Closed and open lines. Point, line, straight line, ray, segment, broken side AB and side BC are adjacent

Lesson duration: 35 minutes

Lesson type: Study and primary consolidation of new material.

Target: Introduce the broken line and its components.

Lesson objectives:

1) Educational:

introduce students to the broken line and its types; mastering the concepts of “broken line”, “link of a broken line”, “vertex of a broken line”;
repeat: segments, lines;
improvement of computing skills.

2) Developmental:

develop logical thinking, spatial imagination, attention, memory, imagination;
improve the level of development of mathematical speech
show the interdisciplinary connection between mathematics and astronomy.

3) Educators:

develop communicative qualities of students
to cultivate pride in one’s homeland, achievements in science, technology, and astronautics.

Materials and equipment:

Multimedia presentation
Computer, projector, screen
"Training route sheet"
Pencils: yellow, blue, red
Spaghetti, a piece of plasticine
Massage mats for feet, SU-JOK (massage set "Chestnut" for hands)

Leading activity: productive, creative, challenging

Working methods: explanatory-illustrative, partially search, verbal, visual, practical.

Teacher function: organizer of cooperation; consultant managing search work.

Pedagogical technologies:

Personally-centered learning;

Explanatory and illustrative teaching;

Pedagogy of cooperation (educational dialogue);

ICT technology (presentation).

Expected Result:

know what a broken line is, what it consists of, how it differs from a segment, ray, straight line, curved line
expanding knowledge about geometric material
increasing student activity in lessons
students’ use of acquired knowledge and skills in practical activities
vocabulary enrichment

List of used literature.

1. Istomina N.B. Mathematics: textbook for 1st grade of general education institutions. - Smolensk: "Association XXI century", 2008.

2. Istomina N.B. Workbook for the textbook "Mathematics" for 1st grade

During the classes

1. Organizational moment

Teacher: Children, 2011 has been declared the year of Russian cosmonautics in our country. How many of you are interested in space? Who wants to fly into space? Today there is such an opportunity for the whole class. We will make a training flight. In order not to make mistakes during the flight, you need to prepare and restore some knowledge. What do you think we need to remember?

Children: Review numbers, addition and subtraction.

Teacher: I agree with you, children. I’ll add: you need to know the geometric shapes you’ve covered.

2. Updating previous knowledge

Teacher: There are “Training Route Sheets” on your desks. We will record all the results of the work in the lesson on these sheets.

Get to know a new word. “Astronomy” (ancient Greek) is derived from the ancient Greek words “astron” - star and “nomos” - law or culture, and literally means “Law of the Stars”.

All scientists - astronomers know mathematics perfectly. Without this knowledge, it is impossible to accurately calculate distances to distant stars during the construction of spaceships, their trajectory, and speed development:

So, the first task: “mathematical dictation”. Listen to the condition, calculate in your head, and write down only the answer.

Of the 9 planets in the solar system, only two have female names. How many male names are there in the names of the planets of the solar system? (7)

The constellation Ursa Major has 7 bright stars. And in the constellation "Cassiopeia" there are 5 bright stars. How many more bright stars are there in the constellation Ursa Major? (2)

To my question at the beginning of the lesson: “Who dreams of flying into space?” 3 girls and 7 boys answered “yes”. How many kids in our class want to fly into space? (10)

Children: write down the answers in their “Training Route Sheets”, and one student - the “commander of the cosmonaut squad” is assigned to write the answers on the board. Then all children check and compare their results with the answers written on the board.

What are the names of the figures? (point, triangle, curved line, straight line, segment)
How does a ray differ from a segment?
How does a straight line differ from a ray?

Why is the second figure called a triangle? (has three vertices and three sides)

Can the sides of a triangle be called segments? Why? (the sides of the triangle are segments, since their forming lines have boundaries)

Teacher: In the “Training Route Sheet”, find the red dot and build a beam. What tool is needed? (Ruler)

Connect the two blue dots. What kind of figure did you get? (Line segment)

Draw a straight line through the yellow dot. Can you do another one? What else? (Yes!)

That's right, countless straight lines can be drawn through one single point.

3. Physical education minute(Guys do exercises while standing at their desks)

One, two!
Speed of light!
Three four!
We are flying!
To distant planets
We want to get there quickly!
To drive ships
To fly into the sky,
There is a lot to know.
You have to know a lot!
And at the same time, and at the same time
Will you notice?
Very important science
Mathematics!

4. Introduction of new material

Today we continue our journey to the country of Geometry.

Look what I have in my hands? (Vermicelli spaghetti)

What geometric figure does it remind you of? (Direct line)

Take the spaghetti that the attendant handed out to you. Break it in the middle, and then break each part in half again.

What geometric shapes remind you of? (Segments, there were 4 of them)

Connect them together with pieces of plasticine. Can the resulting figure now be called a straight line? (No)

What would you call such a geometric figure? (Broken line)

I should correct you a little, it's called a "broken" line.

Look, what does a broken line consist of? (From segments)

Each broken line consists of several segments - links. How many links are there in this broken line? (Four)

The links of the polyline do not lie on the same straight line. The end of one link is the beginning of another. The place where two links join is called an apex.

How many vertices does this broken line have? (Three)

In addition, the polyline has 2 ends.

5. Physical education minute- self-massage of fingers using the SU-JOK massager: Slide No. 4

In order
All planets
Any of us can name:
One - Mercury,
Two - Venus,
Three - Earth,
Four - Mars,
Five - Jupiter,
Six - Saturn.
Seven - Uranus,
Eighth - Neptune.
And then after him,
Called Pluto.

6. Primary consolidation

Teacher: Children, let's remember once again what kind of curved lines are there? (Closed and open)

What do you think, broken lines can be closed or open?

The teacher opens table No. 1 on the board:

What figures are shown in the table? (broken lines)

Which broken line has the most links? (No. 4)

Which broken line has the fewest links? (No. 1)

Which broken line has three vertices? (No. 2)

Which broken line has five vertices? (No. 4)

The teacher opens table No. 2 on the board:

Teacher: These are also broken lines. How do they differ from the broken lines on the first table? (All links are interconnected)

Such broken lines are called “closed” lines, and the lines in the first table are called “open” lines.

Name the closed polyline that has the fewest links. (No. 1)

That's right, but can there be a closed line of two links, think about it. Let's build such a broken line. (No, to “close” the line you need a third link)

Teacher: Find and name the constellations on the star map: open broken lines and closed ones.

Teacher: If your “broken spaghetti line” lying on your desk is turned upside down, it will resemble the constellation “Cassiopeia”. She was named after the queen, who was bewitched by an insidious witch.

7. Physical education minute.

For eyes. Children follow the movement of Kolobok on Slide No. 4

Attention task

For a few seconds I will show you one figure. You must remember it and lay out exactly the same from the counting sticks.

Now work in pairs. Check your classmate's attention.

What kind of figure did you get?

What else can you say about her? Can it be called a broken line?

Can we call it closed? (unclosed?) Why?

8. Summing up the lesson

What geometric figure did you meet? (Broken line)

What elements does a broken line consist of? (From links and vertices)

What types of broken lines are there? (Closed and open)

Turn over the "Training Route Sheet". Trace only broken lines, closed and open, with a colored pencil:

What line did Yu. Gagarin's ship describe in 108 minutes around the Earth? (open curved line)

In the lower right corner of the “Training Route Sheet” an asterisk “smiles” at you. What geometric figure does it resemble? (Closed polyline) Determine the number of vertices? Zvenyev? Are there any endings?

Self-assessment of students’ work in the lesson:

You have 3 colored pencils. Paint the star green if you are completely satisfied with your work in the lesson; yellow - satisfied, but not completely; red - you have to try!

Additional material(Slides 18 - 31): information about planets, stars, space exploration.

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 - with 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 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) Latin letters

line a, line b, line c

a b c

The line may be

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

closed lines

open lines

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.

self-intersecting
without self-intersections

self-intersecting lines

lines without self-intersections

straight
broken
crooked

straight lines

broken lines

curved 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

B A

Direct may be

intersecting if they have a common point. Two lines can intersect only at one point.
- perpendicular if they intersect at right angles (90°).
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

B A

The rays coincide if

located on the same straight line
start at one point
directed in one direction

rays AB and AC coincide

rays CB and CA coincide

C B A

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

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 (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

B A

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.

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

A B C D E 64 62 127 52

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 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 polyline

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

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

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.

A broken line is a special type of geometric figure that is made up of several segments. These segments are connected in series at their ends. The end of each segment, with the exception of the last one, is the starting point of the next one. Adjacent segments should not be on the same straight line.

In contact with

There is another definition of what a broken figure is. According to him, this is a geometric object that is an indirect line and consists of a series of segments sequentially connected to each other. These segments can form angles of different sizes. Even if the angle between them is minimal, it will still break the line and it can already be considered a broken line. This is its main difference from the straight line.

A broken line should be distinguished from a curve. The main difference is that the segments of the polyline are straight lines, but curve segments do not. These concepts will be explained in detail by the school mathematics curriculum for grade 8.

Links, peaks and length

In order to fully understand the essence and properties of this concept, let’s consider what the links of a broken line are in mathematics, as well as what its vertices and length represent:

It is interesting to know: what is convex, its features and symptoms.

Its designation is made up of capital Latin letters that stand at the tops:

Each vertex in the figure is designated by one letter (for example: A, B, C, D or E).
A link is usually denoted by two letters (the ends of the corresponding segment, for example: AB, BC, CD, DE).

In general, such a set is usually called ABCDE or EDCBA.

Varieties

In geometry, it is customary to distinguish several types of structure:

Closed self-intersecting.
Unclosed self-intersecting ones.
Closed without self-intersections.
Unclosed without self-intersections.

As already described above, a closed non-intersecting figure is called a polygon.

If the links of a figure intersect with each other, it is called self-intersecting.

A polygon is a geometric figure that is characterized by the number of angles and links. The angles are made up of pairs of links of a closed broken line, converging at one point. The links are also called the sides of the polygon. The common points of two segments are called the vertices of the polygon.

The number of links or sides in each polygon corresponds to the number of angles in it. A closed polyline of three segments is called triangle. The broken line of four links is called quadrangle. Figure of five segments - pentagon etc.

The part of the plane that is bounded by a closed polyline is called flat polygon. Its other name is polygonal area.

Properties

Below are the basic properties common to all polygons:

If the vertices of a polygon serve as ends of one side, they are called adjacent. If the vertices are not adjacent to the same side, they are non-neighboring.
The smallest number of sides a polygon has is three. However, triangles, being next to each other, can form new shapes.
If a segment connects non-adjacent vertices, it is called a diagonal.
If a figure lies relative to one straight line in any half-plane, it is called convex. In this case, the straight line contains one side of the figure and itself belongs to the half-plane.
An angle adjacent to an interior angle of a polygon at a certain vertex is called an exterior angle.
If all sides and angles of a polygon are equal, it is called regular.

Triangles

In mathematics, a triangle is usually called a flat geometric figure that consists of three points that are not located on the same straight line. These points are connected by three segments.

The points represent the vertices or triangle, and the segments represent its sides. A corner of a triangle is formed near each of the vertices. Thus, this figure has three angles, as is evident from its name.

The following types of triangles are distinguished:

Equilateral - all sides are equal in length.
Versatile - all sides vary in length.
Isosceles - two of the three sides have the same length.
Acute - if all the corners are acute.
Rectangular - if there is a right angle.
Obtuse - if there is one obtuse angle.

Quadrilaterals

A flat geometric figure with four angles and four sides is called a quadrilateral.

If all the angles of a quadrilateral are right angles, it is a rectangle.

A regular quadrilateral is called a square.

There are other types of quadrilaterals - rhombus, trapezoid, parallelogram, etc. All of them obey the general rules described above.

Lesson duration: 35 minutes

Lesson type: Study and primary consolidation of new material.

Target: Introduce the broken line and its components.

Lesson objectives:

1) Educational:

introduce students to the broken line and its types; mastering the concepts of “broken line”, “link of a broken line”, “vertex of a broken line”;
repeat: segments, lines;
improvement of computing skills.

2) Developmental:

develop logical thinking, spatial imagination, attention, memory, imagination;
improve the level of development of mathematical speech
show the interdisciplinary connection between mathematics and astronomy.

3) Educators:

develop communicative qualities of students
to cultivate pride in one’s homeland, achievements in science, technology, and astronautics.

Materials and equipment:

Multimedia presentation
Computer, projector, screen
"Training route sheet"
Pencils: yellow, blue, red
Spaghetti, a piece of plasticine
Massage mats for feet, SU-JOK (massage set "Chestnut" for hands)

Leading activity: productive, creative, challenging

Working methods: explanatory-illustrative, partially search, verbal, visual, practical.

Teacher function: organizer of cooperation; consultant managing search work.

Pedagogical technologies:

Personally-centered learning;

Explanatory and illustrative teaching;

Pedagogy of cooperation (educational dialogue);

ICT technology (presentation).

Expected Result:

know what a broken line is, what it consists of, how it differs from a segment, ray, straight line, curved line
expanding knowledge about geometric material
increasing student activity in lessons
students’ use of acquired knowledge and skills in practical activities
vocabulary enrichment

List of used literature.

1. Istomina N.B. Mathematics: textbook for 1st grade of general education institutions. - Smolensk: "Association XXI century", 2008.

2. Istomina N.B. Workbook for the textbook "Mathematics" for 1st grade

During the classes

1. Organizational moment

Children: Review numbers, addition and subtraction.

Teacher: I agree with you, children. I’ll add: you need to know the geometric shapes you’ve covered.

2. Updating previous knowledge

Teacher: There are “Training Route Sheets” on your desks. We will record all the results of the work in the lesson on these sheets.

Get to know a new word. “Astronomy” (ancient Greek) is derived from the ancient Greek words “astron” - star and “nomos” - law or culture, and literally means “Law of the Stars”.

So, the first task: “mathematical dictation”. Listen to the condition, calculate in your head, and write down only the answer.

Of the 9 planets in the solar system, only two have female names. How many male names are there in the names of the planets of the solar system? (7)

The constellation Ursa Major has 7 bright stars. And in the constellation "Cassiopeia" there are 5 bright stars. How many more bright stars are there in the constellation Ursa Major? (2)

To my question at the beginning of the lesson: “Who dreams of flying into space?” 3 girls and 7 boys answered “yes”. How many kids in our class want to fly into space? (10)

What are the names of the figures? (point, triangle, curved line, straight line, segment)
How does a ray differ from a segment?
How does a straight line differ from a ray?

Why is the second figure called a triangle? (has three vertices and three sides)

Can the sides of a triangle be called segments? Why? (the sides of the triangle are segments, since their forming lines have boundaries)

Teacher: In the “Training Route Sheet”, find the red dot and build a beam. What tool is needed? (Ruler)

Connect the two blue dots. What kind of figure did you get? (Line segment)

Draw a straight line through the yellow dot. Can you do another one? What else? (Yes!)

That's right, countless straight lines can be drawn through one single point.

3. Physical education minute(Guys do exercises while standing at their desks)

One, two!
Speed of light!
Three four!
We are flying!
To distant planets
We want to get there quickly!
To drive ships
To fly into the sky,
There is a lot to know.
You have to know a lot!
And at the same time, and at the same time
Will you notice?
Very important science
Mathematics!

4. Introduction of new material

Today we continue our journey to the country of Geometry.

Look what I have in my hands? (Vermicelli spaghetti)

What geometric figure does it remind you of? (Direct line)

Take the spaghetti that the attendant handed out to you. Break it in the middle, and then break each part in half again.

What geometric shapes remind you of? (Segments, there were 4 of them)

Connect them together with pieces of plasticine. Can the resulting figure now be called a straight line? (No)

What would you call such a geometric figure? (Broken line)

I should correct you a little, it's called a "broken" line.

Look, what does a broken line consist of? (From segments)

Each broken line consists of several segments - links. How many links are there in this broken line? (Four)

The links of the polyline do not lie on the same straight line. The end of one link is the beginning of another. The place where two links join is called an apex.

How many vertices does this broken line have? (Three)

In addition, the polyline has 2 ends.

5. Physical education minute- self-massage of fingers using the SU-JOK massager: Slide No. 4

In order
All planets
Any of us can name:
One - Mercury,
Two - Venus,
Three - Earth,
Four - Mars,
Five - Jupiter,
Six - Saturn.
Seven - Uranus,
Eighth - Neptune.
And then after him,
Called Pluto.

6. Primary consolidation

Teacher: Children, let's remember once again what kind of curved lines are there? (Closed and open)

What do you think, broken lines can be closed or open?

The teacher opens table No. 1 on the board:

What figures are shown in the table? (broken lines)

Which broken line has the most links? (No. 4)

Which broken line has the fewest links? (No. 1)

Which broken line has three vertices? (No. 2)

Which broken line has five vertices? (No. 4)

The teacher opens table No. 2 on the board:

Teacher: These are also broken lines. How do they differ from the broken lines on the first table? (All links are interconnected)

Such broken lines are called “closed” lines, and the lines in the first table are called “open” lines.

Name the closed polyline that has the fewest links. (No. 1)

That's right, but can there be a closed line of two links, think about it. Let's build such a broken line. (No, to “close” the line you need a third link)

Teacher: Find and name the constellations on the star map: open broken lines and closed ones.

7. Physical education minute.

For eyes. Children follow the movement of Kolobok on Slide No. 4

Attention task

For a few seconds I will show you one figure. You must remember it and lay out exactly the same from the counting sticks.

Now work in pairs. Check your classmate's attention.

What kind of figure did you get?

What else can you say about her? Can it be called a broken line?

Can we call it closed? (unclosed?) Why?

8. Summing up the lesson

What geometric figure did you meet? (Broken line)

What elements does a broken line consist of? (From links and vertices)

What types of broken lines are there? (Closed and open)

Turn over the "Training Route Sheet". Trace only broken lines, closed and open, with a colored pencil:

What line did Yu. Gagarin's ship describe in 108 minutes around the Earth? (open curved line)

Self-assessment of students’ work in the lesson:

You have 3 colored pencils. Paint the star green if you are completely satisfied with your work in the lesson; yellow - satisfied, but not completely; red - you have to try!

Additional material(Slides 18 - 31): information about planets, stars, space exploration.

In this lesson we will get acquainted with the concepts of “closed line” and “open line”, learn how to distinguish and construct them. We will also consider such concepts as “links” and “vertices” of a curved line. In the future we will use this knowledge to solve more complex problems.

Subject:Introduction to Basic Concepts

Lesson: Closed and open lines

Exercise 1

In this figure we see that it will be easier for the sheep to get out of the first fence, because it is open - not closed. It will be more difficult to get out from behind the second fence, since it is closed. Let's draw lines that will correspond to the first and second fences.

So, we got two lines, of which the first is closed and the second is open.

Task 2: Determine which lines in Fig. 3 are closed and which are not closed.

In the figure we see that lines No. 1, 3, 6 are open lines. In order to close these lines, it is enough to connect the ends of the lines together. We get:

So, a line whose ends are not connected together is called an open line. A line whose ends are connected together is called closed line.

Each broken line consists of several segments - links . The links of the polyline do not lie on the same straight line. The end of one link is the beginning of another. The place where two links are connected, as well as the ends of an open broken line, is called top .

So, in this lesson we became acquainted with the concepts of “closed line” and “open line”. We learned how to build them, as well as apply knowledge in practice to build such lines.

Bibliography

Alexandrova L.A., Mordkovich A.G. Mathematics 1st grade. - M: Mnemosyne, 2012.
Bashmakov M.I., Nefedova M.G. Mathematics. 1 class. - M: Astrel, 2012.
Bedenko M.V. Mathematics. 1 class. - M7: Russian Word, 2012.

1. Festival of pedagogical ideas ().

3. Festival of pedagogical ideas ().

Homework

1. Determine which lines are shown in the figure.

2. Determine the number of links of each line.

3. Determine the number of vertices of each line.

4. Construct an open line with 4 vertices.

5. Construct a closed line with 6 links.