Many have seen the picture "Mental account in public school"The end of the 19th century, a folk school, a blackboard, an intelligent teacher, poorly dressed children, 9-10 years old, enthusiastically try to solve the problem written on the blackboard in their minds. The first one who decides tells the answer to the teacher in the ear, in a whisper, so that others do not lose interest.
Now look at the problem: (10 squared + 11 squared + 12 squared + 13 squared + 14 squared) / 365 =???
Crap! Crap! Crap! Our children at the age of 9 will not solve such a problem, at least in their minds! Why were grimy and barefoot village children taught so well in a one-room wooden school, while our children are taught so badly?!
Don't be quick to get angry. Take a look at the picture. Don't you think that the teacher looks too intelligent, somehow like a professor, and is dressed with obvious pretense? Why in school class such a high ceiling and an expensive stove with white tiles? Did the village schools and the teachers in them really look like this?
Of course they didn't look like that. The picture is called "Mental counting in the folk school of S.A. Rachinsky." Sergey Rachinsky - professor of botany at Moscow University, a person with certain government connections(for example, a friend of the chief prosecutor of the Synod Pobedonostsev), a landowner - in the middle of his life he abandoned all his affairs, went to his estate (Tatevo in the Smolensk province) and started there (of course, at his own expense) an experimental public school.
The school was one-class, which did not mean that it taught for one year. In such a school they taught then 3-4 years (and in two-class schools - 4-5 years, in three-class schools - 6 years). The word one-class meant that children of three years of study make up a single class, and one teacher deals with them all within the same lesson. It was quite a tricky thing: while the children of one year of study were doing some kind of written exercise, the children of the second year answered at the blackboard, the children of the third year read the textbook, etc., and the teacher alternately paid attention to each group.
Rachinsky's pedagogical theory was very original, and its different parts somehow poorly converged with each other. Firstly, Rachinsky considered the teaching of the Church Slavonic language and the Law of God to be the basis of education for the people, and not so much explanatory as consisting in memorizing prayers. Rachinsky firmly believed that knowing by heart a certain amount of prayers, the child will certainly grow up as a highly moral person, and the very sounds of the Church Slavonic language will already have an effect that improves morality. For practice in the language, Rachinsky recommended that children be hired to read the Psalter over the dead (sic!).
Secondly, Rachinsky believed that it was useful for the peasants and they needed to quickly count in their minds. teaching mathematical theory Rachinsky was not very interested, but he did a very good job of mental counting in his school. The students firmly and quickly answered how much change per ruble should be given to someone who buys 6 3/4 pounds of carrots at 8 1/2 kopecks per pound. The squaring shown in the painting was the most complex mathematical operation studied at his school.
And finally, Rachinsky was a supporter of a very practical teaching of the Russian language - the students were not required to have any special spelling skills or good handwriting, they were not taught theoretical grammar at all. The main thing was to learn to read and write fluently, albeit in a clumsy handwriting and not very competently, but it’s clear that a peasant could come in handy in everyday life: simple letters, petitions, etc. Even at Rachinsky’s school some manual labor was taught, the children sang in chorus, And that's where education ends.
Rachinsky was a real enthusiast. School became his whole life. Rachinsky's children lived in a hostel and were organized into a commune: they performed all the housekeeping work for themselves and the school. Rachinsky, who had no family, spent all the time with the children from early morning until late at night, and since he was a very kind, noble and sincerely attached person to children, his influence on the students was enormous. By the way, Rachinsky gave the first child who solved the problem a gingerbread (in the literal sense of the word, he did not have a whip).
themselves school lessons they took 5-6 months a year, and the rest of the time Rachinsky worked individually with older children, preparing them for admission to various educational institutions of the next level; elementary folk school was not directly related to other educational institutions and after it it was impossible to continue training without additional training. Rachinsky wanted to see the most advanced of his students as teachers. elementary school and priests, so that he prepared children mainly for theological and teacher's seminaries. There were also significant exceptions - first of all, this is the author of the painting himself, Nikolai Bogdanov-Belsky, whom Rachinsky helped to get into the Moscow School of Painting, Sculpture and Architecture. But, oddly enough, to lead peasant children along the main road educated person- gymnasium / university / public service- Rachinsky did not want to.
Rachinsky wrote popular pedagogical articles and continued to enjoy a certain influence in the capital's intellectual circles. The most important was the acquaintance with the ultra-influential Pobedonostsev. Under a certain influence of the ideas of Rachinsky, the spiritual department decided that there would be no sense in the zemstvo school - the liberals would not teach children good - and in the mid-1890s began to develop their own independent network of parochial schools.
In some ways, the parochial schools were similar to the Rachinsky school - they had a lot of Church Slavonic and prayers, and the rest of the subjects were reduced accordingly. But, alas, the dignity of the Tatev school was not transferred to them. The priests showed little interest in school work, ran the schools under duress, did not teach in these schools themselves, and hired the most third-rate teachers, and paid them noticeably less than in zemstvo schools. The peasants took a dislike to the parochial school, because they realized that they almost didn’t teach anything useful there, and prayers were of little interest to them. By the way, it was the teachers of the church school, recruited from pariahs of the clergy, who turned out to be one of the most revolutionized professional groups of that time, and it was through them that socialist propaganda actively penetrated into the village.
Now we see that this is a common thing - any author's pedagogy, designed for the deep involvement and enthusiasm of the teacher, immediately dies with mass reproduction, falling into the hands of uninterested and sluggish people. But for that time it was big bummer. Church-parish schools, which by 1900 accounted for about a third of primary public schools, turned out to be disliked by everyone. When, starting in 1907, the state began to send elementary education a lot of money, there was no question of passing subsidies to church schools through the Duma, almost all the funds went to the Zemstvo.
The more common zemstvo school was quite different from the Rachinsky school. For starters, the Zemstvo considered the Law of God completely useless. It was impossible to refuse his teaching, according to political reasons, so the Zemstvos pushed him into a corner as best they could. The law of God was taught by an underpaid and neglected parish priest, with corresponding results.
Mathematics at the Zemstvo school was taught worse than at Rachinsky, and to a lesser extent. The course ended with operations with simple fractions and non-metric system of measures. Up to raising to a degree, training did not reach, so the students of an ordinary elementary school simply would not understand the task depicted in the picture.
The zemstvo school tried to turn the teaching of the Russian language into world science, through the so-called explanatory reading. The technique consisted in dictating educational text in Russian, the teacher also additionally explained to the students what was said in the text itself. In such a palliative way, the lessons of the Russian language also turned into geography, natural history, history - that is, into all those developing subjects that could not find a place in the short course of a one-class school.
So, our picture depicts not a typical, but a unique school. This is a monument to Sergei Rachinsky, unique personality and teacher, the last representative of that cohort of conservatives and patriots, to which one could not yet attribute famous expression"Patriotism is the last refuge of a scoundrel." The mass public school was economically equipped much poorer, the mathematics course in it was shorter and simpler, and teaching was weaker. And, of course, the students of an ordinary elementary school could not only solve, but also understand the problem reproduced in the picture.
By the way, how do students solve the problem on the board? Only direct, head-on: multiply 10 by 10, remember the result, multiply 11 by 11, add both results, and so on. Rachinsky believed that the peasant did not have writing materials at hand, so he taught only oral methods of counting, omitting all arithmetic and algebraic transformations that required calculations on paper.
For some reason, only boys are depicted in the picture, while all the materials show that children of both sexes studied with Rachinsky. What this means is not clear.
Full title famous painting, which is shown above: Verbal counting. In the folk school of S. A. Rachinsky ". This painting by the Russian artist Nikolai Petrovich Bogdanov-Belsky was painted in 1895, and now hangs in Tretyakov Gallery. In this article, you will learn some details about it. famous work who Sergei Rachinsky was, and most importantly - get the correct answer to the task depicted on the board.
Brief description of the painting
The painting depicts a rural school of the 19th century during an arithmetic lesson. The figure of the teacher has real prototype- Sergei Alexandrovich Rachinsky, botanist and mathematician, professor at Moscow University. Rural schoolchildren decide very interesting example. It is evident that it is not easy for them. In the picture, 11 students are thinking about the problem, but it seems that only one boy figured out how to solve this example in his mind, and quietly speaks his answer in the ear of the teacher.
Nikolai Petrovich dedicated this picture to his school teacher Sergei Alexandrovich Rachinsky, who is depicted on it in the company of his students. Bogdanov-Belsky knew the heroes of his picture very well, since he himself was once in their situation. He was lucky to get into the school of the famous Russian teacher Professor S.A. Rachinsky, who noticed the boy's talent and helped him get an art education.
About Rachinsky
Sergei Alexandrovich Rachinsky (1833-1902) - Russian scientist, teacher, educator, professor at Moscow University, botanist and mathematician. Continuing the undertakings of his parents, he taught at a rural school, even though the Rachinskys were a noble family. Sergei Alexandrovich was a man of versatile knowledge and interests: in the school art workshop, Rachinsky himself conducted classes in painting, drawing and drawing.
IN early period As a teacher, Rachinsky searched in line with the ideas of the German teacher Karl Volkmar Stoya and Leo Tolstoy, with whom he corresponded. In the 1880s, he became the main ideologist in Russia of the parochial school, which began to compete with the Zemstvo school. Rachinsky came to the conclusion that the most important of the practical needs of the Russian people is communication with God.
As for mathematics and mental arithmetic, Sergei Rachinsky left behind his famous problem book " 1001 mental arithmetic tasks ”, some tasks (with answers) from which you can find by.
Read more about Sergei Aleksandrovich Rachinsky on his biography page at.
Chalkboard example solution
There are several ways to solve the expression written on the blackboard in Bogdanov-Belsky's painting. By following this link, you will find four different solutions. If at school you learned the squares of numbers up to 20 or up to 25, then most likely the task on the board will not cause you much difficulty. This expression is: (100+121+144+169+196) divided by 365, which is equal to 730 divided by 365, i.e. "2".
In addition, on our website in the "" section you can get acquainted with Sergei Rachinsky and find out what "" is. And it is the knowledge of these sequences that allows you to solve the problem in a matter of seconds, because:
10 2 +11 2 +12 2 = 13 2 +14 2 = 365
Humor and parodic interpretations
Nowadays, schoolchildren are no longer only solving some of Rachinsky's popular problems, but also writing essays based on the painting “Mental Counting. In the folk school of S. A. Rachinsky ”, which could not but affect the desire of schoolchildren to joke about the work. The popularity of the painting Mental Account is reflected in the many parodies of it that can be found on the Internet. Here are just a few of them:
known to many. The picture shows a village school late XIX centuries during an arithmetic lesson while solving a fraction in your head.
Teacher - a real man, Sergei Alexandrovich Rachinsky (1833-1902), botanist and mathematician, professor at Moscow University. On the wave of populism in 1872, Rachinsky returned to his native village of Tatevo, where he created a school with a hostel for peasant children, developed unique technique learning mental arithmetic, instilling in the village children his skills and the basics of mathematical thinking. Bogdanov-Belsky, himself a former student of Rachinsky, dedicated his work to an episode from the life of a school with a creative atmosphere that reigned in the classroom.
However, with all the fame of the picture, few of those who saw it delved into the content of the "difficult task" that it depicts. It consists in quickly finding the result of the calculation by mental counting:
| 10 2 + 11 2 + 12 2 + 13 2 + 14 2 |
| 365 |
A talented teacher cultivated in his school an oral calculation based on the virtuoso use of the properties of numbers.
The numbers 10, 11, 12, 13 and 14 have a curious feature:
10 2 + 11 2 + 12 2 = 13 2 + 14 2 .
Indeed, since
100 + 121 + 144 = 169 + 196 = 365,
Wikipedia for calculating the value of the numerator suggests the following way:
10 2 + (10 + 1) 2 + (10 + 2) 2 + (10 + 3) 2 + (10 + 4) 2 =
10 2 + (10 2 + 2 10 1 + 1 2) + (10 2 + 2 10 2 + 2 2) + (10 2 + 2 10 3 + 3 2) + (10 2 + 2 10 4 + 4 2) =
5 100 + 2 10 (1 + 2 + 3 + 4) + 1 2 + 2 2 + 3 2 + 4 2 =
500 + 200 + 30 = 730 = 2 365.
For me, it's too smart. It's easier to do otherwise:
10 2 + 11 2 + 12 2 + 13 2 + 14 2 =
= (12 - 2) 2 + (12 - 1) 2 + 12 2 + (12 + 1) 2 + (12 + 2) 2 =
5 12 2 + 2 4 + 2 1 = 5 144 + 10 = 730,
| 730 | = 2. |
| 365 |
The above reasoning is quite possible to carry out orally - 12 2 , of course, you need to remember that the double products of the squares of the binomials to the left and right of 12 2 cancel each other out and can be ignored, but 5 144 \u003d 500 + 200 + 20, - not difficult.
Let's use this trick and verbally find the sum:
48 2 + 49 2 + 50 2 + 51 2 + 52 2 = 5 50 2 + 10 = 5 2500 + 10 = 12510.
Let's complicate:
84 2 + 87 2 + 90 2 + 93 2 + 96 2 = 5 8100 + 2 9 + 2 36 = 40500 + 18 + 72 = 40590.
Rachinsky row
Algebra gives us the means to raise the question of this interesting feature series of numbers
10, 11, 12, 13, 14
more broadly: is it the only row of five consecutive numbers whose sum of the squares of the first three is equal to the sum of the squares of the last two?
Denoting the first of the desired numbers by x, we have the equation
x 2 + (x + 1) 2 + (x + 2) 2 = (x + 3) 2 + (x + 4) 2.
It is more convenient, however, to denote by x not the first, but the second of the desired numbers. Then the equation will have a simpler form
(x - 1) 2 + x 2 + (x + 1) 2 = (x + 2) 2 + (x + 3) 2 .
Opening the brackets and making simplifications, we get:
x 2 - 10x - 11 = 0,
where
x 1 = 11, x 2 = -1.
There are, therefore, two series of numbers that have the required property: the Rachinsky series
10, 11, 12, 13, 14
and row
2, -1, 0, 1, 2.
Indeed,
(-2) 2 +(-1) 2 + 0 2 = 1 2 + 2 2 .
Two!!!
I would like to finish with bright and touching memories of the author's blog V. Iskra in the article On the squares of two-digit numbers and not only about them ...
Once, in the year around 1962, our "mathematician", Lyubov Iosifovna Drabkina, gave this task to us, 7th graders.
I was then very fond of the newly appeared KVN-ohm. He supported the team of the city of Fryazino near Moscow. The “Fryazinians” were distinguished by their special ability to apply logical “express analysis” to solve any problem, “pulling out” the most tricky question.
I couldn't figure it out quickly. However, using the "Fryazin" method, I figured out that the answer should be expressed as an integer. Otherwise, this is no longer an “oral account”! This number could not be one - even if the numerator had the same 5 hundreds, the answer would be clearly more. On the other hand, he clearly did not reach the number "3".
- Two!!! - I blurted out, a second ahead of my friend, Lenya Strukov, the best mathematician in our school.
- Yes, indeed two, - Lenya confirmed.
- What did you think? - asked Lyubov Iosifovna.
- I didn't think so. Intuition - I answered to the laughter of the whole class.
- If you didn’t count, the answer doesn’t count - Lyubov Iosifovna “punned”. Lenya, didn't you count too?
- No, why not, Lenya answered sedately. It was necessary to add 121, 144, 169 and 196. I added the numbers one and three, two and four in pairs. It is more comfortable. It turned out 290 + 340. total amount, including the first hundred - 730. Divide by 365 - we get 2.
- Well done! But for the future, remember - in a series of two-digit numbers - the first five of its representatives - have an amazing property. The sum of the squares of the first three numbers in the series (10, 11 and 12) is equal to the sum of the squares of the next two (13 and 14). And this sum equals 365. Easy to remember! So many days in a year. If the year is not a leap year. Knowing this property, the answer can be obtained in a second. Without any intuition...
* * *
… Years have passed. Our city has acquired its own "Wonder of the World" - mosaic paintings in underground passages. There were many transitions, even more paintings. The topics were very different - the defense of Rostov, space ... In the central passage, under the intersection of Engels (now - Bolshaya Sadovaya) - Voroshilovsky made a whole panorama of the main stages life path Soviet man- maternity hospital - kindergarten school, prom...
On one of the "school" pictures one could see a familiar scene - the solution of the problem ... Let's call it like this: "The Rachinsky Problem" ...
... Years passed, people passed ... Cheerful and sad, young and not very young. Someone recalled their school, someone at the same time "moved their brains" ...
The master tilers and artists, led by Yuri Nikitovich Labintsev, did a wonderful job!
Now the "Rostov miracle" is "temporarily unavailable." Trade came to the fore - directly and figuratively. Nevertheless, let's hope that in this common phrase - the main thing is the word "temporarily" ...
Sources: Ya.I. Perelman. Entertaining Algebra (Moscow, Nauka, 1967), Wikipedia,
Lesson Objectives:
- development of the ability to observe;
- development of the ability to think;
- development of the ability to express thought;
- instilling an interest in mathematics;
- touching the art of N.P. Bogdanov-Belsky.
DURING THE CLASSES
Teaching is the work that educates and shapes a person.
Four pages from the life of a painting
Page one
The painting “Mental Account” was painted in 1895, that is, 110 years ago. This is a kind of anniversary of the picture, which is the creation of human hands. What is shown in the picture? Some boys have gathered around the blackboard and are looking at something. Two boys (these are the ones in front) turned away from the blackboard and remember something, or maybe they count. One boy whispers something into the ear of a man, presumably the teacher, while the other appears to be eavesdropping.
- And why are they in bast shoes?
“Why are there no girls here, only boys?”
Why are they standing with their backs to the teacher?
– What are they doing?
You have probably already understood that students and a teacher are depicted here. Of course, the costumes of the students are unusual: some of the guys are wearing bast shoes, and one of the characters in the picture (the one in the foreground), in addition, has a torn shirt. It is clear that this picture is not from our school life. Here is the inscription on the picture 1895 - the time of the old pre-revolutionary school. The peasants then lived in poverty, they themselves and their children wore bast shoes. The artist depicted peasant children here. Only at that time, few of them could study even in elementary school. Look at the picture: after all, only three of the students are wearing bast shoes, and the rest are in boots. Obviously, guys from rich families. Well, why girls are not depicted in the picture is also not difficult to understand: after all, at that time, girls, as a rule, were not accepted to school. Teaching was “not their business”, and not all of the boys studied.
Page two
This picture is called "Mental Account". See how the boy in the foreground of the picture thinks intently. It is evident that the teacher gave a difficult task. But, probably, this student will soon finish his work, and there should be no mistake: he takes mental counting very seriously. But the student who whispers something in the teacher's ear, apparently, has already solved the problem, only his answer is not quite correct. Look: the teacher listens attentively to the student's answer, but there is no approval on his face, which means that the student did something wrong. Or maybe the teacher patiently waits for others to count correctly, just like the first one, and therefore is in no hurry to approve his answer?
- No, the first one will give the correct answer, the one in front: it is immediately clear that he is the best student in the class.
And what task did the teacher give them? Can't we solve it too?
- But try it.
I will write on the board as you used to write:
(10 10+11 11+12 12+13 13+14 14):365
As you can see, each of the numbers 10, 11, 12, 13 and 14 must be multiplied by itself, the results added up, and the resulting sum divided by 365.
– This is the task (you won’t solve such an example soon, and even in your mind). But still try to count verbally, in difficult places I will help you. Ten ten is 100, everyone knows that. Eleven times eleven is also easy to count: 11 10=110, and even 11 is 121 in total. 144. I also calculated that 13 13=169 and 14 14=196.
But while I was multiplying, I almost forgot what numbers I got. Then I remembered them, and after all, these numbers still need to be added, and then the sum should be divided by 365. No, you yourself will not be able to calculate this.
- I'll have to help a little.
- What numbers did you get?
- 100, 121, 144, 169 and 196 - this was counted by many.
- Now you probably want to add all five numbers at once, and then divide the results by 365?
We will do it differently.
- Well, let's add the first three numbers: 100, 121, 144. How much will it be?
How much should be divided?
– Also on 365!
- How much will it be if the sum of the first three numbers is divided by 365?
- One! - everyone will figure it out.
- Now add the other two numbers: 169 and 196. How much will it be?
– Also 365!
- Here is an example, and quite simple. It turns out that only two!
- Only to solve it, you need to know well that the sum can be divided not all at once, but in parts, each term separately, or in groups of two or three terms, and then add up the resulting results.
Page three
This picture is called "Mental Account". It was painted by the artist Nikolai Petrovich Bogdanov-Belsky, who lived from 1868 to 1945.
Bogdanov-Belsky knew his little heroes very well: he grew up in their environment, was once a shepherd boy. “... I am the illegitimate son of a poor woman, that’s why Bogdanov, and Belsky became the name of the county,” the artist said about himself.
He was lucky to get into the school of the famous Russian teacher Professor S.A. Rachinsky, who noticed the artistic talent of the boy and helped him get an art education.
N.P. Bogdanov-Belsky graduated from the Moscow School of Painting, Sculpture and Architecture, studied under such famous artists like V.D. Polenov, V.E. Makovsky.
Many portraits and landscapes were painted by Bogdanov-Belsky, but he remained in the memory of people, first of all, as an artist who managed to poetically and faithfully tell about the smart rural children eagerly reaching for knowledge.
Which of us is not familiar with the paintings “At the Doors of the School”, “Beginners”, “Composition”, “Village Friends”, “At the Sick Teacher”, “Voice Test”, - these are the names of just some of them. Most often, the artist depicts children at school. Charming, trusting, concentrated, thoughtful, full of lively interest and always marked by a natural mind - Bogdanov-Belsky knew and loved peasant children like that, immortalized in his works like that.
Page Four
The artist depicted non-fictional students and teachers in this picture. From 1833 to 1902, the famous Russian teacher Sergei Aleksandrovich Rachinsky, a remarkable representative of Russian educated people of the century before last, lived. He was a doctor of natural sciences and a professor of botany at Moscow University. In 1868 S.A. Rachinsky decides to go to the people. "He takes the exam" for the title of teacher primary school. At his own expense, he opens a school for peasant children in the village of Tatyevo, Smolensk province, and becomes a teacher there. So, his students counted so well orally that all the visitors to the school were surprised at this. As you can see, the artist depicted S.A. Rachinsky with his students at the lesson of oral problem solving. By the way, the artist N.P. Bogdanov-Belsky was a student of S.A. Rachinsky.
This picture is a hymn to the teacher and the student.
famous Russian artist NIKOLAI PETROVICH BOGDANOV-BELSKY
wrote a unique and incredible life story in 1895.
The work is called "ORAL ACCOUNT",
and in the full version
"VERBAL COUNTING. AT THE PEOPLE'S SCHOOL S.A. RACHINSKY.
The picture is painted in oil on canvas, it depicts a rural school of the 19th century during an arithmetic lesson.
A simple Russian class, the children are dressed in peasant clothes: bast shoes, trousers and shirts. All this very harmoniously and succinctly fits into the plot, unobtrusively bringing to the world the craving for knowledge on the part of the simple Russian people.
Students solve interesting and complex example to solve fractions in your mind. They are in deep thought and searching for the right solution. Someone thinks at the blackboard, someone stands on the sidelines and tries to compare knowledge that will help in solving the problem. Children are completely absorbed in finding the answer to the question posed, they want to prove to themselves and the world that they can do it.
The canvas depicts 11 children and only one boy quietly whispers in the teacher's ear, perhaps the correct answer.
Nearby is a teacher, a real person, Sergei Aleksandrovich Rachinsky, a famous botanist and mathematician, professor at Moscow University. rural children his skills and basics of mathematical thinking.
Warm colors bring kindness and simplicity of the Russian people, there is no envy and falsehood, there is no evil and hatred, children from different families with different incomes came together to make the only right decision.
This is sorely lacking in our modern life where people are used to living in a completely different way, regardless of the opinions of others.
Nikolai Petrovich Bogdanov-Belsky, himself a former student of Rachinsky, dedicated the picture to an episode from the life of the school with a creative atmosphere that reigned in the classroom, to his teacher, the great genius of mathematics, whom he knew and respected well.
Now the picture is in Moscow in the Tretyakov Gallery, if you are there, be sure to take a look at the pen of the great master.
The task depicted in the picture could not be offered to students of a standard elementary school: the program of one-class and two-class elementary public schools did not provide for the study of the concept of a degree.
However, Rachinsky did not follow the typical training course; he was confident in the excellent mathematical abilities of most peasant children and considered it possible to significantly complicate the mathematics program.
SOLUTION
First way
There are several ways to solve this expression. If you learned the squares of numbers up to 20 or up to 25 at school, then most likely it will not cause you much difficulty.
This expression is: (100+121+144+169+196) divided by 365, which eventually becomes the quotient of 730 and 365, which is: 2. intermediate answers.
Second way
If you didn’t learn the squares of numbers up to 20 in school, then a simple method based on the use of a reference number may come in handy. This method allows you to simply and quickly multiply any two numbers less than 20. The method is very simple, you need to add the unit of the second to the first number, multiply this amount by 10, and then add the product of units. For example: 11*11=(11+1)*10+1*1=121. The remaining squares are also found: 12*12=(12+2)*10+2*2=140+4=144
13*13=160+9=169
14*14=180+16=196
Then, having found all the squares, the task can be solved in the same way as shown in the first method.
Third way
Another way involves using a simplification of the numerator of a fraction, based on the use of the formulas for the square of the sum and the square of the difference.
If we try to express the squares in the numerator of the fraction through the number 12, we get the following expression. (12 - 2)2 + (12 - 1)2 + 122 + (12 + 1)2 + (12 + 2)2 . If you know well the formulas for the square of the sum and the square of the difference, then you will understand how this expression can be easily reduced to the form: 5*122+2*22+2*12, which equals 5*144+10=730. To multiply 144 by 5, simply divide this number by 2 and multiply by 10, which equals 720. Then we divide this expression by 365 and get: 2.
The fourth solution
Also, this problem can be solved in 1 second if you know the Rachinsky sequences.
in a series of two-digit numbers - the first five of its representatives - have an amazing property. The sum of the squares of the first three numbers in the series (10, 11 and 12) is equal to the sum of the squares of the next two (13 and 14). And this sum equals 365. Easy to remember! So many days in a year. If the year is not a leap year. Knowing this property, the answer can be obtained in a second. Without any intuition...
It is difficult to say which of the proposed methods of calculation is the simplest: everyone chooses his own based on the characteristics of his own mathematical thinking.
Working in a rural school
Sergei Alexandrovich Rachinsky brought to the people:
Bogdanova I. L. - infectious disease specialist, doctor of medical sciences, corresponding member of the USSR Academy of Medical Sciences;
Vasiliev Alexander Petrovich (September 6, 1868 - September 5, 1918) - archpriest, confessor royal family, pastor-teetotaler, patriot-monarchist;
Sinev Nikolai Mikhailovich (December 10, 1906 - September 4, 1991) - Doctor of Technical Sciences (1956), Professor (1966), Honored Worker of Science and Technology of the RSFSR. In 1941 - Deputy Chief Designer for Tank Building, 1948-61 - Head of Design Bureau at the Kirov Plant. In 1961-91 - Deputy Chairman of the USSR State Committee for the Use of Atomic Energy, laureate of the Stalin and State Prizes(1943, 1951, 1953, 1967) and many others.
S.A. Rachinsky (1833-1902), representative of the ancient noble family, was born and died in the village of Tatevo, Belsky district, and meanwhile was a corresponding member of the Imperial St. Petersburg Academy of Sciences, who devoted his life to creating a Russian rural school. Last May marked the 180th anniversary of the birth of this outstanding Russian man, a true ascetic, a tireless worker, a forgotten rural teacher and an amazing thinker.
Who has L.N. Tolstoy learned to build a village school,
P.I. Tchaikovsky received recordings of folk songs,
and V.V. Rozanov was spiritually instructed in matters of writing.
By the way, the author of the above-mentioned picture, Nikolai Bogdanov - Belsky, came from the poor and was a student of Sergei Alexandrovich, who created about three dozen rural schools and at his own expense helped his brightest students to realize themselves professionally, who became not only rural teachers (about 40 people!) Or professional artists (3 pupils, including Bogdanov), but also the teacher of the king’s children, a graduate of the St. Petersburg Theological Academy, Archpriest Alexander Vasiliev , and a monk of the Trinity-Sergius Lavra, like Titus (Nikonov).
Rachinsky built not only schools, but also hospitals in Russian villages, the peasants of the Belsky district called him nothing more than "father of their own." Through the efforts of Rachinsky, sobriety societies were recreated in Russia, uniting tens of thousands of people throughout the empire by the beginning of the 1900s.
Now this problem has become even more urgent, drug addiction has now grown to it. It is gratifying that the sobriety path of the educator is again picked up, that sobriety societies named after Rachinsky are reappearing in Russia
Russian ascetic teachers looked at teaching as a holy mission, a great service to the noble goals of raising spirituality among the people.
"May Man" Sergei Rachinsky passed away on May 2, 1902. Dozens of priests and teachers, rectors of theological seminaries, writers, and scientists gathered for his burial. In the decade before the revolution, more than a dozen books were written about the life and work of Rachinsky, the experience of his school was used in England and Japan.
