EntranceDrawings on a coordinate plane with house coordinates. Start in science
Russian mathematicians
Keldysh M.
(10.02.1911 - 24.06.1978)
Academician Mstislav Vsevolodovich Keldysh was born into a professorial family with traditions laid down by his grandfathers: on his mother’s side - full general from infantry (infantry) A.N. Skvortsov. and on his father’s side - Keldysh M.F., who graduated from theological seminary, but then chose the medical path and rose to the rank of general.
After graduating from the Physics and Mathematics Department of Moscow State University in 1931, he was sent to work at TsAGI (Central Aero-Hydrodynamic Institute), where he was strongly recommended to the management by his teacher (and later senior comrade, academician), one of the leading employees of the General Theoretical Group of TsAGI M.A. .Lavrentiev.
With his first works (1933), Keldysh attracted the attention of such an outstanding scientist as the scientific director of TsAGI S.A. Chaplygin, who set before the young theorist-mathematician and mechanic a problem with immediate practical application. The scientific value of these works lies not only in the fact that they solved pressing problems of those years, but also laid the foundation for new approaches in the application of mathematical methods to solve problems of hydro-aerodynamics.
In the 1930s, one of these problems in aviation was the problem of overcoming the “flutter” phenomenon, which unexpectedly arose as aircraft speeds increased. The aircraft industry of all advanced countries encountered the phenomenon of flutter, but earlier than others and in the most complete set of all its varieties, flutter was overcome in our country, thanks to the work of M.V. Keldysh and his colleagues. And now, with great interest, we read with great interest the works of that time, where, on the basis of complex mathematical studies, conclusions are very clearly formulated and practical techniques are outlined, following which eliminates the occurrence of self-oscillations of aircraft structures (flutter) in the entire range of flight speeds. Thus, the phenomenon of flutter ceased to be a barrier to the development of high-speed aviation, and by the Patriotic War (1941-1945) our aircraft industry arrived without this disease, which could not be said about the enemy.
In 1938, Keldysh defended his doctoral dissertation on the topic “On the representation of functions of a complex variable and harmonic functions by series of polynomials.” Experts regarded it as a classic, completing a large stage of research in an important branch of mathematics and at the same time opening a new one.
Solving problems on flutter and shimmy “Shimmy of the front wheel of a three-wheeled chassis” (1945) Keldysh continues to study mathematics. The significance of these works for the development of mathematics is no less than those mentioned above for aviation, especially since the latter could hardly have been carried out without fundamental research in the relevant branches of mathematics. Apparently, the fundamental advances in mathematical science that followed from the works of M.V. Keldysh on approximation theory, functional analysis, and differential equations were due to his ability, while preserving the essence of the problem, to formulate the problem being solved in the simplest form. Having perfect knowledge of various branches of mathematics, he knew how to find and build unexpected analogies and thereby effectively use both the existing mathematical apparatus and create a new one. It should be especially emphasized that the seemingly abstract works of Mstislav Vsevolodovich, for example, on the theory of non-self-adjoint operators he deeply developed, are based on specific applied problems, including vibrations of structures with energy dissipation.
The works of M.V. Keldysh on mathematics and mechanics in the mid-40s were recognized by colleagues and scientists, and their author gained fame in the scientific world. In 1943, M.V. Keldysh was elected a corresponding member of the USSR Academy of Sciences, and in 1946 a full member of the Academy.
Since the second half of the forties, the nature of M.V. Keldysh’s activities has changed significantly. The scientific and organizational aspect comes to the fore. “Soon after the war,” recalled Academician I.M. Vinogradov, director of the Steklov Mathematical Institute, “Yu.B. Khariton and other physicists came to me. They asked me to recommend a mathematician who could carry out calculations on atomic topics. I told them to take Keldysh, he He can understand any application of mathematics better than anyone. They liked Keldysh."
The mastery of atomic energy in those years was associated, first of all, with the problem of creating weapons. The problems that needed to be solved here were unprecedented in complexity; humanity had never dealt with them before. The difficulties were aggravated by extremely limited information on the physics of the phenomena themselves that accompany the course of nuclear processes. Therefore, an important method of understanding phenomena was the construction of physical and mathematical models and their subsequent reproduction in calculations.
In 1949, pioneering research on rocket dynamics and applied celestial mechanics (mechanics of space flight) was launched, which had a significant impact on the development of rocket and space technology. In 1953, optimal designs for composite rockets were proposed and analyzed here; ballistic descent of a spacecraft from orbit and the possibility of its use for the return of astronauts is shown; possible stabilization of the apparatus through the use of the earth's gravitational field and many other ideas.
In 1954, M.V. Keldysh, S.P. Korolev and M.K. Tikhonravov submitted a letter to the Government with a proposal to create an artificial Earth satellite (AES). On January 30, 1956, M.V. Keldysh was appointed chairman of the special commission of the Academy of Sciences on artificial satellites.
After the launch of the first satellite in 1957, a new stage in the exploration of outer space began. At the Steklov Institute of Mechanical Engineering, under the leadership of Keldysh, work is underway on tracking satellites and predicting its trajectory, on ballistic design of interplanetary flights of spacecraft (SC) with minimal energy consumption, etc. Examples of brilliant solutions are: the found scheme for accelerating a spacecraft using an artificial intermediate orbit satellite, the use of the planet’s gravitational field to purposefully change the trajectory of movement. These decisions turned out to be fundamental for the design of all subsequent flights.
To solve the atomic problem and rocket and space problems, there were necessary calculations that were practically inaccessible to the computing facilities available at that time. New computing tools - electronic computers (computers) - had to be created and mastered. This was a task of national importance, paramount in solving the problem of mastering atomic energy. M.V. Keldysh himself was not involved in the design of computers, but was the customer of this equipment and its first major consumer. The institute headed by him was supposed to create calculation methods and, on their basis, solve on a computer the entire set of problems falling under atomic problems. Note that the same computers were used by the Keldysh team for calculations on rocket and space topics. All this enormous work, carried out for the first time, on the creation of calculation methods and their implementation on a computer became the basis of a new direction in mathematics, which today has taken shape into its independent section - computational and applied mathematics.
Recognition of the scientist’s merits in solving the defense problem was the awarding of the title of Hero of Socialist Labor to M.V. Keldysh in 1956, and in 1957 the award of the Lenin Prize. In 1961, for special services in the development of rocket technology, in the creation and successful launch of the world's first spaceship "Vostok" with a man on board, M.V. Keldysh was awarded the title of Hero of Socialist Labor for the second time. In 1971, for exceptional services to the state in the development of Soviet science and technology, great scientific and social activities, and in connection with his sixtieth birthday, M.V. Keldysh was awarded for the third time the title of Hero of Socialist Labor and the Hammer and Sickle gold medal. Awarded a gold medal named after. K.E. Tsiolkovsky for his outstanding contribution to the scientific development of problems in the study and exploration of outer space (1972); gold medal named after M.V. Lomonosov for outstanding achievements in the field of mathematics, mechanics and space research (1975).
The name of Mstislav Vsevolodovich Keldysh is immortalized in the names of a research vessel, a minor planet of the solar system, a crater on the Moon, and a square in Moscow. The former NII-1 (now the M.V. Keldysh Research Center) and the Institute of Applied Mathematics, which he created, are named after him. Monuments-busts were erected to him on the Alley of Heroes and Miusskaya Square in Moscow, in Riga; memorial plaques on the buildings where he lived and worked. Gold medal named after. M.V. Keldysh, established by the USSR Academy of Sciences, is awarded for outstanding scientific work in applied mathematics and mechanics and theoretical research in space exploration.
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Introduction
The relevance of research: Why did I choose this topic? While studying the topic “Coordinate Plane” as an elective, I came across some beautiful assignments. They aroused my great interest. All the students in our class enjoyed drawing pictures on the coordinate plane. We learned to understand that abstract dots can be used to create a familiar pattern: we depicted not only individual dots, but also any objects, animals and plants. When my mathematics teacher Natalya Alekseevna gave us homework - to come up with our own drawing in the coordinate plane and write down the coordinates of the points from which this drawing can be constructed, I liked this task so much. And I wanted to come up with my own entertaining tasks for constructing drawings in the coordinate plane.
Hypothesis: I assume that the tasks created by me will be very interesting to my classmates.
Purpose of the study:
create entertaining tasks for constructing drawings for work in mathematics lessons.
Tasks:
- find the necessary information on this topic;
- get acquainted with the history of the origin of coordinates;
- create your own entertaining tasks for constructing drawings in the coordinate plane;
- study the zodiac constellations;
- construct an image of constellations on a coordinate plane;
- conduct astrological research for students in grade 6 “B”;
- conduct a survey among classmates and demonstrate the results of my research.
Objects of study:
- coordinate plane;
- Zodiac signs;
- zodiac constellations;
- students of grade 6 "B".
Subject of study: construction on the coordinate plane.
Expected results:
Create visual aids on the topic under study in the form of cards with tasks that can be used by the teacher in the classroom and a stand to help schoolchildren.
1. Theoretical part:
1.1.Historical background
The history of the origin of coordinates and the coordinate system begins a very, very long time ago. Initially, the idea of the coordinate method arose in the ancient world in connection with the needs of astronomy, geography, and painting. Ancient Greek scientist Anaximander of Miletus (c. 610-546 BC) (Fig. 1) he is considered to be the first compiler of a geographical map. He clearly described the latitude and longitude of a place using rectangular projections.
Rice. 1
In the 2nd century, the Greek scientist Claudius Ptolemy (Fig. 2)- astronomer, astrologer, mathematician, mechanic, optician, music theorist and geographer, used latitude and longitude as coordinates. He left a deep mark in other fields of knowledge - in optics, geography, mathematics, and also in astrology.
Rice. 2
In the 14th century, French mathematician Nicolas Oresme (Fig. 3) entered by analogy with geographical coordinates
on surface. He proposed to cover the plane with a rectangular grid and call latitude and longitude what we now call abscissa and ordinate. This innovation turned out to be very productive. On its basis, the coordinate method arose, connecting geometry with algebra.
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A point on the plane is replaced by a pair of numbers (x; y), i.e. algebraic object. The words “abscissa”, “ordinate”, “coordinates” were first used by Gottfried Wilhelm Leibniz at the end of the 17th century. ( Rice. 4)
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1.2.Rene Descartes
But the main credit for creating the coordinate method belongs to the French mathematician Rene Descartes (Fig. 5).
In 1637, Rene Descartes created his own coordinate system, later named “Cartesian” in his honor.
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Rene Descartes - French mathematician, philosopher, physicist and physiologist, creator of analytical geometry and modern algebraic symbolism, author of the method of radical doubt in philosophy, mechanism in physics.
There are several legends about the invention of the coordinate system.
Such stories have reached our times.
Legend 1: Visiting Parisian theaters, Descartes never tired of being surprised by the confusion, squabbles, and sometimes even challenges to a duel caused by the lack of an elementary order of distribution of the audience in the auditorium. The numbering system he proposed, in which each seat received a row number and a serial number from the edge, immediately removed all reasons for contention and created a real sensation in Parisian high society.
Legend 2: One day, Rene Descartes lay in bed all day, thinking about something, and a fly buzzed around and did not allow him to concentrate. He began to think about how to describe the position of a fly at any given time mathematically in order to be able to swat it without missing. And... came up with Cartesian coordinates, one of the greatest inventions in human history.
After the publication of the work “Geometry”, Rene Descartes’ system won recognition in scientific circles and influenced the development of all areas of mathematical sciences. Thanks to the coordinate system he invented, it was possible to actually interpret the origin of a negative number.
Already at the end of the 17th century, the concept of a coordinate plane began to be widely used in the world of mathematics.
1.3. Other types of coordinate systems
Polar coordinate system.
It is used in cases where the location of a point is determined on a plane.
Such a system is used in navigation, medicine (computed tomography), geodesy, and modeling.
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Oblique coordinate system, most similar to rectangular (Cartesian). It is used in some mechanisms, when calculating in mechanics, when projecting objects.
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Spherical coordinate system.
Used to display the geometric properties of a figure in three dimensions by specifying three coordinates. Used in astronomy.
Rice. 8
Cylindrical coordinate system.
It is an extension of the polar coordinate system by adding a third coordinate, which specifies the height of the point above the plane. Used in geography and military affairs.
Rice. 9
2. Practical part
Stage I: November - December 2017
- collected information about the history of the invention of the coordinate system,
- I learned to mark points in the coordinate plane before we studied this topic in class (date of completion at school: 02/07/2018),
- made drawings on a coordinate plane for my drawings and wrote down their coordinates,
- presented the results of her work to her classmates in January 2018.
In total, I created 13 drawings and wrote out the coordinates of the points from which they could be constructed. These tasks can be used as material in mathematics lessons on the topic “Coordinate plane”. All drawings are in Appendix 1 to the work.
In order to check the coordinates of my drawings, my mathematics teacher Natalya Alekseevna and I conducted three mathematics lessons with my classmates and students 6 “a” and 6 “b”. They were given cards with the coordinates of the points, and they completed the constructions. This experiment confirmed that all the coordinates of the points in my drawings correspond to my drawings. The students really liked the drawings.
Here's the feedback I received:
- Interesting task. Veronica is a good person.
- Veronica, thank you very much for an interesting task.
- I liked it very much. There would be more such tasks. Thank you!
- I liked everything, it was clear and simple! Thank you!
- Everything is very cool! Happened! Thank you!
- Thank you for the interesting and entertaining work, as well as for the cool drawings!
- It was cool and interesting. At first I didn’t understand what it was, but they told me. In fact, everything was cool and the figures were so complicated. I liked everything.
- Cool, big, best.
- Veronica is a good teacher. He will always help and will not leave anyone unattended. I like it!
- This is the top job. The coolest math lesson ever.
Can be done conclusion, that my hypothesis was confirmed - the tasks I created were very interesting to my classmates.
Stage II: January 2018
I didn’t stop only at creating entertaining tasks and drawing pictures in the coordinate plane. I have always liked watching the starry sky. But then I had no idea that in addition to their beautiful location in the sky, you can learn about the zodiac constellations unique, interesting myths and legends, theories of origin and much more about the signs of the zodiac. In the process of working on the project, I decided to research the signs of the Zodiac and associate their location with the coordinate plane, thereby expanding my knowledge not only in mathematics, but also in astronomy. I think that tasks on building constellations will be very interesting to my classmates. Many people know about the zodiac constellations, but not everyone knows what they look like. This part of my work is aimed at constructing the signs of the Zodiac on the coordinate plane.
At this stage of your research:
- collected information about the dates of birth of classmates,
- compiled an astrological characteristic of class 6 “b”,
- found information about these zodiac signs and their constellations,
- made drawings on the coordinate plane for each constellation and wrote out the coordinates of the graphs,
- presented the results of her work to her classmates on 02/09/2018.
To compile the astrological characteristics of grade 6 “b”, I conducted a survey:
- "What is your zodiac sign?",
- “Do you know what your constellation looks like?” and compiled table No. 1 based on the responses.
Table No. 1
|
Last name and first name of the student |
Date of Birth |
Zodiac sign |
Do you know what your constellation looks like? |
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1.Arkhipova Anna |
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2. Baimurzin Arsentiy |
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3. Bugaev Nikita |
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4. Valieva Alina |
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5. Valyavina Veronica |
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6. Voznesensky Pavel |
Twins |
||
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7. Gapichenko Ekaterina |
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8. Zakharov Matvey |
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9. Kovalev Georgy |
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10. Kochetkova Arina |
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11. Kuznetsova Daria |
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12. Materukhin Egor |
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13. Frost Anna |
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14. Nikita Nasonov |
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15. Panova Elena |
Twins |
||
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16. Petrov Mark |
Twins |
||
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17. Razumova Vladislava |
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18. Storozhev Arkhip |
Twins |
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19. Sumbaeva Ksenia |
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20. Tolkueva Maria |
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21. Khoreshko Stepan |
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22. Chereshneva Anastasia |
From which it is clear that (100%) of students do not know what their constellation looks like.
LIBRA (24.09 - 23.10). There are 3 people in our class.
Libras do not look for easy ways and can endlessly argue over the simplest question; they are always very sociable.
Table No. 2
CAPRICORN (22.12 - 20.01). There are 2 people in the class.
People with this zodiac sign are big dreamers. Having set a goal, they clearly move towards it.
Table No. 3
AQUARIUS (21.01 - 20.02). There is 1 person in the class.
Aquarians are absolute realists. People with this zodiac sign are deeply interested in making the world a better place to live. They are kind, curious, calm and reasonable.
Table No. 4
PISCES (21.02 - 20.03). There are 3 people in the class.
Pisces know a lot and demand just as much. Pisces have a very vulnerable character, so they are easily offended.
Table No. 5
ARIES (21.03 - 20.04). There is 1 person in the class.
Aries are generous, kind, honest and optimistic. Aries has unconventional thinking.
Table No. 6
TAURUS (21.04 - 20.05). There are 3 people in the class.
Taurus people love life because they live it. They know how to work.
Table No. 7
GEMINI (21.05 - 21.06). There are 4 people in our class of children who know this. The developed mind of Gemini often leads to exaggeration of events. People with this zodiac sign are excessively stubborn, self-confident, talkative and self-willed.
Table No. 8
CANCER (22.06 - 22.07). There is 1 person in the class.
All Cancers, without exception, have gullibility, gentleness and vulnerability.
Table No. 9
LEO (23.07 - 23.08). There are 4 people in the class.
Leos are hardworking to the point of fanaticism, enterprising and persistent in achieving their goals. They set goals for themselves, trying to achieve their maximum potential in different areas.
Table No. 10
Conclusion: In total there are 9 zodiac signs in our class. Most of all the children were born under the constellations Gemini and Leo, 4 people each, under the constellations Pisces, Libra and Taurus, 3 people each, 2 people were born under the constellations Capricorn, Cancer, Aries and Aquarius, 1 person each. Based on the characteristics of the signs, in general we can say about our class that we are smart, hardworking, persistent, we are interested in everything, we are trusting, optimistic and reasonable, a little talkative and headstrong. We love life and try to understand and learn a lot.
Conclusion
In the course of this research work, I was able to summarize and systematize the studied material on the chosen topic. I got acquainted with the history of the origin of coordinates, learned about different types of coordinate systems and their purpose. While creating tasks for constructing drawings using the coordinates of points, I worked on the topic “Coordinate Plane” in full. These tasks develop students' attentiveness. While working on the project, I learned a lot about the constellations of the zodiac signs. I shared the information I collected with my classmates; they were interested in seeing their zodiac sign and plotting it on a coordinate plane. In the practical part, each card has an image of one of the zodiac signs and gives the coordinates of points (stars) and ways to connect these points. My hypothesis was confirmed - the tasks I created were very interesting to my classmates.
At the end of the work, I believe that my hypothesis has been proven, the set goals and objectives have been accomplished. My classmates and I are pleased with the new knowledge we have received.
Information sources
- Asmus V.F. Ancient philosophy. - M.: Higher School, 1998, p. eleven.
- Asmus V. F. Descartes. - M.: 1956. Reprint: Asmus V. F. Descartes. - M.: Higher School, 2006.
- Bronshten V. A. Claudius Ptolemy. M.: Nauka, 1985. 239 pp. 15,000 copies.
- Grigoriev - Dynamics. — M.: Great Russian Encyclopedia, 2007
- Zhitomirsky S.V. Ancient astronomy and orphism. - M.: Janus-K, 2001.
- Lanskoy G. Yu. Jean Buridan and Nikolai Oresme on the daily rotation of the Earth // Studies in the history of physics and mechanics. 1995 -1997. - M.: Nauka, 1999.
- Wikipedia. Leibniz. Gottfried Wilhelm
- http://v-kosmose.com/sozvezdiya/
- Photos of constellations - http://womanadvice.ru/sozvezdiya-znakov-zodiaka
- http://womanadvice.ru/sozvezdiya-znakov-zodiaka
ANNEX 1:
Tasks for constructing drawings using coordinates
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Drawing |
Coordinates for drawing |
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№1: "Goldfish" Body (7.5;1.5) (8;1) (8.5;1.5) (8;2) (8.5;3) (8;3.5) (7;3) (7 ;4) (6;5.5) (4.5;7) (3;8) (1;8.5) (-1;8.5) (-3;8) (-5;7) ( -6.5;5) (-8.5;3) (-9,5;2) (-11;0,5) (-10;0) (-8;-2) (-6;-3) (-4;-4) (-2;-4,5) (0;-5) (1,5;-4,5) (3;-3,5) (4,5;-2,5) (6;-1) (7,5;1,5) Starting from point (4,5;7) (3;6) (1,5;4) (1;2) (2;-1) (3;-2) (4;-3) Eye (4.5;3.5) Tail (-10.5;1) (-11;2) (-12.5;2.5) (-14;4) (-15;4) (-16;3) (-17;2) (-17;0) (-6,5;-2) (-16;-4) (-15;-6) (-14,5;-8) (-14;-10) (-13,5;-11) (-13,5;-12) (-14;-13) (-14,5;-15) (-16;-17) (-17;-19) (-15;-20) (-14;-20) (-12,5;-18) (-11,5;-19) (-11;-20) (-9;-20) (-7,5;-20) (-7;-19) (-6,5;-18) (-6;-17) (-5;-17,5) (-4;-18) (-3;-18) (-2;-17) (-2;-16) (-2;-14) (-2,5;-12,5) (-3;-11) (-4;-12) (-5;-12) (-7;-11) (-9;-10) (-11;-9) (-12;-7,5) (-13;-6) (-13;-2,5) (-12;-1,5) (-11;-1) (-10;0) Upper fin Starting from point (4,5;7) (4;9) (3;11) (1;13) (-1;14) (-2;14) (-2,5;13) (-3;12,5) (-4;12,5) (-5;13) (-6;13) (-6,5;12,5) (-7;11) (-7,5;9,5) (-8,5;8,5) (-9,5;7,5) (-9,5;6,5) (-9;5) (-9;4) (-9,5;2) Lower fins Starting from point (4;-3) (4;-4) (4;-6) (3.5;-8) (2.5;-9) (1;-8.5) (0;-7) (1;-6) (2;-5) (3;-3,5) Starting from point (-2;-4.5) (-3;-5) (-5.5;-5.5) (-7;-6) (-8;-5) (-8,5;-4) (-8;-3) (-7,5;-2,5) |
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№2: "Mushroom" (-14;-10) 2.(-12,5;-3) 3.(-11;-10) 4.(-8;-6) 5.(-7;-7) 6.(-2;-9) 7.(0;-8) 8.(5;-9) 9.(6;-7) 10.(8;-3) 11.(9;-10) 12.(11;-6) 13.(12;-10) Starting from point (6;-7) 14.(6;-2) 15.(4.5;1.5) 16.(7;1) 17.(9;2) 18.(10;9) 19 .(4;16) 20.(0;18) 21.(-1;18) 22.(-5;16) 23.(-10;9) 24.(-8;3) 25.(-5 ;2) 26.(-2;3) 27.(0;3) 28.(4.5;1.5) Starting from point (-7;-7) 29.(-6;-5) 30.(-5;-2) 1.(-2;18) 2.(-3;17) 3.(-3;15) 4.(-5;13) 5.(-5;11) 6.(-6;12) 7.(-8;10) 8.(-8;11) 9.(-11;8) 1.(6;7) 2.(5;7) 3.(4;6) 4.(4;5) 5.(5;5) 6.(6;6) 7.(6;7) 8.(6;8) 9.(6;7) Paws of a bug. 1.(5;7) 2.(5;7,5) 3.(4,5;7,5) Starting from point (4.5;6.5) 1.(4.5;7) 2.(4;7) Starting from point (4;6) 1.(4;6.5) 2.(3.5;6.5) Starting from point (5;5) 1.(5.5;5) 2.(5.5;4.5) Starting from point (5.5;5.5) 1.(6;5.5) 2.(6;5) Starting from point (6;6) 1.(6.5;6) 2.(6.5;5.5) |
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№3: Rejuvenating apples from the cartoon Wood (-3;-19) (2;-19) (1.5;-17) (1.5;-16) (2;-15) (2;-14) (2;-13) (2,5;-12) (2,5;-11) (3;-10) (3;-9) (3,5;-8) (3,5;-7) (4;-6) (4;-5) (4,5;-4) (4,5;-3) (6;-4) (7,5;-4,5) (9;-5) (11;-4,5) (12;-3) (13;-2) (14;-1) (14;1) (13;3) (12,5;5) (12;6) (11;8) (10,5;10) (9;11) (8,5;12,5) (7,5;13,5) (6,5;14,5) (5,5;15,5) (4;16) (-3,5;16) (-4;15) (-5,5;14) (-7;13) (-8,5;12) (-9,5;10) (10,5;8) (-11,5;6) (-12,5;4) (-13;2) (-13;0) (-12;-2) (-11;-3) (-10;-4) (-9,5;-5) (-8,5;-5) (-7;-4,5) (-6;-4) (-5,5;-5) (-5;-6) (-5;-7) (-4,5;-8) (-4,5;-9) (-4;-10) (-4;-11) (-3,5;-12) (-3;-13) (-3;-14) (-3;-15) (-2,5;-16,5) (-2,5;-17,5) (-3;-19) Starting from point (-5;-4) (-4.5;-3) (-4;-4) (-2;-5) (1;-4) (2;-3.5) (2,5;-3) (4,5;-3) Apple 1 (5.5;13) (5;12) (3;12) (2.5;11) (2.5;9.5) (4;9) (5,5;10,5) (6;10,5) (6;11,5) (5;12) Bullseye 2 (-6;12) (-5;11) (-6;11) (-6.5;10) (-6.5;9) (-5.5;8) (-4;8) (-2,5;8,5) (-2;10) (-2;11) (-3;11,5) (-4;11,5) (-5;11) Bullseye 3 (0;6) (1;5) (0;5) (-1;4) (-0.5;9) (-.5;2) (2;1.5) (3,5;1) (4,5;1,5) (5,5;2,5) (5,5;3,5) (5;5) (4;5,5) (3;5,5) (2;5) Bullseye 4 (-7;2) (-8;1) (-8.5;1.5) (-9.5;2) (-10.5;1.5) (-11.5;0, 5) (-11,5;-1) (-10,5;-2) (-9,5;-2,5) (-8,5;-2) (-7,5;-1) (-7,5;0) Bullseye 5 (8;0) (9;-1) (8;-1) (7;-2) (7.5;-3) (9;-3.5) (10.5;-3) (10,5;-1) (9;-1) |
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№4: The Little Mermaid 1(2;1) 2(1;1) 3(1;2) 4(-1;2) 5(-3;1) 6(-4;-1) 7(-6;-4) 8( -8;-5) 9(-11;-5) 10(-13;-4) 11(-15;-4)12(-17;-5) 13(-16;-5) 14(-11 ;-10) 15(-8;11) 16(-3;-11) 17(-4;-10) 18(-5;-7) 19(-4;-6) 20(1;-3) 21(2;-1) 22(2;1) 23(3;1.5) 24(3;1) 25(3;-2) 26(4;-1) 27(4;10 28(4; 2) 29(4;3) 30(3;3) 31(3;4) 32(2;4) 33(1;4) 34(-1;4) 35(-2;4) 36(-1 ;3) 37(1;3) 38(1.5;3) 39(1;2) 40(3;4) 41(4;5) 42(4;6) 43(5;7) 44(6 ;7) 45(7;6) 46(7;5) 47(6;4) 48(5;4) 49(4;3) 50(5;7) 51(4;7) 52(1;4) ) 53(7;6) 54(7;5) 55(7;4) 56(4;1) eyes and mouth 1(5;6) 2(6;5) 3(5;5) |
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№5: Fantasy flower (-4;-3) (-3,5;-4) (-2,5;-4,5) (-1;-4,5) (0,5;-4) (2;-3) (2;-2) (2;0) (3,5;0,5) (5;1) (6;2) (6,5;3) (6,5;4,5) (6;5,5) (5;6,5) (6;8) (6,5;9,5) (6,5;11,5) (5,5;12,5) (4;13,5) (3;14) (2,5;15,5) (1;16,5) (-1;17) (-3;17) (-4,5;16) (-5;16,5) (-7;17) (-9;17) (-10,5;16,5) (-11,5;15,5) (-12;14) (-14;13,5) (-15,5;12,5) (-16;11) (-16;8,5) (-15;7) (-14;6,5) (-14,5;5,5) (-15;4) (-15;2) (-13;0,5) (-11;0,5) (-11,5;-1) (-11,5;2,5) (-10,5;-3,5) (-8;-4) (-6;-4) (-4,5;-3) Draw straight lines from point (-4;-3) to (-4.5;16) From point (2;0) to (-12;14) From point (5;6.5) to (-14;6.5) From point (3;13.5) to (-11;0.5) Stem (-1;-15) (-0.5;-15) (-3;-4.5) (-2.5;-4.5) Leaf (0;-15) (0.5;-13) (1.5;-11) (3;-9) (4.5;-7.5) (6;-6) (7.5; -4) (9;-2) (10;1) (11;4) (12;1) (12;-2) (12;-4) (10;-6) (8;-8) (6;-10) (4;-12) (2;-14) (2;15) Pot (-8;-15) (-6;-22) (6;-22) (8;-15) (-8;-15) |
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№6: Pencils 1 pencil (9;13.5) (7;13) (5;12) (1;6) (2.5;3.5) (5;4) (9;10) Starting from point (5,12) (6;12) (6;11) (7;11) (7.5;10.5) (8.5;10.5) Starting from point (1;6) (3.5;5.5) (5;4) Point (3;4,5) Pencil 2 (-11;13) (-10.10) (-9;8) (3;-4) (5;-3) (6;-1) (-5.5;10.5) (- 8;12) (-11;13) Draw a straight line from point (-10;10) to (-8;12) Starting from point (-9;8) (-9;9) (-8;9) (-8;10) (-7;10) (-7;11) Starting from point (3;-4) (4;-2) (6;-1) Point (4.5;-2.5) Pencil 3 (-9.5;-1.5) (-9;-3) (-8;-5) (-3;-10) (-1.5;-9.5) (-1;-8) (-6;-3) (-8;-2) (-9,5;-1,5) Draw a straight line from point (-9;-3) to (-8;-2) Starting from point (-8;-5) (-8;-4) (-7;-4) (-7;-3) (-6;-3) Starting from point (-3;-10) (-2.5;-8.5) (-1;-8) Point (-2;-9) Pencil 4 (14;4.5) (12;3.5) (10;2) (3;-10) (4.5;-12.5) (7;-12) (14;0) (14;2,5) (14;4,5) Draw a straight line from point (12;3,5) to (14;2,5) Starting from point (10;2) (11;2) (12;1) (12;0) (13;0.5) (14;0.5) Point (5;-11.5) |
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№7: Scientist Owl Body (0;-7) (2;-7) (3;-6.5) (5;-6) (6;-4) (6.5;-2) (7;0) (7;5 ) (6.5;7) (6;9) (5,5;10,5) (5;12) (4;13,5) (3;15) (2;16) (-2;16) (-4;15) (-5;13,5) (-6;12) (-6,5;10,5) (-7;9) (-7,5;7) (-8;5) (-8;0) (-7,5;-2) (-7;-4) (-6;-6) (-4;-6,5) (-3;-7) (0;-7) Starting from point (2;16) (2.5;17) (5;17.5) (1;20) (-4.5;17.5) (-2,5;17) (-2;16) (2;16) Starting from point (-2.5;17) (0.5;16.5) (2.5;17) Starting from point (-4;15) (-5;16) (-6.5;16.5) (-6.5;15) (-6;13) (-6;12) (3;15) (4;16) (6;16,5) (5,5;15) (5;13) (5;12) Starting from point (0;11) (-1;11.5) (-2;12) (-3;12) (-3.5;11.5) (-4;11) (-4;10) (-3,5;9) (-3;8,5) (-2;8,5) (-1;8,5) (0;9) (1;8,5) (2;8,5) (3;8,5) (3,5;9) (4;10) (4;11) (3;12) (2;12) (1;11,5) From point (-1.5;9.5) circle D=0.5 cm From point (1.5;9.5) circle D=0.5 cm Beak (-1;8) (0;8.5) (1;8) (0;7) (-1;8) Starting from point (-1;8) (-2.7) (-3;6) (-4;4) (-5;2) (-8;0) (-7.5;-2) (-7;-4) (-6;6) (-4;-6,5) (-3;-7) (2;-7) (3;-6,5) (5;-6) (5;2) (4;4) (3;6) (2;7) (1;8) Starting from point (-3;4) (-2.5;3) (-2;2.5) (-1.5;3) (-1;4) (-0.5;3) (0;2,5) (0,5;3) (1;4) (1,5;3) (2;2,5) (2,5;3) (3;4) Starting from point (-4;-2) (-3.5;-3) (-3;-3) (-2.5;-2) (-2;-3) (-1;-3) (-1;-2) (0;-3) (0,5;-30) (1;-2) (1,5;-3) (2;-3) (2,5;-2) (3;-3) (3,5;-3) Paws (-3;-7) (-3;-7.5) (-2.5;-8) (-2.5;-7.5) (-2.5;-7) (-2. 5;-8) (-2;-8,5) (-2;-8) (-2;-7) (-2;-8) (-1,5;-8) (-1,5;-7) (1;-8) (1,5;-8,5) (1,5;-7) (1,5;-8,5) (2;-8,5) (2;-7) (20;-8,5) (2,5;-8) (2,5;-7) |
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№8:Autumn leaf (9;-18) (8;-15) (8;-13,5) (6,5;-12) (6;-11) (8;-12) (9;-13) (11;-13) (9;-11) (8;-9) (7;-8) (8;-8) (10;-9) (12;-9) (10;-7) (9;-5) (8;-3) (7;-1) (7;0) (8;-1) (9;-2) (11;-3) (12,5;-3,5) (14,-3) (13;-2) (12;0,5) (14,5;0) (13;2) (12;3,5) (10;4) (9;5) (15;5) (13,5;6,5) (11;7) (9;8) (8;9) (11;9) (10;10) (9,5;11) (8;12) (7;14) (5;15) (3;15,5) (1;16) (-1,5;15) (-3;14) (-4;13) (-4,5;12) (-4,5;11) (-4,5;9) (;7) (-3;5) (-1,5;3) (-1;1) (0;0) (1;-1) (2;-4) (3;-7) (4;-10) (5;-12) (7;-15) (9;-18) (7;-16,5) (5;-16) (3;-15,5) (1;-15) (-1;-14) (-3;-12) (-5;-10) (-7;-8) (-9;-6) (-9;-7) (-10,5;-6) (-11,5;-4) (-12;-2) (-12,5;-1) (-13;-2) (-14;1) (-14;4,5) (-13,6) (-12;7) (-11;8) (-9;9,5) (-11,5;9) (-11;10) (-9,5;11,5) (-8;12,5) (-7;12,5) (-5;12) (-5,5;13) (-6;14) (-5;15) (-4,5;14) (-4,5;13) (-4,5;12) |
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№9: Torch 1(-2;-11) 2(0;-11) 3(3;2) 4(3;4) 5(2;9) 6(1;7) 7(0;11) 8(-3;7) 9(-4;8) 10(-5;4) 11(-5;2) 12(-2;-11) 13(-5;-2) 14(3;2) 15(3;4) 16(-5;4) |
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№10: Crystal 1(0;-10) 2(10;2) 3(0;-10) 4(3;2) 5(0;-10) 6(-3;2) 7(0;-10) 8(-10;2) 9(10;2) 10(6;5) 11(3;2) 12(0;5) 13(-3;2) 14(-6;5) 15(-10;2) 16(-6;5) 17(6;5) |
From experience working with 6th grade students.
Drawing by coordinates
(drawings made in the “Living Geometry” program
1 ."RHINOCEROS"
Torso
(9;0); (13;2); (16;2) ; (19;4) ; (19;6) ;(17;8); (17;6); (16;6); (15;8); (15;6);(13;8) ; (11;8); (9;10) ; (9;8); (3;6) ;(-5;6) ; (-7;4);(-7;-6);(-2; -6) (-2;-2);(5;-2);(5;-6); (10;-6); (9;0)
2."TOBIK"
(0;-8); (3;-8); (1;-1); (4; -3); (4;-4);(8; -3); (8;2);(7;2), (7;1); (5;3); (6;4); (5;3);(6;4); (4;5);(3;8); (2;6); (1;8);(-1;-1); (-6;-1); (-9;2); (-8; -1);(-8;-8);(-5; -4); (-1;-5); (0;-8)
3. "BAGIRA"
Line 1.(0;-8); (1;-6); (1; -2); (2; -10); (4; -10);(3; -10); (3,5; -4); (4; -9);
(5; -10); (6;-9); (5; -8); (5;-5); (6;0);(6;4);(1;10); (-2;10); (-5; 8); (-4; 8); (-6;7); (-4;7); (-4;6); (-3; 5); (-2;3); (-1;5); (0;4); (-2;2); (-4; -1); (-6; -2);
(-7;-7); (-12;-7); (-13; -10); (-8; -11); (-4; -11); (-5; -10); (-8; -10);(-11;-9)
(-11; -8);(-7; -8); (-4; -10); (0;-10); (1;-9);(0;-8)
Eye:(-3;6); (-2; 7) Mustache: 1)(-2;4); (-4;3). 2)(-2;4);(-4;2). 3)(-2;4);(-3;2)
Made in scale 1:2
4. "Bell".
Line 1 . (3; -5,5); (3;-3); (1,5;-1,5); (3; -5,5); (4,5; -1,5); (3;-3); (3;3,5); (1,5;2,5); (0,5;0); (1; 0,5); (1,5; 0); (2; 0,5); (2,5;0); (1,5; 2,5)
Line 2. (3;1,5); (4.5;3); (3.5; 0.5) ; (4;1); (4.5;0.5); (5;1); (5.5; 0.5); (4.5;3)
5. "Butterfly"
Line 1 . (0,5; 3); (1,5;1,5); (1,5;-1); (2; -1); (2; 1,5); (3;3);
Line 2. (1.5;1); (-1;3); (-1.5; 1); (1.5;0.5);
Line 3. (1.5;-0.5); (-1.5; -1.5); (-1.5; 1) ;
Line 4. (2;1); (4.5; 3); (5; 1) ; (5;-1.5) ; (2;-0.5); (2; 1.5);
6. "Bird"
Line 1 . (-1,5; -1,5); (-2;- 1); (-2,5;-1);
Line 2. (-2; - 1.5); (-2;-1); (eleven); (thirty); (2;3); (2.5;5); (2;6);(1;6); (2;6,5); (1;7); (2;7);(3;8); (3.5;7); (3;5,5); (4;3.5);(4.5;1) (3.5;1.5); (3;0); (3;-5); (2.5;-4.5)
Line 3. (3;-5); (2.5; -5);
Line 4. (3;-5); (2.5; -5.5); Eye: (2.5;7)
7. "Sailboat"
Line 1 . (1; 1); (10,5; 1); (7;-3); (-5;-3); (-8,5;1); (1;1); (1;8); (-3;3);(1;3)
Line 2. (1; 7); (5; 2); (12);
Line 3. (-4;-2);(-3.5;-1.5); (-3 ;-2); (-2;-0.5);
Line 4. (-1.5;-0.5); (-0.5; -0.5); (-0.5;-1); (-1.5;-2);
Line 5. (0.5;-0.5); (1.5; -0.5); (1.5;-1); (0.5;-2)
Line 6. (2 ;-0.5); (3; -0.5); (3;-1); (2;-2)
8. CRUISER "AURORA"
( 0;0), (1; -1), (1;-2), (2; -2) , (2;3), (4; 3), (4; -2) , (5; -2) ,(5;0), (6; -1), (6;-2), (7; -2), (7;2), (9;2), (9; -2), (11; -2),(11; 5), (12;5), (12;- 3), (14; -4), (14; - 6), (-15; -6), (-13; -1),
(-13;-2), (-7; -2), (-8; 0), (-7; 2), (-6; 2), (-6; 7), (-5; 7),(-5; -2), (-3; -2), (-3; 4), (-1;4), (-1; -2), (0; -2),(0;0)
9. "Dwarf".
Line 1. (-3; -1) ; (-20); (-1; 2.5); (-2;3); (-2; 4); (-15) ; (15); (2; 4);
(2; 3); (1; 2,5); (2; 0); (3; -1); (1; -1); (1; 0); (0; 2); (-1; 0); (-1; -1);
Line 2.(0; 5); (-16); (-1; 7.5); (-2; 7); (-1; 8.5); (0; 8.5); (1; 7.5);
Line 3.(-1; 7); (1; 7).
Line 4.(-1; 2.5); (-1; 4.5).
Line 5.(1; 2.5); (1; 4.5).
Eyes: (-0.5;5.5); (0.5;5.5); Nose: (0;6)
10. “Foal.”
Line 1. (-8; 7); (-7; 6); (-4; 4); (- 1; 2); (7; 2); (8; 1); (7; -3); (6; 1); (5; -2); (7; -4); (6; -8); (5; -8); (6; -4); (5; -3); (5; -4); (4; -8); (3; -8); (4; -4); (3; -1); (1; -2); (-1; -2); (0; -5); (-1; -8); (-2; -8); (-1; -5); (-2; -3); (-2; -4); (-3; -8); (-4; -8); (-3; -3); (-5; -1); (-4; 0); (-6; 3); (-9; 2); (-10; 3); (-7; 6).
2.Eye (-7; 4). 
11. "Cheburashka"
| Torso | Legs | Hands |
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| (1;0);(3;1) (4;3); (4;5) | ||||||
| (3;7); (1;8) ,(-1;8); (-3;7) | ||||||
| (-4;5); (-4;3), (-3;1);(-1;0) | ||||||
| (-2;-1);(-3;-2), (-3; -5); | ||||||
| (-1; -8);(1;-8) (2;-7);(3;-5) | ||||||
| Mouth: (0;1); (1;2); (-1;2) | Eyes:( 2;5) | Brows | ||||
| Nose:(1;3); (0;4); (-1;3) | ||||||
12. "Wolf"
| Torso | |||||
| (-2;5);(3;-2), (3;-4);(4;-4) | |||||
| (5;-3);(5;-1),(3;0) | |||||
| (4;1);(5;1), (7;-1);(7;-4) | |||||
| (5;-5);(3;-5), (2;-4);(2;-5) | |||||
13 ."Maple Leaf"
Line 1. (4.5; -0.5) ; (4; -0.5); (4.5; 1); (3;0.5); (4; 3); (3; 3) ; (2.5; 4); (2.5; 5); (1.5;4.5); (1;5); (0;3); (-2;5); (-3.5;4); (-3.5;3);(-4; 3); (-6; 2.8); (-5; 1); (-6; 0);
(-7; -1); (-5,5; -1); (-5; -2); (-3; -2); (-4; -3); (-2; -3); (0;-2,3); (3;-3); (2,5;-2);
Line 2.(0.5, -2); (2.5; 0.5);
Line 3 (0;-1); (-1.5;2) 
Line 4.(-1.5; 0.5); (-3;1.5)
Line 5. (1;-6); (-0.5; - 2.5)
14.Lev.
Line 1 (3; 1); (3; -1,5); (2; -1,5); (2; -2,5); (4; -2,5); (4; 1); (5; 1); (5; 4);
(6; 1,5); (5,5; 1); (7; 0,5); (6,5; 2); (6; 1,5).
Line 2. (5; 4); (-2,5; 4); (-2; 3,5); (-2,5; 3); (-2; 2,5); (-2,5; 2); (-2; 1,5); (-2,5; 1); (-2; 0,5); (-2,5; 0); (-3; 0,5); (-3,5; 0) (-4; 0,5); (-4,5; 0); (-5; 0,5); (-5,5; 0); (-6; 0,5); (-6,5; 0); (-7; 0,5); (-6,5; 1); (-7; 1,5); (-6,5; 2); (-7; 2,5); (-6,5; 3); (-7; 3,5); (-6,5; 4); (-7; 4,5); (-6,5; 5); (-6; 4,5); (-5,5; 5); (-5; 4,5); (-4,5; 5); (-4; 4,5); (-3,5; 5); (-3; 4,5); (-2,5; 5); (-2; 4,5); (-2,5; 4).
Line 3 (-2,5; 0); (-2,5; -1,5); (-3,5; -1,5); (-3,5; -2,5); (-1,5; -2,5); (-1,5; 1).
Line 4 (-5; 3,5); (-5,5; 4,5); (-5,5; 1,5); (-3,5; 1,5); (-3,5; 4,5); (-4; 3,5).
Line 5 (-5,5; 2,5); (-4,5; 2); (-4;2,5)
Line 6 (-4,5; 3); (-4,5; 2,5).
Line 7 (-2,5; 1); (4; 1).
Eyes (-5; 3); (-4; 3).
15. “SABER TOOTH TIGER”
