The right glove goes into the right or left glove. Why gloves are lost: signs and superstitions

Lesson Objectives:

Consolidation of theoretical knowledge on the topic under study;

Improving problem solving skills.

During the classes

I. Organizational moment

II. Actualization of students' knowledge

Frontal work with the class: theoretical survey on the following questions:

1. What is called the movement of space?

2. Give examples of movements.

3. What mapping of space onto itself is called central symmetry?

4. What mapping of space onto itself is called axial symmetry?

5. What is called mirror symmetry?

6. What mapping of space onto itself is called parallel translation?

7. What coordinates does point A have if, with central symmetry with center A, the point, B (1; 0; 2) goes to point C (2; -1; 4). (Answer: A(1.5; -0.5; 3).)

8. How is the plane located with respect to the coordinate axes Ox and Oz, if, with mirror symmetry relative to this plane, the point M (2; 2; 3) goes to the point M1 (2; -2; 3). (Answer: The plane, relative to which the mirror symmetry is considered, at which the point M (2; 2; 3) passes into the point M1 (2; -2; 3), is parallel to the axes Ox and Oz.)

9. Which glove (right or left) does the right glove go into with mirror symmetry? (Answer: to the left), axial symmetry? (Answer: left), central symmetry? (Answer: right).

At the time when frontal work with the class is underway, the student solves problem No. 480 (a) at the blackboard (checking homework).

Problem No. 480 a).

Prove that under central symmetry a plane not passing through the center of symmetry is mapped onto a plane parallel to it.

1) Consider the central symmetry of the space with center O and an arbitrary plane a not passing through the point O (Fig. 1).

Let the line a and b, intersecting at the point A, lie in the plane a. With symmetry with center O, the lines a and b pass, respectively, into parallel lines a1 and b1 (see No. 479 a). In this case, the point A goes to some point A1, which lies both on the line a1 and on the line b1, which means that the lines a1 and b1 intersect.

The intersecting lines define a single plane, that is, the lines a1 and b1 define the plane a1. On the basis of the parallelism of the planes a || a1.

2) Further, it can be proved that under central symmetry with center O, the plane a is mapped onto the plane a1. This can be proved as in problem no. 479 1a), where it was proved that the line AB is mapped onto the line A1B1.

III. Problem solution.

Problem No. 483 a).

With mirror symmetry with respect to the a plane, the β plane is mapped to the β1 plane. Prove that if β || a1, then β1 || a.

Solution: Let us prove the proof by contradiction. Suppose that β || a, but the planes β1 and a intersect. Then they have a common point M. Since M ∈ a, then under the given mirror symmetry the point M is mapped into itself. This implies that the point M, which belongs to the plane β1, also lies in the plane β. But then the planes a and β intersect. The obtained contradiction shows that our proposition is false, therefore, β1 || a.

IV. Independent work (see appendix)

V. Debriefing

Today we consolidated theoretical knowledge on the topic "Movement" and developed the skills to use it in the process of solving problems of various levels of complexity.

Homework

Solve problems: No. 480 (b), 483 (b) (similar ones were considered in the lessons).

Additional tasks:

No. 519 (Indication: consider the linear angles of the dihedral angles formed by the planes a and β, a and β1).

No. 520 (Indication: take two intersecting lines on the plane a and use problem No. 484).

Central symmetry (Fig. 2)

1. Prove that central symmetry is motion.

2. Given a tetrahedron MAVS. Construct a figure centrally symmetrical to this tetrahedron with respect to the point O (Fig. 3).

The slide contains theoretical background material. According to it, you can repeat the theory, conduct a survey of students.

This slide can be used to check the results of independent work (I level).

Mirror symmetry

The a plane coincides with the Oxy plane (Fig. 4).

Points O1 and O2 are the midpoints of segments AA1 and BB1.

1. Prove that mirror symmetry is motion (Fig. 5).

2. Given a tetrahedron MAVS. Construct a figure that is mirror-symmetric to this tetrahedron with respect to the plane β.

Back forward

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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 Weil 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. A 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.

Given points AND, AT and M M relative to the middle of the segment AB.
Which of the following letters have a center of symmetry: A, O, M, X, K?
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 a straight line will always correspond to a point located on the other side of a straight line, and the segments connecting these points will be perpendicular to the axis of symmetry and divide it in half.

Practical task.

Given two points AND 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.
Which of the following letters have an axis of symmetry: A, B, D, E, O?
How many axes of symmetry does: a) a segment; b) straight line; c) beam?
How many axes of symmetry does the drawing have? (see fig. 1)

Mirror symmetry

points AND 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.

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.
Does the right glove go into the right or left glove with mirror symmetry? axial symmetry? central symmetry?
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)
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 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 confusion would reign on Earth if the symmetry in nature were broken!


Rice. four	Rice. five	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 immediately with both hands.
« 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

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.
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.
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. 8	Rice. nine

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?

Base Radius Generators Height Axis Lateral Surface Page

1. The radius of a cylinder is the radius of its base. 2. The bases of a cylinder are its circles. 3. Generators of a cylinder are called segments connecting the points of the circles of its bases. 4. The height of the cylinder is the distance between the bases. 5. The axis of a cylinder is a straight line connecting the centers of its bases. 6. The lateral surface of the cylinder is called its cylindrical surface.

The ends of the segment AB, equal to a, lie on the circles of the base of the cylinder. The radius of the cylinder is r, the height is h, the distance between the straight line AB and the axis OO 1 of the cylinder is d. 1. Explain how to construct a segment whose length is equal to the distance between the intersecting straight lines AB and OO 1 A B O O1O1 ah r C K d 2. Make a plan for finding the value of d for the given values a, h, r. Plan: 1) find AC from ABC, then AK 2) find d from AKO 3. Make a plan for finding the value of h from the given values a, d, r. Plan: 1) find AK from AKO, then AC 2) find BC = h from ABC Task 1.

Problem 2. The plane γ, parallel to the axis of the cylinder, cuts off the arc AmD with the degree measure α from the circumference of the base. The height of the cylinder is h, the distance between the axis of the cylinder and the cutting plane is d. γ D В А С O m α K h 1. Prove that the section of the cylinder by the plane γ is a rectangle. 2. Explain how to construct a segment whose length is equal to the distance between the axis of the cylinder and the cutting plane. 3. Draw up and explain a plan for calculating the cross-sectional area according to α, d, h O1O1

1. A rectangle whose sides are 6 cm and 4 cm rotates around the smaller side. Find the surface area of the body of revolution and the area of its axial section. 2. The axial section of the cylinder is a square, the diagonal of which is 12 cm. Find the surface area of the cylinder.

The height of the cylinder is H, the radius of its base is R. A pyramid is placed in the cylinder, the height of which coincides with the generatrix AA1 of the cylinder, and the base is an isosceles triangle ABC (AB = AC), inscribed in the base of the cylinder. Find the area of the lateral surface of the pyramid if A = 120°. Given: a pyramid is inscribed in a cylinder with height H and radius R, forming AA1 - the height of the pyramid, ABC, AB = AC, ABC - is inscribed in the base of the cylinder, angle A \u003d 120 °. Find: Side of the pyramid. Solution: 1) Draw AD BC and connect the points A 1 and D. According to the theorem, we have A 1 D BC. Since the arc CAB contains 120°, and the arcs AC and AB contain 60° each, then BC = R, AB = R. 2) In ABD we have AD = R/2. Further, from AA 1 D we get A 1 D = ½ Therefore S A1AB = ½ AB AA1 = ½ RH S A1BC = ½ BC A 1 D = ½ R ½ = ¼ R 3) Sside = 2 S A1AB + S A1BC = RH + ¼ R = = R/4(4H +). Answer: R/4(4H+). O O1O1 A A1A1 C B D

The height of the cylinder is 12 cm. A straight line is drawn through the middle of the generatrix of the cylinder, intersecting the axis of the cylinder at a distance of 4 cm from the lower base. This line intersects the plane containing the lower base of the cylinder at a distance of 18 cm from the center of the lower base. Find the radius of the base of the cylinder. M2M2 M1M1 O1O1 O2O2 R BC A Given: cylinder, height O1O2 = 12 cm, B is the middle of the generatrix M1M2, AB intersects O1O2 at point C, CO2 = 4 cm, AO2 = 18 cm. Find: R of the base. Solution: Let's draw a plane through the straight line AB given in the condition of the problem and the axis of the cylinder O 1 O 2. This plane also contains the generatrix M 1 M 2, in which it intersects with the surface of the cylinder. The length of M 1 M 2 is equal to the height of the cylinder, i.e. M 1 M 2 \u003d 12 cm, then according to the condition BM 2 \u003d 6 cm. M 1 M 2 || About 1 About 2, which means that the triangles AVM 2 and ACO 2 also have a common angle A, which means they are similar. From here Answer: 9cm

Topic: Cylinder Tasks 1. The height of the cylinder H, the radius of the base R. The section by a plane parallel to the axis of the cylinder is a square. Find the distance of this section from the axis. 2. The height of the cylinder is 8 cm, the radius is 5 cm. Find the cross-sectional area of the cylinder with a plane parallel to its axis, if the distance between this plane and the axis of the cylinder is 3 cm. ) sides. a) Draw this body of revolution. Give it a definition. b) What does segment BC form during rotation? Segment AB? c) What segments are the radii, height, axis of the cylinder? d) Write a formula for calculating the area of the base and the area of the axial section of the cylinder.

Task on the topic "Symmetry"

"Order, beauty and perfection"

Personally significant cognitive question

“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,” these words belong to the outstanding mathematician Hermann Weyl.

What is axial, central and mirror symmetry. And how do these concepts manifest themselves in the world around us?

Information on this issue, presented in a variety of forms

Text 1.

The concept of symmetry runs through the entire centuries-old history of human creativity.“Once, standing in front of a black board and drawing various figures on it with chalk, I was suddenly struck by the thought: why is symmetry pleasing to the eye? What is symmetry? This is an innate feeling, I answered myself. What is it based on? Is there symmetry in everything in life? L. N. Tolstoy "Boyhood".

New Dictionary of the Russian Language by T.F. Efremova:

SYMMETRY - proportionate, proportional arrangement of parts of smth. in relation to the center, the middle.

Explanatory Dictionary of the Russian Language by D.N. Ushakov:

SYMMETRY - proportionality, proportionality in the arrangement of parts of the whole in space, full correspondence (by location, size) of one half of the whole to the other half.

In general terms, "symmetry" in mathematics means such a transformation of space (plane) in which each point M goes to another point M "with respect to some plane (or line) a, when the segment MM" is perpendicular to the plane (or line) a and splits 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. A 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.

Text 2.Symmetry types.

Central symmetry

Axial symmetry

Mirror symmetry

T glassesAND and ATare called symmetric with respect to the plane α (plane of symmetry) if the plane α passes through the midpoint of the segmentABand perpendicular to this segment. Each point of the plane α is considered symmetrical to itself.

Text 3. This 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".

FROM
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 20th century, the efforts of Russian scientists - V. Beklemishev, V. Vernadsky, V. Alpatov, G. Gause - created a new direction in the study of symmetry - biosymmetry. The study of the symmetry of biostructures at the molecular and supramolecular levels makes it possible to determine in advance the possible symmetry options in biological objects, to strictly describe the external shape and internal structure of any organisms.

Symmetry in inanimate nature.

Observing the world around him, a person historically tried to more or less realistically display it in various types of art, so it is very interesting to consider symmetry in painting, sculpture, architecture, literature, music and dance.

We can already see symmetry in painting in the cave paintings of primitive people. In ancient times, a significant part of the art of drawing was icons, in the creation of which artists used the properties of mirror symmetry. Looking at them today, one is amazed at the amazing symmetry in the images of saints, although sometimes an interesting thing happens - in asymmetric images we feel symmetry as a norm, from which the artist deviates under the influence of external factors.

Elements of symmetry can be seen in the general plans of buildings.

Sculpture and painting also provide many striking examples of the use of symmetry to solve aesthetic problems. Examples are the tomb of Giuliano Medici by the great Michelangelo, the mosaic of the apse of St. Sophia Cathedral in Kyiv, where two figures of Christ are depicted, one communes with bread, the other with wine.

Symmetry, forced out of painting and architecture, gradually occupied new areas of people's lives - music and dance. Thus, a new direction was discovered in the music of the 15th century - imitative polyphony, which is the musical analogue of the ornament, later appeared - fugues, sound versions of a complex pattern. In the modern song genre, in my opinion, the refrain is an example of the simplest translational symmetry along the axis (lyrics of the song).

Literature also did not ignore symmetry. So an example of symmetry in the literature can serve as palindromes, these are parts of the text, the reverse and direct sequence of letters of which coincide. For example, “A rose fell on Azor’s paw” (A. Fet), “I rarely hold a cigarette butt with my hand.” As a special case of palindromes, we know many words in Russian that are shifters: cook, topot, kazak and many others. Riddles are often built on the use of such words - puzzles.

Tasks for working with this information

Familiarization

1. Consider the variety of objects in our school, including furniture, visual aids, and sports equipment that resemble geometric shapes. Which one is symmetrical?

Answer the questions:

What types of symmetry are you familiar with?

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 symmetry in animate and inanimate nature.

-How many axes of symmetry does: a) a segment; b) straight line; c) beam?

Does the right glove go into the right or left glove with mirror symmetry? axial symmetry? central symmetry?

Understanding

AT
Complete the task: Children ran along the beach and left footprints in the sand. Assuming chains of traces to be extended indefinitely in both directions, indicate with arrows for each chain the types of its combinations, i.e. movements that translate it into itself.

Answer the questions:

Which of the following letters have a center of symmetry: A, O, M, X, K?

Which of the following letters have an axis of symmetry: A, B, D, E, O?

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.

Application

Construct a figure symmetrical to the given one with respect to: a) a point; b) straight

Solve problems in groups

1.In a rectangleABCD Ois the intersection point of the diagonals,BH and DE- heights of trianglesAVO and COD respectively, BOH= 60°, AH= 5 cm. Locate OE.

2. In a rhombus ABCDdiagonals intersect at a pointO. OM, OK, OE- perpendiculars dropped to the sidesAB, VS, CDrespectively. Prove thatOM = OK, and find the sum of the anglesMoU and COE.

3. Inside a given acute angle, construct a square with a given side so that two vertices of the square belong to one side of the angle, and the third to the other.

4. In the rectangle MPKH O is the point of intersection of the diagonals, RA and BH are the perpendiculars drawn from the vertices P and H to the straight line MK. It is known that MA = OB. Find the angle ROM.

5. In the rhombus MPKH, the diagonals intersect at a pointO.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.

6. Construct a square along a given diagonal so that two opposite vertices of this square lie on different sides of a given acute angle.

Analyze how many axes of symmetry the image has.

Create a Sketch representatives from the animal and plant world and show in the drawings the center, the axis of symmetry, using mirror symmetry.

Make up palindromes or use such words to build riddles - rebuses.

Suggest possible criteria for evaluating your sketches and literary works in terms of art and literary critics