A point and a straight line are the basic geometric figures on a plane.
The ancient Greek scientist Euclid said: “a point” is something that has no parts.” The word "point" translated from Latin language means the result of an instant touch, a prick. A point is the basis for constructing any geometric figure.
A straight line or simply a straight line is a line along which the distance between two points is the shortest. A straight line is infinite, and it is impossible to depict the entire straight line and measure it.
Points are indicated in capitals with Latin letters A, B, C, D, E, etc., and straight lines are the same letters, but lowercase a, b, c, d, e, etc. A straight line can also be designated by two letters corresponding to the points lying on it. For example, straight line a can be designated AB.
We can say that points AB lie on line a or belong to line a. And we can say that straight line a passes through points A and B.
The simplest geometric figures on a plane are a segment, a ray, a broken line.
A segment is a part of a line that consists of all points of this line, limited by two selected points. These points are the ends of the segment. A segment is indicated by indicating its ends.
A ray or half-line is a part of a line that consists of all points of this line lying on one side of a given point. This point is called the starting point of the half-line or the beginning of the ray. The beam has a starting point, but no end.
Half-straight lines or rays are designated by two lowercase Latin letters: the initial and any other letter, corresponding point belonging to the half-line. In this case, the starting point is placed in the first place.
It turns out that the straight line is infinite: it has neither beginning nor end; a ray has only a beginning, but no end, but a segment has a beginning and an end. Therefore, we can only measure a segment.
Several segments that are sequentially connected to each other so that the segments (neighboring) having one common point are not located on the same straight line, represent broken line.
A broken line can be closed or open. If the end of the last segment coincides with the beginning of the first, we have a closed broken line; if not, it is an open line.
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Lesson topic
What is a geometric figure
Geometric figures are a collection of many points, lines, surfaces or bodies that are located on a surface, plane or space and form a finite number of lines.
The term “figure” is to some extent formally applied to a set of points, but as a rule, a figure is usually called a set that is located on a plane and is limited by a finite number of lines.
A point and a straight line are the basic geometric figures located on a plane.
The simplest geometric figures on a plane include a segment, a ray and a broken line.
What is geometry
Geometry is like this mathematical science, which studies the properties of geometric shapes. If we literally translate the term “geometry” into Russian, it means “land surveying,” since in ancient times the main task of geometry as a science was the measurement of distances and areas on the surface of the earth.
The practical application of geometry is invaluable at all times and regardless of profession. Neither a worker, nor an engineer, nor an architect, nor even an artist can do without knowledge of geometry.
In geometry there is a section that deals with the study various figures on a plane and is called planimetry.
You already know that a figure is an arbitrary set of points located on a plane.
Geometric figures include: point, straight line, segment, ray, triangle, square, circle and other figures that planimetry studies.
Dot
From the material studied above, you already know that the point refers to the main geometric figures. And although this is the smallest geometric figure, it is necessary for constructing other figures on a plane, drawing or image and is the basis for all other constructions. After all, the construction of more complex geometric figures consists of many points characteristic of a given figure.
In geometry, points represent in capital letters Latin alphabet, for example, such as: A, B, C, D....
Now let's summarize, and so, from a mathematical point of view, a point is such an abstract object in space that does not have volume, area, length and other characteristics, but remains one of the fundamental concepts in mathematics. A point is a zero-dimensional object that has no definition. According to Euclid's definition, a point is something that cannot be defined.
Straight
Like a point, a straight line refers to figures on a plane, which has no definition, since it consists of infinite number points located on the same line, which has neither beginning nor end. It can be argued that a straight line is infinite and has no limit.
If a straight line begins and ends with a point, then it is no longer a straight line and is called a segment.
But sometimes a straight line has a point on one side and not on the other. In this case, the straight line turns into a beam.
If you take a straight line and put a point in its middle, then it will split the straight line into two oppositely directed rays. These rays are additional.
If in front of you there are several segments connected to each other so that the end of the first segment becomes the beginning of the second, and the end of the second segment becomes the beginning of the third, etc., and these segments are not on the same straight line and when connected have a common point, then such the chain is a broken line.
Exercise
Which broken line is called unclosed?
How is a straight line designated?
What is the name of a broken line that has four closed links?
What is the name of a broken line with three closed links?
When the end of the last segment of a broken line coincides with the beginning of the 1st segment, then such a broken line is called closed. An example of a closed polyline is any polygon.
Plane
Like a point and a straight line, a plane is a primary concept; it has no definition and one cannot see either a beginning or an end. Therefore, when considering a plane, we consider only that part of it that is limited by a closed broken line. Thus, any smooth surface can be considered a plane. This surface can be a sheet of paper or a table.
Corner
A figure that has two rays and a vertex is called an angle. The junction of the rays is the vertex of this angle, and its sides are the rays that form this angle.
Exercise:
1. How is an angle indicated in the text?
2. What units can you use to measure an angle?
3. What are the angles?
Parallelogram
A parallelogram is a quadrilateral opposing sides which are pairwise parallel.
Rectangle, square and rhombus are special cases of parallelogram.
A parallelogram with right angles equal to 90 degrees is a rectangle.
A square is the same parallelogram; its angles and sides are equal.
As for the definition of a rhombus, it is a geometric figure whose all sides are equal.
In addition, you should know that every square is a rhombus, but not every rhombus can be a square.
Trapezoid
When considering a geometric figure such as a trapezoid, we can say that, in particular, like a quadrilateral, it has one pair of parallel opposite sides and is curvilinear.
Circle and Circle
Circumference - locus points of the plane equidistant from given point, called the center, to a given non-zero distance, called its radius.
Triangle
The triangle you have already studied also belongs to simple geometric figures. This is one of the types of polygons in which part of the plane is limited by three points and three segments that connect these points in pairs. Any triangle has three vertices and three sides.
Exercise: Which triangle is called degenerate?
Polygon
Polygons include geometric shapes different forms, which have a closed broken line.
In a polygon, all points that connect the segments are its vertices. And the segments that make up a polygon are its sides.
Did you know that the emergence of geometry goes back centuries and is associated with the development of various crafts, culture, art and observation of the surrounding world. And the name of geometric figures is confirmation of this, since their terms did not arise just like that, but due to their similarity and similarity.
After all, the term “trapezoid” translated from ancient Greek language from the word “trapezion” means table, meal and other derivative words.
"Cone" comes from Greek word“konos”, which in translation sounds like a pine cone.
“Line” has Latin roots and comes from the word “linum”, translated it sounds like linen thread.
Did you know that if you take geometric figures with the same perimeter, then among them the owner of the most large area turned out to be a circle.
Planimetry is a branch of geometry in which figures on a plane are studied.
Figures studied by planimetry:
3. Parallelogram (special cases: square, rectangle, rhombus)
4. Trapezoid
5. Circumference
6. Triangle
7. Polygon
1) Point:
In geometry, topology and related branches of mathematics, a point is an abstract object in space that has neither volume, area, length, nor any other similar characteristics of large dimensions. Thus, a point is a zero-dimensional object. A point is one of the fundamental concepts in mathematics.
Point in Euclidean geometry:
A point is one of the fundamental concepts of geometry, so “point” has no definition. Euclid defined a point as something that cannot be divided.
A straight line is one of the basic concepts of geometry.
Geometric straight line (straight line) - not closed on both sides, extended and not curved geometric object, cross section which tends to zero, and the longitudinal projection onto the plane gives a point.
In the systematic presentation of geometry, a straight line is usually taken as one of original concepts, which is only indirectly determined by the axioms of geometry.
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 equal to the distance between two points.
3) Parallelogram:
A parallelogram is a quadrilateral whose opposite sides are parallel in pairs, that is, they lie on parallel lines. Special cases of a parallelogram are rectangle, square and rhombus.
Special cases:
Square- a regular quadrilateral or rhombus, in which all angles are right, or a parallelogram, in which all sides and angles are equal.
A square can be defined as: a rectangle whose two adjacent sides are equal;
a rhombus in which all angles are right (any square is a rhombus, but not every rhombus is a square).
Rectangle is a parallelogram in which all angles are right angles (equal to 90 degrees).
Rhombus is a parallelogram in which all sides are equal. A rhombus with right angles is called a square.
4) Trapezoid:
Trapezoid- a quadrilateral with exactly one pair of opposite sides parallel.
1. Trapezium, which sides not equal,
called versatile .
2. A trapezoid whose sides are equal is called isosceles.
3. A trapezoid in which one side makes a right angle with the bases is called rectangular .
The segment connecting the midpoints of the lateral sides of a trapezoid is called midline trapezius (MN). The midline of the trapezoid is parallel to the bases and equal to their half-sum.
A trapezoid can be called a truncated triangle, therefore the names of trapezoids are similar to the names of triangles (triangles can be scalene, isosceles, or right-angled).
5) Circumference:
Circle- the geometric locus of points of the plane equidistant from a given point, called the center, at a given non-zero distance, called its radius.
6) Triangle:
Triangle - simplest polygon having 3 vertices (angles) and 3 sides; part of the plane bounded by three points and three segments connecting these points in pairs.
7) Polygon:
Polygon- this is a geometric figure, defined as a closed broken line. There are three various options definitions:
Flat closed broken lines;
Plane closed polylines without self-intersections;
Parts of the plane bounded by broken lines.
The vertices of the polygon are called the vertices of the polygon, and the segments are called the sides of the polygon.
Basic properties of a line and a point:
1. Whatever the line, there are points that belong to this line and do not belong to it.
Through any two points you can draw a straight line, and only one.
2. Of the three points on a line, one and only one lies between the other two.
3. Each segment has a certain length greater than zero. The length of a segment is equal to the sum of the lengths of the parts into which it is divided by any of its points.
6. On any half-line from its starting point you can plot a segment given length, and only one.
7. From any half-line to a given half-plane you can plot an angle with a given degree measure, less than 180O, and only one.
8. Whatever the triangle, there is an equal triangle in a given location relative to a given half-line.
Properties of a triangle:
Relationships between sides and angles of a triangle:
1) Against larger side lies a larger angle.
2) The larger side lies opposite the larger angle.
3) Against equal sides equal angles lie, and, conversely, equal sides lie opposite equal angles.
The relationship between the internal and external angles of a triangle:
1) The sum of any two internal corners triangle is equal outer corner triangle adjacent to the third angle.
2) The sides and angles of a triangle are also related to each other by relations called the theorem of sines and the theorem of cosines.
The triangle is called obtuse, rectangular or acute-angled , if its largest internal angle is respectively greater than, equal to, or less than 90∘.
Middle line of a triangle is the segment connecting the midpoints of two sides of the triangle.
Properties of the midline of a triangle:
1) The line containing the middle line of the triangle is parallel to the line containing the third side of the triangle.
2) The middle line of the triangle is equal to half of the third side.
3) The midline of a triangle cuts off a similar triangle from a triangle.
Rectangle properties:
1) opposite sides are equal and parallel to each other;
2) the diagonals are equal and bisect at the point of intersection;
3) the sum of the squares of the diagonals is equal to the sum of the squares of all (four) sides;
4) rectangles of the same size can completely cover a plane;
5) a rectangle can be divided into two equal rectangles in two ways;
6) the rectangle can be divided into two equal right triangles;
7) around a rectangle you can describe a circle whose diameter is equal to the diagonal of the rectangle;
8) it is impossible to inscribe a circle in a rectangle (except a square) so that it touches all its sides.
Properties of a parallelogram:
1) The middle of the diagonal of a parallelogram is its center of symmetry.
2) Opposite sides of a parallelogram are equal.
3) Opposite angles of a parallelogram are equal.
4) Each diagonal of a parallelogram divides it into two equal triangles.
5) The diagonals of a parallelogram are bisected by the point of intersection.
6) The sum of the squares of the diagonals of a parallelogram (d1 and d2) is equal to the sum of the squares of all its sides: d21+d22=2(a2+b2)
WITH properties of the square:
1) All angles of a square are right, all sides of a square are equal.
2) The diagonals of a square are equal and intersect at right angles.
3) The diagonals of a square divide its angles in half.
Properties of a rhombus:
1. The diagonal of a rhombus divides it into two equal triangles.
2. The diagonals of a rhombus are divided in half at the point of their intersection.
3. Opposite sides of a rhombus are equal to each other, equal and opposite angles his.
In addition, a rhombus has the following properties:
a) the diagonals of a rhombus are mutually perpendicular;
b) the diagonal of a rhombus divides its angle in half.
Properties of a circle:
1) A straight line may not have common points with a circle; have one common point with the circle (tangent); have two common points with it (secant).
2) Through three points that do not lie on the same line, you can draw a circle, and only one.
3) The point of contact of two circles lies on the line connecting their centers.
Polygon properties:
1) The sum of the internal angles of a plane convex n-gon equal.
2) The number of diagonals of any n-gon is equal.
3).The product of the sides of a polygon and the sine of the angle between them is equal to the area of the polygon.
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Introduction
Geometry is one of essential components mathematics education necessary for acquiring specific knowledge about space and practically significant skills, formation of a language for describing objects of the surrounding world, for the development spatial imagination and intuition, mathematical culture, and also 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 the most important concepts- the concept of parallel lines; new interesting and important properties triangles; one of the the most important theorems in geometry - a theorem on the sum of the angles of a triangle, which allows us to classify triangles according to their 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 problems in geometry classes in which there is a condition problematic 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 research is that the collected material can be used in the process additional classes in geometry, namely at olympiads and geometry competitions.
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
In the world around us, there are many material objects of different shapes and sizes: residential buildings, machine 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 geometry - planimetry, which considers only flat figures. 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 most high 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.
Straight lines in space can occupy different positions; let’s consider some of them and give examples found in the architectural appearance of buildings and structures (Table 1):
Table 1
|
Parallel lines |
Properties of parallel lines |
|
|
If the lines are parallel, then their projections of the same name are parallel: |
Essentuki, mud bath building (photo by the author) |
|
|
Intersecting lines |
Properties of intersecting lines |
Examples in the architecture of buildings and structures |
|
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 |
|
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, dimensions 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 conditions is met: 1. If in a quadrilateral the opposite sides are pairwise equal, 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 in their calculations of areas land plots and astronomers use the properties of triangles to find distances to planets and stars. 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. Is it true, correct formula It was not immediately possible to find it for the area of the 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 on the plane, the distance from which to a given point, called the center of the circle, does not exceed a given one 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 make “shaped” polygons from flat figures, 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 one interesting task, which you can come up with yourself and exchange with other students, and whoever collects the most cut-out figures 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 on final number parts, it is possible, by arranging these parts differently, to form a polygon N 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 note that problems on plane figures are sufficiently represented in various sources, but those that were of interest to me were based on 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 to recreate and creative imagination complete within the specified time period certain actions, and in our case - moves of figures on the 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: Publishing house NSTU, 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 “ 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 get more complex figure. The survey results are summarized in a chart.
Appendix 2
Elements of the game "Tangram" and geometric shapes
Transformation of the "Greek Cross"
2.1. Geometric shapes on a plane
IN last years There has been a tendency towards the inclusion of significant geometric material in initial course mathematics. But in order to introduce students to various geometric shapes and teach them how to depict correctly, he needs an appropriate math training. The teacher must be familiar with the leading ideas of the geometry course, know the basic properties of geometric figures, and be able to construct them.
When depicting a flat figure, no geometric problems arise. The drawing serves either as an exact copy of the original or represents it a similar figure. Looking at the image of a circle in the drawing, we get the same visual impression as if we were looking at the original circle.
Therefore, the study of geometry begins with planimetry.
Planimetry is a branch of geometry in which figures on a plane are studied.
A geometric figure is defined as any set of points.
A segment, a straight line, a circle are geometric shapes.
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 a set, we can say that one figure is included in another; we can consider the union, intersection and difference of figures.
For example, the union of two rays AB and MK is the straight line KB, and their intersection is the segment AM.
There are convex and non-convex figures. A figure is called convex if, together with any two of its points, it also contains a segment connecting them.
Figure F 1 is convex, and figure F 2 is non-convex.
Convex figures are a plane, a straight line, a ray, a segment, and a point. It is not difficult to verify that the convex figure is a circle.
If we continue the segment XY until it intersects with the circle, we get the chord AB. Since the chord is contained in the circle, the segment XY is also contained in the circle, and, therefore, the circle is convex figure.
The basic properties of the simplest figures on the plane are expressed in the following axioms:
1. Whatever the line, there are points that belong to this line and do not belong to it.
Through any two points you can draw a straight line, and only one.
This axiom expresses the basic property of belonging to points and lines on the plane.
2. Of the three points on a line, one and only one lies between the other two.
This axiom expresses the basic property of the location of points on a straight line.
3. Each segment has a certain length greater than zero. The length of a segment is equal to the sum of the lengths of the parts into which it is divided by any of its points.
Obviously, axiom 3 expresses the main property of measuring segments.
This sentence expresses the basic property of the location of points relative to a straight line on a plane.
5. Each angle has a certain degree measure greater than zero. The unfolded angle is 180°. The degree measure of an angle is equal to the sum of the degree measures of the angles into which it is divided by any ray passing between its sides.
This axiom expresses the basic property of measuring angles.
6. On any half-line from its starting point, you can plot a segment of a given length, and only one.
7. From any half-line, into a given half-plane, you can put an angle with a given degree measure less than 180 O, and only one.
These axioms reflect the basic properties of laying out angles and segments.
The basic properties of the simplest figures include the existence of a triangle equal to the given one.
8. Whatever the triangle, there is an equal triangle in a given location relative to a given half-line.
The basic properties of parallel lines are expressed by the following axiom.
9. Through a point not lying on a given line, no more than one straight line parallel to the given one can be drawn on the plane.
Let's look at some geometric shapes that are studied in primary school.
An angle 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 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.
In addition to the concept of an angle given above, in geometry the concept of a plane angle is considered.
A plane angle is a part of a plane bounded by two different rays emanating from one point.
There are two plane angles formed by two rays with common beginning. They are called additional. The figure shows two plane angles with sides OA and OB, one of them is shaded.
Angles can be adjacent or vertical.
Two angles are called adjacent if they have one side in common, and the other sides of these angles are complementary half-lines.
Sum adjacent corners equals 180 degrees.
Two angles are called vertical if the sides of one angle are complementary half-lines of the sides of the other.
Angles AOD and SOV, as well as angles AOS and DOV are vertical.
Vertical angles are equal.
Parallel and perpendicular lines.
Two lines in a plane are called parallel if they do not intersect.
If line a is parallel to line b, then write a II c.
Two lines are called perpendicular if they intersect at right angles.
If line a is perpendicular to line b, then write a b.
Triangles.
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.
Any triangle divides the plane into two parts: internal and external.
In any triangle there is the following elements: sides, angles, heights, bisectors, medians, midlines.
The altitude of a triangle dropped from a given vertex is the perpendicular drawn from this vertex to the line containing the opposite side.
The bisector of a triangle is the segment of the bisector of an angle of a triangle that connects a vertex to a point on opposite side.
The median of a triangle drawn from a given vertex is the segment connecting this vertex with the midpoint of the opposite side.
The midline of a triangle is the segment connecting the midpoints of its two sides.
Quadrilaterals.
A quadrilateral 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 triangle, and the segments connecting them are called its sides.
The sides of a quadrilateral starting from the same vertex are called opposite.
In a quadrilateral ABCD, vertices A and B are adjacent, and vertices A and C are opposite; sides AB and BC are adjacent, BC and AD are opposite; segments AC and WD are the diagonals of this quadrilateral.
Quadrilaterals can be convex or non-convex. Thus, the quadrilateral ABCD is convex, and the quadrilateral KRMT is non-convex.
Among convex quadrilaterals Parallelograms and trapezoids are distinguished.
A parallelogram is a quadrilateral whose opposite sides are parallel.
A trapezoid is a quadrilateral whose only two opposite sides are parallel. These parallel sides are called the bases of the trapezoid. The other two sides are called lateral. The segment connecting the midpoints of the sides is called the midline of the trapezoid.
BC and AD – bases of the trapezium; AB and CD – lateral sides; KM – middle line trapezoids.
Of the many parallelograms, rectangles and rhombuses are distinguished.
A rectangle is a parallelogram whose angles are all right.
A rhombus is a parallelogram in which all sides are equal.
Squares are selected from many rectangles.
A square is a rectangle whose sides are all equal.
Circle.
A circle is a figure that consists of all points of the plane equidistant from a given point, which is called the center.
The distance from the points to its center is called the radius. A segment connecting two points on a circle is called a chord. The chord passing through the center is called the diameter. OA – radius, CD – chord, AB – diameter.
A central angle in a circle is a plane angle with a vertex at its center. The part of the circle located inside the plane angle is called the arc of the circle corresponding to this central corner.
According to new textbooks in new programs M.I. Moreau, M.A. Bantova, G.V. Beltyukova, S.I. Volkova, S.V. In the 4th grade, Stepanova is given construction problems that were not previously included in the elementary school mathematics curriculum. These are tasks such as:
Construct a perpendicular to a line;
Divide the segment in half;
Construct a triangle on three sides;
Build regular triangle, isosceles triangle;
Construct a hexagon;
Construct a square using the properties of the diagonals of a square;
Construct a rectangle using the property of rectangle diagonals.
Let's consider the construction of geometric figures on a plane.
Section of geometry studying geometric constructions, is called constructive geometry. The main concept of constructive geometry is the concept of “constructing a figure.” The main propositions are formed in the form of axioms and are reduced to the following.
1. Each this figure built.
2. If two (or more) figures are constructed, then the union of these figures is also constructed.
3. If two figures are constructed, then you can determine whether their intersection will be empty set or not.
4. If the intersection of two constructed figures is not empty, then it is constructed.
5. If two figures are constructed, then it is possible to determine whether their difference is an empty set or not.
6. If the difference of two constructed figures is not an empty set, then it is constructed.
7. You can draw a point belonging to the constructed figure.
8. You can construct a point that does not belong to the constructed figure.
To construct geometric figures that have some specified properties, use various drawing tools. The simplest of them are: a one-sided ruler (hereinafter simply a ruler), a double-sided ruler, a square, a compass, etc.
Various drawing tools allow you to various formations. The properties of drawing tools used for geometric constructions are also expressed in the form of axioms.
Since in school course Geometry considers the construction of geometric figures using a compass and a ruler; we will also focus on the consideration of the basic constructions performed by these particular drawings with tools.
So, using a ruler you can perform the following geometric constructions.
1. construct a segment connecting two constructed points;
2. construct a straight line passing through two constructed points;
3. construct a ray emanating from the constructed point and passing through the constructed point.
The compass allows you to perform the following geometric constructions:
1. construct a circle if its center and segment have been constructed, equal to radius circles;
2. construct any of two additional arcs of a circle if the center of the circle and the ends of these arcs are constructed.
Elementary construction tasks.
Construction problems are perhaps the most ancient math problems, they help to better understand the properties of geometric shapes and contribute to the development of graphic skills.
The construction task is considered solved if the method for constructing the figure is indicated and it is proven that as a result of execution of these constructions a figure with the required properties is actually obtained.
Let's look at some elementary construction problems.
1. Construct a segment CD on a given straight line equal to this segment AB.
The possibility of construction only follows from the axiom of delaying a segment. This is done using a compass and ruler. in the following way. Let a straight line a and a segment AB be given. We mark a point C on a straight line and construct a circle with a center at a point C with a straight line and denote D. We obtain a segment CD equal to AB.
2. Through this point draw a line perpendicular to a given line.
Let points O and straight line a be given. There are two possible cases:
1. Point O lies on line a;
2. Point O does not lie on line a.
In the first case, we denote a point C that does not lie on line a. From point C as a center we draw a circle of arbitrary radius. Let A and B be its intersection points. From points A and B we describe a circle of the same radius. Let point O be the point of their intersection, different from C. Then the half-line CO is the bisector of the unfolded angle, as well as the perpendicular to the straight line a.
In the second case, from point O as from the center we draw a circle intersecting straight line a, and then from points A and B with the same radius we draw two more circles. Let O be the point of their intersection, lying in a half-plane different from the one in which the point O lies. The straight line OO/ is the perpendicular to the given straight line a. Let's prove it.
Let us denote by C the point of intersection of straight lines AB and OO/. Triangles AOB and AO/B are equal on three sides. Therefore, the angle OAS equal to angle O/AC are equal on both sides and the angle between them. Hence the angles ASO and ASO/ are equal. And since the angles are adjacent, they are right angles. Thus, OS is perpendicular to line a.
3. Through a given point, draw a line parallel to the given one.
Let a line a and a point A outside this line be given. Let's take some point B on line a and connect it to point A. Through point A we draw a line C, forming with AB the same angle that AB forms with a given line a, but on the opposite side from AB. The constructed straight line will be parallel to straight line a, which follows from the equality of the crosswise angles formed at the intersection of straight lines a and with the secant AB.
4. Construct a tangent to the circle passing through a given point on it.
Given: 1) circle X (O, h)
2) point A x
Construct: tangent AB.
Construction.
2. circle X (A, h), where h – arbitrary radius(axiom 1 of the compass)
3. points M and N of the intersection of the circle x 1 and straight line AO, that is (M, N) = x 1 AO (general axiom 4)
4. circle x (M, r 2), where r 2 is an arbitrary radius such that r 2 r 1 (axiom 1 of the compass)
And externally - with his open behavior, and internally - with his mental processes and feelings. Conclusions on the first section For the development of everyone cognitive processes younger schoolchildren must comply following conditions: 1. Educational activities must be purposeful, arouse and maintain constant interest among students; 2. Expand and develop cognitive interests u...
The whole test as a whole, which indicates that their levels of development mental operations comparisons and generalizations are higher than those of low-performing schoolchildren. If we analyze individual data on subtests, then difficulties in answering individual issues talk about poor data skills logical operations. These difficulties are most often encountered among low-performing schoolchildren. This...
Junior schoolboy. Object of study: development imaginative thinking for 2nd grade students high school No. 1025. Method: testing. Chapter 1. Theoretical basis research on imaginative thinking 1.1. The concept of thinking Our knowledge of the surrounding reality begins with sensations and perceptions and moves on to thinking. The function of thinking is to expand the boundaries of knowledge by going beyond...