Computer presentation for math lesson on the topic “Axial symmetry”, 6th grade.
Mathematics teacher: Priyma T.B.
Municipal educational institution secondary school No. 4 with in-depth study of individual subjects
Bataysk
- Introduction.
- The great ones about symmetry.
- Axial symmetry.
- Symmetry in nature.
- Mysterious snowflakes.
- Human symmetry.
- Conclusion.
Symmetry is an idea with which man has tried for centuries to explain and create order, beauty and perfection.
INTRODUCTION
The principles of symmetry play an important role in physics and mathematics, chemistry and biology, technology and architecture, painting and sculpture, poetry and music.
The laws of nature that govern the inexhaustible picture of phenomena in their diversity, in turn, also obey the principles of symmetry.
THE GREATEST ABOUT SYMMETRY…
- Term "symmetry" invented by a sculptor Pythagoras of Rhegium .
- Ancient Greeks believed that the Universe was symmetrical simply because it was beautiful.
- Created the first scientific school in human history Pythagoras of Samos .
- “Symmetry is a kind of “average measure,” - believed Aristotle .
- Roman doctor Galen(2nd century AD) symmetry meant peace of mind and balance.
Pythagoras of Samos
Aristotle
Galen
- Leonardo da Vinci believed that the main role in the picture is played by proportionality and harmony, which are closely connected by symmetry.
- Albrecht Durer(1471-1528) argued that every artist should know how to construct correct symmetrical figures.
Definition
The term "symmetry"(from the Greek Symmetria) - proportionality, proportionality, uniformity in the arrangement of parts.
Symmetry in a broad sense– the immutability of the structure of a material object relative to its transformations.
Symmetry plays a huge role in art and architecture. But it can be seen both in music and poetry. Symmetry is found widely in nature, especially in crystals, plants and animals.
Symmetry can also be found in other areas of mathematics, for example, when constructing graphs of functions.
Axial symmetry
Two points lying on the same perpendicular to a given line on opposite sides and at the same distance from it are called symmetrical with respect to the given line.
A
The figure is said to be symmetrical about a straight line a ,
if for each point of the figure there is a point symmetrical to it relative to the straight line A also belongs to this figure.
Figures with one axis of symmetry
Corner
Isosceles
triangle
Isosceles trapezoid
Figures with two axes of symmetry
Rectangle
Rhombus
Figures having more than two axes of symmetry
Square
Equilateral triangle
Circle
Figures that do not have axial symmetry
Free Triangle
Parallelogram
Irregular polygon
- point symmetrical to this
- segment symmetrical to this
- triangle symmetrical to this
Symmetry in nature
Careful observation shows that the basis of the beauty of many forms created by nature is symmetry .
Mysterious snowflakes
It rains small grains from the sky, flies around the lanterns in huge fluffy flakes,
stands like a pillar in the moonlight with icy needles. It would seem, what nonsense! Just frozen water.
… but how many questions arise in a person looking at snowflakes.
Human symmetry
The beauty of the human body is determined by proportionality and symmetry.
However, the human figure can be asymmetrical.
The structure of human internal organs is not symmetrical.
CONCLUSION
Nature, in its various creations, seemingly very far from each other, can use the same principles.
And man in his creations: painting, sculpture, architecture...
The fundamental principles of beauty are proportions and symmetry.
In everyday life, we often come across objects that have the property of symmetry. Symmetry is also studied in the geometry course, and not even for one hour. There is a whole series of lessons on this topic. In order to understand at least a little about the symmetry that surrounds us, it is necessary to study this topic in a school course. But it is impossible to imagine symmetry without clear examples.
Such examples, of course, can be shown on real objects, but then they need to be found. But for this you will have to spend your time. A good option would be a presentation where you can place both examples and theoretical points. Here, again, it will take time to create the presentation. If you don’t have free and extra time for this, then you can use this presentation, which the author made specifically for teachers teaching mathematics.
slides 1-2 (Presentation topic "Axial and central symmetry", example)
At the very beginning of the presentation, symmetry relative to a straight line is determined. It says here that points are called symmetrical with respect to a certain line if this line intersects the middle of the segment formed by these points at an angle of 90 degrees. For this definition, there is also a drawing that shows what points that are symmetrical about a straight line look like.
slides 3-4 (examples, definition of a symmetrical line)
Then there is a remark on the slide that says that any point on a line is symmetrical to itself. What is shown in the drawing. It also shows examples of two other pairs of symmetrical points that do not lie on the given line.
Next in the presentation, a figure is defined that is symmetrical with respect to a given straight line. It is called symmetrical with respect to this line if any of its points is symmetrical with another point belonging to the same figure with respect to this line. Then this straight line is called the axis of symmetry, and the figure is said to have the property of axial symmetry.
slides 5-6 (examples)
On the next slide, the author gave a wide variety of examples of figures with axial symmetry. This includes an angle with a straight line that is a bisector, a triangle with equal sides with a median, height or bisector, an equilateral triangle that simultaneously has 3 axes of symmetry, a rectangle and a rhombus each have a pair of axes of symmetry, as well as a square with three axes of symmetry and a circle , which has infinitely many such axes.
slides 7-8 (examples)
On the next slide, the author shows two examples where figures do not have axes of symmetry, that is, figures that do not have symmetry. These include an arbitrary triangle and a parallelogram. In fact, there are a lot of such examples, but the author selected for demonstration the most popular ones, which can be found more often than others in a geometry course.
slides 9-10 (examples)
But the theme also stated central symmetry. Therefore, the author further included in the presentation a definition of the concept of symmetry with respect to a point. Here the author defines a figure that is symmetrical with respect to some point O, as one for which each of its points is symmetrical with some point of the same figure with respect to a given point O. It also says that this point O is the center of symmetry, and, therefore, the figure has this case of central symmetry.
slide 11 (examples)
As mentioned above, in everyday life everyone has encountered at least once an object that has any type of symmetry. These could be plants, flowers, animals, insects. Quite often, symmetrical elements can be found in architectural structures. These are the examples depicting symmetrical objects that are presented in the presentation.
This presentation will be useful for both teachers and students. After all, only important information is presented here, which will definitely come in handy in later life, at least even in geometry lessons.
Topic "Axial symmetry"
Oleynikova Galina Mikhailovna,
Municipal state educational institution "Yablochenskaya secondary school"
Khokholsky municipal district of Voronezh region
“Mathematics reveals order, symmetry and certainty, and these are the most important types of beauty.”
Aristotle (384 – 322 BC)
Problem-based learning technology
Subject "Mathematics"
The purpose of the lesson: organization of productive activities of students aimed at achieving the following results:
meta-subject results:
in cognitive activity:
help students understand the social, practical and personal significance of educational material;
use various methods to understand the surrounding world (observation, measurement, experience, experiment, modeling, etc.)
comparison, juxtaposition, classification of items and objects according to one or more proposed criteria;
independent performance of various creative works;
participation in project activities;
in information - communication activities:
creating written statements that adequately convey what was heard and readinformation with a given degree of condensation (briefly, selectively, full)
Bringing exampleditch, selection of arguments, formulation of conclusions;
reflection in oraland written form of the results of its activities;
at the ability to paraphrase a thought (explain “in other words”);
use for solving cognitive and communication problemsvarious sources of information, including encyclopedias, wordsri, Internet resources and other databases;
in reflective activity:
assessing your educational achievements;
conscious determinationareas of your interests and capabilities;
Possession of skills of joint activities: coordination and coordination activities with other participants; objective assessment their contribution to solving the common problems of the team;
evaluating one's activities from a moral point of viewnorms and aesthetic values;
compliance rules of a healthy lifestyle.
personal results:
be able to confidently and easily perform geometric constructions;
be able to express your thoughts in writing;
be able to speak well and express your thoughts easily;
build character;
learn to apply acquired knowledge and skills to solve new problems;
reason logically;
be able to identify your own difficulties, identify their cause, and build ways out of difficulties;
subject results :
be able to construct points and figures symmetrical to the data;
give examples of symmetrical objects in the reality around us;
conduct research on this topic in nature and architecture;
Mastering methods of activity applicable in a mathematics lesson with integration into anatomy, biology, ecology, healthy lifestyle culture, and architecture.
Lesson type: lesson-research.
Forms of work: individual, pair, group, frontal.
Equipment: computer office with Internet access, projector, screen, presentation, token figures, drawings, magnets, colored chalk; Each student has a folder with a set of geometric models, school tools, colored paper, colored pencils, scissors.
Methods: explanatory-illustrative, partially search, research, project.
Forms of cognitive activity of students: frontal, individual.
Pre-students from the first lesson of the topic “Axial Symmetry” are grouped (according to their desire and interests) into 3 groups of equal numbers, so that in each group there are students who have access to the Internet at home. Each group receives a mini-research assignment: symmetry in nature, human anatomy and architecture.
During the lesson, groups are saved. For each correct answer, the team receives a token figure. One figure - one point. The team with the most points receives a score of 5; the other two conduct self-assessments within the group.
Updating.
We live in a rapidly changing high-tech, information society, and we don’t think about why some objects and phenomena around us awaken a sense of beauty, while others do not.
In summer - ladybug. Autumn yellow leaves on trees or leaves that have fallen to the ground are very beautiful. And in winter? - Snowflakes.
We are walking down the street and suddenly slow down when we see a well-proportioned and beautiful building.
Many people pass by, and each of us will pay attention to one and say: “This person is beautiful and harmonious.”
This chain can be continued, but now we are talking about something united: about the beauty, harmony and proportionality of living and inanimate nature.
I invite (I ask a specially trained person to come) a student from this class. Children pay attention to symmetrical hairstyle, earrings, blouse, shawl with a symmetrical pattern.
Today our classmate is visiting us and she’s called...
- “Symmetry”.
And today we will touch upon a wonderful mathematical phenomenon - axial symmetry. (Slide 1-3)
Let's write down the topic of the lesson "Axial symmetry" in our notebook.
Today in class we will try to answer the following questions:
What is symmetry?
What is axial symmetry?
Let's learn to identify symmetrical figures.
Let's repeat the construction of symmetrical points and geometric figures relative to a straight line.
What role does symmetry play in human everyday life (in nature, architecture, everyday life)?
- Is it possible, knowing about the secret of harmony, to make the world a better and more beautiful place?
The teacher and students write down the number, class work, topic of the lesson on the board and in the notebook.
Then he invites students to choose personal goals (or personal results) from those proposed on the screen, to achieve which each of them will try to work as hard as possible in this lesson. Students determine for themselves the personal results (selecting from the list on the screen) that they will strive for in the lesson, and the goal number (in the margins) in the notebook.
Frontal conversation.
What is symmetry? (slide 4-8)
The word symmetry has long been used to mean harmony and beauty.
Euclid, Pythagoras, Leonardo da Vinci, Kepler and many other major thinkers of mankind tried to comprehend the mystery of harmony.
“Symmetry is an idea with the help of which man has tried for centuries to explain and create order, beauty, perfection” G. Weil.
What can you say about the meaning of the words “symmetry” and “axis”?
Symmetry is the sameness, proportionality in the arrangement of parts of something on opposite sides of a point, line or plane.
An axis is a straight line (an imaginary line passing through a geometric figure that has only its inherent properties).
What points are called symmetrical?
Determination of symmetrical points relative to a straight line:
“Two points A and B are called symmetrical with respect to a line p if this line passes through the middle of the segment AB connecting these points and is perpedicular to it.”
Formulate an algorithm for constructing a point symmetrical to a given point relative to a certain line.
Why won’t it be possible to complete a task that sounds like this: “Construct a figure symmetrical to this one”?
This task is incomplete, since it is unclear whether the symmetry is relative to a point or a straight line. This means that to perform axial symmetry it is necessary to know the axis of symmetry.
Fixing the material.
1).Construction of a figure symmetrical to a given one (relay race in groups)
Written work in notebooks and on the board. (Slide 9-12)
Exercise 1. Construct a point symmetrical to the given one relative to the line a.
Task 2. Construct a line symmetrical to the given line with respect to line m.
Task 3. Construct a triangle symmetrical to the given one with respect to line n.
Task 4. Draw a figure by hand, symmetrical to this relatively vertical axis (Christmas tree, bird, cat). (Slide 13)
The figures are drawn on sheets of paper and attached to the board. Everyone comes to the board and makes one element of the image, symmetrical to one figure from those offered to his team. The team that completes the task first wins. Evaluation is carried out according to the following criteria:
Correct execution of construction;
Aesthetic perception;
Participation of each group member.
Exercise 5 (oral work ). Is it true that the following numerical intervals are symm. metric relative to the straight line m, perpendicular to the coordinate line and passing through the origin O:
a) a segment from 3 to 7 and a segment from -7 to -3;
b) a segment from 10 to 25 and an interval from -25 to -10;
c) open rays from 1 to infinity and from minus infinity to 1?
Answer: a) yes; b) no; c) yes.
Task 6. Research work “Find the axes of symmetry of a geometric figure.”
How to determine whether a figure has an axis of symmetry? (Slide 14-18)
Bend it over.
Yes, indeed, if you bend them along the depicted straight line, then its left and right parts will coincide. Such figures are symmetrical with respect to a straight line, and this straight line is the axis of symmetry.
How many axes of symmetry can a figure have? You have geometric shapes on your desks. Your task is to independently determine how many axes of symmetry each figure has. Determine the most “symmetrical” and the most “asymmetrical” figure.
Students find the axes of symmetry of such geometric figures as angles, equilateral, isosceles and scalene triangles, rectangles, rhombuses, squares, trapezoids, parallelograms, circles, and irregular polygons.
Let's find out which geometric figures have one axis of symmetry?
Angle, isosceles triangle, trapezoid.
Two axes of symmetry?
Rectangle, rhombus.
Are the diagonals of a rectangle the axes of symmetry and why?
They are not, because when the rectangle is bent diagonally, the triangles do not coincide.
Students bend the figure diagonally and show that the parts of the rectangle do not coincide, that is, the diagonal of the rectangle is not an axis of symmetry.
Three axes of symmetry?
Equilateral triangle.
Four axes of symmetry?
Square.
How many axes of symmetry does a circle have?
A bunch of. These are straight lines passing through the center of the circle.
So which one the most “symmetrical” and the most “asymmetrical” figure?
The most “symmetrical” is a circle, and the “asymmetrical” are scalene triangle, parallelogram; a polygon whose sides are unequal.
Task 7 ( Orally) . Give examples of symmetrical objects from your surroundings at home and on the street? Do you and I have symmetry?
Task 8 (Research and “local history” work - 10 points).
I propose to conduct mini-research in pairs or small groups, followed by a discussion about the presence of symmetry in the external and internal structure of humans, animals, and plants; in the architecture of buildings around the world, our city and school.
When preparing messages, students use the Internet.
Mini-study results represented by the students of the class. Each group of students presents research results on the following topics:
Axial symmetry and nature.
Axial symmetry and man.
Axial symmetry in architecture.
Create their own written product and presentation.
Protection is assessed by:
Optimally selected material,
Laconic presentation, logical reasoning,
Aesthetic perception
Application in human life.
- “Axial symmetry in nature."(Slide 19-22)
Careful observation shows that the basis of the beauty of many forms created by nature is symmetry. Leaves, flowers, and fruits have pronounced symmetry.
Research by ecologists is closely related to the plants and trees around us.
Based on the symmetry of birch leaves, we can talk about the healthy ecological situation of the microdistrict. If the birch leaves are not symmetrical, then the environmental situation is unfavorable, this indicates the presence of radiation or chemical pollution. We examine birch leaves collected in the microdistrict of western Bataysk. Based on the handouts, we conclude that the ecological situation of the microdistrict is favorable.
It rains small grains from the sky, flies around the lanterns in huge fluffy flakes, and stands like a pillar in the moonlight with icy needles. It would seem, what nonsense! Just frozen water. ...but how many questions arise in a person looking at snowflakes.
Snowflake is a group of crystals formed from more than two hundred ice particles.
Symmetry – this is the property of crystals to be combined with each other in different positions through rotations, parallel transfers, reflections.
Count the axes of symmetry of your snowflake model.
- “Axial symmetry and the animal world.” (Slide 23)
Students note the symmetry of the external structure of animals, give examples of symmetrical color, but argue that the internal structure of animals is not symmetrical.
- “Axial symmetry and man.” (Slide 24-25)
The beauty of the human body is determined by proportionality and symmetry. The structure of the internal organs is not symmetrical.However, the human figure can be asymmetrical. One such example is scoliosis - a curvature of the spine acquired, among other things, by incorrect posture.
Scoliosis - a lateral curvature of the spine - most often occurs between the ages of 5 and 16 years. Among five-year-olds, approximately 5-10% of children suffer from scoliosis, and by the end of school, scoliosis is detected in almost half of adolescents.
One of the main reasons is incorrect posture during training sessions, which causes an uneven load on the spine and muscles. Why is scoliosis dangerous and what diseases can it lead to in the future?
Most organs of the human body are directly controlled from the spinal cord through the spinal nerves. Infringement of the nerve roots extending from the spinal cord leads to disruption of the functioning of internal organs. Hippocrates pointed out the existence of a connection between the condition of the spine and the functioning of internal organs. Preventing scoliosis is better than treating it.
At the first signs of scoliosis, you need to consult with a specialist, follow a regimen that eases the load on the spine, provide a diet rich in vitamins and minerals (the spine urgently needs microelements such as calcium, zinc, copper), you need to do morning exercises and physical therapy. It is important to learn how to sit correctly at a desk: the back of your head should be slightly raised and slightly back, and your chin should be slightly lowered. With this position of the head, the entire spine straightens and blood supply to the brain improves. Feet should be on the floor, and the angle at the knee joints should be approximately 90 degrees.
The spine is one of the most important parts of the human body. Thanks to him, we can walk, run, jump, and squat. The beauty and charm of a person largely depend on posture.
80% of Russian children suffer from various types of posture disorders, from flat feet to scoliosis. The formation of the curves of the spine ends at 6-7 years and is fixed by 14-17 years. This means that it is at this age that it is important for a teenager to develop correct posture and thereby lay a reliable foundation for health for many years to come.
Poor posture is not a disease, but a condition that needs to be corrected. They say that until the age of 21, while the body is growing, many diseases of the musculoskeletal system can be cured. I suggest that all participants in our lesson monitor correct posture.
- “Axial symmetry in the architecture of buildings in cities around the world, the city of Bataysk.”(Slide 26-32)
Symmetry is most clearly visible in architecture. In the minds of ancient Greek architects, symmetry became the personification of regularity, expediency, and beauty. Examples of such structures are the Pyramid of Cheops in Egypt, Notre Dame Cathedral and the Eiffel Tower in France, Big Ben in Great Britain, and the Taj Mahal Mosque in Turkey.
The architecture of Russian Orthodox churches and cathedrals indicates that since ancient times, architectsThey knew mathematical proportion and symmetry well and used them in the construction of architectural structures in Rus': the Kremlin, the Cathedral of Christ the Savior in Moscow, the Kazan and St. Isaac's Cathedrals in St. Petersburg, the cathedrals in Pskov, Nizhny Novgorod and others.
We asked ourselves another question: “Do modern architects know the secret of creating beauty?” Our hometown is of interest to us. For example, the symbol of Bataysk, which is located in the Central Park, is loved by many citizens; we explain its aesthetic perception by the symmetry of its arch. We see symmetry in administrative, residential buildings, and cultural leisure buildings.
The appearance of the Holy Trinity Church - the main attraction of the city, according to the architectural canons of the construction of Russian cathedrals, is an example of symmetry and proportionality. While studying the Oath of Generations memorial and monuments, we found out that they are based on symmetry. The building of the railway station of our city is also an example of a symmetrical building. Thus, most of the buildings that form the face of our city are harmonious and comply with the laws of beauty.
- “Axial symmetry and our schoolyard.” (Slide 33)
Examining the size of our own school, we see that the facade of the building, the porch, the section of the school fence, small architectural forms, and flower beds comply with the rules of symmetry. Therefore, the overall appearance of the school yard looks harmonious.
Reflection. (Slide 34-37)
- The presentation slides present examples of symmetrical and asymmetrical objects in the surrounding world (3 slides). Students are asked to identify examples of symmetrical and asymmetrical objects and analyze why?
Homework:
- creative assignments on the topic “Statements of great scientists about symmetry”;
- mini-presentations, photo reports about the symmetry of the surrounding reality;
- create models with symmetry using colored paper, scissors, felt-tip pens;
Yourscreative task.
conclusions. (Slide 38)
Axial symmetry is a mathematical concept.
Learned to identify symmetrical figures.
We learned how to construct symmetrical points and geometric figures relative to a straight line.
Symmetry is harmony.
The great thinkers of mankind tried to comprehend the mystery of harmony. Today in class we also plunged into solving this mystery. We found out that symmetry plays one of the main directions in human everyday life: in household items, in architecture, in nature.Knowing about the secrets of harmony, one of which is axial symmetry, you can make the world a better and more beautiful place.
Do you know the famous phrase: “Beauty will save the world?” It is difficult to disagree with Fyodor Mikhailovich Dostoevsky. We all want to make our lives more harmonious and beautiful. Guys, do you think maybe we have found the secret to creating beauty?
Lesson summary.
Was an answer given to the problematic situation of the lesson, what new things were learned in the lesson, what were learned, what caused difficulties and were they resolved in the lesson?
Grades are posted in student journals and diaries. The team with the most points and students from other groups with high personal results receive a grade of 5; second place team - score 4.
