis and synthesis and make inferences; apply previously acquired knowledge in changed conditions; listen to the interlocutor and conduct a dialogue; listen to the teacher and fulfill his requirements; evaluate yourself, the boundaries of your knowledge and ignorance; work in pairs and evaluate a friend.

During the classes

1.Organizational moment

Math is calling

First-graders to class,

Numbers lead us forward

We will know everything by heart

2.Updating knowledge

Today the cat Tishka came to visit us at our lesson with unfamiliar friends, and what kind of friends are you going to name them a little later?

a) Count forward and backward within 10.

Individual survey.

b) Problems in verse:

Tishka is such a stupid cat

Tishka loves fish very much.

Went fishing

Caught two minnows

Two pike and two ruff.

Tishka's life is good!

Who counted faster?

How many fish did the cat catch? (6)

A rooster flew onto the fence

Met two more there.

How many roosters are there? (3)

Along the path to the forest

The bun rolled.

I met a gray bunny

I met a wolf, I met a bear,

Yes the cheating fox

He met in the forest

Answer quickly

How many animals did the bun meet? (4)

Game "Silence"

(The teacher shows the pass, students show the corresponding number on the fan of numbers.)

4 - □ = 2 5 - □= 2

4 - □ = 3 5 - 1 = □

1 + 3 = □ □ - 3=1

□ -4=1 1 + □ = 2

3. Physical education minute

4. Self-determination for activity

In the land of Geometry there lived a dot. She was small. It was left by a pencil when it stepped on a piece of notebook paper, and no one noticed it. This is how she lived until she came to visit the lines. (There is a drawing on the board.) (Math tablet)

Look what those lines were. (Straight and curved.)

Straight lines are like stretched ropes, and the ropes

those that are not tensioned are crooked lines.

How many straight lines? (2.)

How many curves? (3.)

Straight line started boasting: “I’m the longest!” I have neither beginning nor end! I am endless!

It became very interesting to look at her. The point itself is tiny. She came out and was so carried away that she didn’t notice how she stepped on a straight line. And suddenly the straight line disappeared. In her place a beam appeared.

It was also very long, but still not as long as a straight line. He got a start.

The dot got scared: “What have I done!” She wanted to run away, but as luck would have it she stepped on the beam again.

And in place of the beam a segment appeared. He didn't brag about how big he was, he already had a beginning and an end.

This is how a small dot was able to change the life of large lines.

So who guessed who came to visit us with the cat? ?(straight line, ray, segment and point)

That's right, along with the cat, a straight line, a ray, a segment and a point came to our lesson.

Who guessed what we will do in this lesson? (Learn to recognize and draw a straight line, ray, segment.)

5. Work on the topic of the lesson

Practical work

What lines did you learn about? (About a line, ray, segment.)

What did you learn about the straight line? (It has neither beginning nor end. It is endless.)

(The teacher takes two spools of thread, pulls them, depicting a straight line, and unwinding first one, then the other, demonstrates that the straight line can be continued in both directions indefinitely.)

What did you learn about the beam? (U it has a beginning, but no end.)(The teacher takes scissors, cuts the thread. Shows that now the line can only be continued in one direction.)

What did you learn about the segment? (It has both a beginning and an end.)(The teacher cuts the other end of the thread and shows that the thread

doesn't stretch. It has both a beginning and an end.)

6.Work according to the textbook

- Look at the picture on p. 40. Explain how a straight line differs from a curve. (A straight line is stretched, a curve is not.)

What do you remember about a straight line, ray, segment? (Children's answers.)

How to draw a straight line? ( Draw a line along the ruler.)

How to draw a line segment? (Put two points and connect them.)

7. Physical education minute

On Monday I swam

(Arm movements performed when swimming.)

And on Tuesday I painted,

(Image drawing.)

On Wednesday I took a long time to wash my face,

(Pretend washing.)

And on Thursday I played football.

(Running in place.)

On Friday I ran, jumped,

(Jumping in place.)

I danced for a very long time.

(Spin around.)

And on Saturday, Sunday

(Clap your hands.)

I rested the whole day.

(Squat down, hands under cheeks.)

8. Consolidation of the studied material

Work in a notebook with a printed base

Open your notebook to p. 15. Consider the lines. What groups can they be divided into? (Straight lines - 2.3, 5 and curves -1.4.)

Complete the following task.

How many lines can be drawn through two points? (One.)

How many curves can be drawn through two points? (A lot of.)

Read the next task.

Color the pictures yourself.

9. Finger gymnastics

Working in a notebook

Tishka wants to learn how to draw a line, a segment, a ray.

Now draw in your notebook a straight line, a segment, a ray and a curved line along which the cat Tishka will run.

Discuss the lines drawn in pairs.

10.Work according to the textbook

Read the assignment in the margin on p. 40. How do you know which segment is the longest? (Count how many cells make up the length of each segment.)

Count and tell which segment is the longest. (Blue.)

Which segment is the shortest? (Red.)

Look at the picture on p. 41. Tell your desk neighbor what lines you see.

(Work in pairs.)

Look at the pictures and notes below.

Which entries go with the pictures?

Explain their meaning.

(4 + 1 = 5 - another one came running to 4 chickens.

There are now 5 chickens. 5-2 = 3- 5 ducklings swam, 2 ducklings left.

3 ducklings left.

Entries 4- 1 = 3 and 5- 1 = 4 are not suitable.)

I liked the lesson

It was difficult but interesting

I didn't like the lesson

Summing up the lesson

What new things have you learned about lines?

Where are straight lines found in life? crooked lines?

What can a dot, a straight line, a curved line mean to a cat?

(The dot is like a ball - it can play, roll;

Beam – letting in “bunnies”

Direct line to the road – where you need to follow traffic rules;

A curved line leads to a winding path where he can play tag with his friends)

We will look at each of the topics, and at the end there will be tests on the topics.

Point in mathematics

What is a point in mathematics? A mathematical point has no dimensions and is designated by capital letters: A, B, C, D, F, etc.

In the figure you can see an image of points A, B, C, D, F, E, M, T, S.

Segment in mathematics

What is a segment in mathematics? In mathematics lessons you can hear the following explanation: a mathematical segment has a length and ends. A segment in mathematics is the set of all points lying on a straight line between the ends of the segment. The ends of the segment are two boundary points.

In the figure we see the following: segments ,,,, and , as well as two points B and S.

Direct in mathematics

What is a straight line in mathematics? The definition of a straight line in mathematics is that a straight line has no ends and can continue in both directions indefinitely. A line in mathematics is denoted by any two points on a line. To explain the concept of a straight line to a student, you can say that a straight line is a segment that does not have two ends.

The figure shows two straight lines: CD and EF.

Beam in mathematics

What is a ray? Definition of a ray in mathematics: a ray is a part of a line that has a beginning and no end. The name of the beam contains two letters, for example, DC. Moreover, the first letter always indicates the starting point of the beam, so letters cannot be swapped.

The figure shows the rays: DC, KC, EF, MT, MS. Beams KC and KD are one beam, because they have a common origin.

Number line in mathematics

Definition of a number line in mathematics: a line whose points mark numbers is called a number line.

The figure shows the number line, as well as the OD and ED rays