Origami is the Japanese art of creating models of various objects, animals, birds, flowers by bending a sheet of paper. The only working material is paper. The only tool is your hands. The skills acquired during folding can be used in mathematics and design lessons.

Origami is an art accessible to everyone. An incredible variety of different shapes can be folded from a sheet of paper. By folding paper several times and watching how it gains volume, we not only have fun, but also learn the art of harmoniously combining space, shape, volume and proportions. The easy-to-learn origami technique opens up a world full of surprises and wonders.

My name is Vasily Petukhov, I’m 13 years old and I’m in 6th grade. I have been interested in origami since the 5th grade. At first, not everything worked out for me, but my love for origami and diligence allowed me not only to overcome the first difficulties, but also to achieve my first, and I hope not my last, successes. After all, I can do my favorite thing until I’m very old.

“This winter, one mother taught me to make cockerels out of paper, folding and turning it inside out in a well-known way, which, when you pull them by the tail, flap their wings. This invention is from Japan. I made these cockerels for children many times, and not only the children, but always all the big ones present, who did not know these cockerels, both the gentlemen and the servants, were amused and became close to these cockerels, everyone smiled and rejoiced: how similar these cockerels flap their wings to a bird. The one who invented this cockerel rejoiced from the bottom of his heart that he managed to make the likeness of a bird, and this feeling is transmitted, and therefore, strange as it may be to say, the work of such a cockerel is real art.”

11. The large ball consists of 16 rows (including the first row). During assembly, the spherical shape does not immediately form. But the whole figure is quite elastic. Imagine that you are holding a clay pot in your hands. You need to plunge your fingers inside and gradually bend the walls, giving the desired shape. Make the last row 4 modules smaller. To reduce it by two modules, you need to put a couple of modules on not 4, but 6 corners.

Dorzhieva Sofia

Relevance research is that it allows us to expand our knowledge of origami and geometry, the ability to solve geometric problems using a regular sheet.

Purpose of the study – master the origami technique and use it in life.
Tasks:
- study the history of origami;
- become familiar with the origami technique;
- learn to create origami products;
- identify the connection between origami and geometry;
- find ways to use origami in everyday life.

Find out how much students like origami classes and draw conclusions.
Object of study is the art of origami.
Subject of research is the use of origami in everyday life.
Hypothesis : Origami develops creativity, intelligence, and teaches work.
Practical significance. The art of origami helps you create different products and apply this knowledge in your studies.

Research methods:

• search for material
• survey analysis

Methods and techniques:

• research
• informational - educational

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Municipal government institution of the education department of the Dzhidinsky district

Municipal budgetary educational institution

"Botsinskaya secondary school"

Topic: "Geometry and origami."

Section: Geometry

Head: Nadezhda Rafikovna Zinnatova,

Mathematic teacher.

2014

CONTENT

Introduction…………………………………………………………….……..…..…1-2

1. History of origami………………………………….………..……3

2. Origami is mathematics? …………………………………………..…………4-5

3. Modular origami……………………………..…………………………. 6

4. Solving problems using origami………………………………………….7-8

Conclusion…………………………………………………………………….…………….…..…… 9

Application.

Since ancient times, Japanese wisdom says:
"The Great Square has no limits."
Try to fold a simple figure,
And you will instantly be captivated by an interesting matter.

Introduction.

Origami is the Japanese art of creating models of various objects, animals, birds, flowers by folding a sheet of paper. The only working material is this is paper , the only instrument is hands.

A unique activity to create beautiful toys and geometric shapes with your own hands. When making origami figures, imagination, fine motor skills of the hands, spatial thinking are developed, taste, accuracy, and hard work are cultivated, which is what learning to use origami does relevant for research. The skills acquired during folding can be used in mathematics and design lessons. Having taken up this art, I saw how strongly origami is connected with geometry.You can get acquainted with origami and repeat the basic geometric shapes: triangle, rectangle, square, rhombus, quadrangle. Concepts: side, angle, vertex of an angle, diagonal, center of a figure and their properties.

Currently, we can say that mathematical education is basic for people in many professions, so much attention is paid to learning the basics of arithmetic, geometry and algebra. Of particular relevance is the problem of teaching the elements of geometry and increasing the level of mathematical knowledge in general.

Geometric material has a lot in common with the artistic perception of the world, since a large place in geometry belongs to figurative thinking. The art of origami is ideally suited for solving these problems. Back in the 19th century, the German teacher F. Froebel founded an integrated course in teaching mathematics using origami, on the basis of which geometric knowledge and skills can be improved and strengthened.

The relevance of research is that pos.wants to expand our knowledge about origami and geometry, skill solve geometric problems using a regular worksheet.

Purpose of the study– master the origami technique and use it in life.
Tasks:
- study the history of origami;
- become familiar with the origami technique;
- learn to create origami products;
- identify the connection between origami and geometry;
- find ways to use origami in everyday life.

- find out how much students like origami classes and draw conclusions.
Object of studyis the art of origami.
Subject of researchis the use of origami in everyday life.
Hypothesis: Origami develops creativity, intelligence, and teaches work.
Practical significance. The art of origami helps you create different products and apply this knowledge in your studies.

Research methods:

• search for material
• survey analysis

Methods and techniques:

• research
• informational - educational

1. The history of origami and geometry.

The official date of the “appearance” of paper in China is considered to be 105 BC. And in 610, the Japanese were already producing their own paper, which was superior in quality to Chinese paper. The first large paper mill in Tokyo opened in 1870.

In ancient times, in China, paper was used in religious ceremonies. It is no coincidence that the first origami appears in temples. One of the rituals using them was to make boxes sanbo , which were filled with pieces of fish and vegetables for sacrifice to the gods. This art in Japan has become a tradition that has been passed down from generation to generation, mainly through the female line.. The first Japanese publication on origami is considered to be the book “Senbazuru Orikata” (1797, “how to fold a thousand cranes”). The author is considered to be the abbot of the Rokan Temple.In the second half of the 19th century, origami expanded beyond the borders of Japan. In Russia it so happens that most adult origamists are teachers, and young folders are their students. Origami is spreading among teachers and is seen as a technique for teaching and developing students. The largest origami centers in Russia are located in Moscow and St. Petersburg.

Geometry in its original meaning it was understood as the science of figures, the relative position and sizes of their parts, as well as the transformations of figures. The history of geometry is lost in ancient times, but its cradle is undoubtedly the East. Geometric information and facts were mainly reduced to rules for calculating areas and volumes, and it must be assumed that these rules were more empirical than logical in nature. In the 7th century BC e. geometric information was, according to Greek historians, transferred from Egypt and Babylonia to Greece. Greek philosophers began to become acquainted with Egyptian and Babylonian wisdom. From this time on, the period of development of geometry began - the period of systematic presentation of geometry as a science, where all proposals were proven. The name “origami” itself comes from two Japanese words “ori” and “kami” and literally means “folded paper”. And this is not accidental, since bending is the simplest operation with a piece of paper, which does not require any skills from a person other than imagination and, oddly enough, this simplest operation gives rise to a unique geometry, no less rich than the geometry of a compass and ruler. At first it seems that the bending capabilities include the entire geometry of one ruler, but this is a mistake. In some cases, folds also contain compass capabilities, although they do not allow direct drawing of circular arcs. And the further you dive into this unusual geometry, the more you become amazed by the variety of funny and serious problems that can be solved. With the help of origami it is possible to show thatmathematics is not a dry science, but beauty and harmony.

1. Origami is mathematics?

Nowadays, in Japan, the USA and other developed countries, teaching geometry using origami is practiced in many schools.If in the past mathematics was used in a fairly limited number of areas of human life, being, therefore, necessary for a relatively small number of specialists, then in the modern era mathematics has penetrated into all those areas in which rational thinking is practiced, and this process, which is in constant development, requires appropriate mathematical training.

Simple Basic Shapes

Triangle	Book	Door	Kite
Medium basic forms
Crap	Fish	Double triangle	Double square

Seeing these forms, you understand that in mathematics classes you can repeat the following concepts using origami:

– horizontal, vertical, oblique lines;

– fold the square in different ways, show adjacent sides, diagonal;

– squares;

– all types of triangles.

At the first stage While studying geometry using origami, we get acquainted with the basic geometric shapes (triangle, rectangle, square, rhombus, quadrilateral), concepts (side angle, vertex of an angle, diagonal, center of the figure), their properties and learn the basics of origami technique. At the second stage We develop a system of knowledge, skills and abilities acquired at the initial stage of training. Working according to diagrams, the process of folding planar figures. This activity is aimed at developing perception, which is associated with various thinking operations. At this stage, it is important to pay attention to becoming familiar with the ornaments, which are very beautiful in their shape. By adding them in various combinations, you can get polyhedra. At the same time, spatial imagination develops, which contributes to the successful mastery of stereometry in high school. The importance of this stage lies in the fact that a special place here is occupied by the method of solving construction problems without the help of a compass and ruler. The special value of this method is that it allows you to construct regular polygons, the construction of which using a compass and ruler is difficult, and in some cases impossible. Thus, the skills accumulated during training in the first and second stages will allow us to successfully study in the third stage.

At the third stageIn studying geometry using origami, targeted work is being carried out to develop meaningful logical and mathematical thinking. Of great importance for the development of imagination is the production of geometric figures, in which a certain pattern of arrangement of modules can be traced, the visual perception of which helps to understand this feature and cope with geometric tasks. This includes the construction of regular, semi-regular and irregular polyhedra, their sections, finding the areas of lateral surfaces and volumes of geometric bodies.

1. Modular origami.

Modular origami- folding techniqueorigami, which, unlike classic origami, uses several sheets in the folding processpaper. Each individual sheet is folded intomoduleaccording to the rules of classic origami, and then the modules are connected by nesting them into each other, the resulting forceelasticityprevents the structure from falling apart. Removing the limit on the number of sheets makes it easier to create largemodelswith a complex structure. Modular origami has come into fashion in the last 20 years. Origami is a truly universal construction tool, since you can make anything from just one piece (a sheet or square of paper). In this case, a separate paper part is called a module (from the Latin word modulus - tact, rhythm, measure, module). The modules can be identical and together form an ornament, a star, or a regular polyhedron. Different modules are usually used to compose complex structures. In this case, they talk about heteromodular origami.

Today there are many transforming toys and mosaic pictures, all the creativity of working with which lies in the mechanical connection of ready-made elements. Such games are assembled once and for all, and the child himself does not participate in their development and production. A game that a child makes himself will undoubtedly have a greater educational and aesthetic impact than anything bought in a store.

Modular origamiThis is a type of three-dimensional origami. Identical modules are prepared, which are then nested one within the other. No glue is used.

Scheme:

Conclusion:

The diagram shows what geometric figures and concepts we will use: rectangle, isosceles triangle, height, bisector, parallel lines, bending at an angle of 45 0 .

You can also make any motifs from the modules - small and large shapes. The modules are nested inside each other without glue. Depending on how they are connected to each other and what sizes of modules are used, you can get one design or another.

1. Solving problems using origami.

Let's look at examples of problems solved using origami methods. As a rule, they are simpler and more intuitive, and relative simplicity helps to verify the correctness of classical statements and theorems and encourages further research. How many curious secrets lie hidden in an ordinary piece of paper that is always at hand! For example, when studying the topic “Remarkable points of a triangle,” students are convinced that each trio of bisectors, medians, altitudes, and perpendicular bisectors of a triangle intersect at one point, and then they try to confirm their beliefs mathematically. The possibilities of folding a sheet of paper are great, which makes it possible to solve a wide variety of problems.

When solving problems using origami techniquesthe role of straight lines is played by the edges of the sheet and the fold lines formed when it is folded, and the role of points is played by the vertices of the corners of the sheet and the points of intersection of the fold lines with each other or with the edges of the sheets.

Any Origami task consists of:

1. From the problem statement.

2. From an Origami solution, test or construction method.

3. From a mathematical justification, that is, proof that the result really is a figure with the required properties.

As an example, let's solve a simple problem.

Task. Using the origami method, divide one of the corners of the square into three equal angles.

When solving this problem using the origami method, you need to know some of the conventions adopted in origami. They are given in the following table:

	Fold line "valley", "toward"
	Bend arrow "valley", "toward"
	Combine marked points
	Bend and Unbend

Origama solution

1. Mark a fold that divides the top side of the square in half.
2. Align the top of the lower right corner of the square with some point on the intended fold line.
3. Bend the upper left part of the figure and return to the original square position.
4. Check the result. The top of the lower left corner of the square is divided into three equal angles by fold lines.

Mathematical justification

Using the drawing fig. 5, we can write: VAS – equilateral, which means ABC=60 0 .

OVA=90 0 -60 0 =30 0 , ABN=30 0 , OVA= ABN= NBC=30 0 .

So, using this method we divided the corner of the square into three equal parts. A continuation of this problem is the problem of constructing an equilateral triangle in a square.

Dividing a sheet of paper into 5 parts.

Dividing the side of a square into four equal parts. To do this, it is enough to divide them in half, and then, each of the halves in half again. This is exactly what happens when we fold the basic shape door.

TO As you might guess, dividing a square into five parts by folding is a much more difficult task. Her solution is shown in the figure. Try to prove for yourself that in this way we really divide the square into five parts.

Conclusion.

While working on this topic, I highlighted the most significant points. Origami, as the basis of various areas of art, is the most logical and harmonious form of studying geometry. Logic here acts as a means of confirming clarity and practical significance. By making geometric figures using the origami technique, they become familiar with new geometric concepts, basic definitions and visually study the patterns of behavior of a two-dimensional plane in three-dimensional space. The great Chinese wisdom says correctly: I hear and forget, I see and remember, I do and understand. Only the knowledge that is applied in practice is retained in the head.

Different techniques can be used to make paper products - such an activity will not get boring and will not seem boring.

Making your own paper products is a great way of self-expression that is accessible to everyone. This hobby develops artistic taste, fantasy and imagination, fine motor skills, perseverance and accuracy.

The pace of modern life is very fast: one task is quickly replaced by another during the day. And in the hustle and bustle of everyday life, it is very important to find time for your favorite activities. Making paper products is creativity, an opportunity to express yourself in magnificent works, to find spiritual harmony

My work is relevant, predisposes to the development of positive personality traits, and also promotes communication between people united in solving assigned problems.

Bibliography.

1. O. V. Vesnovskaya Origami: ornaments, kusudama, polyhedra. -Cheb.: ed. “Russika”, 2003, 52 p.

2. V. A. Gusev Methods of teaching geometry. - M.: ed. "Academy", 2004, 376 p.

3. //Does the 21st century school need Geometry? (I. F. Sharygin) Mathematical

Education. No. 3, issue. 8.-M.: MNNMO, 2004 -264 pp., P37-52.

4. V. V. Nurkova and N. B. Berezanskaya Psychology. -M.: ed. "Urayt", 2004, 498 p.

5. S. N. Belim Geometry problems solved by origami methods. – M.: ed. “Akim”, 1998, 66 p.

6. Kolyagin Yu.M., Tarasova O.V. Visual geometry and its role, place, history of origin. - Magazine “Primary School” No. 4, 2000.

7. Glazer G.D. What should a school geometry course be like / G.D. Glazer // Mathematics at school. – 1991. - No. 1. - P. 68 – 71

8. Beskin N.M. Methodology of geometry with the application of the chapter “Methodology of teaching visual geometry by A.M. Astryaba” / N.M. Beskin. – M.: Uchpedgiz, 1947. - 274 p.

9. http://www.trozo.ru/archives/34530#more-34530

APPLICATION.

Questioning.

1. Do you know what origami is?

2. Have you tried origami?

____________________________________________________________________

3. Do you know what modular origami is?

____________________________________________________________________

4. Did you know that you can solve geometric problems using origami?

____________________________________________________________________

Survey results.

I conducted a survey among students in grades 5-11 of the Botsinskaya Secondary School. Based on the results of the survey, we can say the following: all students have an idea of ​​what origami is, but not everyone knows what modular origami is. Many people tried to do origami, but not everyone succeeded. 25 students did not know that geometric problems can be solved using origami. Everyone liked my project, everyone was interested in it. Now we can handle geometry problems!

Municipal budgetary educational institution secondary school in the working village of Mukhen, municipal district named after Lazo, Khabarovsk Territory

Research

Theme: The Magic World of Origami

Completed by: Mironova Marina,

student of grade 5 "a" MBOU secondary school district. p. Mukhen

Head: Gornostaeva Valentina Ivanovna,

librarian MBOU secondary school district p. Mukhen

2016

Content

1. Introduction………………………………………………………………………………3-4

2. Main part

2.1. Origami - the art of folding paper………………………….5

2.2. History of origami……………………………………………………...6-7

2.3. Cranes of Hiroshima……………………………………………..8-9

2.4. Origami ABC……………………………………………………….10

2.5. Folding technique……………………………………………………………11

2.6. Origami and basic geometric concepts……………………..12-13

2.7. Study of origami paper…………………………………..14

2.8. Origami in everyday life……………………………………...15

3. Conclusion………………………………………………………………………………16-17

4. Literature……………………………………………………………..18

5. Applications……………………………………………………………………………….19-24

Introduction

Origami is an amazing, mysterious word. Almost every person is familiar with origami, although not all of us have heard this foreign word. Origami is the Japanese art of creating models of various objects, animals, birds, flowers by bending a sheet of paper.

The only working material is paper. The only tool is your hands. This is a unique activity of folding beautiful toys and geometric shapes with your own hands. Our grandparents also made a variety of things from paper and cardboard, although the concept itself was not familiar to them. These included paper toys: boats, airplanes, paper hats and caps that protected the head from the sun’s rays, and original paper souvenirs (Annex 1).

The first time I heard this word was in 1st grade during a technology lesson. But then I didn’t pay much attention to it. At the end of 4th grade, during a lesson we were shown paper crafts made using a triangular module using the origami technique(Appendix 2). The teacher showed us how to connect these modules together. This was the only lesson and I wanted to learn more about this interesting, fascinating and useful art.

My mother, seeing my hobby, bought me the book "Modular Origami". By doing various crafts according to the book's diagrams, I felt that I was gaining experience in folding paper, which was useful to me in mathematics lessons. While playing, I learned the basics of geometry. When making origami figures, imagination, fine motor skills of the hands, spatial thinking are developed, taste, accuracy, and hard work are cultivated, which is what learning to use origami doesrelevant for research.

An object: origami as a form of applied art.Subject of study: use of origami in everyday life.Target: consider the possibilities of using origami techniques for the education and development of students.

Hypothesis: Origami classes contribute to the development of creative abilities, fine motor skills, spatial thinking, and successful learning of mathematics.

Tasks:

Study the history of origami;

Get acquainted with the origami technique;

Learn to create origami products;

Identify the connection between origami and mathematics;

Find ways to use origami in everyday life.

Research methods:

1. Method of mathematical statistics and data processing.

2. Analysis of popular science literature.

2. Main part 2.1. Origami is the art of paper folding.

The concept of "origami" comes from the Japanese language. The Japanese themselves were the founders of this art form. Literally translated, it means “folded paper,” since “ori” translates to “folded” and kama means “paper.” The Japanese understand origami as “the art of a whole sheet”, i.e. its initial condition is the continuity of a sheet of paper, its integrity without any kind of interference, additions or other actions in relation to it. Only bending or folding the sheet.

Today, many people all over the world are interested in the art of origami. Paper figures are made by children and adults, artists and designers. It is even taught in schools, books are written about it and magazines are published with interesting articles and descriptions of various models.

Now the original philosophical meaning of this toy has been forgotten. In addition to traditional square crafts, many other ways of creating paper figures have been invented. These can be models made up of a regular triangle and half a square, torn off vertically or diagonally, or even from pentagons, hexagons, or octagons. The latest “fashion” is to fold origami from a sheet of regular standard-sized writing paper.

They also make models woven from endless tape - this is where origami turns into macrame. The method of making models from many identical parts (modules) has also completely departed from the traditional “origami”. So, from several squares you can make a beautiful bracelet or even a tiara(Appendix 3).

Nowadays, origami has ceased to be just a toy. Scientists and designers became interested in this art. Scientific symposiums on origami are held. The most complex technical structures - paper models - are already being created.

But children, of course, are most interested in origami as an opportunity to create a new toy. Or you can create new, your own three-dimensional paper figures.

2.2. History of origami

Getting acquainted with origami should begin with ancient history. It was there, in Ancient China, in 105 AD, that the first prerequisites for the emergence of origami appeared - the art of folding any figures from a square sheet of paper without the use of scissors and glue. As history shows, in that significant year, the official Tsai Lun made an official report to the emperor that a paper production technology had been created. For many decades, under pain of death, the Chinese kept the secret of creating a white sheet. But over time, when the monks of China began their travels to Japan, some of the secrets of this country began to travel with them. In the 7th century, the wandering Buddhist monk Dan-Ho, whom contemporaries said was rich in knowledge and knew how to make ink and paper, made his way to Japan and taught monks how to make paper using Chinese technology. Very soon, Japan managed to establish its own mass production of paper, largely ahead of China(Appendix 4).

The first pieces of paper folded into unusual figures appear first in monasteries. It couldn't be any other way. Indeed, in Japanese, the concepts “God” and “Paper” sound the same, although they are denoted by different hieroglyphs. Paper figures had a symbolic meaning. Initially, paper figurines and crafts remained only objects of rituals and ceremonies of the peoples of Asian countries. They were made for wedding celebrations, they were used to decorate the house during feasts or other festive events and traditional rituals. But it was not art yet. Just a piece of paper, very valuable and expensive, bearing the name of God, became an integral part of the life of the Japanese. Later, the art of origami began to be used by samurai. They invented a way to fold a sheet of paper in such a way that only a person initiated into the secrets of origami could unfold it without spoiling it.

By the beginning of the 14th century. the fashion for origami has so captured the inhabitants of the countries

East, that almost every wealthy family had its own teacher - an origamist who taught children this skill. The inability to make paper crafts was considered the height of bad manners. Any education was now unthinkable without knowledge and mastery of the origami technique.

Japan, which created the entire origami alphabet, the basis of paper craftsmanship, officially remained the birthplace of this art. It was here that all the classic models of paper figurines and crafts that are used all over the world to this day were invented.

Origami came to Rus' much later. In the 19th century One of the first who learned to bend various figures from a sheet of paper were the children of Tsar Nicholas II. They were taught this skill by a philologist from Cambridge, who was invited to teach the heir to the throne. The children immediately liked this activity. For a long time, origami in Russia remained a “children’s art.”

Russian writer Leo Tolstoy also knew how to fold paper figures.(Appendix 5). In the draft for the article “What is Art,” he writes: “This winter, one mother taught me to make cockerels out of paper, folding and turning it inside out in a well-known way, which, when you pull them by the tail, flap their wings. This invention is from Japan. I love it a lot.” Once I made these cockerels for children, and not only the children, but always all the big ones present, who did not know these cockerels, both the gentlemen and the servants, were amused and became close to these cockerels, everyone was surprised and happy: how similar to birds these cockerels flap their wings. whoever invented this cockerel rejoiced from the bottom of his heart that he managed to make the likeness of a bird, and this feeling is transmitted, and therefore, strange as it may seem, the work of such a cockerel is real art.”

The world owes the restoration and return of origami to the famous origami master Akira Yoshizawa(Appendix 6). It was he who, after World War II, devoted himself to restoring what had been lost and inventing new things in the art of origami. He recorded all his knowledge and experience and reflected it in precise drawings, developing origami symbols.

Modular origami has now been added to the traditional ones. A module is a part, detail, element of a figure. This type of origami is similar to assembly, when many identical modules are combined to form one or another paper structure. In general, this type of origami resembles children's blocks, from which you can build castles, palaces, cars and other things. Multifaceted blanks in the form of cubes, balls, boxes and other small identical parts made of paper are connected by origamists in a certain sequence. Most often, this method is used to create large structures (castles or palaces), as well as complex figures of animals, flowers and other plants, birds, and fish. The more experienced the origamist, the more complex the figure(Appendix 7). Modular origami is often used by craftsmen, as it allows them to create complex origami structures from various geometric parts. However, to practice modular origami, it is important that the master knows how to make very complex geometric calculations. Otherwise, the craft will have an unsightly warped or inverted appearance.

2.3. Hiroshima Cranes

The international symbol of origami is the Japanese paper crane.The crane is a symbol of happiness and good luck in Japan(Appendix 8). There is one sad legend story associated with the paper crane, which gave it additional meaning and turned it into a sacred bird of the world.During World War II, there lived a little girl named Sadako Sasaki in Hiroshima.(Appendix 9). She was born in 1943 and was just a baby when her father and mother were killed in the bombing of Hiroshima. Sadako herself survived, but suffered from radiation sickness. The girl fought the disease as best she could, but she got worse every year. While lying in the hospital, Sadako folded paper cranes. She believed that if she folded a thousand cranes, her wish would come true. It became more and more difficult to work on the cranes, and, realizing that she would never recover, Sadako began to dream of peace for all the people of the Earth, so that there would be no more wars and innocent people would not die. But she did not have time to make a thousand cranes. After making 644 cranes, Sadako died of illness in hospital on October 25, 1955. Having learned about this, children from all countries began to send origami paper cranes they had made to Hiroshima in the hope that Sadako’s dream would come true. A monument to Sadako was erected in Japan, and children still send paper cranes to the Peace Museum in Hiroshima as a symbol of peace and memory.

The monument depicts a bomb, on top and on the sides of which there are figures of children with their hands raised to the sky.(Appendix 10).

Making a crane:

1. We begin to fold the crane from the basic shape of a Square

2. Move the layers of paper apart on the sides and make three folds: bend and unbend

right and left edges, after which we bend and unbend the top of the figure.

Turn over and repeat the same with the other side;

3. Carefully lift the top layer of the diamond and bend it upward.

We do this by pressing on the sides.

4. Turn the figure over and repeat the same with the other side;

5. Spread the layers of paper apart on the sides and fold the sides of the front layer of the figure towardscenter;

6. Turn the figure over to the other side and do the same as in the previous point;

7. Now we spread the layers of paper along the sides of the figure and wrap them up

sharp bottom ends. Press on the sides, align the shape and bend it

to the sides, you guessed it, the nose and tail of a crane;

8. Bend the crane’s nose, spread its wings and voila.

2.4. Origami ABC

The simplest methods of constructing crafts are based on the ability to fold a square in half, vertically or horizontally, and sequentially bending the paper, first along and then across, aligning the sides to opposite corners. All these actions are marked with symbols, arrows, developed and approved by the Japanese Origami Association.In the international literature on origami, a certain set of conventional signs has long been formed, necessary in order to sketch the folding diagram of even the most complex product. In addition to signs, there is a small set of techniques that occur quite often. International conventional signs, together with a set of simple techniques, constitute a unique"ABC" origami.Most of the conventional signs were put into practice in the middle20th century famous Japanese masterAkira Yoshizawa .

Twisting techniques A technique sometimes used in origami is turning a figurine or part of it inside out. Naturally, this can only be done when this part consists of at least two layers of paper.The technique and result of turning paper inside out can be seen on models of kusudama - “balls”(Appendix 11).

2.5. Folding technique While researching the folding technique, I paid attention to the folds. They must be “sharp” and the layers of paper must fit tightly to each other so that there is no displacement and the product turns out beautiful and neat. You need to connect the corners evenly and accurately.

Whether the paper is folded correctly or not depends onskill and eye , which can be developed using origami.Origami helps to developobservation . After all, in order to make, for example, a dog, I need to imagine its shape, movements and habits: otherwise, my figurine will not look like a real dog. If this is not done, then the work remains simply a repetition of movements.

Since in origami the main tool is the fingers and they need to be trained, which allows you to improvefine motor skills . To do this, I suggest doing simple basic forms first.

2.6. Origami and basic geometric concepts

Now there are three main trends in origami.First current – traditional origami, where a square is used as the basis.Second current – models are made up of sheets of triangular, rectangular, five-, six-, and octagonal shapes.Third current – modular origami, models are made from a certain, sometimes quite large number of modules of the same type.That is, all the figures in origami are made from geometric shapes, which means this is one of the points of contact between origami and mathematics. But in origami, figures can be built without drawing tools, using several folds.When working with a square, we become familiar with the concepts: angle, side, diagonal, center, midline, vertex, dividing a segment into parts, an angle into parts, with methods of folding a square and folding other geometric shapes from a square. Thus, with the help of origami, geometric problems on a plane are solved.Continuing my research, putting together modular structures and traditional kusudama, I came to the conclusion that they resemble geometric bodies. And I plunged into origametry.Origametry is a combination of origami and geometry, which brings with it the originality of a different approach to geometric problems.In origametry it is considered:a) the role of straight lines will be played by the edges of the sheet and the fold lines formed when it is folded;b) the role of points - the vertices of the corners of the sheet and the points of intersection of fold lines with each other or with the edges of the sheets.

Back in the 19th century, the German teacher F. Febel founded an integrated course in teaching mathematics using origami(Appendix 12). Origami can be used in mathematics classes in tasks such as:a) find horizontal, vertical, oblique lines;b) find all the squares;c) find all triangles;

d) give a name to the images;c) folding a square, adjacent sides, diagonal.

Froebel's ideas are still very interesting today. Origami promotes the activity of both the left and right hemispheres of the brain, as it requires simultaneous control of the movements of both hands.

Folding polyhedra is a fascinating activity, but at the same time it is not easy. It requires accuracy, precision and high concentration. In high school we will study a section of mathematics - STEREOMETRY. She studies the properties of figures in space. Among them are the Platonic solids(Appendix 13).

There are five regular polyhedra, which are called Platonic solids.They are composed of regular polygons (tetrahedra - 4 triangles, octahedron - 8 triangles, icosahedron - 20 triangles, cube - 6 squares, dodecahedron - 12 pentagons).

There are many other polyhedra in the world, but they will be either semi-regular or irregular. The best way to get acquainted with the Platonic solids and other polyhedra is to practice origami, since such figures are folded from paper easily and quickly.

2.7. Origami paper research

Lovers of the art of origami among all prefer and most actively use such types of paper as writing, newspaper, wrapping, wallpaper, printing, tracing paper, cardboard, velvet and others. If we consider each type separately, then each of them will have its own advantages and disadvantages for origami products.

So I decided to research the paper and choose the one that is best for origami

Type of paper

Advantages

Flaws

Origami paper

Easy to fold and holds wrinkles well.

-

Napkins

Suitable for special products.

It is difficult to fold, over time the figure unfolds on its own.

Wrapping

Dense, but thin, does not tire your fingers when folding. Its size is not limited.

Coloring is missing.

Office paper

Holds folds well and is durable.

-

Paper for labor lessons

Very thin. This allows you to fold the figures with a large number of folds.

It dries out and white streaks appear on the folds.

From this study we can conclude that for origamiany paper is suitable . I just need to find suitable paper for the figure I want to fold.

2.8. Origami in everyday life

We are connected with paper from our very first days to the end of our lives. By creating varied and complex paper products, we make our creations part of everyday life. Napkins - the art of origami can also be used in table setting(Appendix 14). It is now difficult to imagine our life and leisure without origami art products. It also helps in the design of halls and halls, stands for exhibitions, expositions, etc.
How nice it is to receive gifts! But it’s no less pleasant to give them. A gift made with your own hands, and not bought in a store, is something unique, original and especially pleasant. Thanks to the art of origami, we can give our little sisters and brothers funny toys. Whatever we cook, our family and loved ones will be pleasantly surprised. Nowadays, such a trend in origami as gift wrapping has become widely known. The box can be an ordinary square, star-shaped, matchbox-shaped, unusually shaped, sanbo, zunako - there can be countless types of boxes... A small gift can be put in a beautifully decorated box, and a larger gift can be decorated with a rose or a star made of paper(Appendix 15). The art of origami is a great way to make a gift unusual and interesting.

Flowers occupy a very important place in our lives. We give them to our loved ones on holidays. We use them to decorate our apartments, outfits, and gifts for loved ones. Paper flowers can be made both voluminous and flat. Volumetric flowers can be placed in any decorative or paper vase. Flat flowers are mainly used for making panels and decorating gift boxes.(Appendix 16).

3 . Conclusion
Origami is, first of all, an art designed to give people joy.
Many entrepreneurs order paper figurines from craftsmen to use as a company symbol. Paper figures are used to create advertising videos and posters.

Origami is both a children's game, a design element, and an integral attribute of folk holidays in many countries around the world. There are theaters where the characters and decorations are paper figures. But origami also contributes to the successful study of geometry. In the process of folding origami figures, I can easily navigate in space and on a sheet of paper, I can divide the whole into parts, find the vertical, horizontal, and diagonal. All this helps the development of the first drawing skills. In the process of working on the project, I learned a lot of new things related to geometry and mathematics.

My first primary school teacher, Irina Aleksandrovna, from her I learned that working with your hands is also useful for the development of the cerebral hemispheres, since the right hand is responsible for the left hemisphere, and the left hand for the right. It turns out that when folding origami, both hemispheres of the brain develop, since both hands work simultaneously. It is not without reason that speech therapists use this activity in their practice. It helps people with musculoskeletal disorders and mental disorders. My hypothesis was confirmed that origami classes contribute to the development of creative abilities, fine motor skills, spatial thinking, and successful learning of mathematics.

Conclusion : origami

- teaches various techniques for working with paper;

- develops the ability to work with hands;

- teaches you to perform consistent actions;

- stimulates memory development;

- teaches you to concentrate;

- introduces in practice basic geometric concepts;

- develops spatial imagination, artistic taste and

Creative skills.

- develops confidence in one’s strengths and abilities;

Nomination: “Mathematics”

"The Great Square knows no limits"

Japanese folk proverb

The art of origami fascinated me at an early age. My grandmother and I made simple paper shapes when I was only three years old! Of course, this was not learning, but a game - the magical transformation of a simple piece of paper into a toy! Origami is an ideal construction set that consists of one part (sheet), with the help of which an endless variety of shapes are created, thousands and thousands of different figures are folded.

I noticed that the art of origami combines beautiful shapes and surprisingly regular lines. And in my lessons at school, I always liked mathematics the most... I wondered how closely the art of origami is related to mathematics? Perhaps it is precisely because of this that origami masters say that when folding figures, “the head works with the hands” and very successfully.

Hypothesis: The art of origami is closely related to mathematics and can be a good basis for its study.

Target: establish the relationship between the art of origami and the science of mathematics.

Tasks:

• Introduction to the main stages of learning origami.
• Analysis of the relationship between the fundamentals of origami and mathematics.
• Search for historical facts.
• Introduction to the concept of polyhedron.
• Studying the types of polyhedra.
• A study of the possibility of origami techniques for creating regular polygons and polyhedrons.

Object of study- connection between the art of origami and mathematics

Subject of study - paper.

Research methods: searching for information from different sources (special literature, Internet resources);practical work.

Progress

Here in front of us is an ordinary sheet of paper, most often rectangular in shape (standard sheet with dimensions 21-29.6). To turn a piece of paper into a figurine, you can turn to the art of origami.

Now there are three main trends in origami.

First current-traditional origami, where a square is used as a base.

Second current-models are made up of sheets of triangular, rectangular, five-, six-, and octagonal shapes.

Third current- modular origami, models are made from a certain, sometimes quite large number of modules of the same type.

That is, all the figures in origami are made from geometric shapes, which means this is one of the points of contact between origami and mathematics. But in origami, figures can be built without drawing tools, using several folds.

When working with a square, we become familiar with the concepts: angle, side, diagonal, center, midline, vertex, dividing a segment into parts, an angle into parts, with methods of folding a square and folding other geometric shapes from a square. Thus, with the help of origami, geometric problems on a plane are solved.

Continuing my research, putting together modular structures and traditional kusudama, I came to the conclusion that they resemble geometric bodies.

And I plunged into origametry. Folding polyhedra- a fascinating activity, but at the same time not easy. It requires accuracy, precision and high concentration.

There are five regular polyhedra, which are called Platonic solids. They are composed of regular polygons (tetrahedra - 4 triangles, octahedron - 8 triangles, icosahedron - 20 triangles, cube - 6 squares, dodecahedron - 12 pentagons).

Origami can provide invaluable assistance in making polyhedra. You can make a polyhedron of any size without any pattern. You just need to choose the size of the sheet of paper. In addition, the origami polyhedron can always be disassembled, and its modules will not take up much space.

Based on the results of my research, we can conclude that the hypothesis was confirmed.

Conclusion: the art of origami is closely related to mathematics and can be a good basis for its study. By doing origami, I went beyond the boundaries of the standard mathematics curriculum in elementary school and became acquainted in practice with the elements of geometry on a plane and in space.

Project "Origami and Mathematics"

Koroleva Dasha, 11 years old, “Origami” association, Municipal Educational Establishment of Children’s Educational Institution Syut No. 2, Krasnoyarsk. Dasha has been interested in the art of origami for several years, and has repeatedly become a winner and prize-winner of regional, city and regional events. I am always happy to share my knowledge and skills, introducing children and adults to the great art that gives food for the head and joy for the heart. Head: Nina Anatolyevna Tarasova, teacher of additional education of the highest category, work experience 29 years, Municipal Educational Institution of Children's Education and Training Syut No. 2, Krasnoyarsk.

Business card

MBOU "Nizhneoshminsk secondary school"

Mamadysh municipal district

Republic of Tatarstan.

Research work

"My Footprint in Science VII"

Theme: Origami

Section: creative works and scientific and technical modeling

Work completed by: 6th grade student

Gilfanova Milyausha M.

Head: Mukhametzyanova Gulnar G.

1. Introduction……………………………………………………………..... 3

2. Stories of the origin of origami…….……………….……...................3

3. Theoretical part: manufacturing of modules…………….. ………..…..4

4. Practical part: making crafts…….………………………...5

5. Conclusion……………………………………………………………..10

6. Sources of information………………………………………………………………....11

7. Appendix (my works)…………………………………………….....12

Goal of the work: explore the possibilities of modular origami techniques for making crafts.

Tasks: 1. Consider origami as a type of decorative and applied art: the history of its origin, the necessary material, techniques and technology of execution. 2. Make samples of products using the modular origami technique.

I, Gilfanova Milyausha, I really like making toys with my own hands. I have a lot of them. This year I found a new kind of hobby. These are toys... origami. Modular origami.

The magician who invented colored paper
Red, yellow and blue,
I probably believed that the guys could
Make figures from different squares.
These figures are all over the world
Only Japanese children knew.

The white crane has become a symbol of peace,
The symbol of happiness is a paper boat.
Fairytale butterflies, pink hares
This can be done using your fingers.
I suggest trying with you
Learn the origami technique.

Every person has probably at least once in his life created the simplest product from a sheet of paper - a boat or an airplane. And in those days, when stores did not have such a selection of straw hats and panama hats, people in the summer often made themselves a “cap” from a newspaper. Both the paper boats and the cap are made using the origami principle.
Origami is the traditional Japanese art of paper folding.
The Japanese can create miracles from ordinary paper. The paper figures they make decorate temples and homes. In Japan, paper balls are called kusudama and cranes are mascots and bring good luck. Therefore, they are often given as gifts and hung as decorations during folk festivals.
Japanese magicians, traveling around Europe, introduced the Western world to the art of origami. They were true masters of their craft and in a few seconds they could make a bird, insect, or animal out of paper for the amusement of numerous spectators.
And really, isn’t it a miracle: you can make anything you want without scissors and glue, without any improvised means, from a simple sheet of paper. Did you know that many famous people not only admired the art of origami, but also folded various paper figures with great pleasure.
Among these people were the famous Italian artist and inventor Leonardo da Vinci, writer Lewis Carroll, author of the world famous book “Alice in Wonderland” and others. Even the great Leo Tolstoy described in his article “What is Art” a case when he was taught “to make out of paper, folding and turning it in a certain way, cockerels, which, when you pull them by the tail, flap their wings.”
It's no secret that paper was invented in China, and it was brought to Japan six centuries later. And not only the Japanese used paper for folding - the Chinese had already done this long before them. For a number of reasons, in the West the art of paper folding became known in its Japanese version - origami. But, nevertheless, there is a specific folding style, to which the name “Chinese modular origami” has been assigned.

Features of this technique: the use of a fairly simple triangular module; a typical method of connecting modules (modules are connected by inserting them into each other, the friction force that appears in this case prevents the structure from falling apart); a very large number of modules, which makes it easier to create large models with complex structures.

Modular origami gained particular popularity in 1993, when a ship carrying illegal Chinese immigrants arrived in the United States. The poor fellows ended up in prison and, to pass the time, they collected paper models - fortunately, paper can be obtained even in prison. And, thanks to this, the world learned about this folding method. At first there was an opinion that this was a completely new folding technique that the prisoners themselves had invented. But later it turned out that this technique has long been popular in China. Since then, modular origami has been popularized and developed widely and is now represented by thousands of works.

Today, many people all over the world are interested in the art of origami.

A Japanese proverb says:

“Tell me - I will hear,

Show me - I'll remember

Let me do it myself -

I will understand!

And I would like to present to you my creative works made from paper using the example of “The Hare”. We will need triangles, from which this modular origami is assembled. The hare turns out to be very lively, beautiful and looks incredibly impressive.

But before you start making a three-dimensional origami hare, you need to learn how to make triangular origami modules.

How to make triangular origami modules for crafts

1. Fold the rectangle in half.

2. Bend and straighten to mark the middle line. Turn the slide towards you.

3. Fold the edges towards the middle.

4. Turn it over.

5. Raise the edges up.

6. Fold the corners, bending them over the large triangle.

7. Straighten up.

8. Fold the small triangles again along the marked lines and lift the edges up.

9. Fold in half.

10. The resulting module has two corners and two pockets.

In total, the bunny will require 522 triangular modules. You can take only white modules (office paper is best) or make 402 white and 120 color modules. The bunny sweater can be made in one color or striped. There are five rows in total (five stripes), each row has 24 modules. For this model we took 48 blue, 48 yellow and 24 blue modules. Let's take 48 white modules to assemble the first two rows. Let's arrange the three modules as shown in the figure, and put the corners of the first two modules in the pockets of the third.

Let's take two more modules and connect them to the first ones in the same way. Let's assemble all the modules in a chain. The last module will close the ring. It turned out to be two rows of 24 modules each.

Let's complete the third row, putting on the modules in a checkerboard pattern.

Carefully turn the circle out, turning it into something like a bowl. The first row then becomes a stand and is practically invisible from the side. Let's take 24 blue modules and complete the fourth row. Please note that the blue modules need to be strengthened a little higher, slightly moving the lower end. Then the sweater will move slightly away from the body.

Let's complete the entire fourth row and align the modules.

Let's do four more rows in the same way.

Take 24 white modules and put them on the blue ones, turning the short side outward and the long side inward.

This row should be narrower than the previous ones due to the fact that the white modules are turned on the other side and placed at the ends of the blue ones.

The next row must be increased by 6 modules. To do this, we put not one, but two modules on every fourth module. The modules are turned with the long side outward.

Attention! The first two rows of the head have a rather unstable connection. The following rows will secure the entire structure, but these two rows must be assembled carefully and carefully.

Now there will be 30 modules in each row. We continue to collect row after row. In total, the head consists of eight rows (one of 24 modules, the others of 30).

In the last row we bring all the modules together, giving the head a rounded shape.

For the ear, take 6 modules and strengthen them in the same way as in the first row of the head.

In the second row we will put on 5 modules, in the third - 6 again. Moreover, the outer modules will be put on the outer corners of the first and second rows. Let's complete 7 rows in this way. In the eighth row it is necessary to make a narrowing. We put the two outer modules on the three outer ends of the previous row at once. There will be 5 modules in the eighth row. In the ninth we will put on only 4 modules, and we will place the two middle ones above the others.

Let's skip one module, i.e. leave 4 corners free, and make the second eyelet in the same way.

Let's draw, cut out and glue the details: eyes, muzzle, arms, collar. We cut out the handles from rectangles of thick paper measuring 2x3 cm. We round the rectangles on one side, and on the other we glue strips of paper of the same color as the clothes. Handles can simply be inserted between the modules or glued. You can cut a 4x8 cm rectangle with fringe, curl it and give the bunny a bang. That's all. The bunny turned out to be very funny and bright.

Conclusion

These toys became my friends. Now I have started assembling the boat. Try it too. Very exciting and interesting. As the great Leonardo da Vinci said, “The acquisition of any knowledge is always useful for the mind, for it can subsequently reject the useless and preserve the good. After all, not a single thing can be loved or hated if you don’t first know it.” Folding figures from triangular origami modules is such an exciting and enjoyable activity that having mastered one model, you immediately want to try something else and learn new possibilities. Having completed several voluminous crafts (see Appendix 1, 2,3,4.5), I was able to assemble a flower basket without a diagram (see Appendix 6). And everything doesn’t always work out right away... And having started a new craft, you want to finish it faster and see the result. And I want to say a huge thank you to my mom, dad, and sisters for helping me create these crafts. After all, they require a lot of modules and paper. For example, two bunnies required more than a thousand modular triangles, and about 65 A4 sheets of paper were needed. For several evenings, the whole family makes modules for crafts, and then the most interesting, most mysterious thing comes - “the paper comes to life”! Try it too. As the great Leonardo da Vinci said: “The acquisition of any knowledge is always useful for the mind, because it can subsequently reject the useless and preserve the good. After all, not a single thing can be loved or hated unless you first know it.”

Economic assessment

Calculation of the cost of a vase.

4.yourorigami.info/2008/01/28/statya -posvyashhennaya -origami.html

5. Prosnyakova T.N. Funny figures. Modular origami.

My works

1. Owl 2. Snowman 3. Moonwalker

4. Swan Lake 5. Cat and kitten

6. Snowman with gifts 7. Me and my works.