Math game to add a shape from triangles. Tangram: schemes and shapes
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Equal shapes are added using both blue triangles and triangles of other colors. The help of the black lines is not great; blue figures and figures of other colors are arranged by placing next to each other; figures of equal area are placed next to each other and surround them with a border of ribbon, thus separating them from other figures. The matching ribbons are in baskets in boxes with structural triangles; find other shapes by moving and overturning figures; add arbitrary geometric shapes from all triangles; fold a geometric figure of the largest possible area; form as few quadrangles as possible. Lessons with triangles provide ample opportunities for knowledge due to the numerous relationships of individual figures with each other;
- drawing, coloring, cutting shapes; ordering figures that have equal area; ordering shapes that have the same color and shape; colored triangles are placed on a nearby table, blue ones are on the carpet. The child leaves a mark next to a blue triangle and brings the corresponding colored triangle.
- the well-known collective game with other tasks, for example: "I see what you do not see. It is triangular, it is square, it is rectangular"; the child chooses a quadrangle and looks for an object of a similar shape in his environment, for example, he takes a rectangle and finds a rectangular table surface; flat figure with the help of ribbons breaks into triangles.
- add one large equilateral triangle from all triangles; fold other large figures, for example, a trapezoid, rhombus, parallelogram; put the compound triangle on the colored one and circle it, then removing the small triangles of which it consists one by one. Run a pencil along the vacant sides each time. Cut the resulting triangles; circle and cut out the gray equilateral triangle. Individual parts, for example, red triangles, can be circled and cut out. Experiment with them and find shapes that have equal areas, but different shapes. Large hexagonal box.
- fold large figures, for example, a triangle, a trapezoid; combinations with figures from a triangular box; by turning and overlapping, find figures that have equal areas, but different shapes. Small hexagonal box
- the bodies lie in a covered basket. The child puts his hand into it, feels some body, says whether this body rolls or overturns, and pulls it out; the child closes his eyes. The teacher gives him some kind of body. The child feels it and returns it to the teacher, who places it among other bodies. The child opens his eyes and must now recognize this body again without feeling; the child forms a set (groups) of bodies that can only roll, which can stand, which can stand and roll. A game that clarifies the concept of sets. The dividing set!
- the child looks for objects from his environment that roll or overturn, and arranges them in accordance with these properties; on two rugs there is one geometric body each time. The child is looking for an object of a similar shape: for example, a ball, a bead, a ball of yarn look like a ball; on a cube - a children's cube, some kind of box.
- put on one foundation all the bodies that correspond to it; find a set of bodies with a rectangular base or side face. A game in which ideas about sets are clarified; find a body with rectangular and square side faces; build a row of all bodies so that two standing next to the bodies had something in common; bodies are distributed to children. One child pronounces their names, other children bring bodies; bodies, the names of which are known to the child, are placed in a basket and covered with a handkerchief. The child feels the body, names it and takes it out of the basket; name the body and find it in a closed basket.
- the child examines the properties of the fabrics from which his clothes are sewn (smooth - rough, thick - thin, etc.); the child checks what material his clothes are made of; the child is trying to determine the properties of other textiles in the room.
- the teacher shows the child how to weigh several plates at the same time. Each time the child compares an equal number of tablets from each series. The difference in weight is stronger and clearer; the child exercises with two series that have less difference, for example, with series 1 and 2-f; with series 2 and 3; - to master the middle series. The teacher takes a tablet from it and compares all the other tablets with it. He puts the lighter ones on one side, the heavier ones on the other, and the equal in weight in the middle.
Regular polygons have been considered a symbol of beauty and perfection since ancient times. Of all polygons with a given number of sides, the most pleasing to the eye is a regular polygon, in which all sides are equal and all angles are equal. One of these polygons is a square, or in other words, a square is a regular quadrilateral.
There are several ways to define a square: a square is a rectangle with all sides are equal and a square is a rhombus with all angles are straight.
From school course geometry is known:
1 the square has all sides equal,
2 all corners are straight,
The 3 diagonals are equal, mutually perpendicular, the intersection point is halved and the corners of the square are halved.
4 The square has a symmetry that gives it simplicity and a certain perfection of form: the square serves as a standard when measuring the areas of all figures.
This is a small part of what can be revealed in this question, because modern mathematics knows a lot of interesting and useful properties square. Therefore, the goal of this abstract is an:
1 to study in more detail the properties of the square,
2 consider geometric methods cutting a square,
3 justify the possibilities of transforming figures by cutting a square,
4 find various options for constructions that can be reproduced by folding a square sheet of paper, and identify the advantages in this type of constructions.
When studying this topic, we used articles from books and magazines devoted to specific issues of metatics.
VF Kagan "On the transformation of polyhedra". This book provides a proof of F. Bolai's theorem using the example of a square.
In the book "Amazing Square" B.A. Kordemsky and N.V. Rusalev, the proofs of some properties of a square are presented in detail, an example of a "perfect square" and the solution of one problem on cutting a square by the Arab mathematician of the 10th century Abul Vefa are given.
In the book by I. Lehmann "Fascinating Mathematics" collected several dozen problems, among which there are those whose age is estimated in thousands of years. From this book in the abstract were used problems for cutting a square.
Books by Ya. I. Perelman are among the most accessible of the books on entertaining math... The book "Entertaining Geometry" popularly expounds the question of figures with the largest area for a given perimeter or with the smallest perimeter for a given area.
For a complete understanding of the construction by bending a square square of a sheet of paper, the book by I.N. Sergeeva "Apply Mathematics".
CHAPTER Ι. 1.1 REMARKABLE PROPERTIES OF THE SQUARE
The square has two practical properties:
The perimeter of a square is less than the perimeter of any rectangle of the same size,
The area of a square is larger than the area of any rectangle with the same perimeter.
Fig. 1
In his book "Amazing Square" B.A. Kordemsky and N.V. Rusalev describe in detail the evidence for these properties.
To prove the first property, the perimeter of the square ABCD, with side x, of a given area (Fig. 1) was compared with any rectangle BEFG, with a larger side y, of the same area. Obviously, y is greater than x,; then the other side of z is certainly less than x. The drawing shows that ABEK is a common part for both a square and a rectangle; there remain two equal rectangles AKFG and KESD, i.e. AG.FG = DC.KD. But since FGKD or y-x> x-z. Hence, y + z> 2x and 2y + 2z> 4x, that is, the perimeter of any rectangle equal to the square is greater than the perimeter of the square. This means that among all rectangles of equal size, the square has the smallest perimeter.
To prove the second property, the authors of the book used a method when they prove converse theorems - by contradiction.
Given a square whose perimeter is p and whose area is q. Suppose there is a rectangle whose perimeter is also p and whose area is Q> q. Then the authors built a new square, equal in size to this rectangle, that is, with an area that is also equal to Q, and, therefore, greater than the area of this square. But according to the previous theorem, the perimeter of the new square p These properties can be considered practical, because they can be used in life situations... For example, if you need to enclose a piece of land with a hedge, fence or trellis a certain area so that the length of the fence is as small as possible, and the fenced area should be rectangular, but with any aspect ratio. Translated into exact, mathematical language, this means: which of the rectangles of a given area has the smallest perimeter?
In the book "Entertaining geometry" Ya.I. Perelman provides examples and popular questions about figures with the largest area for a given perimeter or with the smallest perimeter for a given area
1.2 SQUARE TO SQUARE
A square inscribed in a square has some peculiarities.
a) 
b) 
v) 
Rice. 2.
If we connect in series the midpoints of the sides of the square ABCD (Fig. 2, a) with segments, we get a new square EFKL, the area of which is half the area of this square ABCD.
If you cut off four right-angled triangles located at the corners of the square ABCD. The sum of their areas is also half the area of the square ABCD. If we take the area of the square ABCD as a unit, then the sum of the areas of the cut triangles is equal to Ѕ.
If you inscribe the square A B C D in the remaining square EPKL in the same way (Fig. 2, b) and again cut off the four triangular corners. The sum of the areas of the cut triangles is Ѕ the area of the square
EFKL and, therefore, ј the area of the square ABCD. Repeating this technique (Fig. 2, c), four more triangles are obtained, the sum of the areas of which will be ⅛ of the area of the square ABCD.
Applying this technique any number of times, all new fours of right triangles will be obtained, with which you can again lay out the original square. The sums of the areas of the fours of triangles represent an infinite series of numbers.
Ѕ, ј ,⅛…
1.3 PERFECT SQUARING
This curious problem was not solved for a long time, and many thought that it was impossible to solve it.
In terms of content, this is the problem of making a square from several squares, but this time without cutting them into pieces and further complicated by the requirement that the sides of the squares be expressed by non-repeating integers. The number of these squares is irrelevant. 
Fig. 3
Dividing a square into a finite number of non-overlapping squares, no two of which are equal, is called perfect squaring, and a square made up of non-repeating squares is called a perfect square.
Some mathematicians have suggested that perfect squaring of the square is impossible. One of these mathematicians was G. Steinhaus, who argued in his book "Mathematical Kaleidoscope" that "it is not known whether it is possible to break a square into non-repeating squares."
Since this was only allowed by mathematicians, but was not proved, the search for a solution continued, and a little more than ten years ago, in foreign mathematical journals, finally, squares composed of non-repeating squares appeared. In his book "Amazing Square" Kordemsky BA and Rusalev N.V. presented a square, consisting of 26 unequal squares (Fig. 3). (The numbers in the figure indicate the lengths of the sides of the corresponding squares). Kordemsky and Rusalev write that you can also make a square from 28 non-repeating squares, etc.
The question of whether 26 is the smallest possible number of squares to form a perfect square remains not entirely clear.
CHAPTER ΙΙ. 2.1 SQUARE CUTTING PROBLEM
A square is very similar to a mechanism with well-fitted parts, which can be disassembled and a new mechanism assembled from the same parts.
In order to compose it again from the finished parts of the square or to compose several other, pre-specified figures, no calculations and constructions are needed.
From the finished parts of the square, you can add not only polygons, but also make a right-angled or equilateral triangle, regular pentagon or hexagon, three or five squares, etc.
In the language of geometry, this means: to find those geometric constructions with the help of which the square is cut, and to prove that the required figure can be made from the obtained parts.
This formulation of the question immediately turns each puzzle into a more interesting, but also more difficult geometric problem of "cutting" figures. The originality of this kind of problems is in their certain uncertainty. For example, let us formulate the puzzle from the book "Fascinating Mathematics" by I. Lehmann as the following geometric problem: to show how to divide a given square with straight-line cuts so that by arranging the parts obtained, it would be possible to make three solid squares equal to each other.
In this problem, nothing is said about how to cut a given square and into how many parts - hence the uncertainty.
It is desirable that the number of cuts be as small as possible, although this number is not known in advance, and it is also unknown whether it can be established by any preliminary calculations. Usually, the number of divisions depends on the cutting method, that is, on those geometric constructions that were used to solve the problem.
In search of the smallest number of divisions, you can use a variety of construction techniques and thereby obtain different solutions to the same problem of cutting a given figure. Thus, when solving such problems, a wide opportunity opens up for the manifestation of resourcefulness and initiative, the development of geometric intuition.
2.2 HOW ABUL WEFA MADE A SQUARE FROM THREE EQUAL SQUARES
The tasks of transforming one figure into another by rearranging the cut parts have been dealt with in ancient times. They arose from the needs of practitioners, land surveyors and builders architectural structures the ancient world... Practical techniques and rules appeared that were not substantiated by evidence, and it is natural that many of them were incorrect, erroneous.
One of the most remarkable Arab mathematicians, Abul Vefa, who lived in the 10th century, solved a number of problems related to the geometric transformation of figures. In the essay "The Book of geometric constructions”, Which has come down to us not completely, in the lists of his students, Abul Vefa writes:
“In this book we will deal with the decomposition of figures; This question is necessary for many practitioners and is the subject of their special research. We come to such questions when we need to decompose the squares so that we get smaller squares, or when we need to make a large square out of several squares. In view of this, we will give the basic principles that relate to these issues, since all the methods used by the workers, not based on any principles, are not trustworthy and very erroneous; meanwhile, on the basis of such methods, they perform different actions. "
At one of the meetings of geometers and practitioners Abul Vefe, a problem was proposed:
Make a square of three equal squares.
Abul Vefa cut squares I and II along the diagonals and attached each of the halves to square III, as shown in Fig. 4. 
Fig. 4
Then he connected the vertices E, F, G and H by line segments. The resulting quadrilateral EFGH turned out to be the required square.
The proof immediately follows from the equality of the formed small triangles HLK, EKD and the rest of the same (HL = ED; the angles HLK and EDK are at 45є and the angles HKL and EKD are equal).
The above solution, according to Abul Vefa, "accurately and at the same time satisfies the practitioners."
2.3 POSSIBILITY OF SQUARE TRANSFORMATIONS
Solving puzzles and tasks on transforming a square into another figure of the same size by cutting or, conversely, some polygon into a square, thereby establishing the possibility of such a transformation.
Questions arise as to how far this ability of a square can be reshaped into another figure without any loss of area.
Is it possible to reshape a square into any desired polygon of the same area, or, which is the same, - can any polygon be reshaped into a square of the same size?
The answer to these questions is given by the following theorem:
Any polygon can be turned into a square of the same size. This theorem is considered only for simple polygons.
In the book by V.F. Kagan "On the Transformation of Polyhedra" describes in detail the proof of F. Bol'an's theorem.
The main steps in the proof of the theorem on the possibility of transforming a polygon into a square can be formulated in the form of several lemmas:
1. Any polygon can be cut into a certain number of triangles.
2. Any triangle is scissor-congruent with some parallelogram (two polygons are said to be scissor-congruent if one of them can be cut into parts that, when folded differently, give the second polygon.
Thus, each of the triangles into which the polygon is cut can be turned into a parallelogram.
Further:
3. Any parallelogram can be turned into a square.
4. If two polygons separately can be turned into the third, then the first can be turned into the second ("transitivity property").
Lemmas 2, 3, and 4 imply the fifth:
5. Any triangle can be turned into a square of the same size.
6. Every two squares can be turned into one.
Turning every two squares into one, you end up with one square, which will be equidistant with this polygon.
This is the proof of the possibility of transforming a polygon into a square, which is described in the book by V.F. Kagan.
CHAPTER ΙΙΙ. 3.1 CONSTRUCTIONS BY BENDING A SQUARE PAPER SHEET
Among the many possible actions with paper, a special place is occupied by the operation of folding it. One of the advantages of this operation is that it can be performed without having any additional tools at hand - no ruler, no compass, or even a pencil. By folding the paper, you can not only make funny or interesting toys, but also get a visual idea of many figures on a plane, as well as their properties.
The practical properties of paper give rise to a peculiar geometry. The role of lines in this geometry will be played by the edges of the sheet and folds formed at its folds, and the role of points will be the vertices of the corners of the sheet and the points of intersection of the folds with each other or with the edges of the sheet. It turns out that the possibilities of folding the sheet are very great. There is no doubt that they include all the geometry of one ruler, but to a certain extent they also conceal the capabilities of a compass, although they do not allow direct circular arcs. 
a) b)
Let's explore some properties of the square. The fold line passing through two opposite corners of a square is the diagonal of that square. The other diagonal is obtained by folding the square through another pair of opposite corners, as shown in Figure 5a (the lines inside the square are the fold lines). Each diagonal divides the square into two coinciding triangles when superimposed, the apexes of which are in opposite corners of the square. These triangles are isosceles and rectangular, since each of them has a right angle.
If you fold the paper square in half so that one side coincides with the opposite to it. You will get a fold passing through the center of the square (Figure 5b). The line of this fold has the following properties:
1) it is perpendicular to the other two sides of the square,
2) divides these sides in half,
3) parallel to the first two sides of the square,
4) itself is divided in half in the center of the square,
5) divides the square into two coinciding rectangles when superimposed, 6) each of these rectangles is equal in size (that is, equal in area) to one of the triangles into which the square is divided by the diagonal.
If you fold the square again so that the other two sides coincide, then the resulting fold and the one made earlier will divide the square into 4 matching squares when overlapping.
Using these properties, you can perform various constructions and transformations. For example, get a regular hexagon. Figure 6a shows a sample of an ornament of equilateral triangles and hexagons obtained by folding a square sheet of paper. These many other constructions are described in detail in the book "Apply Mathematics" by I.N. Sergeeva. 
a) b)
Fig. 6.
It is possible to divide a hexagon into equal regular hexagons and equilateral triangles, making bends through the points dividing its sides into three equal parts. The result is a beautiful symmetrical ornament. Also, by folding a square sheet of paper, you can build the bisector of the corner. 
Fig. 7
Bend the paper along straight BC and AB (not on the front side), and then bend the folded edge BC with the folded edge AB. The resulting fold BD will be the bisector of the angle ABC. (Fig. 7)
By folding a square piece of paper, rather complex constructions can be made. For example, produce “ golden ratio»The sides of a given square piece of paper using only folds.
By the way, on the basis of folding a square piece of paper, the art of origami arose - folding paper figures (Fig. 8). Ancient art came from China, from where Japan drew spiritual wealth. The square acts as an original constructor; it is transformed endlessly. 
CHAPTER ΙV. 4.1 TANGRAM AND OTHER PUZZLES,
RELATED TO SQUARE.
History of the Tangram puzzle: 
Puzzle "Tangram" - a square cut into 7 parts of which various silhouettes are formed. It appeared in China at the end of the eighteenth century (figure). The first image of it (1780) was found on a woodcut by the Japanese artist Utamaro, where two girls put together figurines "chi chao tu" - this is the name of the tashram in his homeland (translated as a mental puzzle of seven parts "). The name tangram originated in Europe more likely all from the word "tan" (in Cantonese - Chinese) and the often encountered Greek root "gram" (letter). a legend from start to finish invented by inventive puzzle author Sam Loyd.
Probably, these parts of the square originally served to demonstrate figures, because you can easily make up a rectangle, parallelogram, trapezoid, etc. from parts of a square. Over time, it was noticed that many silhouette figures can be made from these parts (Fig. 9) the most bizarre shape, using all seven parts of the square to compose each figure. The image is schematic, but the image is easily guessed from the main characteristic features object, its structure, proportional to the ratio of parts and form. Silhouettes are difficult to compose. First you need to find the similarity of elements with objects, letters, etc. Then you can create silhouettes of toys, furniture, vehicles, animals.
This is how the fascinating tangram puzzle game was created, which has become widespread, especially in its homeland - in China. There this game is known as widely as, for example, we have chess. Even special drafting competitions are arranged with the least amount of time.
Drawings composed of parts of a tangram: 
Fig. 9
Pentamino This game was invented in the 50s of the twentieth century. American mathematician S. Golomb. It consists in adding different shapes from a given set of pentominoes. The set contains 12 figures, each of which is made up of 5 identical squares.
CONCLUSION
The square is an inexhaustible figure that is used in many fields and has properties that are interesting for anyone who wants to expand the scope of their geometric representations.
As a result of the work done, several conclusions can be formulated:
1) the perimeter of a square is less than the perimeter of any rectangle of the same size;
2) the area of a square is greater than the area of any rectangle with the same perimeter;
3) with the help of cuts, you can make the transformation of various polygons into a square. It was found that exercises in cutting a square and constructing figures from the resulting parts are not only useful geometric fun, but also have practical meaning: they can help future and present innovators of production, in rational cutting of materials, in the use of scraps of leather, fabric, wood, etc. etc., to turn them into useful things;
4) by bending a square sheet of paper, you can perform various constructions without having at hand any tools - not a ruler, not a compass, not even a pencil;
5) there are entertaining games that use a square.
LIST OF USED LITERATURE
1) B.A. Kordemsky, N.V. Rusalev "Amazing Square". Moscow-Leningrad, 1952
2) V.F. Kagan "On the transformation of polyhedra". Gostekhizdat, 1933
3) G. Steinhaus "Mathematical Kaleidoscope". Gostekhizdat, 1949
4) E.I. Ignatiev "In the kingdom of ingenuity." Moscow "Science", 1981
5) Z.A. Mikhailova "Game entertaining tasks for preschoolers ". Moscow "Education", 1990
6) I. Lehman "Fascinating mathematics". Moscow "Science" 1978
7) I.N. Sergeev "Apply Mathematics". Moscow "Science", 1989
8) "Kvant" 1989. No. 5 - P. 40.
9) R. Honsberger "Mathematical highlights". Moscow "Science", 1992
10) Ya.I. Perelman "Living Mathematics". Moscow "Science", 1977
11) Ya.I Perelman "Entertaining geometry". Moscow "AST", 2003
It is noteworthy that the very word "tangram" is actually old English word, composed of two parts - "tan" - Chinese and "gram" - in Greek "letter". In China, the game is called Chi-Chao-Tu (7 cunning figures).
The essence of this puzzle is the addition of 7 geometric shapes tanrama of various silhouettes, as well as in inventing new ones. Imagine, it is calculated that 7000 different combinations can be made from tangram elements. When solving the puzzle, only 2 rules must be observed: first, all 7 tangram figures must be used, and second, the figures must not overlap each other.
What is the use of tangram?
Folding in tangram patterns contributes to the development of perseverance, attention, imagination, logical thinking, helps to create a whole from parts and foresee the result of their activities, teaches to follow the rules and act according to the instructions. All these skills are necessary for a child during school, and in adulthood.
Tangram: schemes for younger students
Small children are better off offering simple and interesting schemes tangram, such as animal silhouettes. We offer to bring a cat, carp, camel, fox, turkey and duck together with the children. Please note that one picture can be changed quite a bit by moving several figures, and the assembled animal changes position, that is, it seems to come to life.
Kitty
Carp and camel
Chanterelle
Duck and turkey
For you detailed description schemes of tangram with the image of a hare.
1. The first figurine of our hare will start from the head - a square. Attach the ears to the head: a medium-sized triangle and a parallelogram. We will make the body from 2 large triangles, and the legs from small ones.
2. Our bunny got scared of something and changed its shape: pressed its ears, folded its legs. Lay out the body from 2 large triangles, connecting them in the form of a parallelogram. We attach a head from a square to the body, and ears from a parallelogram to the head. It remains to make legs from 2 small and 1 medium triangle.
3. The hare stopped being afraid and decided to look out from behind the bush: he pricked up his ears (parallelogram and middle triangle), and he also had a tail - a small triangle.
And this is what a fox looks like chasing a hare.
Tangram schemes for high school students
A fifth-grader can already boldly tackle more complex tangram schemes - images of people in motion. Also, children of this age will surely like the intricate silhouettes of numbers and letters.
Tangram develops abstract thinking well, therefore it will be useful for preschoolers who are preparing for school and.
Tangram in design
Adults can not only play tangram with children but also go further - use the technique of this puzzle in design. You can decorate the interior in an original and beautiful way bookshelves in the form of tangram figures.
Embody your most interesting ideas, it all depends only on your imagination.