Puzzles to put the shapes. Do-it-yourself tangram (game schemes, figures). The pedagogical meaning of the tangram

Tangram - an old oriental puzzle of figures obtained by cutting a square into 7 parts in a special way: 2 large triangles, one medium, 2 small triangles, a square and a parallelogram. As a result of folding these parts with each other, flat figures are obtained, the contours of which resemble all kinds of objects, ranging from humans, animals and ending with tools and household items. These types of puzzles are often referred to as "geometric construction sets", "cardboard puzzles" or "cut puzzles".

With a tangram, a child will learn to analyze images, highlight geometric shapes in them, learn to visually break an entire object into parts, and vice versa - to compose a given model from elements, and most importantly - to think logically.

How to make a tangram

A tangram can be made from cardboard or paper by printing out a template and cutting along the lines. You can download and print the tangram square diagram by clicking on the picture and selecting "print" or "save picture as...".

It is possible without a template. We draw a diagonal in a square - we get 2 triangles. Cut one of them in half into 2 small triangles. We mark the middle on each side of the second large triangle. We cut off the middle triangle and the rest of the figures at these marks. There are other options for how to draw a tangram, but when you cut it into pieces, they will be exactly the same.

A more practical and durable tangram can be cut from a rigid office folder or a plastic DVD box. You can complicate your task a little by cutting out tangrams from pieces of different felt, overcasting them around the edges, or even from plywood or wood.

How to play tangram

Each figure of the game must be made up of seven parts of the tangram, and at the same time they must not overlap.

The easiest option for preschool children 4-5 years old is to assemble figures according to diagrams (answers) drawn into elements, like a mosaic. A little practice, and the child will learn to make figures according to the contour pattern and even invent their own figures according to the same principle.

Schemes and figures of the game tangram

AT recent times tangram is often used by designers. The most successful use of tangram, perhaps, as furniture. There are tangram tables, and transformable upholstered furniture, and cabinet furniture. All furniture, built on the principle of tangram, is quite comfortable and functional. It can be modified depending on the mood and desire of the owner. How many different options and combinations can be made from triangular, square and quadrangular shelves. When buying such furniture, along with instructions, the buyer is given several sheets with pictures on various topics that can be folded from these shelves.In the living room you can hang shelves in the form of people, in the nursery you can put cats, hares and birds out of the same shelves, and in the dining room or library - the drawing can be on a construction theme - houses, castles, temples.

Here is such a multifunctional tangram.

How to make Pentomino

You can make a pentomino from cubes, but then you will need to glue and glue 60 cubes with colored film - it's difficult. We propose to make elements of their thick cardboard.

• We draw each element on a solid cardboard, cut it out, check that the element is included in the “U” element. Trim if needed. We drew details from 2.5x2.5 cm squares.

• We circle the finished cardboard element on colored paper folded in half and cut out two colored parts at once. It is better to make colored parts smaller than cardboard ones, and they stick better, and the corners will be more even.

• We glue colored paper with glue-pencil on both sides of the cardboard.

• We find a box for storing parts, where we will also put the schemes and tasks for the game.

Games and tasks with Pentomino

Fold a rectangle.

The most common pentomino task is to fold all the figures, without overlaps and gaps, into a rectangle. Since each of the 12 figures includes 5 squares, the rectangle must have an area of ​​60 unit squares. Rectangles 6x10, 5x12, 4x15 and 3x20 are possible.
There are exactly 2339 different arrangements of pentominoes in a 6x10 rectangle, but there are only 2 variants of a 3x20 rectangle.

One of two ways to fold a 3x20 rectangle

To be honest, I tried to put it together all evening - it didn’t work out, so it’s better not to offer the child such a task.

It is better for children to train on small rectangles of several parts.
Here we have drawn options for folding rectangles from three parts.

Fold the figure

Their elements can be combined with various shapes, symmetrical patterns, letters of the alphabet, numbers.
For young children, it is better to fold the figures according to the pattern, like a mosaic.
Figures can be printed or redrawn on a piece of paper in a box.

Figure "Duck", folded according to the model.

Games with kids.

It’s better to play with kids in a completely different way, you shouldn’t give them complex logic tasks right away, let them play with pentominoes like puzzles.

• My daughter (3.5 years old) folds them one into the other, looks for a suitable color or shape, and in the resulting collected figure looks for signs of resemblance to an animal or familiar object. For example, if the figure looks like an elephant, then you can try to make the trunk longer or enlarge the ears, and then remove a couple of elements and turn the figure into a mouse or someone else.

• Show your child how to fold a small rectangle. Then break, as if by accident. Before you break it, you can draw the child's attention to where which parts are. Ask for help to collect it again, otherwise you can’t.

Yes, you can come up with many more games with pentominoes, the main thing is that the child and you would be interested.

Pentomino from Lego

By the way, if you have a lot of standard Lego bricks at home, you can try to make a pentomino from them. Figurines folded from Lego turn out to be voluminous, and it will be possible to assemble, in addition to ordinary, planar models, voluminous figures.

The assembly scheme is quite simple: two rows of bricks stacked on top of each other with an offset.

The new class of games with pentominoes, which we will now consider, can be characterized as problems of "combining" figures, that is, problems of folding two or more equal figures from pentominoes. Here are some examples:

1. Try to make two identical 5×6 rectangles out of 12 different pentominoes (6 pentominoes will be spent on each). On fig. Figure 21 shows the sets of pentominoes corresponding to these rectangles, and it is curious that the above division of our figures into two sets of six pentominoes is the only possible one. However, it does not follow from this that the problem has a unique solution. Indeed, for the set of figures shown in the figure on the right, we can connect the F- and N-pentominoes in different ways, thus obtaining the same figure (how?).

Rice. 21. Two sets of 6 pentominoes to form 5×6 rectangles

Note, by the way, that the solution to this problem simultaneously serves as a solution to the problem of covering 12 pentomino rectangles of sizes 5×12 and 6×10. In order to verify this, it is enough to attach our 5 × 6 rectangles to each other in two ways.

2. Find such a cover with 12 different pentominoes chessboard 8x8 with a 2x2 hole in the center of the board so that the board can be split into two identical pieces, each covered with six pentominoes. Three typical solutions to this problem are shown in fig. 22.

Rice. 22. A typical solution to the problem of covering an 8×8 chessboard with a central "hole" 2×2, and the covering is divided into two congruent parts

3. Divide the 12 pentominoes into three groups of four pieces each so that there is a 20-cell "board" that can be covered by four pentominoes forming any of the groups. The solution shown in fig. 23, is by no means the only one; the reader can try to find his own solution.

4. Again divide our 12 pentominoes into three groups of four pentominoes; divide each group in turn into pairs of pentominoes and come up with three 10-cell "boards" (one for each group), covered by any of the pairs of polyominoes included in the corresponding group. One of the solutions is shown in Fig. 24. Try to find other solutions, in particular those where none of the three "boards" has holes (similar solutions exist).

5. Divide the 12 pentominoes into three groups of four polyominoes again. If we now add monominoes to all sets, we can try to add three 3 × 7 rectangles out of them. The solution of the problem is shown in fig. 25. It is known that there are no other solutions, except for the fact that monominoes and Y-pentominoes can be rearranged in the leftmost rectangle in such a way that they make up the same figure as a whole.

Rice. 25. Solving the Problem of Covering Three 3×7 Rectangles

The proof of the uniqueness of the solution of the last problem was suggested by the engineer C. S. Lawrence of the Aerospace Corporation (Los Angeles). in fig. 26. Finishing the first rectangle, we obviously can no longer use either the F- or the W-pentamino. It is also easy to see that the last two figures must obviously belong to different rectangles of size 3×7; in other words, of our three 3×7 rectangles, one will contain an X and a U pentomino, another a W pentomino, and finally a third an F pentomino. We give the reader the opportunity to complete the solution of the problem on their own and, with the help of a simple, albeit rather boring analysis of all possible remaining options for the arrangement of figures, show that the solution shown in Fig. 25, in fact, is the only one.

Rice. 26. The only possible position of X-pentamino in a 3×7 rectangle

6. Divide our 12 pentominoes into four groups of three pieces each and come up with such a 15-cell "board" that it can be covered with all the pentominoes of any of the groups.

This problem has not yet been solved, but at the same time it has not been proven that such a "board" does not exist.

7. Cut out from the chessboard a figure of the smallest possible area, consisting of a certain number of adjoining cells of the board, so that any pentomino can be placed on this figure.

The minimum area of ​​such a figure is 9 squares (cells); two 9-cell solutions of the problem are shown in fig. 27. Indeed, it is easy to check that any pentomino will fit on each of the "boards" shown in the figure. On the other hand, it can be proved that the smallest possible area of ​​the required figure is an area of ​​9 squares. Indeed, if there were less than 9-cell figure that satisfies the required conditions, then by placing I-, X- and V-pentominoes on it, we would combine them so that they together cover an area of ​​no more than 8 cells. It is clear that I- and X-pentamino will be combined in this case in three cells: otherwise we will either immediately get a figure of 9 cells, or (if the central cell of X-pentamino coincides with the outer cell of I-pentamino) we will come to a figure of 9 cells - if we require that V-pentamino could also be placed on this figure. But this condition is met only by the two shown in Fig. 28 configurations of 8 cells, such that the V-pentomino is placed on the "board" in question. However, it is easy to see that both "boards" do not fit, for example, U-pentamino; in order to ensure that the U-pentamino is placed on the "board" as well, it will be necessary to increase any of the figures shown in Fig. 28 pieces for at least one more square. Thus, an area of ​​8 cells will not be enough to solve the problem, while 9-cell figures that satisfy the condition of the problem, as we saw above, exist.

A few years ago, modern electronic computers were used to solve various polyomino problems. So, in the message of a well-known American specialist in mathematical logic Dan Stuart Scott, professor at Stanford University (see bibliography at the end of the book), talked about two problems solved using the Stanford University computer MANIAC. The first of these, already familiar to us, consisted of folding 12 different pentominoes into a 3x20 rectangle. It turned out that her two solutions listed on page 24 were the only possible ones. The second task was to enumerate all possible coverings of 12 different pentominoes on an 8x8 chessboard with a 2x2 square cut out in the center (a square tetramino). It turned out that the last problem has 65 different (that is, not obtained from each other by rotations and reflections of the board) solutions.

When compiling the program, D. Scott used a very simple and ingenious idea, which was as follows: X-pentamino can be placed on a chessboard with only three essential different ways shown in fig. 29; The electronic computer MANIAC found 20 solutions for the first X-pentamino arrangement, 19 for the second and 26 for the third arrangement. Three of the most interesting solutions among these 65 are shown in fig. 30, and in fig. Figure 31 shows three impossible situations - they are impossible simply because they are not on Scott's list.

Rice. 29. Three possible X-pentomino positions on an 8×8 chessboard with the central 2×2 square removed

Rice. 30. Three interesting solutions to the problem of covering an 8×8 board with a 2×2 central square removed

Rice. 31. Impossible coverings of polyomino chessboard 8×8

Professor of the University of Manchester S. B. Haselgrove, an English astronomer, also known for his results in number theory, not so long ago, using a computer, calculated the number of possible ways to add from all 12 pentominoes of a 6 × 10 rectangle. Here is his result: not counting the turns and reflections of the chessboard, the computer found 2339 fundamentally different solutions! At the same time, Hazelgrove checked and confirmed the two results of Dan Scott mentioned above.

In conclusion, here are three more undoubtedly noteworthy problems related to the composition of figures from pentominoes:

1. Cover the "64 cell pyramid" shown in fig. 32, 12 different pentominoes and a square tetramino (however, the latter can be replaced by any other tetramino). One of the solutions is shown in Fig. 32.

Rice. 32. "Triangle" of 64 squares

2. Cover with 12 pentominoes the elongated cross shown in fig. 33.

3. Professor R. M. Robinson (who also first pointed out the "jagged square" given in Chapter VI) has a very simple proof that the 60-cell figure shown in fig. 34, you can't cover 12 different pentominoes. Indeed, from the edges this figure is limited to 22 cells (including four corner ones), and if we count how many squares of each of the 12 pentominoes can be on the edge of our figure, then in total we get only 21 cells - one less than required:

T-pentamino - 1; W-pentamino - 3; Z-pentamino - 1; L-pentamino - 1; U-pentamino - 1; X-pentamino - 3; F-pentamino - 3; P-pentamino - 2; V-pentamino - 1; Y-pentamino - 2; 1-pentamino - 1; N-pentamino - 2 Total: 21 cells.

Arguments of this kind, where the internal and "boundary" cells of the board are considered separately, are very useful when folding "zigzag" pieces.

Other interesting pentomino puzzles will be discussed in Chap. VI.

We collect tangram

According to one of the legends, tangram appeared almost two and a half thousand years ago in Ancient China. The long-awaited son and heir was born to the elderly emperor. Years passed. The boy grew up healthy and quick-witted beyond his years. But the old emperor was worried that his son, the future ruler of a vast country, did not want to study. The boy liked to play with toys more. The emperor called three wise men to himself, one of whom was known as a mathematician, the other became famous as an artist, and the third was a famous philosopher, and ordered them to come up with a game, having fun with which, his son would comprehend the beginnings of mathematics, learned to look at the world around him with the gaze of an artist , would become patient, like a true philosopher, and would understand that often complex things are made up of simple things. And the three wise men came up with "Shi-Chao-Chu" - a square cut into seven parts.

Parfenova Valentina Nikolaevna, teacher kindergarten

One of constituent parts methodological support for the section “Elementary mathematical representations in kindergarten" is the game "Tangram", through which you can solve mathematical, speech and correctional problems.

The game "Tangram" is one of the simplest math games. The game is easy to make. A square 10 by 10 cm made of cardboard or plastic, equally colored on both sides, is cut into 7 parts, which are called tans. The result is 2 large, 2 small and 1 medium triangles, a square and a parallelogram. Each child is given an envelope with 7 tanas and a sheet of cardboard on which they lay out a picture from the sample. Using all 7 dances, tightly attaching them one to another, children make up a lot of different images according to samples and according to their own design.

The game is interesting for both children and adults. Children are fascinated by the result - they are involved in active practical activities to select the method of arranging the figures in order to create a silhouette.

The success of mastering the game in preschool age depends on the level of sensory development of children. While playing, children memorize the names geometric shapes, their properties, distinctive features, examine the forms in a visual and tactile-motor way, freely move them in order to obtain a new figure. Children develop the ability to analyze simple images, highlight geometric shapes in them and in surrounding objects, practically modify the figures by cutting and compose them from parts.

At the first stage of mastering the Tangram game, a series of exercises are carried out aimed at developing children's spatial representations, elements of geometric imagination, and at developing practical skills in composing new figures by attaching one of them to another.

Children are offered different tasks: to make figures according to a model, an oral task, a plan. These exercises are preparatory to the second stage of mastering the game - drawing up figures according to dissected samples.<Приложение №1 >.

The ability to visually analyze the shape of a planar figure and its parts is necessary for the successful reconstruction of figures. Children often make mistakes in connecting figures on the sides and in proportion.

Then follow the exercises in drawing up the figures. In case of difficulty, the children turn to the sample. It is made in the form of a table on a sheet of paper of the same size silhouette figure as the sets of figures that children have. This makes it easier in the first lessons to analyze and check the recreated image with a sample.<Рисунок №1>.

The third stage of mastering the game is the compilation of figures according to patterns of a contour character, undivided<Приложение №1>. This is available to children 6-7 years old subject to training. The pattern making games are followed by exercises in making pictures according to one's own design.

The stages of work on the introduction of the game "Tangram" with children of senior preschool age with general speech underdevelopment (OHP) were as follows.

At first, the Tangram game was played as part of a math class for 5-7 minutes. Observations of the children during the game confirmed the fact that the children liked the game. After that, an element of competition was introduced, and the one who posted the picture faster than the others received a chip award.

The kids were even more interested. They began to ask to leave more time for the game "Tangram". This made it possible to conduct mathematical leisure activities, quizzes, where children played up to 20-40 minutes.

To enrich the theme of the game, it became necessary to diversify this material, it was found in magazines “ Primary School”, “Preschool education”, in the books by Z.A. Mikhailova, T.I. Tarabarina, N.V. Elkina. and etc.

Many pictures were developed by the teacher. A number of pictures invented by children preparatory group. Children's observations confirmed that this game develops mental and speech abilities in children.

There were guys diagnosed general underdevelopment speech”, with poor memory, with a small vocabulary, closed. They often played alone. With such children, the teachers played individually, offered pictures for the whole family to play at home. The results were unexpected, the children began to level out, some faster, some slower, but they no longer lag behind their peers in posting pictures and even outperformed some. Having overcome their shyness, isolation, these children began to master the alphabet, reading, mathematics faster and left kindergarten with a clear speech, being able to read and count well.

The next step in complicating this game was the selection of speech material for pictures: riddles, funny short poems, tongue twisters, tongue twisters, counting rhymes, physical minutes. In a speech therapy kindergarten, this speech material for children with impaired sound pronunciation and speech has become especially useful. While playing "Tangram", the children memorized this material, consolidated and automated the sounds in tongue twisters and tongue twisters. Speech was enriched in children, memory was trained.

During the game "Tangram" the skills of quantitative counting were consolidated in children. (Total 5 triangles, 2 large triangles, 2 small triangles, 1 medium-sized triangle. There are 7 tans in the game).

Children practically mastered the ordinal account. So, if you count the thanas of the “Rocket” picture from top to bottom, then the square is in fifth place, small triangles are in first and fourth place, the middle triangle is in third, large triangles are in sixth and seventh place<Приложение №1 >.

Counting tanas from top to bottom, from left to right, children practice orientation on a sheet of paper.

Compiling this or that picture, the children compare the size of the triangles, determine the place for small, large and medium triangles in the pictures of the Tangram game.

The children's knowledge of geometric shapes in this game (triangle, square and quadrangle) is constantly consolidated.

Playing, rearranging small cardboard figurines-tans, children train the small muscles of the hands and fingers.

In the speech therapy groups of the kindergarten, work is carried out on lexical and grammatical topics, within which children's knowledge of the world around them is clarified and consolidated. On many topics, pictures for the game "Tangram" were developed (wild and domestic animals and birds, trees, houses, furniture, toys, dishes, transport, people, families, flowers, mushrooms, insects, fish, etc.). On the topic “Wild Animals”, pictures have been developed: a hare, a fox, a wolf, a bear, a squirrel, a lion, a kangaroo<Приложение №1 >. Playing with pictures, laying them out, children memorize a variety of speech material, as well as consolidate and automate the sounds set by the speech therapist.

Often dads ask themselves: what to play with the child at home? Yes, so that the game would be beneficial for the development of the baby. Especially if this kid is already running and talking at full speed.

At a time when mothers are more fond of playing games to develop the child's creative abilities (sing, draw, sculpt with the baby), fathers are more likely to take care of the logical and mathematical development of their child. So what to play?

We offer you the Tangram puzzle game, which you, dear dads, can easily make for your children yourself. This game is often referred to as “cardboard puzzle” or “geometric construction set”. "Tangram" is one of the simple puzzles that a child from 3.5-4 years old can do, and by complicating tasks, it can be interesting and useful for children 5-7 years old.

How to make "Tangram"?

Making a puzzle is very easy. You need a square 8x8 cm. You can cut it out of cardboard, from smooth ceiling tiles (if left over after repair) or from a plastic box from DVD movies. The main thing is that this material should be the same color on both sides. Then the same square is cut into 7 parts. It should be: 2 large, 1 medium and 2 small triangles, a square and a parallelogram. Using all 7 parts, tightly attaching them to each other, you can make a lot of different figures according to samples and according to your own design.

How useful is play for a child?

Initially, "tangram" is a puzzle. It is aimed at the development of logical, spatial and constructive thinking, ingenuity.

As a result of these game exercises and tasks, the child will learn to analyze simple images, highlight geometric shapes in them, visually break the whole object into parts, and vice versa, compose a given model from elements.

So where do you start?

Stage 1

To begin with, you can compose images from two or three elements. For example, from triangles to make a square, a trapezoid. The child can be offered to count all the details, compare them in size, find triangles among them.

Then you can simply attach the parts to each other and see what happens: a fungus, a house, a Christmas tree, a bow, a candy, etc.

Stage 2

A little later, you can move on to exercises for folding figures according to a given example. In these tasks, you need to use all 7 elements of the puzzle. It is better to start by drawing up a hare - this is the simplest of the figures below.

Stage 3

A more complex and interesting task for the children is to recreate images according to contour samples. This exercise requires the visual division of the form into its component parts, that is, into geometric shapes. Such tasks can be offered to children 5-6 years old.

This is already more complicated - the figures of a man running and sitting.

These are the most difficult pieces in this puzzle. But having trained, we think that your guys will be able to do it too.

Here, children can already collect images according to their plans. The picture is first conceived mentally, then the individual parts are assembled, after which the whole picture is created.

Dear dads, it is not necessary to spend money on expensive toys. Remember that the most expensive of all toys for a child can be those that you make for him yourself. And, of course, with whom you will play together.

More tasks with answers to the puzzle:

To organize classes, the following tools and accessories are needed: a ruler, square, compasses, scissors, a simple pencil, cardboard.

- "tangram"

"Tangram" is a simple game that will be interesting for children and adults. The success of mastering the game at preschool age depends on the level of sensory development of the child. Children should know not only the names of geometric shapes, but also their properties, distinguishing features.

A square measuring 100x100 mm, pasted over on both sides with colored paper, is cut into 7 parts. The result is 2 large, 1 medium and 2 small triangles, a square and a parallelogram. Various silhouettes are formed from the resulting figures.

Puzzle "Pythagoras"

Cut a 7x7 cm square into 7 pieces. From the resulting figures, harmonize various silhouettes.

"Magic Circle"

The circle is cut into 10 parts. The rules of the game are the same as in others similar games: use all 10 parts to create a silhouette, without overlapping one another. The cut circle should be colored the same on both sides.

Tangram (Chinese 七巧板, pinyin qī qiǎo bǎn, lit. "seven boards of skill") is a puzzle consisting of seven flat figures that are folded in a certain way to get another, more complex figure (depicting a person, animal, household item, letter or number, etc.). The figure to be obtained is usually specified in the form of a silhouette or an external contour. When solving the puzzle, two conditions must be met: first, all seven tangram figures must be used, and second, the figures must not overlap.

figures

Dimensions are given relative to a large square, the sides and area of ​​\u200b\u200bwhich are taken equal to 1.

5 right triangles

2 small (with hypotenuse, equal and legs)

1 medium (hypotenuse and legs)

2 large (hypotenuse and legs)

1 square (with a side)

1 parallelogram (with sides and and angles and)

Among these seven parts, the parallelogram stands out for its lack of mirror symmetry (it has only rotational symmetry), so that its mirror image can only be obtained by flipping it upside down. This is the only part of the tangram that needs to be turned over in order to fold certain shapes. When using a one-sided set (in which it is forbidden to flip the pieces), there are pieces that can be folded, while their mirror image cannot.

The pedagogical meaning of the tangram

Promotes the development in children of the ability to play by the rules and follow instructions, visual-figurative thinking, imagination, attention, understanding of color, size and shape, perception, combinatorial abilities.

The author of the book, known to many readers for his speeches in the press about the upbringing of children, talks about the experience of using and using educational games in his family, which allow him to successfully solve the problem of developing the child's creative abilities.

The book contains a description of games that are a kind of "mental gymnastics", detailed description methods of their implementation and method of manufacture.

INTRODUCTION

CHAPTER 1. WHAT ARE DEVELOPING GAMES?

Educational games Nikitins. Golden mean. creators and performers. What games does Nikitin have. How many games do you need to have? "Monkey"

CHAPTER 2

When and how to start. Drawing tasks. Errors, help and hints. Not only patterns. The same, not the same. Same color. Dimensions. Check. One, many, several. Account in order. More, less, equally. As many. Guess how much. Count down. The composition of the number. Meet ten. Let's get to know the numbers. Plus, minus, equal. Make-believe. We share equally. Hide and seek with an account. We train and remember. Orientation in space. Paths and houses. Dictation cubes. Looking for treasure. Sequences. What changed? As it was? Perimeter and area. Figures and their sides. Introduction to the perimeter. Introduction to the area. Both perimeter and area. Combinatorics. Symmetry.

CHAPTER 3. MONTESSORI FRAMES AND INSERTS

Introduction to the game. Learning to close the "windows". We close the "windows" ourselves. Outline the frames and learn to paint over. Draw frames and play. Circle the liners. We paint over. We shade. "Know the figure by touch." Insert by touch. Sort. Compare. Compliance. "Beads". "House". We train mindfulness.

CHAPTER 4. "UNICUB", "FOLD THE SQUARE" AND OTHER GAME SETS "Unicube". "Fold the square."

Color, shape, size. Find similar. Angles. Length. What does it look like? We play Monkey. "Find the mistake." Draw figurines. Reduced copy. initial geometry. Complete the silhouette. What changed? As it was? Symmetry. "Bricks". "Cubes for Everyone"

CHAPTER 5. NOW ATTENTION! "Attention". "Attention! Guess"

CHAPTER 6. PLANS AND MAPS

puppet plans. Plan of the room and apartment. Plan for the little ones. Neighborhood plan. My city. Games with real geographical maps. Games with a map hanging on the wall. Games with a card lying on the floor. Map in pieces. Travel games. Game "I know!". Guess what it is?

CHAPTER 7. WHAT TIME IS IT?

Introduction to watches. Half an hour. How much was? Five minutes. How to say? Schedule.

CHAPTER 8. MATHEMATICS WITH NIKITIN'S GAMES

"Fractions". We play with circles. Same and different. Big and small. From big to small. We play Monkey. As it was? Learning to count. Equally. The composition of the number. Let's get to know fractions. Numerator and denominator. From writing down the number to counting in the mind. What part is colored? How much is missing? A whole and a half. Compare fractions. Not only fractions. And again symmetry. THERMOMETER AND KNOTS

APPENDIX BIBLIOGRAPHY.

The text of the book itself is 104 pages long. The rest of the appendix book is game materials. Below is a photo of individual pages of the book. For example, a page from the "fold the pattern" chapter and a page from the appendix to this game.

Photo of a couple of pages from the chapters "fractions" and "Montessori frames and inserts"

If you evaluate the book on the content and style of presentation, I personally would put "5+".

As can be seen from the content, the book discusses the techniques of playing with the Nikitin games. Before buying this book, I already had Nikitin's book "Intellectual Games". Then I thought, is there still a need for a book, if there is a primary source. Having bought the book, I answered myself unambiguously “yes”, because.

1. The book discusses not only the games recommended by Nikitin, but also other games invented by Lena Danilova. It turns out that, having several games, you can play for a long time and in a variety of ways.

2. Applications are very useful. We ourselves have so far only used the applications for the game “fold the pattern”. It is not so easy to start making Nikitin's patterns right away. The appendix gives examples of drawings, starting with one cube and then in increasing complexity. There are apps for other games as well.

3. The book gives recommendations on how to interest the child if it is not possible to play right away (both general recommendations and specific games are given). Not all children want to play by the rules, and not all children are willing to show interest just at the sight of new game parents of such children will find a lot of useful advice in the book.

Tangram in Chinese has a literal meaning as "seven tablets of skill." It is believed that this is one of the oldest puzzles in the history of human civilization, although for the first time about this intellectual game was mentioned in a Chinese book during the reign of the seventh Manchu emperor of the Qing state, who ruled under the motto "Jiaqing - Beautiful and joyful." And in the European lexicon, the word "tangram" first appeared in 1848 in the brochure "Puzzles for Teaching Geometry" written by Thomas Hill, later president of Harvard University.

Considered a classic tangram, it consists of seven flat geometric figures - two large, one medium and two small triangles, a square and a parallelogram. These figures are added to obtain another, more complex, figure. Often these figures depict a person in various movements, any animal or object, letter or number. The figure that needs to be folded is given in the form of a silhouette or contour, and the task is to find a solution how to place the geometric shapes included in the tangram to get the desired one.

When finding a Tangram solution, two conditions must be observed: the first is that all seven tangram figures must be used, and the second is that the figures must not overlap (overlap each other).

As you can see from history, very respected and smart people attributed such a very simple-looking game to a method of developing intelligence worthy of the closest attention. Try it and you - buy a tangram and add a few figures of these seven polygons.

In addition to this type, there are other types of tangrams. All of them are interesting and exciting in finding a solution. Try it yourself.

Puzzle "Tangram"

One of the most famous fans of the tangram is the world-famous writer and mathematician Lewis Carroll, the one to whom humanity owes the appearance of the various adventures of the girl Alice. He adored the game and often offered his friends problems from a Chinese book he had with 323 problems.

He also wrote the book "Chinese Fashion Puzzle", in which he claimed that Napoleon Bonaparte, after his defeat and imprisonment on the island of St. Helena, spent time at the tangram "exercising his patience and resourcefulness." He had classic set of this logic game made of ivory and a book with tasks. Confirmation of this occupation of Napoleon is in the book by Jerry Slocum "The Tangram Book".

Edgar Allan Poe was no less famous for thinking about putting together a puzzle of seven separate figures. This popular writer of detective stories with interesting plots often solved the problems of the Tangram puzzle.

We talked about only a few well-known personalities who were fascinated by this interesting logic game. We hope that it will be more interesting to buy a Tangram puzzle now. It is worth adding that the great variety of possible figures from the seven geometric figures is amazing - there are several thousand of them, Perhaps you can add a few more to them.

Tangram puzzle "Stomachion"(Archimedes game)

The great thinker and mathematician Archimedes mentions this logical task in his work, which is now called the Palimpsest of Archimedes. It contains the treatise of the same name "Stomachion", which tells about such a concept as absolute infinity, as well as about combinatorics and mathematical physics. About everything that in our modern era is an important section of computer science.

It is believed that Archimedes attempted to find out the number of combinations with which it is possible to add up a perfect square from 14 segments. And only in 2003, with the help of a specially designed computer program, the American Bill Butler was able to calculate all possible solutions. The mathematician came to the conclusion that in total this game has 17152 combinations, and provided that the square cannot rotate and it cannot have a mirror reflection, then “only” 536 options.

The puzzle game "Stomachion" is very similar to the tangram and the main difference is the number and shape of the elements it consists of. For all its simplicity, this logic game is worthy of attention. The ancient Greeks and Arabs attached great importance to tasks and learning with it.

In addition to the task of finding 536 variants of the ideal square of Archimedes, this logic game offers to add various shapes from its 14 geometric shapes. Try to put together the figures of a person, animals and objects. This is actually not an easy task as it might seem at first glance. The rules are simple: all elements of the Stomachion puzzle can be turned to either side, and all of them must be used.

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Polyomino

In this article, we will consider polyominoes - figures composed of one-celled squares so that each square adjoins at least one neighboring one that has a common side with it.

Tasks with polyominoes are very characteristic of combinatorial geometry - a branch of mathematics dealing with the mutual arrangement and combination of geometric shapes. This is a very beautiful, but still almost undeveloped branch of mathematics, since there are apparently very few general methods in it, and the methods known today are so primitive that they cannot be improved. Many important engineering problems encountered in practice, primarily those related in one sense or another to the optimal arrangement of figures of a given shape, essentially belong to combinatorial geometry.

In the following combinatorial problems it is assumed that polyominoes can be rotated (that is, rotated by 90, 180, or 270) and mirrored (flipped) without changing the shape of the shapes themselves.

Dominoes

Rice. one

Dominoes consists of two squares and can have only one shape - the shape of a 1 × 2 rectangle (see Fig. 1). First associated with dominoes the problem is probably familiar to many: given a chessboard with a pair of opposite corner squares cut out, and a box of dominoes, each of which covers exactly two squares of the chessboard (see Fig. 2). Is it possible to completely cover the board with 31 dominoes (without free cells and overlays)? The answer to this question is "NO" and has a remarkable proof. The chessboard contains 64 alternating cells of white and black coloring (meaning the usual chess coloring of the board). Each domino placed on such a board and covering two adjacent cells will cover one white and one black field, and n domino bones - n whites and n black fields, i.e. equally for both. But the chessboard shown in the figure contains more black cells than white ones, and therefore it cannot be covered with dominoes. This result is a typical theorem of combinatorial geometry.

Rice. 2

Trimino

Rice. 3

Trimino (or triomino) - polyomino of the third order, that is, a polygon obtained by combining three equal squares connected by sides. If turns and mirror reflections are not considered different forms, then there are only two “free” forms of tromino (see Fig. 3): straight (I-shaped) and angular (L-shaped).

Tetramino

Rice. four

FROM tetramino many tasks are connected to compose different shapes from them. It is proved that to fold any rectangle from the complete set tetramino impossible. The proof uses checkerboard coloring. All tetramino , except for the T-shaped, contain 2 black and 2 white cells, and the T-shaped tetramino - 3 cells of one color and 1 cell of another. Therefore, any figure from the complete set tetramino (see Fig. 4) will contain two more cells of one color than another. But any rectangle with an even number of cells contains an equal number of black and white cells.

Pentomino

Rice. 5

Polyominoes covering five squares of a chessboard are called pentominoes. There are 12 types pentomino , which can be denoted in capital Latin letters, as shown in the figure (see Fig. 5). As a technique that makes it easy to remember these names, we indicate that the corresponding letters make up the end of the Latin alphabet (TUVWXYZ) and enter the name FiLiPiNo. Since there are 12 different pentomino and each of these figures covers five squares, then together they cover 60 squares.

The most common task pentomino - fold from all the figures, without overlaps and gaps, a rectangle. Since each of the 12 figures includes 5 squares, the rectangle must have an area of ​​60 unit squares. Rectangles 6x10, 5x12, 4x15 and 3x20 are possible (see Fig. 6).

Rice. 6

For the 6×10 case, this problem was first solved in 1965 by John Fletcher. There are exactly 2339 different styles pentomino into a 6 × 10 rectangle, not counting the rotations and reflections of the whole rectangle, but counting the rotations and reflections of its parts (sometimes a symmetrical combination of shapes is formed inside the rectangle, by rotating which you can get additional solutions).

For a 5×12 rectangle there are 1010 solutions, 4×15 - 368 solutions, 3×20 - only 2 solutions (which differ in the rotation described above). In particular, there are 16 ways to add two 5x6 rectangles, which can be used to make both a 6x10 and a 5x12 rectangle.

Another interesting pentomino problem is Pentomino tripling problem (See Fig. 7). This problem was proposed by Professor R. M. Robinson of the University of California. Having chosen one of the 12 pentomino figures, it is necessary to build from any 9 of the 11 remaining pentomino a figure similar to the one chosen, but 3 times the length and width. A solution exists for any of the 12 pentomino , and not the only one (from 15 solutions for X to 497 for P). There is a variant of this problem, in which it is allowed to use the original figure itself to construct a tripled figure. In this case, the number of solutions is from 20 for X to 9144 for P-pentamino.

Rice. 7