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Lesson type: combined.

Lesson Objectives:

• Consider axial, central and mirror symmetries as properties of some geometric shapes.
• Learn to build symmetrical points and recognize shapes that have axial symmetry and central symmetry.
• Improve problem solving skills.

Lesson objectives:

• Formation of spatial representations of students.
• Developing the ability to observe and reason; development of interest in the subject through the use of information technology.
• Raising a person who knows how to appreciate the beautiful.

Lesson equipment:

• Use of information technologies (presentation).
• Drawings.
• Homework cards.

During the classes

I. Organizational moment.

Inform the topic of the lesson, formulate the objectives of the lesson.

II. Introduction.

What is symmetry?

The outstanding mathematician Hermann Weyl highly appreciated the role of symmetry in modern science: "Symmetry, no matter how broadly or narrowly we understand this word, is an idea with which a person tried to explain and create order, beauty and perfection."

We live in a very beautiful and harmonious world. We are surrounded by objects that please the eye. For example, a butterfly, a maple leaf, a snowflake. Look how beautiful they are. Did you pay attention to them? Today we will touch this beautiful mathematical phenomenon - symmetry. Let's get acquainted with the concept of axial, central and mirror symmetries. We will learn to build and define figures that are symmetrical about the axis, center and plane.

The word "symmetry" in Greek sounds like "harmony", meaning beauty, proportionality, proportionality, the sameness in the arrangement of parts. Since ancient times, man has used symmetry in architecture. It gives harmony and completeness to ancient temples, towers of medieval castles, modern buildings.

In the most general form, "symmetry" in mathematics means such a transformation of space (plane) in which each point M goes to another point M" relative to some plane (or line) a, when the segment MM" is perpendicular to the plane (or line) a and split it in half. The plane (straight line) a is called the plane (or axis) of symmetry. The fundamental concepts of symmetry include the plane of symmetry, the axis of symmetry, the center of symmetry. The plane of symmetry P is a plane that divides the figure into two mirror equal parts, located relative to each other in the same way as an object and its mirror reflection.

III. Main part. Symmetry types.

Central symmetry

Symmetry about a point or central symmetry is such a property of a geometric figure, when any point located on one side of the center of symmetry corresponds to another point located on the other side of the center. In this case, the points are on a straight line segment passing through the center, dividing the segment in half.

Practical task.

1. Given points BUT, AT and M M relative to the middle of the segment AB.
2. Which of the following letters have a center of symmetry: A, O, M, X, K?
3. Do they have a center of symmetry: a) a segment; b) beam; c) a pair of intersecting lines; d) square?

Axial symmetry

Symmetry with respect to a straight line (or axial symmetry) is such a property of a geometric figure, when any point located on one side of the straight line will always correspond to a point located on the other side of the straight line, and the segments connecting these points will be perpendicular to the axis of symmetry and divide it in half.

Practical task.

1. Given two points BUT and AT, symmetric with respect to some straight line, and a point M. Construct a point symmetrical to a point M about the same line.
2. Which of the following letters have an axis of symmetry: A, B, D, E, O?
3. How many axes of symmetry does: a) a segment; b) straight line; c) beam?
4. How many axes of symmetry does the drawing have? (see fig. 1)

Mirror symmetry

points BUT and AT are called symmetric with respect to the plane α (plane of symmetry) if the plane α passes through the midpoint of the segment AB and perpendicular to this segment. Each point of the plane α is considered symmetrical to itself.

Practical task.

1. Find the coordinates of the points into which the points A (0; 1; 2), B (3; -1; 4), C (1; 0; -2) pass with: a) central symmetry about the origin; b) axial symmetry about the coordinate axes; c) mirror symmetry with respect to coordinate planes.
2. Does the right glove go into the right or left glove with mirror symmetry? axial symmetry? central symmetry?
3. The figure shows how the number 4 is reflected in two mirrors. What will be seen in place of the question mark if the same is done with the number 5? (see fig. 2)
4. The figure shows how the word KANGAROO is reflected in two mirrors. What happens if you do the same with the number 2011? (see fig. 3)

Rice. 2

It is interesting.

Symmetry in nature.

Almost all living beings are built according to the laws of symmetry, it is not without reason that the word "symmetry" translated from Greek means "proportion".

Among colors, for example, rotational symmetry is observed. Many flowers can be rotated so that each petal takes the position of its neighbor, the flower is aligned with itself. The minimum angle of such a rotation for different colors is not the same. For iris, it is 120°, for bluebell - 72°, for narcissus - 60°.

In the arrangement of leaves on the stems of plants, helical symmetry is observed. Being located by a screw along the stem, the leaves, as it were, spread out in different directions and do not obscure each other from the light, although the leaves themselves also have an axis of symmetry. Considering the general plan of the structure of any animal, we usually notice a well-known regularity in the arrangement of parts of the body or organs that repeat around a certain axis or occupy the same position in relation to a certain plane. This correctness is called the symmetry of the body. The phenomena of symmetry are so widespread in the animal world that it is very difficult to point out a group in which no symmetry of the body can be noticed. Both small insects and large animals have symmetry.

Symmetry in inanimate nature.

Among the infinite variety of forms of inanimate nature, such perfect images are found in abundance, whose appearance invariably attracts our attention. Observing the beauty of nature, one can notice that when objects are reflected in puddles, lakes, mirror symmetry appears (see Fig. 4).

Crystals bring the charm of symmetry to the world of inanimate nature. Each snowflake is a small crystal of frozen water. The shape of snowflakes can be very diverse, but they all have rotational symmetry and, in addition, mirror symmetry.

It is impossible not to see the symmetry in faceted gemstones. Many cutters try to shape their diamonds into a tetrahedron, cube, octahedron, or icosahedron. Since the garnet has the same elements as the cube, it is highly prized by gem connoisseurs. Garnet art objects were found in the tombs of ancient Egypt dating back to the pre-dynastic period (over two millennia BC) (see Fig. 5).

In the collections of the Hermitage, the gold jewelry of the ancient Scythians enjoys special attention. Unusually fine art work of gold wreaths, diadems, wood and decorated with precious red-violet garnets.

One of the most obvious uses of the laws of symmetry in life are the structures of architecture. This is what we see most often. In architecture, symmetry axes are used as a means of expressing architectural intent (see Figure 6). In most cases, patterns on carpets, fabrics, and room wallpapers are symmetrical about the axis or center.

Another example of a person using symmetry in his practice is technique. In engineering, axes of symmetry are most clearly indicated where deviation from zero is required, such as on the steering wheel of a truck or on the steering wheel of a ship. Or one of the most important inventions of mankind, having a center of symmetry, is a wheel, also a propeller and other technical means have a center of symmetry.

"Look in the mirror!"

Should we think that we see ourselves only in a "mirror image"? Or, at best, can we find out how we “really” look only on photos and film? Of course not: it is enough to reflect the mirror image a second time in the mirror in order to see your true face. Trills come to the rescue. They have one large main mirror in the center and two smaller mirrors on the sides. If such a side mirror is placed at a right angle to the average, then you can see yourself exactly in the form in which others see you. Close your left eye, and your reflection in the second mirror will repeat your movement with your left eye. Before trellis, you can choose whether you want to see yourself in a mirror image or in a direct image.

It is easy to imagine what confusion would reign on Earth if the symmetry in nature were broken!

Rice. four	Rice. 5	Rice. 6

IV. Fizkultminutka.

• « lazy eights» – activate the structures that provide memorization, increase the stability of attention.
  Draw the number eight in the air in a horizontal plane three times, first with one hand, then with both hands at once.
• « Symmetrical drawings » - improve hand-eye coordination, facilitate the process of writing.
  Draw symmetrical patterns in the air with both hands.

V. Independent work of a verification nature.

Ι option

ΙΙ option

1. In the rectangle MPKH O is the intersection point of the diagonals, RA and BH are the perpendiculars drawn from the vertices P and H to the line MK. It is known that MA = OB. Find the angle ROM.
2. In the rhombus MPKH, the diagonals intersect at a point O. On the sides MK, KH, PH, points A, B, C are taken, respectively, AK = KV = PC. Prove that OA = OB and find the sum of the angles ROS and MOA.
3. Construct a square along a given diagonal so that two opposite vertices of this square lie on different sides of a given acute angle.

VI. Summing up the lesson. Evaluation.

• What types of symmetry did you get acquainted with in the lesson?
• What two points are said to be symmetrical about a given line?
• Which figure is said to be symmetrical with respect to a given line?
• What two points are said to be symmetrical with respect to the given point?
• Which figure is said to be symmetrical with respect to a given point?
• What is mirror symmetry?
• Give examples of figures that have: a) axial symmetry; b) central symmetry; c) both axial and central symmetry.
• Give examples of symmetry in animate and inanimate nature.

VII. Homework.

1. Individual: complete by applying axial symmetry (see fig. 7).

Rice. 7

2. Construct a figure symmetrical to the given one with respect to: a) a point; b) straight line (see Fig. 8, 9).

Rice. eight	Rice. 9

3. Creative task: "In the world of animals." Draw a representative from the animal world and show the axis of symmetry.

VIII. Reflection.

• What did you like about the lesson?
• What material was the most interesting?
• What difficulties did you encounter while completing the task?
• What would you change during the lesson?

Goals:

• educational:
  • give an idea of ​​\u200b\u200bsymmetry;
  • introduce the main types of symmetry in the plane and in space;
  • develop strong skills in constructing symmetrical figures;
  • expand ideas about famous figures by introducing them to the properties associated with symmetry;
  • show the possibilities of using symmetry in solving various problems;
  • consolidate the acquired knowledge;
• general education:
  • learn to set yourself up for work;
  • teach to control oneself and a neighbor on the desk;
  • to teach how to evaluate yourself and a neighbor on your desk;
• developing:
  • activate independent activity;
  • develop cognitive activity;
  • learn to summarize and systematize the information received;
• educational:
  • educate students "a sense of shoulder";
  • cultivate communication;
  • inculcate a culture of communication.

DURING THE CLASSES

In front of each are scissors and a sheet of paper.

Exercise 1(3 min).

- Take a sheet of paper, fold it in half and cut out some figure. Now unfold the sheet and look at the fold line.

Question: What is the function of this line?

Suggested answer: This line divides the figure in half.

Question: How are all the points of the figure located on the two resulting halves?

Suggested answer: All points of the halves are at an equal distance from the fold line and at the same level.

- So, the fold line divides the figure in half so that 1 half is a copy of 2 halves, i.e. this line is not simple, it has a remarkable property (all points relative to it are at the same distance), this line is the axis of symmetry.

Task 2 (2 minutes).

- Cut out a snowflake, find the axis of symmetry, characterize it.

Task 3 (5 minutes).

- Draw a circle in your notebook.

Question: Determine how the axis of symmetry passes?

Suggested answer: Differently.

Question: So how many axes of symmetry does a circle have?

Suggested answer: A lot of.

- That's right, the circle has many axes of symmetry. The same wonderful figure is the ball (spatial figure)

Question: What other figures have more than one axis of symmetry?

Suggested answer: Square, rectangle, isosceles and equilateral triangles.

– Consider three-dimensional figures: a cube, a pyramid, a cone, a cylinder, etc. These figures also have an axis of symmetry. Determine how many axes of symmetry a square, rectangle, equilateral triangle and the proposed three-dimensional figures have?

I distribute the halves of plasticine figures to the students.

Task 4 (3 min).

- Using the information received, finish the missing part of the figure.

Note: the figurine can be both flat and three-dimensional. It is important that students determine how the axis of symmetry goes and fill in the missing element. The correctness of the execution is determined by the neighbor on the desk, evaluates how well the work has been done.

A line is laid out from a lace of the same color on the desktop (closed, open, with self-crossing, without self-crossing).

Task 5 (group work 5 min).

- Visually determine the axis of symmetry and, relative to it, complete the second part from a lace of a different color.

The correctness of the work performed is determined by the students themselves.

The students are presented with elements of drawings

Task 6 (2 minutes).

Find the symmetrical parts of these drawings.

To consolidate the material covered, I propose the following tasks, provided for 15 minutes:

Name all equal elements of the triangle KOR and KOM. What are the types of these triangles?

2. Draw in a notebook several isosceles triangles with a common base equal to 6 cm.

3. Draw a segment AB. Construct a line perpendicular to segment AB and passing through its midpoint. Mark points C and D on it so that the quadrilateral ACBD is symmetrical with respect to the line AB.

- Our initial ideas about the form belong to a very distant era of the ancient Stone Age - the Paleolithic. For hundreds of thousands of years of this period, people lived in caves, in conditions that differed little from the life of animals. People made tools for hunting and fishing, developed a language to communicate with each other, and in the late Paleolithic era, they decorated their existence by creating works of art, figurines and drawings, which reveal a wonderful sense of form.
When there was a transition from simple gathering of food to its active production, from hunting and fishing to agriculture, humanity enters a new stone age, the Neolithic.
Neolithic man had a keen sense of geometric form. The firing and coloring of clay vessels, the manufacture of reed mats, baskets, fabrics, and later metal processing developed ideas about planar and spatial figures. Neolithic ornaments were pleasing to the eye, revealing equality and symmetry.
Where is symmetry found in nature?

Suggested answer: wings of butterflies, beetles, tree leaves…

“Symmetry can also be seen in architecture. When constructing buildings, builders clearly adhere to symmetry.

That's why the buildings are so beautiful. Also an example of symmetry is a person, animals.

Homework:

1. Come up with your own ornament, depict it on an A4 sheet (you can draw it in the form of a carpet).
2. Draw butterflies, mark where there are elements of symmetry.

If you think for a moment and imagine any object in your imagination, then in 99% of cases the figure that comes to mind will be of the correct form. Only 1% of people, or rather their imagination, will draw an intricate object that looks completely wrong or disproportionate. This is rather an exception to the rule and refers to unconventionally thinking individuals with a special view of things. But returning to the absolute majority, it is worth saying that a significant proportion of the correct items still prevail. The article will deal exclusively with them, namely, the symmetrical drawing of those.

Image of the right subjects: just a few steps to the finished drawing

Before you start drawing a symmetrical object, you need to select it. In our version, it will be a vase, but even if it does not in any way resemble what you decided to depict, do not despair: all the steps are absolutely identical. Follow the sequence and you'll be fine:

1. All regularly shaped objects have a so-called central axis, which, when drawing symmetrically, should definitely be highlighted. To do this, you can even use a ruler and draw a straight line in the center of the album sheet.
2. Next, carefully look at your chosen object and try to transfer its proportions to a piece of paper. It is not difficult to do this if, on both sides of the line drawn in advance, outline light strokes, which will subsequently become the outlines of the object being drawn. In the case of a vase, it is necessary to highlight the neck, bottom and the widest part of the body.
3. Do not forget that symmetrical drawing does not tolerate inaccuracies, so if there are some doubts about the intended strokes, or you are not sure about the correctness of your own eye, double-check the pending distances with a ruler.
4. The last step is to connect all the lines together.

Symmetric drawing available to computer users

Due to the fact that most of the objects around us have the correct proportions, in other words, are symmetrical, the developers of computer applications have created programs in which absolutely everything can be easily drawn. You just need to download them and enjoy the creative process. However, remember, the machine will never be a substitute for a sharpened pencil and album sheet.

Today we will talk about a phenomenon that each of us constantly encounter in life: about symmetry. What is symmetry?

Approximately we all understand the meaning of this term. The dictionary says: symmetry is the proportionality and full correspondence of the arrangement of parts of something relative to a line or point. There are two types of symmetry: axial and radial. Let's look at the axis first. This is, let's say, "mirror" symmetry, when one half of the object is completely identical to the second, but repeats it as a reflection. Look at the halves of the sheet. They are mirror symmetrical. The halves of the human body (full face) are also symmetrical - the same arms and legs, the same eyes. But let's not be mistaken, in fact, in the organic (living) world, absolute symmetry cannot be found! The halves of the sheet do not copy each other perfectly, the same applies to the human body (look at it for yourself); the same is true of other organisms! By the way, it is worth adding that any symmetrical body is symmetrical relative to the viewer in only one position. It is necessary, say, to turn the sheet, or raise one hand, and what? - see for yourself.

People achieve true symmetry in the products of their labor (things) - clothes, cars ... In nature, it is characteristic of inorganic formations, for example, crystals.

But let's move on to practice. It’s not worth starting with complex objects like people and animals, let’s try to finish the mirror half of the sheet as the first exercise in a new field.

Draw a symmetrical object - lesson 1

Let's try to make it as similar as possible. To do this, we will literally build our soul mate. Do not think that it is so easy, especially the first time, to draw a mirror-corresponding line with one stroke!

Let's mark several reference points for the future symmetrical line. We act like this: we draw with a pencil without pressure several perpendiculars to the axis of symmetry - the middle vein of the sheet. Four or five is enough. And on these perpendiculars we measure to the right the same distance as on the left half to the line of the edge of the leaf. I advise you to use the ruler, do not really rely on the eye. As a rule, we tend to reduce the drawing - it has been noticed in experience. We do not recommend measuring distances with your fingers: the error is too large.

Connect the resulting points with a pencil line:

Now we look meticulously - are the halves really the same. If everything is correct, we will circle it with a felt-tip pen, clarify our line:

The poplar leaf has been completed, now you can swing at the oak one.

Let's draw a symmetrical figure - lesson 2

In this case, the difficulty lies in the fact that the veins are marked and they are not perpendicular to the axis of symmetry, and not only the dimensions but also the angle of inclination will have to be exactly observed. Well, let's train the eye:

So a symmetrical oak leaf was drawn, or rather, we built it according to all the rules:

How to draw a symmetrical object - lesson 3

And we will fix the topic - we will finish drawing a symmetrical leaf of lilac.

He also has an interesting shape - heart-shaped and with ears at the base you have to puff:

Here is what they drew:

Look at the resulting work from a distance and evaluate how accurately we managed to convey the required similarity. Here's a tip for you: look at your image in the mirror, and it will tell you if there are any mistakes. Another way: bend the image exactly along the axis (we have already learned how to bend correctly) and cut the leaf along the original line. Look at the figure itself and at the cut paper.

TRIANGLES.

§ 17. SYMMETRY RELATIVELY DIRECT.

1. Figures symmetrical to each other.

Let's draw some figure on a sheet of paper with ink, and with a pencil outside it - an arbitrary straight line. Then, without letting the ink dry, fold the sheet of paper along this straight line so that one part of the sheet overlaps the other. On this other part of the sheet, the imprint of this figure will thus be obtained.

If you then straighten the sheet of paper again, then there will be two figures on it, which are called symmetrical relative to this straight line (Fig. 128).

Two figures are called symmetrical with respect to some straight line if they are combined when the plane of the drawing is folded along this straight line.

The line with respect to which these figures are symmetrical is called their axis of symmetry.

It follows from the definition of symmetrical figures that all symmetrical figures are equal.

You can get symmetrical figures without using the bending of the plane, but with the help of a geometric construction. Let it be required to construct a point C", symmetrical to a given point C with respect to the straight line AB. Let us drop the perpendicular from point C
CD to the straight line AB and on its continuation we set aside the segment DC "= DC. If we bend the plane of the drawing along AB, then the point C will coincide with the point C": points C and C "are symmetrical (Fig. 129).

Suppose now it is required to construct a segment C "D" symmetrical to a given segment CD with respect to the straight line AB. Let's build points C "and D", symmetrical to points C and D. If we bend the plane of the drawing along AB, then points C and D will coincide with points C "and D" (Fig. 130), respectively. Therefore, the segments CD and C "D" will coincide , they will be symmetrical.

Let us now construct a figure symmetrical to the given polygon ABCD with respect to the given axis of symmetry MN (Fig. 131).

To solve this problem, we drop the perpendiculars A a, AT b, FROM With, D d and E e on the axis of symmetry MN. Then, on the extensions of these perpendiculars, we set aside the segments
a A" = A a, b B" = B b, With C" \u003d Cs; d D""=D d and e E" = E e.

The polygon A "B" C "D" E "will be symmetrical to the polygon ABCD. Indeed, if the drawing is bent along the straight line MN, then the corresponding vertices of both polygons will coincide, which means that the polygons themselves will also coincide; this proves that the polygons ABCD and A" B"C"D"E" are symmetrical with respect to the straight line MN.

2. Figures consisting of symmetrical parts.

Often there are geometric figures that are divided by some straight line into two symmetrical parts. Such figures are called symmetrical.

So, for example, an angle is a symmetrical figure, and the bisector of the angle is its axis of symmetry, since when it is bent along it, one part of the angle is combined with the other (Fig. 132).

In a circle, the axis of symmetry is its diameter, since when bending along it, one semicircle is combined with another (Fig. 133). In the same way, the figures in the drawings 134, a, b are symmetrical.

Symmetrical figures are often found in nature, construction, and jewelry. The images placed on the drawings 135 and 136 are symmetrical.

It should be noted that symmetrical figures can be combined by simple movement along the plane only in some cases. To combine symmetrical figures, as a rule, it is necessary to turn one of them upside down,