Municipal educational budgetary institution -

Secondary school No. 51

Orenburg.

Project on:

mathematic teacher

Egorcheva Victoria Andreevna

2017

Hypothesis : If graph theory is brought closer to practice, then the most beneficial results can be obtained.

Target: Get acquainted with the concept of graphs and learn how to apply them in solving various problems.

Tasks:

1) Expand knowledge about methods of constructing graphs.

2) Identify types of problems whose solution requires the use of graph theory.

3) Explore the use of graphs in mathematics.

“Euler calculated, without any visible effort, how a person breathes or how an eagle soars above the earth.”

Dominic Arago.

I. Introduction. p.

II . Main part.

1. The concept of a graph. Problem about the Königsberg bridges. p.

2. Properties of graphs. p.

3. Problems using graph theory. p.

Sh. Conclusion.

The meaning of graphs. p.

IV. Bibliography. p.

I . INTRODUCTION

Graph theory is a relatively young science. “Graphs” has the root of the Greek word “grapho,” which means “I write.” The same root is in the words “graph”, “biography”.

In my work, I look at how graph theory is used in various areas of people's lives. Every math teacher and almost every student knows how difficult it is to solve geometric problems, as well as algebra word problems. Having explored the possibility of using graph theory in a school mathematics course, I came to the conclusion that this theory greatly simplifies understanding and solving problems.

II . MAIN PART.

1. The concept of a graph.

The first work on graph theory belongs to Leonhard Euler. It appeared in 1736 in publications of the St. Petersburg Academy of Sciences and began with a consideration of the problem of the Königsberg bridges.

You probably know that there is such a city as Kaliningrad; it used to be called Koenigsberg. The Pregolya River flows through the city. It is divided into two branches and goes around the island. In the 17th century there were seven bridges in the city, arranged as shown in the picture.

They say that one day a city resident asked his friend if he could walk across all the bridges so as to visit each of them only once and return to the place where the walk began. Many townspeople became interested in this problem, but no one could come up with a solution. This issue has attracted the attention of scientists from many countries. The famous mathematician Leonhard Euler managed to solve the problem. Leonhard Euler, a native of Basel, was born on April 15, 1707. Euler's scientific achievements are enormous. He influenced the development of almost all branches of mathematics and mechanics, both in the field of fundamental research and in their applications. Leonhard Euler not only solved this specific problem, but also came up with a general method for solving these problems. Euler did the following: he “compressed” the land into points, and “stretched” the bridges into lines. The result is the figure shown in the figure.

Such a figure, consisting of points and lines connecting these points, is calledcount. Points A, B, C, D are called the vertices of the graph, and the lines that connect the vertices are called the edges of the graph. In a drawing of vertices B, C, D 3 ribs come out, and from the top A - 5 ribs. Vertices from which an odd number of edges emerge are calledodd vertices, and vertices from which an even number of edges emerge areeven.

2. Properties of the graph.

While solving the problem about the Königsberg bridges, Euler established, in particular, the properties of the graph:

1. If all the vertices of the graph are even, then you can draw a graph with one stroke (that is, without lifting the pencil from the paper and without drawing twice along the same line). In this case, the movement can start from any vertex and end at the same vertex.

2. A graph with two odd vertices can also be drawn with one stroke. The movement must begin from any odd vertex and end at another odd vertex.

3. A graph with more than two odd vertices cannot be drawn with one stroke.

4.The number of odd vertices in a graph is always even.

5. If a graph has odd vertices, then the smallest number of strokes that can be used to draw the graph will be equal to half the number of odd vertices of this graph.

For example, if a figure has four odd numbers, then it can be drawn with at least two strokes.

In the problem of the seven bridges of Königsberg, all four vertices of the corresponding graph are odd, i.e. You cannot cross all the bridges once and end the journey where it started.

3. Solving problems using graphs.

1. Tasks on drawing figures with one stroke.

Attempting to draw each of the following shapes with one stroke of the pen will result in different results.

If there are no odd points in the figure, then it can always be drawn with one stroke of the pen, no matter where you start drawing. These are figures 1 and 5.

If a figure has only one pair of odd points, then such a figure can be drawn with one stroke, starting drawing at one of the odd points (it doesn’t matter which). It is easy to understand that the drawing should end at the second odd point. These are figures 2, 3, 6. In figure 6, for example, drawing must begin either from point A or from point B.

If a figure has more than one pair of odd points, then it cannot be drawn with one stroke at all. These are figures 4 and 7, containing two pairs of odd points. What has been said is enough to accurately recognize which figures cannot be drawn with one stroke and which ones can be drawn, as well as from what point the drawing should begin.

I propose to draw the following figures in one stroke.

2. Solving logical problems.

TASK No. 1.

There are 6 participants in the table tennis class championship: Andrey, Boris, Victor, Galina, Dmitry and Elena. The championship is held in a round robin system - each participant plays each of the others once. To date, some games have already been played: Andrey played with Boris, Galina, Elena; Boris - with Andrey, Galina; Victor - with Galina, Dmitry, Elena; Galina - with Andrey, Victor and Boris. How many games have been played so far and how many are left?

SOLUTION:

Let's build a graph as shown in the figure.

7 games played.

In this figure, the graph has 8 edges, so there are 8 games left to play.

TASK #2

In the courtyard, which is surrounded by a high fence, there are three houses: red, yellow and blue. The fence has three gates: red, yellow and blue. From the red house, draw a path to the red gate, from the yellow house to the yellow gate, from the blue house to the blue one so that these paths do not intersect.

SOLUTION:

The solution to the problem is shown in the figure.

3. Solving word problems.

To solve problems using the graph method, you need to know the following algorithm:

1.What process are we talking about in the problem?2.What quantities characterize this process?3.What is the relationship between these quantities?4.How many different processes are described in the problem?5.Is there a connection between the elements?

Answering these questions, we analyze the condition of the problem and write it down schematically.

For example . The bus traveled for 2 hours at a speed of 45 km/h and for 3 hours at a speed of 60 km/h. How far did the bus travel during these 5 hours?

S
¹=90 km V ¹=45 km/h t ¹=2h

S=VT

S ²=180 km V ²=60 km/h t ²=3 h

S ¹ + S ² = 90 + 180

Solution:

1)45 x 2 = 90 (km) - the bus traveled in 2 hours.

2)60 x 3 = 180 (km) - the bus traveled in 3 hours.

3)90 + 180 = 270 (km) - the bus traveled in 5 hours.

Answer: 270 km.

III . CONCLUSION.

As a result of working on the project, I learned that Leonhard Euler was the founder of graph theory and solved problems using graph theory. I concluded for myself that graph theory is used in various areas of modern mathematics and its numerous applications. There is no doubt about the usefulness of introducing us students to the basic concepts of graph theory. Solving many mathematical problems becomes easier if you can use graphs. Data presentation V the form of a graph gives them clarity. Many proofs are also simplified and become more convincing if you use graphs. This especially applies to such areas of mathematics as mathematical logic and combinatorics.

Thus, the study of this topic has great general educational, general cultural and general mathematical significance. In everyday life, graphic illustrations, geometric representations and other visual techniques and methods are increasingly used. For this purpose, it is useful to introduce the study of elements of graph theory in primary and secondary schools, at least in extracurricular activities, since this topic is not included in the mathematics curriculum.

V . BIBLIOGRAPHY:

2008

Review.

A project on the theme “Graphs around us” was completed by Nikita Zaytsev, a student of class 7 “A” at Municipal Educational Institution No. 3, Krasny Kut.

A distinctive feature of Nikita Zaitsev’s work is its relevance, practical orientation, depth of coverage of the topic, and the possibility of using it in the future.

The work is creative, in the form of an information project. The student chose this topic to show the relationship of graph theory with practice using the example of a school bus route, to show that graph theory is used in various areas of modern mathematics and its numerous applications, especially in economics, mathematical logic, and combinatorics. He showed that solving problems is greatly simplified if it is possible to use graphs; presenting data in the form of a graph gives them clarity; many proofs are also simplified and become convincing.

The work addresses issues such as:

1. The concept of a graph. Problem about the Königsberg bridges.

2. Properties of graphs.

3. Problems using graph theory.

4. The meaning of graphs.

5. School bus route option.

When performing his work, N. Zaitsev used:

1. Alkhova Z.N., Makeeva A.V. "Extracurricular work in mathematics."

2. Magazine “Mathematics at school”. Appendix “First of September” No. 13

2008

3. Ya.I.Perelman “Entertaining tasks and experiments.” - Moscow: Education, 2000.

The work was done competently, the material meets the requirements of this topic, the corresponding drawings are attached.

Third city scientific

student conference

Computer Science and Mathematics

Research

Euler circles and graph theory in problem solving

school mathematics and computer science

Valiev Airat

Municipal educational institution

"Secondary school No. 10 with in-depth study

individual subjects", 10 B class, Nizhnekamsk

Scientific supervisors:

Khalilova Nafise Zinnyatullovna, mathematics teacher

IT-teacher

Naberezhnye Chelny

Introduction. 3

Chapter 1. Euler circles. 4

1.1. Theoretical foundations about Euler circles. 4

1.2. Solving problems using Euler circles. 9

Chapter 2. About columns 13

2.1.Graph theory. 13

2.2. Solving problems using graphs. 19

Conclusion. 22

Bibliography. 22

Introduction

“All our dignity lies in thought.

It's not space, it's not time that we can't fill,

elevates us, namely it, our thought.

Let us learn to think well.”

B. Pascal,

Relevance. The main task of the school is not to provide children with a large amount of knowledge, but to teach students to acquire knowledge themselves, the ability to process this knowledge and apply it in everyday life. The given tasks can be solved by a student who not only has the ability to work well and hard, but also a student with developed logical thinking. In this regard, many school subjects contain various types of tasks, which develop logical thinking in children. When solving these problems, we use various solution techniques. One of the solution methods is the use of Euler circles and graphs.

Purpose of the study: study of material used in mathematics and computer science lessons, where Euler circles and graph theory are used as one of the methods for solving problems.

Research objectives:

1. Study the theoretical foundations of the concepts: “Eulerian circles”, “Graphs”.

2. Solve the problems of the school course using the above methods.

3. Compile a selection of material for use by students and teachers in mathematics and computer science lessons.

Research hypothesis: the use of Euler circles and graphs increases clarity when solving problems.

Subject of study: concepts: “Euler circles”, “Graphs”, problems of a school course in mathematics and computer science.

Chapter 1. Euler circles.

1.1. Theoretical foundations about Euler circles.

Euler circles (Euler circles) are a method of modeling accepted in logic, a visual representation of the relationships between volumes of concepts using circles, proposed by the famous mathematician L. Euler (1707–1783).

The designation of relationships between the volumes of concepts by means of circles was used by a representative of the Athenian Neoplatonic school - Philoponus (VI century), who wrote commentaries on Aristotle's First Analytics.

It is conventionally accepted that a circle visually depicts the volume of one concept. The scope of a concept reflects the totality of objects of one or another class of objects. Therefore, each object of a class of objects can be represented by a point placed inside a circle, as shown in the figure:

The group of objects that makes up the appearance of a given class of objects is depicted as a smaller circle drawn inside a larger circle, as is done in the figure.

https://pandia.ru/text/78/128/images/image003_74.gif" alt="overlapping classes" width="200" height="100 id=">!}

This is precisely the relationship that exists between the scope of the concepts “student” and “Komsomol member”. Some (but not all) students are Komsomol members; some (but not all) Komsomol members are students. The unshaded part of circle A reflects that part of the scope of the concept “student” that does not coincide with the scope of the concept “Komsomol member”; The unshaded part of circle B reflects that part of the scope of the concept “Komsomol member” that does not coincide with the scope of the concept “student”. The shaded part, which is common to both circles, denotes students who are Komsomol members and Komsomol members who are students.

When not a single object displayed in the volume of concept A can simultaneously be displayed in the volume of concept B, then in this case the relationship between the volumes of concepts is depicted by means of two circles drawn one outside the other. Not a single point lying on the surface of one circle can be on the surface of another circle.

https://pandia.ru/text/78/128/images/image005_53.gif" alt=" concepts with the same volumes - coinciding circles" width="200" height="100 id=">!}

Such a relationship exists, for example, between the concepts “the founder of English materialism” and “the author of the New Organon.” The scope of these concepts is the same; they reflect the same historical figure - the English philosopher F. Bacon.

It often happens like this: one concept (generic) is subordinated to several specific concepts at once, which in this case are called subordinate. The relationship between such concepts is depicted visually by one large circle and several smaller circles, which are drawn on the surface of the larger circle:

https://pandia.ru/text/78/128/images/image007_46.gif" alt="opposite concepts" width="200" height="100 id=">!}

At the same time, it is clear that between opposite concepts a third, average, is possible, since they do not completely exhaust the scope of the generic concept. This is exactly the relationship that exists between the concepts “light” and “heavy”. They are mutually exclusive. It is impossible to say about the same object, taken at the same time and in the same relation, that it is both light and heavy. But between these concepts there is a middle ground, a third: objects are not only light and heavy weight, but also medium weight.

When there is a contradictory relationship between concepts, then the relationship between the volumes of concepts is depicted differently: the circle is divided into two parts as follows: A is a generic concept, B and non-B (denoted as B) are contradictory concepts. Conflicting concepts exclude each other and belong to the same genus, which can be expressed by the following diagram:

https://pandia.ru/text/78/128/images/image009_38.gif" alt="subject and predicate of definition" width="200" height="100 id=">!}

The diagram of the relationship between the volumes of the subject and the predicate in a general affirmative judgment, which is not a definition of a concept, looks different. In such a judgment, the scope of the predicate is greater than the scope of the subject; the scope of the subject is entirely included in the scope of the predicate. Therefore, the relationship between them is depicted by means of large and small circles, as shown in the figure:

School libraries" href="/text/category/shkolmznie_biblioteki/" rel="bookmark">school library, 20 - in the district. How many of the fifth graders:

a) are not readers of the school library;

b) are not readers of the district library;

c) are readers only of the school library;

d) are readers only of the district library;

e) are readers of both libraries?

3. Each student in the class learns either English or French, or both. 25 people study English, 27 people study French, and 18 people study both. How many students are there in the class?

4. On a sheet of paper, draw a circle with an area of ​​78 cm2 and a square with an area of ​​55 cm2. The area of ​​intersection of a circle and a square is 30 cm2. The part of the sheet not occupied by the circle and square has an area of ​​150 cm2. Find the area of ​​the sheet.

5. There are 52 children in the kindergarten. Each of them loves either cake or ice cream, or both. Half of the children like cake, and 20 people like cake and ice cream. How many children love ice cream?

6. There are 86 high school students in the student production team. 8 of them do not know how to operate either a tractor or a combine. 54 students mastered the tractor well, 62 - the combine. How many people from this team can work on both a tractor and a combine?

7. There are 36 students in the class. Many of them attend clubs: physics (14 people), mathematics (18 people), chemistry (10 people). In addition, it is known that 2 people attend all three circles; Of those who attend two circles, 8 people are involved in mathematical and physical circles, 5 are in mathematical and chemical circles, 3 are in physical and chemical circles. How many people do not attend any clubs?

8. 100 sixth-graders at our school took part in a survey to find out which computer games they liked best: simulators, quests or strategies. As a result, 20 respondents named simulators, 28 - quests, 12 - strategies. It turned out that 13 schoolchildren give equal preference to simulators and quests, 6 students - to simulators and strategies, 4 students - to quests and strategies, and 9 students are completely indifferent to these computer games. Some of the schoolchildren answered that they were equally interested in simulators, quests, and strategies. How many of these guys are there?

Answers

https://pandia.ru/text/78/128/images/image012_31.gif" alt="Oval: A" width="105" height="105">1.!}

A – chess 25-5=20 – people. know how to play

B – checkers 20+18-20=18 – people play both checkers and chess

2. Ш – many visitors to the school library

P – many visitors to the district library

https://pandia.ru/text/78/128/images/image015_29.gif" width="36" height="90">.jpg" width="122 height=110" height="110">

5. 46. P – cake, M – ice cream

6 – children love cake

6. 38. T – tractor, K – combine

54+62-(86-8)=38 – able to work on both a tractor and a combine

graphs" and systematically study their properties.

Basic concepts.

The first of the basic concepts of graph theory is the concept of a vertex. In graph theory it is taken as primary and is not defined. It is not difficult to imagine it on your own intuitive level. Usually the vertices of the graph are visually depicted in the form of circles, rectangles and other figures (Fig. 1). At least one vertex must be present in each graph.

Another basic concept in graph theory is arcs. Typically, arcs are straight or curved segments connecting vertices. Each of the two ends of the arc must coincide with some vertex. The case when both ends of the arc coincide with the same vertex is not excluded. For example, in Fig. 2 there are acceptable images of arcs, and in Fig. 3 they are unacceptable:

In graph theory, two types of arcs are used - undirected or directed (oriented). A graph containing only directed arcs is called a directed graph or digraph.

Arcs can be unidirectional, with each arc having only one direction, or bidirectional.

In most applications, it is possible without loss of meaning to replace an omnidirectional arc with a bidirectional arc, and a bidirectional arc with two unidirectional arcs. For example, as shown in Fig. 4.

As a rule, the graph is either immediately constructed in such a way that all arcs have the same directional characteristic (for example, all are unidirectional), or is brought to this form through transformations. If the arc AB is directed, then this means that of its two ends, one (A) is considered the beginning, and the second (B) is the end. In this case, they say that the beginning of the arc AB is vertex A, and the end is vertex B, if the arc is directed from A to B, or that the arc AB comes from vertex A and enters B (Fig. 5).

Two vertices of a graph connected by some arc (sometimes, regardless of the orientation of the arc) are called adjacent vertices.

An important concept in the study of graphs is the concept of path. A path A1,A2,...An is defined as a finite sequence (tuple) of vertices A1,A2,...An and arcs A1, 2,A2,3,...,An-1, n sequentially connecting these vertices.

An important concept in graph theory is the concept of connectivity. If for any two vertices of a graph there is at least one path connecting them, the graph is called connected.

For example, if you depict the human circulatory system as a graph, where the vertices correspond to internal organs and the arcs correspond to blood capillaries, then such a graph is obviously connected. Is it possible to say that the circulatory system of two arbitrary people is a disconnected graph? Obviously not, since the so-called are observed in nature. "Siamese twins".

Connectedness can be not only a qualitative characteristic of a graph (connected/disconnected), but also a quantitative one.

A graph is called K-connected if each of its vertices is connected to K other vertices. Sometimes they talk about weakly and strongly connected graphs. These concepts are subjective. A researcher calls a graph strongly connected if for each of its vertices the number of adjacent vertices, in the researcher’s opinion, is large.

Sometimes connectivity is defined as a characteristic not of each, but of one (arbitrary) vertex. Then type definitions appear: a graph is called K-connected if at least one of its vertices is connected to K other vertices.

Some authors define connectivity as the extreme value of a quantitative characteristic. For example, a graph is K-connected if there is at least one vertex in the graph that is connected to K adjacent vertices and no vertex that is connected to more than K adjacent vertices.

For example, a child’s drawing of a person (Fig. 6) is a graph with a maximum connectivity of 4.

Another graph characteristic studied in a number of problems is often called graph cardinality. This characteristic is defined as the number of arcs connecting two vertices. In this case, arcs having the opposite direction are often considered separately.

For example, if the vertices of the graph represent information processing nodes, and the arcs are unidirectional channels for transmitting information between them, then the reliability of the system is determined not by the total number of channels, but by the smallest number of channels in any direction.

Cardinality, like connectivity, can be determined both for each pair of vertices of the graph, and for some (arbitrary) pair.

An essential characteristic of a graph is its dimension. This concept is usually understood as the number of vertices and arcs existing in a graph. Sometimes this quantity is defined as the sum of the quantities of elements of both types, sometimes as a product, sometimes as the number of elements of only one (one or another) type.

Types of graphs.

Objects modeled by graphs are of a very diverse nature. The desire to reflect this specificity led to the description of a large number of varieties of graphs. This process continues to this day. Many researchers, for their specific purposes, introduce new varieties and carry out their mathematical study with greater or lesser success.

At the heart of all this diversity are several fairly simple ideas, which we will talk about here.

Coloring

Graph coloring is a very popular way to modify graphs.

This technique allows you to increase the clarity of the model and increase the mathematical workload. Methods for introducing color can be different. Both arcs and vertices are colored according to certain rules. The coloring can be determined once or change over time (that is, when the graph acquires any properties); colors can be converted according to certain rules, etc.

For example, let the graph represent a model of human blood circulation, where the vertices correspond to internal organs, and the arcs correspond to blood capillaries. Let's color the arteries red and the veins blue. Then the following statement is obviously true - in the graph under consideration (Fig. 8) there are, and only two, vertices with outgoing red arcs (the red color is shown in bold in the figure).

Length

Sometimes the object elements modeled by vertices have significantly different characters. Or, during the formalization process, it turns out to be useful to add some fictitious elements to the elements that actually exist in the object. In this and some other cases, it is natural to divide the vertices of the graph into classes (shares). A graph containing vertices of two types is called bipartite, etc. In this case, rules regarding the relationships between vertices of different types are included in the graph restrictions. For example: “there is no arc that would connect vertices of the same type.” One of the varieties of graphs of this kind is called a “Petri net” (Fig. 9) and is quite widespread. Petri nets will be discussed in more detail in the next article in this series.

The concept of valleys can be applied not only to vertices, but also to arcs.

2.2. Solving problems using graphs.

1. Problem about the Königsberg bridges. In Fig. 1 shows a schematic plan of the central part of the city of Koenigsberg (now Kaliningrad), including two banks of the Pergola River, two islands in it and seven connecting bridges. The task is to go around all four parts of the land, crossing each bridge once, and return to the starting point. This problem was solved (it was shown that there was no solution) by Euler in 1736. (Fig. 10).

2. The problem of three houses and three wells. There are three houses and three wells, somehow located on a plane. Draw a path from each house to each well so that the paths do not intersect (Fig. 2). This problem was solved (it was shown that there is no solution) by Kuratovsky in 1930. (Fig. 11).

3. The four color problem. A division of a plane into non-overlapping areas is called a map. Areas on a map are called adjacent if they have a common border. The task is to color the map in such a way that no two adjacent areas are painted with the same color (Fig. 12). Since the end of the century before last, the hypothesis has been known that four colors are enough for this. In 1976, Appel and Heiken published a solution to the four-color problem, which was based on a computer search. The solution to this problem “programmatically” was a precedent that gave rise to a heated discussion, which is by no means over. The essence of the published solution is to try a large but finite number (about 2000) types of potential counterexamples to the four-color theorem and show that not a single case is a counterexample. This search was completed by the program in about a thousand hours of supercomputer operation. It is impossible to check the resulting solution “manually” - the scope of enumeration goes far beyond human capabilities. Many mathematicians raise the question: can such a “program proof” be considered a valid proof? After all, there may be errors in the program... Methods for formally proving the correctness of programs are not applicable to programs of such complexity as the one being discussed. Testing cannot guarantee the absence of errors and in this case is generally impossible. Thus, we can only rely on the programming skills of the authors and believe that they did everything right.

4.

Dudeney's tasks.

1. Smith, Jones and Robinson work on the same train crew as a driver, conductor and fireman. Their professions are not necessarily named in the same order as their surnames. There are three passengers with the same last names on the train served by the brigade. In the future, we will respectfully call each passenger “Mr.”

2. Mr. Robinson lives in Los Angeles.

3. The conductor lives in Omaha.

4. Mr. Jones has long forgotten all the algebra that he was taught in college.

5. The passenger, the conductor’s namesake, lives in Chicago.

6. The conductor and one of the passengers, a famous expert in mathematical physics, although they go to the same church.

7. Smith always wins over the fireman when they happen to meet at a game of billiards.

What is the driver's last name? (Fig. 13)

Here 1-5 are the numbers of moves, in brackets are the numbers of points of the problem on the basis of which the moves (conclusions) were made. It further follows from paragraph 7 that the fireman is not Smith, therefore, Smith is the machinist.

Conclusion

Analysis of theoretical and practical material on the topic under study allows us to draw conclusions about the success of using Euler circles and graphs for the development of logical thinking in children, instilling interest in the material being studied, using visual aids in lessons, as well as reducing difficult problems to easy ones for understanding and solving.

Bibliography

1. “Entertaining tasks in computer science”, Moscow, 2005

2. “Scenarios for school holidays” by E. Vladimirova, Rostov-on-Don, 2001

3. Tasks for the curious. , M., Education, 1992,

4. Extracurricular work in mathematics, Saratov, Lyceum, 2002.

5. The wonderful world of numbers. , ., M., Education, 1986,

6. Algebra: textbook for 9th grade. , and others, ed. , - M.: Enlightenment, 2008

Nomination "Fatherland's Glorious Sons"

Topic: “Alexey Petrovich Chulkov - Hero of the Soviet Union”

Galiullin Ravil

MBOU "Yukhmachinskaya secondary school named after Hero of the Soviet Union Aleksey Petrovich Chulkov"

7th grade student

Moskvina G.A.

1. Introduction.

2. Main part

2.1. Life and feat of A.P. Chulkova

2.2. Memory - perpetuation of the name of the Hero of the Soviet Union in memorial objects

3.Conclusion

4. List of references used

1. Introduction

The Great Patriotic War is one of the most terrible trials that befell our people. The severity and bloodshed of the war left a big imprint on people's minds. Patriotism has always been a national character trait in the Russian state.

Every town and village has its own heroes who glorified our country. Unfortunately, recently it has been said that the younger generation has begun to forget about the exploits of our grandfathers and great-grandfathers. And all around there are information surges, seeking once again to denigrate the feat of the Soviet people. Therefore, this topic of research work is relevant for solving such a problem as the education of a moral and patriotic personality. Our task is to remember the heroes, cherish this memory and pass it on to subsequent generations.

Memory of the past... No, this is not just a property of human consciousness, its ability to preserve traces of the past.

Memory is the link between the past and the future. No matter how many years have passed, no matter how many centuries have passed, we must remember with gratitude those who saved the world from the brown plague, and our people from destruction. And don't let history be rewritten.

Now, when in the West, in the former Soviet republics of the Baltic states and in Ukraine, the exploits of Red Army soldiers are put on a par with service on the side of the Nazis, and monuments are erected to SS men, we must remember again and again those who laid down their lives on the altar of the Fatherland.

Objective of the project: study the military path and feat of the Hero of the Soviet Union, whose name our school bears.

Tasks:- get acquainted with the algorithm for working on the project;

Study all available literature and media publications on the research topic;

Analyze the information received and draw conclusions

The work is devoted to the study of the biography of Aleksey Petrovich Chulkov, a hero of the Soviet Union, born in the village of Yukhmachi, Tatar Autonomous Soviet Socialist Republic.

Hero of the Soviet Union Alexey Petrovich Chulkov is our fellow countryman, our school in the village of Yukhmachi bears his name. Who is he, how did he live, what did he dream about, why was he awarded the title of Hero of the Soviet Union?

More than 70 years have passed since the end of the Great Patriotic War. In the vastness of our Motherland there are obelisks to the fallen, to those who did not return from the battlefields. They were young. When did they manage to do so much that they were nominated for the highest award of the Motherland? Why did they sacrifice themselves? Didn't they really want to survive?

The topic of my research work: The fate of my fellow countryman.

I decided to cover this question in more detail. To do this, I visited the school museum, where a section is dedicated to Alexei Petrovich. Also in my work I relied on the memoirs of the Hero of the Soviet Union, General - Colonel Vasily Vasilyevich Reshetnikov, Wikipedia, as well as the book by Yu.N. Khudov "The Winged Commissar".

Methods: During the implementation of the project, I became acquainted with the algorithm for conducting research work, studied local history literature, looked through the available literature, Internet materials, and the memories of a colleague.

Significance of the study: this material can be used in history lessons, during extracurricular activities dedicated to memorable dates and anniversaries, and museum lessons.

2. Main part

2.1. Life and feat of A.P. Chulkova

Chulkov Alexey Petrovich was born on April 30, 1908 in the village of Yukhmachi of the Russian Empire, now the Alkeevsky district of Tatarstan, into a working-class family. Russian by nationality. In 1920, after being wounded at the front, his father dies. Four children were left orphans. The eldest Sergei, even earlier, left for Karabanovo, to visit his relatives, where he gets a job at a factory. Together with ten-year-old Alexei, his mother left two younger sisters - Olya and Polina. This year, a terrible drought broke out in the Volga region. A great famine began. Lyosha gets a job as a farm laborer for a kulak, tending his flock for meager food. One day the owner beat Lesha. And the boy, having said goodbye to his mother and sisters, decides to go to his brother in Karabanovo. Money for travel and food - not a penny. With a gang of the same street children, Lyosha makes his way towards Moscow. At the station in Kostroma we were caught in another raid. So Alexey ended up in the Kostroma orphanage, where he completed the remaining two classes and, with a certificate of completion of primary school, arrived at the age of 14 and came to Karabanovo

Since 1925 - resident of the village of Karabanovo (now a city) in the Vladimir region. Here Alexey worked at the weaving factory of the 3rd International from 1927 to 1933. Here at the factory he met his future wife Vera. With whom Alexei Petrovich had four sons.

Member of the CPSU(b)/CPSU since 1931. Graduated from the workers' faculty and 1st year of the Moscow Pedagogical Institute. Worked in Moscow.

Drafted into the Red Army in 1933, he graduated from the Lugansk Military Aviation School in 1934. He made his first combat missions during the Soviet-Finnish War of 1939-1940, and successfully participated in the bombing and air attack of the fortifications of the Mannerheim Line. The combat skill and skillful fruitful political work of the pilot, senior political instructor Alexei Chulkov were highly appreciated by the command. He was awarded the Order of the Red Banner and was given the military rank of battalion commissar.

In the battles of the Great Patriotic War from the first days. By November 1942, the deputy squadron commander for political affairs of the 751st Air Regiment, Major Alexey Chulkov, made 114 combat missions to bomb military-industrial facilities deep behind enemy lines and his troops on the front line.

On November 7, 1942, while returning from a combat mission near the city of Orsha, his plane was hit by anti-aircraft fire and crashed in the Kaluga area.

In 2004, a book by Vasily Vasilyevich Reshetnikov, Hero of the Soviet Union, Colonel General, was published.

During the war, he was a pilot of the 751st regiment of the 17th long-range bomber air division. In 1942 he fought in the squadron, of which Chulkov was the commissar. He repeatedly flew under his leadership on combat missions. Vasily Vasilyevich remembers his commissar this way: That night, from the seventh to the eighth of November 1942, the crew of commissar Alexei Petrovich Chulkov did not return from a combat mission. Although he was the commissar of the Uruta squadron, the entire regiment revered him as their commissar, causing involuntary jealousy among others, including the regimental, but non-flying political workers.

This is a subtle thing - authority, especially commissar authority. The criteria for official position do not work here at all, even if they successfully provide the entire complex of external signs of veneration. In the fixed price of respect, only the moral and intellectual scale of an individual is quoted. Precisely individuals, not positions. In war, deeds were valued, and even if the word was a living one, not a dead, official one.

Alexey Petrovich was far from being a textbook commissar - he was outwardly completely unassuming, and certainly not tribune-like. He was more famous as an excellent combat pilot, and, as I remember, he did not fool anyone with reports or edifications. He was given a strong natural mind, a kind soul and a strong fighting spirit. He went through the Soviet-Finnish war, like a faithful soldier of his Fatherland, and did not hesitate on the first day of the Great Patriotic War. Now the count of his combat missions was in his second hundred. He flew along with us, like an ordinary ship commander, but he liked to take off first, or maybe he didn’t like it, not seeing any tactical advantages in it, but he apparently considered the place in front of the squadron his own.

Chulkov, after the bombing of the Orsha airfield, was already walking home and was half an hour away from his own people, when suddenly they came under fire, a shell hit the right engine. It began to smoke, gurgled, coughed, and had to be turned off. The propeller, unfortunately, continued to rotate, sliding became inevitable, and the car began to decline slightly. There was very little altitude left to the front line, but Alexey Petrovich and his constant navigator Grigory Chumash along the way found a base for our fighters in the Kaluga region and decided to land on the move.

At night, such airfields do not operate and do not even have night landing facilities, but the duty “T” lights were on, and Alexey Petrovich made a successful landing along the landing strip, perhaps with some overshoot. The airfield was tiny, for camouflage it was furnished with haystacks and models of animals, and when the plane was at the very edge of it, the radio gunners, seeing this “rural landscape,” shouted in one voice: “False airfield!” Alexey Petrovich gave in to the scream, and although the next moment Chumash shouted: “Sit down!” - It was too late. The left engine pulled the car further at full throttle, but it was unable to regain the lost speed and altitude, and even with one landing gear not retracted. While turning around, outside the airfield, the plane hit the pine trees with its wing, fell to the ground and caught fire. The flames from the tanks crawled towards the pilot cabin. Chulkov was wounded and could not get up himself. It burned there. Radio operator Dyakov also died in the fire. Overcoming the pain from bruises and abrasions, shooter Glazunov climbed out through the turret ring, but was unable to get through the fire to the commander. Grisha Chumash was thrown out of his broken navigator's shell and during the fall he broke his leg in two places. He crawled away from the fire, bandaged his bleeding wounds with scraps of linen and began to wait for help. She came from the airfield. After numerous operations, the leg was noticeably shortened, and I had to say goodbye to flying work.

This is how our legendary commissar died.

In just over a year of the war, he made 119 combat missions, 111 of them at night.

Bombed Berlin and other cities and military installations in Germany. Carrying out bombing strikes, he supported our ground troops on the front line. At the cost of his life, bringing the hour of Victory closer.

In December, during the formation of the regiment, the order was read out. There are these words:

For boundless devotion to the Motherland, for the good organization of the combat work of the squadron, for personal courage and heroism in battle, despising death, battalion commissar Chulkov is worthy of the highest government award of the title of “Hero of the Soviet Union” with the presentation of the Order of Lenin and the Gold Star medal - Posthumously

He was buried in the city of Kaluga.

Awards

By Decree of the Presidium of the Supreme Soviet of the USSR of December 31, 1942 For the feat and excellent performance of combat missions of the command, Major Alexei Petrovich Chulkov was posthumously awarded the title of Hero of the Soviet Union.

Awarded two Orders of Lenin and two Orders of the Red Banner.

From the award list:

Major Chulkov works as deputy commander of the air squadron for political affairs. Flying on an Il-4 aircraft as part of a night crew, where the navigator is captain Chumash, the gunner-radio operator foreman Kozlovsky and the air gunner senior sergeant Dyakov.

He has been in the active army since the first days of World War II. During this period, he carried out 114 combat sorties, 111 of them at night and all with excellent performance of the combat mission. He flew to bomb enemy military-industrial facilities and political centers deep in the rear: Berlin - 2 times, Budapest - 1 time, Danzig - 1 time, Koenigsberg - 1 time, Warsaw - 2 times.

For the excellent performance of combat missions of the command to defeat German fascism, he was awarded the Order of Lenin and the Order of the Red Banner. After the award, he carried out 55 combat missions. While working as a military commissar of an air squadron, he established himself as an educator of personnel in the spirit of devotion to the Motherland and hatred of the enemy. His squadron flew 951 sorties against the enemy during combat operations. Comrade Chulkov, by his personal example, inspires subordinate personnel to achieve heroic deeds. Disciplined, demanding of himself and his subordinates. He enjoys well-deserved authority among the personnel. He is devoted to the cause of Lenin's party and the socialist Motherland.

For the excellent performance of the command’s combat missions to defeat German fascism and the courage and heroism displayed, Major Chulkov is worthy of the government award of the Order of Lenin.

Commander 751 AP DD Hero of the Soviet Union
Lieutenant Colonel TIKHONOV November 4, 1942.

Conclusion of the Military Council.

Worthy of the government award of the title of Hero of the Soviet Union.

Air Commander Member of the Military Council
long-range aviation
General of Aviation GOLOVANOV
Divisional Commissioner GURYANOV
November 30, 1942

2.2. Memory - perpetuation of the name of the Hero of the Soviet Union in memorial objects

Memorial of Glory on Poklonnaya Hill in Moscow

Memorial complex of Kaluga

A street in the city of Karabanovo, Vladimir region, bears the name of the Hero.

In 2004, V.V. Reshetnikov’s book, “What Was, Was,” was published, which talks about Chulkov.

Documentary story “The Winged Commissar” by Yu.N. Khudova

In 2000, our school was named after the Countryman Hero.

The director of our school is a relative of Chulkov Alexey Petrovich Chulkov Petr Alexandrovich. It is largely thanks to his activities that our school bears the name of the Hero. Pyotr Alexandrovich himself is a worthy son of the Fatherland. In 1983 he was drafted into the Armed Forces of the USSR. Served in the Republic of Afghanistan, commander of a security platoon of a separate motorized rifle escort. He and his comrades accompanied convoys of KAMAZ trucks with cargo. One day the column came under fire, and Pyotr Alexandrovich was wounded.

Chulkov Pyotr Aleksandrovich was awarded: the star “Participant in the Afghan War”, the order badge “Warrior – Internationalist”, the medal “From the Grateful Afghan People”, the Certificate of the Presidium of the Supreme Soviet of the USSR “For Courage and Military Valor”.

He is distinguished by modesty, responsibility, rigor, and elegance. He is a talented leader and organizer of teaching and student teams. Under his leadership, the school is one of the best schools in the area.

Exhibition in the school museum of the village of Yukhmachi

Victory Park in Kazan

Monument dedicated to Chulkov A.P. in the village of Yukhmachi, in the Hero’s Homeland.

V.V. Reshetnikov with granddaughter A.P. Chulkova Elena Shusharina. Moscow 2007.

3.Conclusion

Life and feat, we often hear these words. A simple man from the outback, who was 34 years old, turned out to be a real hero of the war, of bloody battles. A.P. Chulkov became a Hero for a reason, he was a real person, raised by his family, his Motherland.

Work on materials about the Hero contributed to the determination of spiritual guidelines, moral values, universal human priorities, and the formation of patriotic consciousness as one of the most important values ​​and foundations of spiritual and moral unity.

And the need to participate in the affairs of the Russian schoolchildren movement, of which I am a member, becomes clear. This is a public-state children's and youth organization, formed by the decision of the constituent meeting of March 28, 2016 at Moscow University named after M.V. Lomonosov. In accordance with the Decree of the President of the Russian Federation of October 29, 2015. RDS works in the following areas: - military-patriotic - “Youth Army”

Personal development

Civic activism (volunteering, search work, studying history, local history)

Information and media.

4. References:

1.V.V. Reshetnikov “What happened, what happened”, M., 2004.

2. Yu.N. Khudov "Winged Commissar"

3. Materials from the school museum of the village of Yukhmachi

4. Photo from the personal archive of Chulkov P.A.

5.http://ru.wikipedia.org

Participant Application Form

Republican project competition “Glorious pages of history.

School of Heroes" for students in grades 5-7 of general education

Organizations of the Republic of Tatarstan bearing the name of the Hero

Territory RT, Alkeevsky district, Yukhmachi village

Nomination "Glorious Sons of the Fatherland"

First name, last name of the participant Ravil Galiullin

Date of Birth 05. 01.2005

Age group 7th grade

Full name of the educational organization MBOU "Yukhmachinskaya secondary school named after Hero of the Soviet Union Aleksey Petrovich Chulkov"Yukhmachi village, st. Shkolnaya, house 10 a

Phone number 89276781352

E-mail [email protected]

Teacher's full name (in full) Moskvina Galina Alexandrovna

Teacher's contact phone number 89270389187

Consent to the processing of personal data

I, Shubina Tatyana Nikolaevna, passport 9200097914 , issued ATC of the Aircraft Construction District of Kazan, 01.11.2002__________________________________________________________
(when, by whom)

RT, Alkeevsky district, Yukhmachi village, st. School 4.

____________________________________________________________________________________________________________________

I consent to the processing of my child’s personal data Galiullin Ravil Rashitovich

RT, Alkeevsky district, Yukhmachi village, st. School 4.

operator of the Ministry of Education and Science of the Republic of Tatarstan to participate in the competition.

List of personal data for the processing of which consent is given: last name, first name, patronymic, school, class, home address, date of birth, telephone number, email address, results of participation in the final stage of the competition.

The operator has the right to collect, systematize, accumulate, store, clarify, use, transfer personal data to third parties - educational organizations, educational authorities of districts (cities), the Ministry of Education and Science of the Republic of Tatarstan, the Ministry of Education of the Russian Federation, other legal entities and individuals responsible for organizing and conducting various stages of the competition, depersonalization, blocking, and destruction of personal data.

With this statement, I authorize the following personal data of my child to be considered publicly available, including on the Internet: last name, first name, class, school, preschool, the result of the final stage of the competition, as well as the publication in the public domain of a scanned copy of the work.

Processing of personal data is carried out in accordance with the norms of the Federal Law of the Russian Federation dated July 27, 2006 No. 152-FZ “On Personal Data”.

This Agreement comes into force from the date of its signing and is valid for 3 years.

______________________ _____________________________ (personal signature, date)

Kuchin Anatoly Nikolaevich

Project Manager:

Kuklina Tatyana Ivanovna

Institution:

MBOU "Basic secondary school" Troitsko-Pechorsk Rep. Komi

In his research work in mathematics "In the world of graphs" I will try to find out the features of using graph theory in solving problems and in practical activities. The result of my mathematics research work on graphs will be my family tree.

In my research work in mathematics, I plan to get acquainted with the history of graph theory, study the basic concepts and types of graphs, and consider methods for solving problems using graphs.

Also, in a research project on mathematics about graphs, I will show the application of graph theory in various areas of human activity.

Introduction
Chapter 1. Getting to know graphs
1.1. History of graphs.
1.2. Types of graphs
Chapter 2. Possibilities of applying graph theory in various areas of everyday life
2.1. Application of graphs in various areas of people's lives
2.2. Application of graphs in problem solving
2.3. Family tree is one of the ways to apply graph theory
2.4. Description of the research and compilation of the family tree of my family
Conclusion
References
Applications

“In mathematics, it is not the formulas that should be remembered,
but the process of thinking."
E.I. Ignatyeva

Introduction

Counts are everywhere! In my research paper on mathematics on the topic “In the world of graphs” we will talk about graphs that have nothing to do with the aristocrats of the past. "" have the root of the Greek word " grapho", What means " writing" The same root in the words " schedule», « biography», « holography».

For the first time with the concept “ graph” I met while solving math olympiad problems. Difficulties in solving these problems were explained by the absence of this topic in the compulsory school curriculum. The problem that arose was the main reason for choosing the topic of this research work. I decided to study in detail everything related to graphs. How widely the graph method is used and how important it is in people's lives.

There is even a special section in mathematics, which is called: “ Graph theory" Graph theory is part of both topology, so combinatorics. The fact that this is a topological theory follows from the independence of the properties of the graph from the location of the vertices and the type of lines connecting them.

And the convenience of formulating combinatorial problems in terms of graphs has led to the fact that graph theory has become one of the most powerful tools of combinatorics. When solving logical problems, it is usually quite difficult to keep in memory numerous facts given in the condition, establish connections between them, express hypotheses, draw particular conclusions and use them.

Find out the features of using graph theory in solving problems and in practical activities.

Object of study is mathematical graphs.

Subject of research are graphs as a way to solve a number of practical problems.

Hypothesis: If the graph method is so important, then it will certainly be widely used in various fields of science and human activity.

To achieve this goal, I put forward the following tasks:

1. get acquainted with the history of graph theory;
2. study the basic concepts of graph theory and types of graphs;
3. consider ways to solve problems using graphs;
4. show the application of graph theory in various areas of human life;
5. create a family tree of my family.

Methods: observation, search, selection, analysis, research.

Study:
1. Internet resources and printed publications were studied;
2. areas of science and human activity in which the graph method is used are outlined;
3. solution of problems using graph theory is considered;
4. I studied the method of compiling the family tree of my family.

Relevance and novelty.
Graph theory is currently an intensively developing branch of mathematics. This is explained by the fact that many objects and situations are described in the form of graph models. Graph theory is used in various areas of modern mathematics and its numerous applications, especially in economics, technology, and management. Solving many mathematical problems becomes easier if you can use graphs. Presenting data in the form of a graph makes it clearer and simpler. Many mathematical proofs are also simplified and become more convincing if graphs are used.

To make sure of this, the teacher and I proposed to students in grades 5-9, participants in the school and municipal rounds of the All-Russian Olympiad for Schoolchildren, 4 problems in which graph theory can be applied to solve ( Annex 1).

The results of solving the problems are as follows:
A total of 15 students (5th grade - 3 students, 6th grade - 2 students, 7th grade - 3 students, 8th grade - 3 students, 9th grade - 4 students) applied graph theory in 1 problem - 1, in 2 problem - 0, in Problem 3 – 6, problem 4 – 4 students.

Practical significance research is that the results will undoubtedly be of interest to many people. Haven't any of you tried to build your family tree? How to do this correctly?
It turns out that they can be easily solved using graphs.