Lesson duration: 35 minutes
Lesson type: Studying and primary consolidation of new material.
Target: Introduce the polyline 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", "top of a broken line";
- repeat: segments, lines;
- improvement of computational skills and abilities.
2) Developing:
- develop logical thinking, spatial imagination, attention, memory, fantasy;
- improve the level of development of mathematical speech
- show the interdisciplinary connection between mathematics and astronomy.
3) Educators:
- educate the communication skills of students
- to cultivate pride in their homeland, achievements in science, technology, astronautics.
Materials and equipment:
- multimedia presentation
- Computer, projector, screen
- "Study 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, problematic
Working methods: explanatory and illustrative, partially search, verbal, visual, practical.
Teacher function: organizer of cooperation; prospecting consultant.
Pedagogical technologies:
student-centered learning;
Explanatory and illustrative teaching;
Pedagogy of cooperation (learning dialogue);
ICT technology (presentation).
Expected Result:
- know what a broken line is, what it consists of, how it differs from a segment, a ray, a straight line, a curved line
- expanding knowledge of geometric material
- increasing the activity of students in the classroom
- use by students of acquired knowledge and skills in practical activities
- vocabulary enrichment
List of used literature.
1. Istomina N.B. Mathematics: a textbook for the 1st grade of educational institutions. - Smolensk: "Association XXI century", 2008.
2. Istomina N.B. Workbook for the textbook "Mathematics" for grade 1
During the classes
1. Organizing moment
Teacher: Children, 2011 has been declared the year of Russian cosmonautics in our country. Who among you is interested in space? Who wants to fly into space? Today 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, restore some knowledge. What do you think we need to remember?
Children: Review numbers, addition and subtraction.
Teacher: I agree with you children. I will add: you need to know the geometric shapes passed.
2. Actualization of previous knowledge
Teacher: There are "Study Route Sheets" on your tables. All the results of the work in the lesson will be recorded on these sheets.
Get to know a new word. "Astronomy" (ancient Greek) is formed 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 the distances to distant stars, during the construction of spacecraft, their trajectory of movement, development of speed:
So, the first task: "mathematical dictation". Listen to the condition, calculate in your mind, write down only the answer.
Of the 9 planets in the solar system, only two have female names. And how many male names are in the names of the planets of the solar system? (7)
The constellation Ursa Major has 7 bright stars. And in the constellation "Cassiopeia" 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 detachment" is instructed to write the answers on the board. Then all the children check, compare their results with the answers written on the board.
- What are the shapes called? (point, triangle, curved line, straight line, segment)
- What is the difference between a ray and a segment?
- What is the difference between a straight line and 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, because the lines that form them have borders)
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 figure did you get? (Line segment)
Draw a straight line through the yellow dot. Can you do another one? What else? (Yes!)
It is true that an infinite number of straight lines can be drawn through a single point.
3. Physical education(The guys do the exercises, standing at their desks)
One, two!
The speed of light!
Three four!
We are flying!
To distant planets
We want to get there soon!
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
You notice,
Very important science
Mathematics!
4. Introduction of new material
Today we continue our journey to Geometry.
See what I have in my hands? (Vermicelli Spaghetti)
What geometric shape does it remind you of? (straight line)
Pick up the spaghetti handed out to you by the attendant. Break in the middle, and then break each part in half again.
What geometric shapes remind you of? (Segments, they turned out 4)
Connect them with pieces of plasticine to each other. Is it now possible to call the resulting figure a straight line? (No)
What would you call such a geometric figure? (broken line)
I have to correct you a little, it's called a "broken" line.
See what a broken line consists of? (From segments)
Each broken line consists of several segments - links. How many links are 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 meet is called a vertex.
How many vertices does this polyline have? (Three)
In addition, a broken line has 2 ends.
5. Physical education- self-massage of the fingers with the help of a SU-JOK massager: Slide No. 4
In order
All planets
Call any of us:
Once - Mercury,
Two - Venus,
Three - Earth,
Four - Mars,
Five - Jupiter,
Six - Saturn.
Seven - Uranus,
The eighth is Neptune.
And after him, later
called Pluto.
6. Primary fastening
Teacher: Children, let's remember again, what are the curved lines? (closed and open)
What do you think, broken lines can be closed and open?
The teacher opens table number 1 on the board:
What figures are shown in the table? (broken lines)
Which broken line has the most links? (No. 4)
Which polyline has the fewest links? (No. 1)
Which polyline has three vertices? (No. 2)
Which polyline has five vertices? (No. 4)
The teacher opens table number 2 on the board:
Teacher: These are also broken lines. How do they differ from the broken lines in the first table? (All links are interconnected)
Such broken lines are called "closed" lines, and the lines on the first table are called "open" lines.
Name the closed broken line that has the fewest links. (#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 starry sky map: open broken lines and closed ones.
Teacher: If your "broken line of spaghetti" lying on the desk is turned over, it will resemble the constellation "Cassiopeia". It was named after the queen, who was bewitched by an insidious sorceress.
7. Physical education.
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 memorize it and lay out exactly the same from the counting sticks.
Now work in pairs. Check your classmate's attention.
What figure did you get?
What else can you say about her? Can it be called a broken line?
Can it be called closed? (open?) Why?
8. Summing up the lesson
What geometric figure are you familiar with? (broken line)
What elements does a broken line consist of? (From links and peaks)
What are broken lines? (Closed and open)
Turn over the Study Route Sheet. Circle with a colored pencil only broken lines, closed and open:
What line was described by Yuri Gagarin's ship in 108 minutes around the Earth? (open curved line)
In the lower right corner of the "Educational Route Sheet" an asterisk "smiles" at you. What geometric figure does it resemble? (Closed polyline) Determine the number of vertices? Links? Are there any ends?
Self-assessment of the work of students in the lesson:
You have 3 colored pencils. Color the star in 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.
Lesson duration: 35 minutes
Lesson type: Studying and primary consolidation of new material.
Target: Introduce the polyline 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", "top of a broken line";
- repeat: segments, lines;
- improvement of computational skills and abilities.
2) Developing:
- develop logical thinking, spatial imagination, attention, memory, fantasy;
- improve the level of development of mathematical speech
- show the interdisciplinary connection between mathematics and astronomy.
3) Educators:
- educate the communication skills of students
- to cultivate pride in their homeland, achievements in science, technology, astronautics.
Materials and equipment:
- multimedia presentation
- Computer, projector, screen
- "Study 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, problematic
Working methods: explanatory and illustrative, partially search, verbal, visual, practical.
Teacher function: organizer of cooperation; prospecting consultant.
Pedagogical technologies:
student-centered learning;
Explanatory and illustrative teaching;
Pedagogy of cooperation (learning dialogue);
ICT technology (presentation).
Expected Result:
- know what a broken line is, what it consists of, how it differs from a segment, a ray, a straight line, a curved line
- expanding knowledge of geometric material
- increasing the activity of students in the classroom
- use by students of acquired knowledge and skills in practical activities
- vocabulary enrichment
List of used literature.
1. Istomina N.B. Mathematics: a textbook for the 1st grade of educational institutions. - Smolensk: "Association XXI century", 2008.
2. Istomina N.B. Workbook for the textbook "Mathematics" for grade 1
During the classes
1. Organizing moment
Teacher: Children, 2011 has been declared the year of Russian cosmonautics in our country. Who among you is interested in space? Who wants to fly into space? Today 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, restore some knowledge. What do you think we need to remember?
Children: Review numbers, addition and subtraction.
Teacher: I agree with you children. I will add: you need to know the geometric shapes passed.
2. Actualization of previous knowledge
Teacher: There are "Study Route Sheets" on your tables. All the results of the work in the lesson will be recorded on these sheets.
Get to know a new word. "Astronomy" (ancient Greek) is formed 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 the distances to distant stars, during the construction of spacecraft, their trajectory of movement, development of speed:
So, the first task: "mathematical dictation". Listen to the condition, calculate in your mind, write down only the answer.
Of the 9 planets in the solar system, only two have female names. And how many male names are in the names of the planets of the solar system? (7)
The constellation Ursa Major has 7 bright stars. And in the constellation "Cassiopeia" 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 detachment" is instructed to write the answers on the board. Then all the children check, compare their results with the answers written on the board.
- What are the shapes called? (point, triangle, curved line, straight line, segment)
- What is the difference between a ray and a segment?
- What is the difference between a straight line and 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, because the lines that form them have borders)
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 figure did you get? (Line segment)
Draw a straight line through the yellow dot. Can you do another one? What else? (Yes!)
It is true that an infinite number of straight lines can be drawn through a single point.
3. Physical education(The guys do the exercises, standing at their desks)
One, two!
The speed of light!
Three four!
We are flying!
To distant planets
We want to get there soon!
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
You notice,
Very important science
Mathematics!
4. Introduction of new material
Today we continue our journey to Geometry.
See what I have in my hands? (Vermicelli Spaghetti)
What geometric shape does it remind you of? (straight line)
Pick up the spaghetti handed out to you by the attendant. Break in the middle, and then break each part in half again.
What geometric shapes remind you of? (Segments, they turned out 4)
Connect them with pieces of plasticine to each other. Is it now possible to call the resulting figure a straight line? (No)
What would you call such a geometric figure? (broken line)
I have to correct you a little, it's called a "broken" line.
See what a broken line consists of? (From segments)
Each broken line consists of several segments - links. How many links are 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 meet is called a vertex.
How many vertices does this polyline have? (Three)
In addition, a broken line has 2 ends.
5. Physical education- self-massage of the fingers with the help of a SU-JOK massager: Slide No. 4
In order
All planets
Call any of us:
Once - Mercury,
Two - Venus,
Three - Earth,
Four - Mars,
Five - Jupiter,
Six - Saturn.
Seven - Uranus,
The eighth is Neptune.
And after him, later
called Pluto.
6. Primary fastening
Teacher: Children, let's remember again, what are the curved lines? (closed and open)
What do you think, broken lines can be closed and open?
The teacher opens table number 1 on the board:
What figures are shown in the table? (broken lines)
Which broken line has the most links? (No. 4)
Which polyline has the fewest links? (No. 1)
Which polyline has three vertices? (No. 2)
Which polyline has five vertices? (No. 4)
The teacher opens table number 2 on the board:
Teacher: These are also broken lines. How do they differ from the broken lines in the first table? (All links are interconnected)
Such broken lines are called "closed" lines, and the lines on the first table are called "open" lines.
Name the closed broken line that has the fewest links. (#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 starry sky map: open broken lines and closed ones.
Teacher: If your "broken line of spaghetti" lying on the desk is turned over, it will resemble the constellation "Cassiopeia". It was named after the queen, who was bewitched by an insidious sorceress.
7. Physical education.
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 memorize it and lay out exactly the same from the counting sticks.
Now work in pairs. Check your classmate's attention.
What figure did you get?
What else can you say about her? Can it be called a broken line?
Can it be called closed? (open?) Why?
8. Summing up the lesson
What geometric figure are you familiar with? (broken line)
What elements does a broken line consist of? (From links and peaks)
What are broken lines? (Closed and open)
Turn over the Study Route Sheet. Circle with a colored pencil only broken lines, closed and open:
What line was described by Yuri Gagarin's ship in 108 minutes around the Earth? (open curved line)
In the lower right corner of the "Educational Route Sheet" an asterisk "smiles" at you. What geometric figure does it resemble? (Closed polyline) Determine the number of vertices? Links? Are there any ends?
Self-assessment of the work of students in the lesson:
You have 3 colored pencils. Color the star in 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 broken line is a special kind of geometric figure, which is composed of several segments. These segments are connected in series with each other at their ends. The end of each segment, except for 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, which is an indirect line and consists of a series of segments connected in series to each other. These segments can form angles of various 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 polyline segments are straight lines, but the segments of the curve do not. These concepts will be explained in detail in the school curriculum in mathematics for grade 8.
Links, peaks and length
To fully grasp the essence and properties of this concept, consider what the links of a broken line in mathematics are, as well as what its vertices and length are:
It is interesting to know: what is convex, its features and signs.
Its designation is made up of capital Latin letters that stand at the tops:
- Each vertex in the figure is denoted by one letter (for example: A, B, C, D or e).
- The 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 called ABCDE or EDCBA.
Varieties
In geometry, it is customary to distinguish several varieties by structure:
- Closed self-intersecting.
- Non-closed self-intersecting.
- Closed without self-intersections.
- Open without self-intersections.
As already described above, a closed non-intersecting figure is called a polygon.
If the links of the figure have intersections 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. Common points of two segments are called polygon vertices.
The number of links or sides in each polygon corresponds to the number of angles in it. A closed broken line of three segments is called triangle. The broken line of four links was called quadrilateral. A 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
The following are the main properties common to all polygons:
- If the vertices of the polygon serve as the ends of one side, they are called adjacent. If the vertices are not adjacent to the same side, they are not adjacent.
- The smallest number of sides for a polygon is three. However, triangles, being next to each other, can form new shapes.
- If a segment connects non-neighboring vertices, it is called a diagonal.
- If a figure lies with respect 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 some vertex is called an exterior angle.
- If all sides and angles of a polygon are equal, it is called regular.
triangles
A triangle in mathematics is called a flat geometric figure, which consists of three points that are not located on one straight line. These points are connected by three lines.
The points represent the vertices or triangle, and the segments represent its sides. Near each of the vertices, a corner of the triangle is formed. Thus, this figure has three corners, as can be seen from its name.
There are the following types of triangles:
- Equilateral - all sides are equal in length.
- Versatile - all sides vary in length.
- Isosceles - two of the three sides are the same length.
- Acute - if all angles are acute.
- Rectangular - if there is a right angle.
- Obtuse - if there is one obtuse angle.
Quadrangles
A flat geometric figure with four corners and four sides is called a quadrilateral.
If all corners of a quadrilateral are right angles, then it is a rectangle.
A regular quadrilateral is called a square.
There are other varieties of quadrilaterals - a rhombus, a trapezoid, a parallelogram, etc. All of them obey the general rules described above.
1. How to measure the distance to the damage site with a REIS reflectometer
cable line, consisting of several cables of different types?
Any of the REIS reflectometers allows you to perform these measurements. In this case, two cases are possible.
1st case
with the same reduction factors.
In this case, the measurement of the distance to the damage site is carried out in the usual way. First, a shortening factor is set in the REIS reflectometer, which is the same for all pieces of the cable. Then one of the cursors is set to the beginning of the front of the probing pulse, and the other - to the beginning of the pulse reflected from the damage site. The distance between the cursors will correspond to the distance to the damage site.
An example of this case is shown in the figure.
The figure indicates:
L1 - length of the first piece of cable (shortening factor g 1),
L2 - length of the second piece of cable (shortening factor g 1),
L3 - distance from the beginning of the third piece of cable to the fault (shortening factor g 1),
L is the distance from the beginning of the cable to the point of damage,
A - signal reflected from the junction of the first and second pieces of cable,
B - signal reflected from the junction of the second and third pieces of cable,
C - signal reflected from the damage site.
The amplitude of the signals A and B depends on the ratio of the wave impedances W1, W2 and W3 of the individual pieces of the cable. If the wave impedances of neighboring cable pieces are equal, then the reflection from their junction has a minimum amplitude. And vice versa. In the above trace, the wave impedance W2 of the second piece of cable is less than the wave impedance W1 of the first piece of cable (W2< W1). Волновое сопротивление третьего и второго кусков кабеля также не равны, причем W3 >W2.
2nd case. The cable line consists of several pieces
with different reduction factors.
The measurement of the distance to the damage in this case is carried out in stages. Consider the sequence of measurements on the example of the reflectogram shown in the figure.
First, in the REIS reflectometer, the shortening factor g 1 is set for the first piece of cable and the length of this piece is measured. To do this, the zero cursor is set to the beginning of the front of the probing pulse (in Position 1), and the measuring cursor is set to the beginning of the front of the pulse reflected from the junction of the first and second pieces of the cable (in Position 2). The resulting length of the first piece of cable L1 is recorded.
Next, set the coefficient of shortening g 2 for the second piece of cable and measure the length of the second piece. To do this, leaving the measuring cursor in place, move the zero cursor to the beginning of the pulse reflected from the junction of the second and third pieces of the cable (to Position 3). The resulting length of the second piece of cable is recorded.
Then set the shortening factor g 3 for the third piece of cable and measure the distance from the beginning of the third piece of cable to the fault. To do this, leaving the zero cursor in place (in Position 3), move the measuring cursor to the beginning of the pulse reflected from the damage site (in Position 4). The resulting distance L3 from the beginning of the third piece of cable to the fault is recorded.
The distance to the damage point L is determined as the sum of the measured values: L = L1 + L2 + L3.
Similarly, it is possible to determine the distance to the point of damage of a cable line, consisting of any number of pieces of cables of different types, having different coefficients of shortening.
2. Why is sometimes the length of the power cable on the reel specified by the manufacturer
cable differs from the length measured by the reflectometer? When measuring
the velocity factor has been set correctly. What is the length data
cables are more accurate?
Such a difference can be observed when the manufacturer measures the length of the cable using the bridge method according to the resistance of the conductors. The cores in the power cable are twisted, so their length is always slightly longer than the length of the cable itself. Measuring the length of the cable by the resistance of the conductors (electrical length) gives an overestimated value compared to the actual, geometric length of the cable.
The difference can also be in the case when the factory measures the length of the manufactured cable using mechanical devices that have rollers that can slip when the cable passes through them.
If the length of the power cable is measured with a reflectometer, then the discrepancy between the electrical and geometric lengths of the cable is taken into account in the shortening factor. Therefore, with a properly set velocity factor, length measurements made with a reflectometer are more accurate than measurements made with a bridge method.
Note: The above length discrepancy can be observed not only for the power cable, but also for any other cable.
3. Why, when measuring with a reflectometer over long (more than a few kilometers)
multi-pair telephone lines, such as the CCI type, zero line
trace curves and does not allow you to set
Is there a high gain in the reflectometer?
The indicated curvature of the zero line of the reflectogram, due to its characteristic appearance, is also called “skiing”. An example of such a “ski” is shown in the figure.
The figure shows the case in which in the “ski” area there is a signal reflected from the cable defect, in particular, a leak. When measuring with an OTDR on a cable, it is usually necessary to increase the gain due to the effect of attenuation. An increase in gain in the presence of a "ski" leads to further distortion of the reflectogram, which greatly complicates and may make analysis of the reflectogram completely impossible.
The reason for the appearance of the “ski” is the distributed capacitance of the cable (capacitance between the cores and between the core and the ground) and the longitudinal ohmic resistance of the cable cores.
At the moment of impact on the cable of the probing pulse from the reflectometer, the specified distributed capacitance of the cable is charged. At the end of the probing pulse, the distributed capacitance of the cable begins to gradually discharge, a “ski” appears.
To reduce the influence of the “ski” on the results of measurements with REIS-105, REIS-205 or REIS-305 reflectometers, you need to turn on the compensation pulse and select its duration.
The degree of compensation can be set by the operator depending on the line, since the “ski” depends on many cable parameters: the number and diameter of the cores, the length of the cable, the type of insulation, etc.
4. When measuring the length of an armored cable with a reflectometer, we get
the following incomprehensible results: if you connect the reflectometer according to the scheme
core-core, then the cable length is less than when connecting
according to the scheme lived-armor. What's the matter here?
In fact, no matter how you connect the reflectometer to the cable when measuring its length, the length of the cable remains the same.
The different values of the cable lengths you measured for different connection schemes are due to the fact that the coefficients of shortening of the wave channels core-core and core-armor differ from each other.
