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Lesson type: combined.

Lesson objectives:

• Consider axial, central and mirror symmetries as properties of some geometric figures.
• Teach to construct symmetrical points and recognize figures with axial symmetry and central symmetry.
• Improve problem solving skills.

Lesson objectives:

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

Lesson equipment:

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

During the classes

I. Organizational moment.

Inform the topic of the lesson, formulate the goals 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 the word, is an idea with the help of which man has 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. Have you paid attention to them? Today we will touch on this wonderful mathematical phenomenon - symmetry. Let's get acquainted with the concept of axial, central and mirror symmetries. We will learn to build and identify figures that are symmetrical relative to the axis, center and plane.

The word “symmetry” translated from Greek sounds like “harmony”, meaning beauty, proportionality, proportionality, uniformity in the arrangement of parts. Man has long used symmetry in architecture. It gives harmony and completeness to ancient temples, towers of medieval castles, and modern buildings.

In the most general view"symmetry" in mathematics is understood as 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 is divided in half by it . The plane (straight line) a is called the plane (or axis) of symmetry. The fundamental concepts of symmetry include plane of symmetry, axis of symmetry, center of symmetry. A plane of symmetry P is a plane that divides a figure into two mirror-like equal parts, located relative to each other in the same way as an object and its mirror image.

III. Main part. Types of symmetry.

Central symmetry

Symmetry about a point or central symmetry is 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 located on a straight line segment passing through the center, dividing the segment in half.

Practical task.

1. Points are given A, IN 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 about a line (or axial symmetry) is a property of a geometric figure when any point located on one side of the line will always correspond to a point located on the other side of the line, and the segments connecting these points will be perpendicular to the axis of symmetry and divided by it in half.

Practical task.

1. Given two points A And IN, symmetrical with respect to some line, and a point M. Construct a point symmetrical to the point M relative to 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 have? b) straight; c) beam?
4. How many axes of symmetry does the drawing have? (see Fig. 1)

Mirror symmetry

Points A And IN are called symmetrical relative to the plane α (plane of symmetry) if the plane α passes through the middle of the segment AB and perpendicular to this segment. Each point of the α plane is considered symmetrical to itself.

Practical task.

1. Find the coordinates of the points to which points A (0; 1; 2), B (3; -1; 4), C (1; 0; -2) go with: a) central symmetry relative to the origin; b) axial symmetry relative to the coordinate axes; c) mirror symmetry relative 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 visible in place of the question mark if the same is done with the number 5? (see Fig. 2)
4. The picture 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

This is interesting.

Symmetry in living nature.

Almost all living beings are built according to the laws of symmetry; it is not for nothing that the word “symmetry” means “proportionality” when translated from Greek.

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

There is helical symmetry in the arrangement of leaves on plant stems. Positioned like a screw along the stem, the leaves seem to 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 certain regularity in the arrangement of body parts or organs, which are repeated around a certain axis or occupy the same position in relation to a certain plane. This regularity is called body symmetry. The phenomena of symmetry are so widespread in the animal world that it is very difficult to indicate 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, you can notice that when objects are reflected in puddles and 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.

One cannot help but see symmetry in faceted gemstones. Many cutters try to give diamonds the shape of a tetrahedron, cube, octahedron or icosahedron. Since the garnet has the same elements as the cube, it is highly valued by experts. precious stones. Art products of garnets were found in graves Ancient Egypt, dating back to the predynastic period (over two millennia BC) (see Fig. 5).

In the Hermitage collections special attention used gold jewelry of the ancient Scythians. Extraordinarily thin artwork golden wreaths, tiaras, wood and decorated with precious red-violet garnets.

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

Another example of a person using symmetry in his practice is technology. In engineering, symmetry axes are most clearly designated where it is necessary to estimate the deviation from the zero position, for example, on the steering wheel of a truck or on the steering wheel of a ship. Or one of the most important inventions of mankind that has a center of symmetry is the wheel; the propeller and other technical means also have a center of symmetry.

"Look in the mirror!"

Should we think that we only see ourselves in a “mirror image”? Or in best case scenario Only in photographs and film can we find out what we “really” look like? Of course not: it is enough to reflect the mirror image a second time in the mirror to see your true face. Trellis come to the rescue. They have one large main mirror in the center and two smaller mirrors on the sides. If you place such a side mirror at right angles to the middle one, 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 the 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 kind of confusion would reign on Earth if the symmetry in nature were broken!

Rice. 4	Rice. 5	Rice. 6

IV. Physical education minute.

• « Lazy Eights» – activate structures that ensure memorization, increase 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 and facilitate the writing process.
  Draw symmetrical patterns in the air with both hands.

V. Independent testing work.

Ι option

ΙΙ option

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

VI. Summing up the lesson. Assessment.

• What types of symmetry did you learn about in class?
• Which two points are called symmetrical with respect to a given line?
• Which figure is called symmetrical with respect to a given line?
• Which two points are said to be symmetrical about a given point?
• Which figure is called symmetrical about 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 living and inanimate nature.

VII. Homework.

1. Individual: complete it 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 (see Fig. 8, 9).

Rice. 8	Rice. 9

3. Creative task: “In the animal world.” 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 most interesting?
• What difficulties did you encounter when completing this or that task?
• What would you change during the lesson?

This pair of means determines the location of the elements of the composition relative to the main axis. If it is the same, then the composition appears as symmetrical; if there is a slight deviation to the side, then the composition is disymmetrical. With such a significant deviation, it becomes asymmetrical.

Very often, symmetry, like asymmetry, is expressed in the juxtaposition of several compositional axes. The simplest case is the relationship between the main axis and its subordinate axes, which determine the position of the secondary parts of the composition. If the secondary axes diverge significantly from the main axis, the composition may collapse. To achieve its integrity, various techniques are used: bringing the axes closer together, merging them, adopting a common direction. Figure 17 shows formal compositions (schemes) built on their basis.

Figure 17 - Compositions with different axes of symmetry

Practical task

1 Create a symmetrical composition (different types of symmetry) (Appendix A, Figures 15-16).

2 Create an asymmetrical composition (Appendix A, Figure 17).

Requirements:

7-10 search variants of the composition are performed;

pay close attention to the arrangement of elements; When implementing the main idea, take care of the accuracy of execution.

Pencil, ink, watercolor, colored pencils. Sheet format – A3.

Equilibrium

A correctly constructed composition is balanced.

Equilibrium- this is the placement of composition elements in which each item is in a stable position. There is no doubt about its location and no desire to move it along the pictorial plane. This does not require an exact mirror match between the right and left sides. The quantitative ratio of tonal and color contrasts of the left and right parts of the composition should be equal. If in one part there are more contrasting spots, it is necessary to strengthen the contrast ratios in the other part or weaken the contrasts in the first. You can change the outlines of objects by increasing the perimeter of contrasting relationships.

To establish balance in the composition, the shape, direction, and location of the visual elements are important (Figure 18).

Figure 18 - Balance of contrasting spots in the composition

An unbalanced composition looks random and unreasonable, causing a desire to further work on it (rearrange elements and their details) (Figure 19).

Figure 19 - Balanced and unbalanced composition

A properly constructed composition cannot cause doubts or feelings of uncertainty. It should have a clarity of relationships and proportions that soothes the eye.

Let's consider the simplest schemes for constructing compositions:

Figure 20 – Schemes of composition balance

Image A is balanced. In the combination of his squares and rectangles of various sizes and proportions, life is felt, you don’t want to change or add anything, there is a compositional clarity of proportions.

You can compare the stable vertical line in Figure 20, A with the oscillating one in Figure 20, B. The proportions in Figure B are based on small differences that make it difficult to determine their equivalence, to understand what is depicted - a rectangle or a square.

In Figure 20, B, each disc individually appears unbalanced. Together they form a pair that is at rest. In Figure 20, D, the same pair looks completely unbalanced, because shifted relative to the axes of the square.

There are two types of equilibrium.

Static balance occurs when figures are symmetrically arranged on a plane relative to the vertical and horizontal axes of the format of a composition of symmetrical shape (Figure 21).

Figure 21 - Static equilibrium

Dynamic equilibrium occurs when figures are asymmetrically arranged on a plane, i.e. when they are shifted to the right, left, up, down (Figure 22).

Figure 22 - Dynamic equilibrium

In order for the figure to appear depicted in the center of the plane, it needs to be moved slightly upward relative to the format axes. The circle located in the center appears to be shifted downwards, this effect is enhanced if the bottom of the circle is painted in dark color(Figure 23).

Figure 23 – Balance of the circle

A large figure on the left side of the plane is able to balance a small contrasting element on the right, which is active due to its tonal relationship with the background (Figure 24).

Figure 24 – Balance of large and small elements

Practical task

1 Create a balanced composition using any motifs (Appendix A, Figure 18).

2 Perform an unbalanced composition (Appendix A, Figure 19).

Requirements:

perform search options (5-7 pcs.) in achromatic design with finding tonal relationships;

the work must be neat.

Material and dimensions of the composition

Mascara. Sheet format – A3.

If you think for a minute and imagine any object in your mind, then in 99% of cases the figure that comes to mind will be of the correct shape. 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 correct items still prevails. The article will talk exclusively about them, namely about symmetrical drawing of them.

Drawing the right objects: just a few steps to the finished drawing

Before you start drawing symmetrical object, you need to select it. In our version it will be a vase, but even if it doesn’t in any way resemble what you decided to depict, don’t despair: all the steps are absolutely identical. Follow the sequence and everything will work out:

1. All objects of regular shape have a so-called central axis, which should definitely be highlighted when drawing symmetrically. To do this, you can even use a ruler and draw a straight line down the center of the landscape sheet.
2. Next, look carefully at the item you have chosen and try to transfer its proportions onto a sheet of paper. This is not difficult to do if you mark light strokes on both sides of the line drawn in advance, which will later 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 of the correctness of your own eye, double-check the laid down distances with a ruler.
4. The last step is connecting all the lines together.

Symmetrical drawing is available to computer users

Due to the fact that most of the objects around us have the correct proportions, in other words, they are symmetrical, computer application developers have created programs in which you can easily draw absolutely everything. Just download them and enjoy creative process. However, remember, a machine will never be a substitute for a sharpened pencil and a sketchbook.

Goals:

• educational:
  • give an idea of ​​symmetry;
  • introduce the main types of symmetry on the plane and in space;
  • develop strong skills in constructing symmetrical figures;
  • expand your understanding of famous figures by introducing properties associated with symmetry;
  • show the possibilities of using symmetry in solving various problems;
  • consolidate acquired knowledge;
• general education:
  • teach yourself how to prepare yourself for work;
  • teach how to control yourself and your desk neighbor;
  • teach to evaluate yourself and your desk neighbor;
• developing:
  • intensify independent activity;
  • develop cognitive activity;
  • learn to summarize and systematize the information received;
• educational:
  • develop a “shoulder sense” in students;
  • cultivate communication skills;
  • instill a culture of communication.

DURING THE CLASSES

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

Exercise 1(3 min).

- Let's take a sheet of paper, fold it into pieces and cut out some figure. Now let's unfold the sheet and look at the fold line.

Question: What function does this line serve?

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 on equal distance from the fold line and at the same level.

– This means that 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 an 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 goes?

Suggested answer: Differently.

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

Suggested answer: A lot of.

– That’s right, a circle has many axes of symmetry. An equally remarkable figure is a ball (spatial figure)

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

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

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

I distribute halves of plasticine figures to students.

Task 4 (3 min).

– Using the information received, complete the missing part of the figure.

Note: the figure can be both planar and three-dimensional. It is important that students determine how the axis of symmetry runs and complete the missing element. The correctness of the work is determined by the neighbor at the desk and evaluates how correctly the work was done.

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

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.

Elements of drawings are presented to students

Task 6 (2 minutes).

– Find the symmetrical parts of these drawings.

To consolidate the material covered, I suggest the following tasks, scheduled for 15 minutes:

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

2. Draw several isosceles triangles in your notebook with a common base of 6 cm.

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

– Our initial ideas about form date back to the 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 little different from the life of animals. People made tools for hunting and fishing, developed a language to communicate with each other, and during the late Paleolithic era they embellished their existence by creating works of art, figurines and drawings that reveal a remarkable sense of form.
When there was a transition from simple gathering of food to its active production, from hunting and fishing to agriculture, humanity entered a new stone Age, in the Neolithic.
Neolithic man had a keen sense of geometric form. Firing and painting clay vessels, making 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 does symmetry occur in nature?

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

– Symmetry can also be observed in architecture. When constructing buildings, builders strictly adhere to symmetry.

That's why the buildings turn out so beautiful. Also an example of symmetry is humans and animals.

Homework:

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