There are an infinite number of forms. Shape is the external outline of an object.
The study of shapes can begin from early childhood, drawing your child’s attention to the world around us, which consists of shapes (a plate is round, a TV is rectangular).
From the age of two, a child should know three simple figures– circle, square, triangle. At first he should just show them when you ask. And at three years old, you can already name them yourself and distinguish a circle from an oval, a square from a rectangle.
The more exercises a child does to consolidate shapes, the more new shapes he will remember.
The future first-grader must know all the simple geometric shapes and be able to make applications from them.
What do we call a geometric figure?
A geometric figure is a standard with which you can determine the shape of an object or its parts.
Figures are divided into two groups: flat figures, three-dimensional figures.
We call plane figures those figures that are located in the same plane. These include circle, oval, triangle, quadrangle (rectangle, square, trapezoid, rhombus, parallelogram) and all kinds of polygons.
Three-dimensional figures include: sphere, cube, cylinder, cone, pyramid. These are those shapes that have height, width and depth.
Follow two simple tips when explaining geometric shapes:
- Patience. What seems simple and logical to us, adults, will seem simply incomprehensible to a child.
- Try drawing shapes with your child.
- A game. Start learning shapes in a playful way. Good exercises for consolidation and learning flat shapes– applications from geometric shapes. For voluminous ones, you can use ready-made store-bought games, and also choose applications where you can cut out and glue a voluminous shape.
Geometric figure- a set of points on a surface (often on a plane) that forms a finite number of lines.
The main geometric figures on the plane are dot And straight line. A segment, a ray, a broken line are the simplest geometric shapes on a plane.
Dot- the smallest geometric figure that is the basis of other figures in any image or drawing.
Each one is more complex geometric figure there are many points that have a certain property, characteristic only for this figure.
Straight line, or straight - this is an infinite set of points located on the 1st line, which has no beginning and end. On a sheet of paper you can only see part of a straight line, because... it has no limit.
The straight line is depicted like this:
A part of a straight line that is bounded on both sides by points is called segment straight or segment. He is depicted like this:
Ray is a directed half-line that has a starting point and has no end. The beam is depicted like this:
If you put a point on a straight line, then this point will split the straight line into 2 oppositely directed rays. These rays are called additional.
broken line- several segments that are connected to each other in such a way that the end of the 1st segment turns out to be the beginning of the 2nd segment, and the end of the 2nd segment is the beginning of the 3rd segment, and so on, with neighboring ones (which have 1 thing in common) point) the segments are located on different straight lines. When the end of the last segment does not coincide with the beginning of the 1st, then this broken line will be called open:
When the end of the last segment of a broken line coincides with the beginning of the 1st, it means that this broken line will be closed. An example of a closed polyline is any polygon:
Four-link closed broken line - quadrilateral (rectangle):
Three-link closed broken line -
Little children are ready to learn everywhere and always. Their young brain is able to capture, analyze and remember so much information that is difficult even for an adult. What parents should teach their children has generally accepted age limits.
Children should learn basic geometric shapes and their names between the ages of 3 and 5 years.
Since all children are learning differently, these boundaries are only conditionally accepted in our country.
Geometry is the science of shapes, sizes and arrangement of figures in space. It may seem like it's difficult for kids. However, the objects of study of this science are all around us. This is why having basic knowledge in this area is important for both children and elders.
To get children interested in learning geometry, you can use funny pictures. Additionally, it would be nice to have aids that the child can touch, feel, trace, color, and recognize with his eyes closed. The main principle of any activities with children is to keep their attention and develop a craving for the subject using gaming techniques and a relaxed, fun atmosphere.
The combination of several means of perception will do its job very quickly. Use our mini-tutorial to teach your child to distinguish geometric shapes and know their names.
The circle is the very first of all shapes. In nature, many things around us are round: our planet, the sun, the moon, the core of a flower, many fruits and vegetables, the pupils of the eyes. A volumetric circle is a ball (ball, ball)
It is better to start studying the shape of a circle with your child by looking at drawings, and then reinforce the theory with practice by letting the child hold something round in his hands.
A square is a shape in which all sides have the same height and width. Square objects - cubes, boxes, house, window, pillow, stool, etc.
It is very easy to build all sorts of houses from square cubes. It’s easier to draw a square on a checkered piece of paper.
A rectangle is a relative of a square, which differs in that it has equal opposite sides. Just like a square, a rectangle's angles are all 90 degrees.
You can find many objects shaped like a rectangle: cabinets, household appliances, doors, furniture.
In nature, mountains and some trees have a triangle shape. From the immediate environment of children, we can cite as an example the triangular roof of a house and various road signs.
Some ancient structures, such as temples and pyramids, were built in the shape of a triangle.
An oval is a circle elongated on both sides. For example, eggs, nuts, many vegetables and fruits, a human face, galaxies, etc. have an oval shape.
An oval in volume is called an ellipse. Even the Earth is flattened at the poles - elliptical.
Rhombus
A rhombus is the same square, only elongated, that is, it has two obtuse angles and a pair of acute ones.
You can study a rhombus with the help of visual aids - a drawn picture or a three-dimensional object.
Memorization techniques
Geometric figures It's easy to remember the names. You can turn their study into a game for children by applying the following ideas:
- Buy a children's picture book that has fun and colorful drawings of shapes and their analogies from the world around them.
- Cut out a lot of different figures from multi-colored cardboard, laminate them with tape and use them as construction sets - you can create a lot of interesting combinations by combining different figures.
- Buy a ruler with holes in the shape of a circle, square, triangle and others - for children who are already familiar with pencils, drawing with such a ruler is a very interesting activity.
You can think of many ways to teach kids to know the names of geometric shapes. All methods are good: drawings, toys, observations of surrounding objects. Start small, gradually increasing the complexity of the information and tasks. You will not feel how time flies, and the baby will definitely please you with success in the near future.
Geometric figure defined as any set of points.
If all the points of a geometric figure belong to one plane, it is called flat. For example, a segment, a rectangle are flat figures. There are figures that are not flat. This is, for example, a cube, a ball, a pyramid.
Since the concept of a geometric figure is defined through the concept of set, we can say that one figure is included in another (or contained in another), we can consider the union, intersection and difference of figures.
A point is an undefined concept. A dot is usually introduced by drawing it or piercing it with the tip of a pen in a piece of paper. It is believed that a point has neither length, nor width, nor area.
Line– an indefinable concept. The line is introduced by modeling it from a cord or drawing it on a board or on a sheet of paper. The main property of a straight line: a straight line is infinite. Curved lines can be closed or open.
Ray- this is a part of a straight line, limited on one side.
Line segment- part of a line enclosed between two points - the ends of a segment.
Broken- a line of segments connected in series at an angle to each other. The link of the broken line is a segment. The connection points of the links are called the vertices of the broken line.
Corner is a geometric figure that consists of a point and two rays emanating from this point. The rays are called the sides of the angle, and their common beginning is its vertex. An angle is designated in different ways: either its vertex, or its sides, or three points are indicated: the vertex and two points on the sides of the angle.
An angle is called developed if its sides lie on the same straight line. An angle that is half a straight angle is called a right angle. An angle less than a right angle is called acute. An angle greater than a right angle but less than a straight angle is called an obtuse angle.
Two angles are called adjacent if they have one common side, and the other sides of these angles are additional half-lines.
Triangle- one of the simplest geometric figures. A triangle is a geometric figure that consists of three points that do not lie on the same line and three pairwise segments connecting them. In any triangle, the following elements are distinguished: sides, angles, altitudes, bisectors, medians, midlines.
A triangle is called acute if all its angles are acute. Rectangular - a triangle that has a right angle. A triangle that has an obtuse angle is called obtuse. Triangles are called congruent if their corresponding sides and corresponding angles are equal. In this case, the corresponding angles must lie opposite the corresponding sides. A triangle is called isosceles if its two sides are equal. These equal sides are called lateral, and the third side is called the base of the triangle.
Quadrangle is a figure that consists of four points and four consecutive segments connecting them, and no three of these points should lie on the same line, and the segments connecting them should not intersect. These points are called the vertices of the quadrilateral, and the segments connecting them are called sides.
A diagonal is a line segment connecting opposite vertices of a polygon.
Rectangle is a quadrilateral whose angles are all right angles.
Quadrato m is a rectangle whose sides are all equal.
Polygon A simple closed broken line is called if its neighboring links do not lie on the same straight line. The vertices of the broken line are called the vertices of the polygon, and its links are called its sides. Segments connecting non-adjacent ones are called diagonals.
Circumference called a figure that consists of all points of the plane equidistant from a given point, which is called the center. But since in primary school it's not a given classic definition, acquaintance with the circle is carried out by demonstration, linking it with the direct practical activity of drawing a circle using a compass. The distance from the points to its center is called the radius. A line connecting two circle points, is called a chord. The chord passing through the center is called the diameter.
Circle-part of a plane bounded by a circle.
Parallelepiped– a prism whose base is a parallelogram.
Cube is a rectangular parallelepiped, all edges of which are equal.
Pyramid- a polyhedron in which one face (it is called the base) is some kind of polygon, and the remaining faces (they are called lateral) are triangles with a common vertex.
Cylinder- a geometric body formed by segments of all parallel straight lines enclosed between two parallel planes, intersecting a circle in one of the planes, and perpendicular to the planes of the bases. A cone is a body formed by all the segments connecting a given point - its top - with points of a certain circle - the base of the cone.
Ball– a set of points in space located from a given point at a distance not greater than some given positive distance. This point is the center of the ball, and this distance is the radius.
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Introduction
Geometry is one of the most important components of mathematical education, necessary for the acquisition of specific knowledge about space and practically significant skills, the formation of a language for describing objects in the surrounding world, for the development of spatial imagination and intuition, mathematical culture, as well as for aesthetic education. The study of geometry contributes to the development logical thinking, formation of proof skills.
The 7th grade geometry course systematizes knowledge about the simplest geometric figures and their properties; the concept of equality of figures is introduced; the ability to prove the equality of triangles using the studied signs is developed; a class of problems involving construction using a compass and ruler is introduced; one of the most important concepts is introduced - the concept of parallel lines; new interesting and important properties triangles; one of the most important theorems in geometry is considered - the theorem on the sum of the angles of a triangle, which allows us to classify triangles by angles (acute, rectangular, obtuse).
During classes, especially when moving from one part of the lesson to another, changing activities, the question arises of maintaining interest in classes. Thus, relevant The question arises about using tasks in geometry classes that involve the condition of a problem situation and elements of creativity. Thus, purpose This study is to systematize tasks of geometric content with elements of creativity and problem situations.
Object of study: Geometry tasks with elements of creativity, entertainment and problem situations.
Research objectives: Analyze existing geometry tasks aimed at developing logic, imagination and creative thinking. Show how you can develop interest in a subject using entertaining techniques.
Theoretical and practical significance of the research is that the collected material can be used in the process of additional lessons in geometry, namely at Olympiads and competitions in geometry.
Scope and structure of the study:
The study consists of an introduction, two chapters, a conclusion, a bibliography, contains 14 pages of main typewritten text, 1 table, 10 figures.
Chapter 1. FLAT GEOMETRIC FIGURES. BASIC CONCEPTS AND DEFINITIONS
1.1. Basic geometric figures in the architecture of buildings and structures
There are many material objects in the world around us. different forms and sizes: residential buildings, car parts, books, jewelry, toys, etc.
In geometry, instead of the word object, they say geometric figure, while dividing geometric figures into flat and spatial. In this work, we will consider one of the most interesting sections of geometry - planimetry, in which only plane figures are considered. Planimetry(from Latin planum - “plane”, ancient Greek μετρεω - “measure”) - a section of Euclidean geometry that studies two-dimensional (single-plane) figures, that is, figures that can be located within the same plane. A flat geometric figure is one in which all points lie on the same plane. Any drawing made on a sheet of paper gives an idea of such a figure.
But before considering flat figures, it is necessary to get acquainted with simple but very important figures, without which flat figures simply cannot exist.
The simplest geometric figure is dot. This is one of the main figures of geometry. It is very small, but it is always used for building various forms on surface. The point is the main figure for absolutely all constructions, even the highest complexity. From a mathematical point of view, a point is an abstract spatial object that does not have such characteristics as area or volume, but at the same time remains a fundamental concept in geometry.
Straight- one of the fundamental concepts of geometry. In a systematic presentation of geometry, a straight line is usually taken as one of the initial concepts, which is only indirectly determined by the axioms of geometry (Euclidean). If the basis for constructing geometry is the concept of distance between two points in space, then a straight line can be defined as a line along which the path is equal to the distance between two points.
Lines in space can occupy various provisions, let's look at some of them and give examples found in the architectural appearance of buildings and structures (Table 1):
Table 1
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Parallel lines |
Properties of parallel lines |
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If the lines are parallel, then their projections of the same name are parallel: |
Essentuki, mud bath building (photo by the author) |
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Intersecting lines |
Properties of intersecting lines |
Examples in the architecture of buildings and structures |
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Intersecting lines have a common point, that is, the intersection points of their projections of the same name lie on a common connection line: |
"Mountain" buildings in Taiwan https://www.sro-ps.ru/novosti_otrasli/2015_11_11_pervoe_zdanie_iz_grandioznogo_proekta_big_v_tayvane |
|
|
Crossing lines |
Properties of skew lines |
Examples in the architecture of buildings and structures |
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Straight lines that do not lie in the same plane and are not parallel to each other are intersecting. None is a common communication line. If intersecting and parallel lines lie in the same plane, then intersecting lines lie in two parallel planes. |
Robert, Hubert - Villa Madama near Rome https://gallerix.ru/album/Hermitage-10/pic/glrx-172894287 |
1.2. Flat geometric shapes. Properties and Definitions
Observing the forms of plants and animals, mountains and river meanders, landscape features and distant planets, man borrowed from nature its correct forms, sizes and properties. Material needs prompted people to build houses, make tools for labor and hunting, sculpt dishes from clay, and so on. All this gradually contributed to the fact that man came to understand the basic geometric concepts.
Quadrilaterals:
Parallelogram(ancient Greek παραλληλόγραμμον from παράλληλος - parallel and γραμμή - line, line) is a quadrilateral whose opposite sides are pairwise parallel, that is, they lie on parallel lines.
Signs of a parallelogram:
A quadrilateral is a parallelogram if one of the following holds true: following conditions: 1. If in a quadrilateral the opposite sides are equal in pairs, then the quadrilateral is a parallelogram. 2. If in a quadrilateral the diagonals intersect and are divided in half by the point of intersection, then this quadrilateral is a parallelogram. 3. If two sides of a quadrilateral are equal and parallel, then this quadrilateral is a parallelogram.
A parallelogram whose angles are all right angles is called rectangle.
A parallelogram in which all sides are equal is called diamond
Trapezoid— It is a quadrilateral in which two sides are parallel and the other two sides are not parallel. Also, a trapezoid is a quadrilateral in which one pair of opposite sides is parallel, and the sides are not equal to each other.
Triangle is the simplest geometric figure formed by three segments that connect three points that do not lie on the same straight line. These three points are called vertices triangle, and the segments are sides triangle. It is precisely because of its simplicity that the triangle was the basis of many measurements. Land surveyors, when calculating land areas, and astronomers, when finding distances to planets and stars, use the properties of triangles. This is how the science of trigonometry arose - the science of measuring triangles, of expressing the sides through its angles. The area of any polygon is expressed through the area of a triangle: it is enough to divide this polygon into triangles, calculate their areas and add the results. True, it was not immediately possible to find the correct formula for the area of a triangle.
The properties of the triangle were especially actively studied in XV-XVI centuries. Here is one of the most beautiful theorems of that time, due to Leonhard Euler:
A huge amount of work on the geometry of the triangle, carried out in the XY-XIX centuries, created the impression that everything was already known about the triangle.
Polygon - it is a geometric figure, usually defined as a closed polyline.
Circle- the geometric locus of points in the plane, the distance from which to a given point, called the center of the circle, does not exceed a given non-negative number, called the radius of this circle. If radius equal to zero, then the circle degenerates into a point.
Exists a large number of geometric shapes, they all differ in parameters and properties, sometimes surprising with their shapes.
In order to better remember and distinguish flat figures by properties and characteristics, I came up with a geometric fairy tale, which I would like to present to your attention in the next paragraph.
Chapter 2. PUZZLES FROM FLAT GEOMETRIC FIGURES
2.1.Puzzles for constructing a complex figure from a set of flat geometric elements.
After studying flat shapes, I wondered if there were any interesting problems with flat shapes that could be used as games or puzzles. And the first problem I found was the Tangram puzzle.
This is a Chinese puzzle. In China it is called "chi tao tu", or a seven-piece mental puzzle. In Europe, the name “Tangram” most likely arose from the word “tan”, which means “Chinese” and the root “gram” (Greek - “letter”).
First you need to draw a 10 x 10 square and divide it into seven parts: five triangles 1-5 , square 6 and parallelogram 7 . The essence of the puzzle is to use all seven pieces to put together the figures shown in Fig. 3.
Fig.3. Elements of the game "Tangram" and geometric shapes
Fig.4. Tangram tasks
It is especially interesting to compose from flat figures“shaped” polygons, knowing only the outlines of objects (Fig. 4). I came up with several of these outline tasks myself and showed these tasks to my classmates, who happily began to solve the tasks and created many interesting polyhedral figures, similar to the outlines of objects in the world around us.
To develop imagination, you can also use such forms of entertaining puzzles as tasks for cutting and reproducing given figures.
Example 2. Cutting (parqueting) tasks may seem, at first glance, to be quite diverse. However, most of them use only a few basic types of cuts (usually those that can be used to create another from one parallelogram).
Let's look at some cutting techniques. In this case, we will call the cut figures polygons.
Rice. 5. Cutting techniques
Figure 5 shows geometric shapes from which you can assemble various ornamental compositions and create an ornament with your own hands.
Example 3. Another interesting task that you can come up with on your own and exchange with other students, and whoever collects the most cut pieces is declared the winner. There can be quite a lot of tasks of this type. For coding, you can take all existing geometric shapes, which are cut into three or four parts.
Fig. 6. Examples of cutting tasks:
------ - recreated square; - cut with scissors;
Basic figure
2.2. Equal-sized and equally-composed figures
Let's consider another interesting technique for cutting flat figures, where the main “heroes” of the cuts will be polygons. When calculating the areas of polygons, a simple technique called the partitioning method is used.
In general, polygons are called equiconstituted if, after cutting the polygon in a certain way F into a finite number of parts, it is possible, by arranging these parts differently, to form a polygon H from them.
This leads to the following theorem: Equilateral polygons have the same area, so they will be considered equal in area.
Using the example of equipartite polygons, we can consider such an interesting cutting as the transformation of a “Greek cross” into a square (Fig. 7).
Fig.7. Transformation of the "Greek Cross"
In the case of a mosaic (parquet) composed of Greek crosses, the parallelogram of the periods is a square. We can solve the problem by superimposing a mosaic made of squares onto a mosaic formed with the help of crosses, so that the congruent points of one mosaic coincide with the congruent points of the other (Fig. 8).
In the figure, the congruent points of the mosaic of crosses, namely the centers of the crosses, coincide with the congruent points of the “square” mosaic - the vertices of the squares. By moving the square mosaic in parallel, we will always obtain a solution to the problem. Moreover, the problem has several possible solutions if color is used when composing the parquet ornament.
Fig.8. Parquet made from a Greek cross
Another example of equally proportioned figures can be considered using the example of a parallelogram. For example, a parallelogram is equivalent to a rectangle (Fig. 9).
This example illustrates the partitioning method, which consists in calculating the area of a polygon by trying to divide it into a finite number of parts in such a way that these parts can be used to create a simpler polygon whose area we already know.
For example, a triangle is equivalent to a parallelogram having the same base and half the height. From this position the formula for the area of a triangle is easily derived.
Note that the above theorem also holds converse theorem: if two polygons are equal in size, then they are equivalent.
This theorem, proven in the first half of the 19th century. Hungarian mathematician F. Bolyai and German officer and a mathematics lover P. Gervin, can be represented in this way: if there is a cake in the shape of a polygon and a polygonal box of a completely different shape, but the same area, then you can cut the cake into a finite number of pieces (without turning them cream side down), that they can be placed in this box.
Conclusion
In conclusion, I would like to note that there are quite a lot of problems on flat figures in various sources, but those that were of interest to me were the ones on the basis of which I had to come up with my own puzzle problems.
After all, by solving such problems, you can not only accumulate life experience, but also acquire new knowledge and skills.
In puzzles, when constructing actions-moves using rotations, shifts, translations on a plane or their compositions, I got independently created new images, for example, polyhedron figures from the game “Tangram”.
It is known that the main criterion for the mobility of a person’s thinking is the ability, through reconstructive and creative imagination, to perform certain actions within a set period of time, and in our case, moves of figures on a plane. Therefore, studying mathematics and, in particular, geometry at school will give me even more knowledge to later apply in my future professional activities.
Bibliography
1. Pavlova, L.V. Non-traditional approaches to teaching drawing: tutorial/ L.V. Pavlova. - Nizhny Novgorod: NSTU Publishing House, 2002. - 73 p.
2. encyclopedic Dictionary young mathematician / Comp. A.P. Savin. - M.: Pedagogy, 1985. - 352 p.
3.https://www.srops.ru/novosti_otrasli/2015_11_11_pervoe_zdanie_iz_grandioznogo_proekta_big_v_tayvane
4.https://www.votpusk.ru/country/dostoprim_info.asp?ID=16053
Annex 1
Questionnaire for classmates
1. Do you know what a Tangram puzzle is?
2. What is a “Greek cross”?
3. Would you be interested to know what “Tangram” is?
4. Would you be interested to know what a “Greek cross” is?
22 8th grade students were surveyed. Results: 22 students do not know what “Tangram” and “Greek cross” are. 20 students would be interested in learning how to use the Tangram puzzle, consisting of seven flat figures, to obtain a more complex figure. The survey results are summarized in a diagram.
Appendix 2
Elements of the game "Tangram" and geometric shapes
Transformation of the "Greek Cross"