INTRODUCTION

At all times, mathematics has been and remains one of the main subjects in school, because mathematical knowledge is necessary for all people. Not every student, studying at school, knows what profession he will choose in the future, but everyone understands that mathematics is necessary for solving many life problems: calculations in a store, payment for utilities, calculation family budget etc. In addition, all schoolchildren need to take exams in the 9th grade and in the 11th grade, and for this, starting from the 1st grade, it is necessary to master mathematics with high quality, and above all, you need to learn how to count.

Is it possible to imagine a world without numbers? Without numbers, you won’t make a purchase, you won’t know the time, you won’t dial a phone number. And spaceships, lasers and all the others technical achievements?! They would simply be impossible if it were not for the science of numbers.

Two elements dominate mathematics - numbers and figures with their infinite variety of properties and relationships. In my work, preference is given to the elements of numbers and actions with them.

Now, at the stage of the rapid development of informatics and computer technology, modern schoolchildren do not want to bother themselves with mental arithmetic. So I decidedshow not only that the process of performing an action can be important, but also an interesting activity.

Target: to study the methods of fast counting, to show the need for their application to simplify calculations.

In accordance with the goal, the tasks:

Investigate whether students use quick counting techniques.
Learn quick counting techniques that you can use to make calculations easier.
Make a memo for students in grades 5-6 to use quick counting techniques.

Object of study:quick counting techniques.

Subject of study: calculation process.

Research hypothesis:if it is shown that the use of fast counting techniques facilitates calculations, then it can be achieved that the computational culture of students will increase, and it will be easier for them to solve practical problems.

The following were used in the work tricks and methods : survey (questionnaire), analysis (statistical data processing), work with information sources, practical work, observations.

This work refers toapplied research, because it shows the role of applying fast counting techniques for practical activities.

While working on a report, Iused the following methods:

search method using scientific and educational literature, as well as searching for the necessary information on the Internet;
practical method of performing calculations using non-standard counting algorithms;
analysis data obtained during the study.

Relevance my research is that in our time more and more often calculators come to the aid of students, and all large quantity students cannot count orally. But the study of mathematics develops logical thinking, memory, flexibility of the mind, accustoms a person to accuracy, to the ability to see the main thing, provides the necessary information for understanding challenging tasks arising in various fields of activity modern man. Therefore, in my work, I want to show how you can count quickly and correctly and that the process of performing actions can be not only useful, but also interesting. It is the use of non-standard techniques in the formation of computational skills that enhances students' interest in mathematics and contributes to the development of mathematical abilities.

Per simple actions addition, subtraction, multiplication and division are hidden secrets of the history of mathematics. Accidentally heard the words "multiplication by a lattice", "chess way" intrigued. I wanted to know these and other methods of calculation, as well as compare them with today's ones.

Can you count? The question, perhaps even offensive for a person older than three years of age. Who can't count? Everyone will answer that for this, special art is not required. And he will be right. But the question is how to count? You can count on a calculator, you can count as a column in a notebook, or you can count verbally using quick counting techniques. I count very quickly verbally, I almost never solve in a column, in writing, all because I know and apply various methods of fast counting. Of my classmates, few people can count quickly orally, and I wanted to find out if they know the tricks of quick counting, if not, then help them master these tricks, for this purpose, compose a memo for them with quick counting tricks.

In order to find out whether modern schoolchildren know other ways to perform arithmetic operations, except for multiplication, addition, subtraction by a column and division by a “corner” and would like to learn new ways, a test survey was conducted.

To begin with, I conducted a survey in the 6th grade of our school. Asked the guys simple questions. Why do you need to know how to count? What school subjects require correct arithmetic? Do they know how to count quickly? Would you like to learn how to count quickly orally? (Appendix I).

61 people took part in the survey. After analyzing the results, I concluded that the majority of students believe that the ability to count is useful in life and is necessary at school, especially when studying mathematics, physics, chemistry, computer science and technology. Several students know how to count quickly, and almost everyone would like to learn how to count quickly. (The results of the survey are reflected in the diagrams) (Appendix II).

After statistical processing of the data, I concluded that not all students know quick counting techniques, so it is necessary to make quick counting techniques for students in grades 5-6 in order to use them when performing calculations.

Survey results:

Question	5th grade	6 classes	Total
	Yes	No	don't know	Yes	No	don't know



Would you like to know?

Summary table of the survey:

Question	5, 6 grades
	Yes	No	don't know
Do modern people need to be able to perform arithmetic operations with natural numbers?
Can you multiply, add, subtract numbers in a column, divide by a “corner”?
Do you know other ways to do arithmetic?
Would you like to know?

Based on the results of the survey, we can conclude that in most cases, modern schoolchildren do not know other ways to perform actions other than multiplication, addition, subtraction by a column and division by a “corner”, since they rarely refer to material that is outside school curriculum.

Chapter I. HISTORY OF THE ACCOUNT

1. HOW THE NUMBERS ARISED

People learned to count objects back in the ancient Stone Age - the Paleolithic, tens of thousands of years ago. How did it happen? At first, people only compared different quantities of the same objects by eye. They could determine which of the two piles had more fruit, which herd had more deer, and so on. If one tribe exchanged caught fish for stone knives made by people of another tribe, it was not necessary to count how many fish they brought and how many knives. It was enough to put a knife next to each fish for the exchange between the tribes to take place.

In order to be successful agriculture, arithmetic knowledge was required. Without counting days, it was difficult to determine when to sow the fields, when to start watering, when to expect offspring from animals. It was necessary to know how many sheep were in the flock, how many sacks of grain were put in the barns.
And more than eight thousand years ago, the ancient shepherds began to make mugs of clay - one for each sheep. To find out if at least one sheep was lost during the day, the shepherd put aside a mug each time the next animal entered the pen. And only after making sure that the same number of sheep returned as there were circles, he calmly went to sleep. But in his flock were not only sheep - he grazed cows, and goats, and donkeys. Therefore, other figures had to be made of clay. And farmers with the help of clay figurines kept records harvested crop, noting how many sacks of grain are put in the barn, how many jugs of oil are squeezed out of olives, how many pieces of linen are woven. If the sheep bore offspring, the shepherd added new mugs to the mugs, and if some of the sheep went for meat, several mugs had to be removed. So, still not knowing how to count, ancient people were engaged in arithmetic.

Then numerals appeared in the human language, and people were able to name the number of objects, animals, days. Usually there were few such numerals. For example, the Murray River tribe in Australia had two prime numbers: enea (1) and petcheval (2). They expressed other numbers with compound numerals: 3 = “petcheval-enea”, 4 “petcheval-petcheval”, etc. Another Australian tribe, the Camiloroi, had simple numerals mal (1), bulan (2), guliba (3). And here other numbers were obtained by adding smaller ones: 4="bulan-bulan", 5="bulan-guliba", 6="guliba-guliba", etc.

For many peoples, the name of the number depended on the items being counted. If the inhabitants of the Fiji Islands counted boats, then the number 10 was called "bolo"; if they counted coconuts, then the number 10 was called "karo". The Nivkhs living on Sakhalin near the banks of the Amur did the same. Back in the 19th century, they called the same number with different words if they counted people, fish, boats, nets, stars, sticks.

We still use different indefinite numerals with the meaning "a lot": "crowd", "herd", "flock", "heap", "bundle" and others.

With the development of production and trade, people began to better understand what three boats and three axes, ten arrows and ten nuts have in common. The tribes often engaged in item-for-item exchanges; for example, they exchanged 5 edible roots for 5 fish. It became clear that 5 is the same for both roots and fish; so it can be called with one word.

Similar counting methods were used by other peoples. So there were numberings based on counting by fives, tens, twenties.

So far, I have talked about mental counting. How were the numbers written? At first, even before the advent of writing, they used notches on sticks, notches on bones, knots on ropes. The found wolf bone in Dolni-Vestonice (Czechoslovakia) had 55 cuts made more than 25,000 years ago.

When writing appeared, there were also numbers for writing numbers. At first, the numbers resembled notches on sticks: in Egypt and Babylon, in Etruria and Dates, in India and China, small numbers were written down with sticks or dashes. For example, the number 5 was written with five sticks. The Aztecs and Mayans used dots instead of sticks. Then special signs appeared for some numbers, such as 5 and 10.

At that time, almost all numbering was not positional, but similar to Roman numbering. Only one Babylonian sexagesimal numbering was positional. But for a long time there was also no zero in it, as well as a comma separating the integer part from the fractional one. Therefore, the same figure could mean 1, 60, and 3600. One had to guess the meaning of the number according to the meaning of the problem.

A few centuries before the new era, they invented new way writing numbers, in which the letters of the ordinary alphabet served as digits. The first 9 letters denoted the numbers tens 10, 20, ..., 90, and another 9 letters denoted hundreds. This alphabetical numbering was used until the 17th century. To distinguish “real” letters from numbers, a dash was placed above the letters-numbers (in Russia this dash was called “titlo”).

In all these numberings, it was very difficult to perform arithmetic operations. Therefore, the invention in the VI century by the Indians of decimal positional numbering is rightfully considered one of the greatest achievements of mankind. Indian numbering and Indian numerals became known in Europe from the Arabs and are usually referred to as Arabic.

When writing fractions for a long time the whole part was written in the new decimal numbering, and the fractional part - in sexagesimal. But at the beginning of the XV century. Samarkand mathematician and astronomer al-Kashi began to use decimal fractions in calculations.

The numbers we work with are positive and negative numbers. But it turns out that these are not all the numbers that are used in mathematics and other sciences. And you can find out about them without waiting high school, and much earlier, if you study the history of the emergence of numbers in mathematics.

Chapter II. OLD METHODS OF CALCULATION

2.1. RUSSIAN PEASANT METHOD OF MULTIPLICATION

In Russia, several centuries ago, among the peasants of some provinces, a method was spread that did not require knowledge of the entire multiplication table. It was only necessary to be able to multiply and divide by 2. This method was called PEASANT (there is an opinion that it originates from the Egyptian).

Example: multiply 47 by 35,

write the numbers on one line, draw a vertical line between them;
we will divide the left number by 2, multiply the right number by 2 (if a remainder occurs during division, then we discard the remainder);
the division ends when a unit appears on the left;
we cross out those lines in which there are even numbers on the left;35 + 70 + 140 + 280 + 1120 = 1645
then add the remaining numbers to the right - this is the result.

2.2. GRID METHOD

The outstanding Arab mathematician and astronomer Abu Abdalah Mohammed Ben Mussa al-Khwarizmi lived and worked in Baghdad. The scientist worked in the House of Wisdom, where there was a library and an observatory, almost all major Arab scientists worked here.

There is very little information about the life and work of Muhammad al-Khwarizmi. Only two of his works have survived - on algebra and on arithmetic. In the last of these books, four rules of arithmetic are given, almost the same as those used today.


	1	3
	0	1

In his "The Book of Indian Counting"the scientist described a method invented in ancient india, and later called"GRID METHOD". This method is even simpler than the one used today.

Example: multiply 25 and 63.

Let's draw a table in which two cells in length and two in width, we write one number in length and another in width. In the cells we write the result of multiplying these numbers, at their intersection we separate the tens and ones with a diagonal. We add the resulting numbers diagonally, and the result can be read along the arrow (down and to the right).

I have considered a simple example, however, any multi-valued numbers can be multiplied in this way.

Let's consider another example: multiply 987 and 12:

draw a 3 by 2 rectangle (according to the number of decimal places for each factor);
then we divide the square cells diagonally;
at the top of the table we write the number 987;
on the left of the table the number 12;
now in each square we enter the product of numbers located in the same line and in the same column with this square, tens below the diagonal, ones above;
after filling in all the triangles, the numbers in them are added along each diagonal on the right side;
the result is read by the arrow.

This algorithm for multiplying two natural numbers was common in the Middle Ages in the East and Italy.

I would like to note the inconvenience of this method in the laboriousness of preparing a rectangular table, although the calculation process itself is interesting and filling in the table resembles a game.

2.3. MULTIPLICATION ON FINGERS

The ancient Egyptians were very religious and believed that the soul of the deceased in the afterlife was subjected to an exam by counting on the fingers. This already speaks of the importance that the ancients attached to this method of performing the multiplication of natural numbers (it was calledFINGER ACCOUNT).

They multiplied single-digit numbers from 6 to 9 on the fingers. To do this, they extended as many fingers on one hand as the first multiplier exceeded the number 5, and on the second they did the same for the second multiplier. The rest of the fingers were bent. After that, they took as many tens as the fingers extended on both hands, and added to this number the product of the bent fingers on the first and second hands.

Example: 8 ∙ 9 = 72

Later, the finger count was improved - they learned to show numbers up to 10,000 with the help of fingers.

finger movement - this is another way to help memory: with the help of fingers, remember the multiplication table for 9. Putting both hands side by side on the table, we number the fingers of both hands in order as follows: the first finger on the left will be denoted by 1, the second after it will be denoted by the number 2, then 3 , 4 ... up to the tenth finger, which means 10. If you need to multiply by 9 any of the first nine numbers, then for this, without moving your hands from the table, you need to lift up the finger whose number means the number by which nine is multiplied; then the number of fingers to the left of the raised finger determines the number of tens, and the number of fingers to the right of the raised finger indicates the number of units of the resulting product (see for yourself).

So, the old multiplication methods we have considered show that the algorithm for multiplying natural numbers used in school is not the only one and it was not always known.

However, it is quite fast and most convenient.

Chapter III. ORAL COUNTING - GYMNASTICS OF THE MIND

3.1. DIFFERENT WAYS OF ADDITION AND SUBTRACTION

ADDITION

The basic rule for doing mental addition is:

To add 9 to a number, add 10 to it and subtract 1; to add 8, add 10 and subtract 2; to add 7, add 10 and subtract 3, and so on. For example:

56+8=56+10-2=64;

65+9=65+10-1=74.

ADDITION IN THE MIND OF TWO-DIGITAL NUMBERS

If the number of units in the added number is greater than 5, then the number must be rounded up, and then subtract the rounding error from the resulting amount. If the number of units is less, then we add tens first, and then units. For example:

34+48=34+50-2=82;

27+31=27+30+1=58.

ADDITION OF THREE-DIGIT NUMBERS

We add from left to right, that is, first hundreds, then tens, and then ones. For example:

359+523= 300+500+50+20+9+3=882;

456+298=400+200+50+90+6+8=754.

SUBTRACTION

To subtract two numbers in your head, you need to round the subtracted, and then correct the resulting answer.

56-9=56-10+1=47;

436-87=436-100+13=349.

SUBTRACT A NUMBER LESS THAN 100 FROM A NUMBER OVER 100

If the subtrahend is less than 100 and the minuend is greater than 100 but less than 200, there is an easy way to calculate the difference in your mind. 134-76=58

76 is 24 less than 100. 134 is 34 more than 100. Add 24 to 34 and get the answer: 58.

152-88=64

88 is 12 less than 100, and 152 is more than 100 by 52, so

152-88=12+52=64

3.2. DIFFERENT WAYS OF MULTIPLICATION AND DIVISION

After studying the literature on this topic, I made a selection, from a variety of quick counting techniques, I chose multiplication and division techniques that are easy to understand and use for any student. I included these techniques in the memo (Appendix III), which will be useful for students in grades 5-6.

Multiplying and dividing a number by 4.

To multiply a number by 4, you need to multiply it by 2 twice.

For example:

26 4=(26 2) 2=52 2=104;

417 4=(417 2) 2=834 2=1668.

To divide a number by 4, you need to divide it twice by 2.

For example:

324:4=(324:2):2=162:2=81.

Multiplying and dividing a number by 5.

To multiply a number by 5, you need to multiply it by 10 and divide by 2.

For example:

236 5=(236 10):2=2360:2=1180.

To divide a number by 5, you need to multiply 2 and divide by 10, i.e. separate the last digit with a comma.

For example:

236:5=(236 2):10=472:10=47.2.

Multiplying a number by 1.5.

To multiply a number by 1.5, you need to add half of it to the original number.

For example: 34 1.5=34+17=51;

146 1.5=146+73=219.

Multiplying a number by 9.

To multiply a number by 9, add 0 to it and subtract the original number.

For example: 72 9=720-72=648.

Multiply by 25 a number divisible by 4.

To multiply by 25 a number that is divisible by 4, you need to divide it by 4 and multiply the resulting number by 100.

For example: 124 25=(124:4) 100=31 100=3100.

Multiplying a two-digit number by 11

When multiplying a two-digit number by 11, you need to enter the sum of these digits between the units digit and the tens digit, and if the sum of the digits is more than 10, then one must be added to the most significant digit (first digit).

For example:
23 11=253, because 2+3=5, so between 2 and 3 we put the number 5;
57 11=627, because 5+7=12, put the number 2 between 5 and 7, and add 1 to 5, write 6 instead of 5.

“Fold the edges, put them in the middle” - these words will help you remember easily this way multiplication by 11.

This method is only suitable for multiplying two-digit numbers.

Multiplying a two-digit number by 101.

In order to multiply a number by 101, you need to attribute this number to itself.

For example: 34 101 = 3434.

To clarify, 34 101 = 34 100+34 1=3400+34=3434.

Squaring a two-digit number ending in 5.

To square a two-digit number ending in 5, you need to multiply the tens digit by the digit greater by one, and add the number 25 to the resulting product on the right.
For example: 35 2 =1225, i.e. 3 4 \u003d 12 and we attribute 25 to 12, we get 1225.

Squaring a two-digit number starting with 5.

To square a two-digit number starting with five, you need to add the second digit of the number to 25 and assign the square of the second digit to the right, and if the square of the second digit is a single-digit number, then the number 0 must be assigned before it.

For example:
52 2 = 2704, because 25+2=28 and 2 2 =04;
58 2 = 3364, because 25+8=33 and 82=64.

3.3. GAMES

Guessing the received number.

Think of a number. Add 11 to it; multiply the amount received by 2; subtract 20 from this product; multiply the resulting difference by 5 and subtract a number from the new product that is 10 times the number you intended.I guess you got 10. Right?
Think of a number. Treat him. Subtract 1 from the result. Multiply the result by 5. Add 20 to the result. Divide the result by 15. Subtract the intended result from the result.You got 1.
Think of a number. Multiply it by 6. Subtract 3. Multiply by 2. Add 26. Subtract twice what you thought. Divide by 10. Subtract what you thought.You got 2.
Think of a number. Triple it. Subtract 2. Multiply by 5. Add 5. Divide by 5. Add 1. Divide by what you thought.You got 3.
Think of a number, double it. Add 3. Multiply by 4. Subtract 12. Divide by what you thought.You got 8.

Guessing the given numbers.

Invite your friends to think of any numbers. Let everyone add 5 to their intended number.
Let the resulting sum be multiplied by 3.
Let subtract 7 from the product.
Let's subtract 8 more from the result.
Let everyone give you a sheet with the final result. Looking at the sheet, you immediately tell everyone what number he has in mind.

(To guess the conceived number, the result, written on a piece of paper or told to you orally, is divided by 3).

CONCLUSION

We have entered the new millennium! Grandiose discoveries and achievements of mankind. We know a lot, we can do a lot. It seems something supernatural that with the help of numbers and formulas you can calculate the flight spaceship, "the economic situation" in the country, the weather for "tomorrow", describe the sound of notes in the melody. We know the saying ancient Greek mathematician, a philosopher who lived in the 4th century BC. - Pythagoras - "Everything is a number!".

Describing ancient ways of computing and modern techniques quick calculation, I tried to show that both in the past and in the future, one cannot do without mathematics, a science created by the human mind.

The study of ancient methods of calculation showed that these arithmetic operations were difficult and complex due to the variety of methods and their cumbersome execution.

Modern methods of computing are simple and accessible to everyone.

When getting acquainted with the scientific literature, I discovered faster and more reliable methods of calculation.

It is possible that the first time many will not be able to quickly, on the go, perform these or other calculations. Let at first fail to use the technique shown in the work. No problem. Constant computational training is needed. Lesson after lesson, year after year. It will help to acquire useful oral counting skills.

The German scientist Karl Gauss was called the king of mathematicians. His mathematical talent manifested itself already in childhood. Once at school (Gauss was 10 years old), the teacher asked the class to add up all the numbers from 1 to 100. While he was dictating the task, Gauss already had an answer ready. On his slate board it was written: 101 50=5050. How did he calculate? It's very simple - he applied the quick counting technique, he added the first number to the last, the second to the penultimate one, and so on. There are only 50 such sums and each is equal to 101, so he was able to give the correct answer almost instantly.

1+2+…+50+51+...+99+100=(1+100)+(2+99)+…+(50+51)=101 50=5050. This example shows best of all that it is possible to count quickly and correctly orally almost to all schoolchildren, for this you just need to know the methods of quick counting.

I designed the results of my work in a memo that I will offer to all my classmates, and I will also place it on the school thematic stand “It's interesting!”. It is possible that from the first time not everyone will be able to quickly, on the move, perform calculations using these techniques, even if at first you cannot use the technique shown in the memo, it's okay, you just need constant computational training. It will help you to acquire useful skills of quick counting.

After statistical processing of the data, the following results were obtained. results:

You need to be able to count, because it will come in handy in life, 93% of students believe that in order to study well at school - 72%, to quickly decide - 61%, to be literate - 34% and it is not necessary to be able to count - only 3%.
Good counting skills are necessary when studying mathematics, according to 100% of students, as well as when studying physics - 90%, chemistry - 80%, computer science - 44%, technology - 36%.
16% (many tricks), 25% (several tricks) know fast counting tricks, 59% of students do not know quick counting tricks.
21% of students use fast counting techniques, sometimes they are used by 15%.
93% of students would like to learn how to quickly count.

Conclusions:

Knowledge of fast counting techniques allows you to simplify calculations, save time, develop logical thinking and flexibility of mind.
There are practically no quick counting techniques in school textbooks, so the result of this work - a quick counting guide will be very useful for students in grades 5-6.

LIST OF USED LITERATURE

Vantsyan A.G. Mathematics: Textbook for grade 5. - Samara: Fedorov Publishing House, 1999.
Kordemsky B.A., Akhadov A.A. amazing world numbers: Book of students, - M. Enlightenment, 1986.
Minskykh E.M. "From game to knowledge", M., "Enlightenment", 1982
Svechnikov A.A. Numbers, figures, tasks. M., Enlightenment, 1977. Yes No Don't know https://accounts.google.com

December 23, 2013 at 03:10 pm

Effective counting in the mind or warm-up for the brain

Maths

This article was inspired by the topic and is intended to spread the techniques of S.A. Rachinsky for oral counting.
Rachinsky was a wonderful teacher who taught in rural schools in the 19th century and showed own experience that it is possible to develop the skill of fast mental counting. It wasn't much of a problem for his students to calculate a similar example in their minds:

Using round numbers

One of the most common methods of mental counting is that any number can be represented as the sum or difference of numbers, one or more of which is “round”:

Because on the 10 , 100 , 1000 and other round numbers to multiply faster, in the mind you need to reduce everything to such simple operations as 18x100 or 36x10. Accordingly, it is easier to add by “splitting off” a round number, and then adding a “tail”: 1800 + 200 + 190 .
Another example:
31 x 29 = (30 + 1) x (30 - 1) = 30 x 30 - 1 x 1 = 900 - 1 = 899.

Simplify multiplication by division

When calculating mentally, it is more convenient to operate with a dividend and a divisor than with an integer (for example, 5 present in the form 10:2 , a 50 as 100:2 ):
68 x 50 = (68 x 100) : 2 = 6800: 2 = 3400; 3400: 50 = (3400 x 2) : 100 = 6800: 100 = 68.
Similarly, multiplication or division by 25 , after all 25 = 100:4 . For example,
600: 25 = (600: 100) x 4 = 6 x 4 = 24; 24 x 25 = (24 x 100) : 4 = 2400: 4 = 600.
Now it doesn't seem impossible to multiply in the mind 625 on the 53 :
625 x 53 = 625 x 50 + 625 x 3 = (625 x 100) : 2 + 600 x 3 + 25 x 3 = (625 x 100) : 2 + 1800 + (20 + 5) x 3 = = (60000 + 2500): 2 + 1800 + 60 + 15 = 30000 + 1250 + 1800 + 50 + 25 = 33000 + 50 + 50 + 25 = 33125.

Squaring a two-digit number

It turns out that to simply square any two-digit number, it is enough to remember the squares of all numbers from 1 before 25 . Good, squares up 10 we already know from the multiplication table. The remaining squares can be seen in the table below:

Reception Rachinsky is as follows. In order to find the square of any two-digit number, you need the difference between this number and 25 multiply by 100 and to the resulting product add the square of the complement of the given number to 50 or the square of its excess over 50 -Yu. For example,
37^2 = 12 x 100 + 13^2 = 1200 + 169 = 1369; 84^2 = 59 x 100 + 34^2 = 5900 + 9 x 100 + 16^2 = 6800 + 256 = 7056;
In general ( M- two-digit number):

Let's try to apply this trick when squaring a three-digit number, first breaking it into smaller terms:
195^2 = (100 + 95)^2 = 10000 + 2 x 100 x 95 + 95^2 = 10000 + 9500 x 2 + 70 x 100 + 45^2 = 10000 + (90+5) x 2 x 100 + + 7000 + 20 x 100 + 5^2 = 17000 + 19000 + 2000 + 25 = 38025.
Hmm, I wouldn't say it's much easier than stacking, but maybe you can get used to it with time.
And, of course, you should start training with squaring two-digit numbers, and there you can already reach disassembly in your mind.

Multiplication of two-digit numbers

This interesting technique was invented by a 12-year-old student of Rachinsky and is one of the options for adding up to a round number.
Let two two-digit numbers be given, in which the sum of units is equal to 10:
M = 10m + n, K = 10a + 10 - n.
Compiling their product, we get:

For example, let's calculate 77x13. The sum of the units of these numbers is equal to 10 , because 7 + 3 = 10 . First put the smaller number in front of the larger one: 77 x 13 = 13 x 77.
To get round numbers, we take three units from 13 and add them to 77 . Now let's multiply the new numbers 80x10, and to the result we add the product of the selected 3 units to the difference of the old number 77 and a new number 10 :
13 x 77 = 10 x 80 + 3 x (77 - 10) = 800 + 3 x 67 = 800 + 3 x (60 + 7) = 800 + 3 x 60 + 3 x 7 = 800 + 180 + 21 = 800 + 201 = 1001.
This technique has a special case: everything is greatly simplified when two factors have the same number of tens. In this case, the number of tens is multiplied by the number following it, and the product of the units of these numbers is attributed to the result. Let's see how elegant this technique is with an example.
48x42. Number of tens 4 , the next number: 5 ; 4 x 5 = 20 . Product of units: 8x2= 16 . So 48 x 42 = 2016.
99x91. Number of tens: 9 , the next number: 10 ; 9 x 10 = 90 . Product of units: 9 x 1 = 09 . So 99 x 91 = 9009.
Yeah, that is, to multiply 95x95, it is enough to calculate 9 x 10 = 90 and 5 x 5 = 25 and the answer is ready:
95 x 95 = 9025.
Then the previous example can be calculated a little easier:
195^2 = (100 + 95)^2 = 10000 + 2 x 100 x 95 + 95^2 = 10000 + 9500 x 2 + 9025 = 10000 + (90+5) x 2 x 100 + 9000 + 25 = = 10000 + 19000 + 1000 + 8000 + 25 = 38025.

Instead of a conclusion

It would seem, why be able to count in the mind in the 21st century, when you can simply give a voice command to your smartphone? But if you think about what will happen to humanity if it loads not only physical work, but also any mental? Is it degrading? Even if you do not consider mental counting as an end in itself, it is quite suitable for tempering the mind.

References:
“1001 tasks for mental arithmetic at the school of S.A. Rachinsky.

Methods of teaching in the last century such professions as an economist, a salesman, a merchandiser, a teacher of arithmetic elementary school, erased from the memory of society, as remnants of the Soviet past. But they were very useful. In particular, such exercises, which activated brain activity, developed logical thinking, using both hemispheres of the brain to find optimal solutions math problems and be able to count in your mind quickly.

Individual elements methods formed the basis of modern courses in mental mathematics and training programs for rapid mental counting. Today it is a luxury - the ability to quickly calculate in the mind, and in the distant past, it was necessary condition social adaptation and survival.

Why you need to be able to count in your mind

The human brain is an organ that needs a constant load, otherwise the mechanism of atrophy is triggered.

Another feature is that all neural processes in the brain occur simultaneously and are interconnected. So, insufficient physical and mental activity, the predominance of a static load, lead to absent-mindedness, inattention and irritability. In the worst case, it may develop stressful condition the consequences of which are difficult to predict.

Knowledge of the world and laws public life, comes to the child as they grow up and learn, and mathematics plays an important role in this, since it is she who teaches to build logical connections, algorithms and parallels.

Psychologists and experienced teachers identify different reasons why a child needs to learn to count in his mind:

Increasing concentration and observation.
Short term memory training.
Activation of thought processes and development competent speech.
Ability to think creatively and abstractly.
Training the ability to recognize patterns and analogies.

Counting Techniques and Exercises for Adults

The range of tasks and problems solved by an adult is much wider than that of a child. In a number of professions and in everyday life, people daily have to deal with mathematical problems a hundred times a day:

How much profit will it bring me.
Did they cheat me in the store?
Did the reseller overestimate the margin on the purchased goods.
Cheaper to get a loan monthly payment percent or every three months.
Which is better - an hourly salary of 150 rubles or a monthly salary of 18,000 rubles.

The list goes on, but the need for oral counting skills is undeniable.

The preparatory stage - realizing the need for oral counting

Mental mathematics and any other technique designed to teach counting at home in the mind faster and more efficiently teaches adults and children.

Their only difference is the scope of knowledge application. The developers of MM courses are trying to select tasks for adults in such a way that they are in demand in their work.

☞ Example:

You have a futures contract in your hands with an expiration date of January 1, 2019, and you set out to calculate what day of the week this event will fall on (suddenly Friday). All operations are carried out with the last two digits of the year, in our case it is 19. First, you need to add a quarter to 19, this can be done by simple division: 19:2 = 8.5, then 8.5:2 = 4.25. The numbers after the decimal point are discarded. We add: 19 + 4 = 23. The day of the week is determined simply: from the received figure, it is necessary to subtract the product with the number 7 closest to it. In our case, this is 7 * 3 = 21. Therefore, 23 - 21 = 2. The futures expiration date is the second day or Tuesday.

It’s easy to check by looking at the calendar, but if it’s not at hand, this technique can be useful and raise you in the eyes of others.

Video plot

Techniques for fast addition, subtraction, multiplication and division of different numbers

Examples with varying degrees of difficulty require varying amounts of time, although with constant practice the amount of effort expended decreases.

Addition and subtraction in mental mathematics tend to be simplified. Complex and global tasks are divided into smaller and simpler ones. Big numbers are rounded off.

☞ Addition example:

17 996 + 2676 + 3592 = 18 000 + 3600 + 2680 – 4 – 8 — 4 = 21600 + 2000 + 600 + 80 – 10 – 6 = 23600 + 600 + 70 – 6 = 24200 + 70 – 6 = 24270 – 6 = 24264.

At first, it will be difficult to keep such a long chain in your head and you will have to mentally pronounce all the numbers so as not to go astray, but as your short-term memory improves, the process will become easier and clearer.

☞ Subtraction example:

For subtraction, the process is identical. First, subtract the rounded number, and then add the excess. Simple example: 7635 - 5493 = 7635 - 5500 + 7 = 2135 + 7 = 2142

Multiplication and division have their own little tricks, including those previously mentioned in the date example. In practice, most often there are examples with percentages or proportions. The essence of their solution also comes down to fragmentation and simplification of the problem. Some can be solved with just one click.

☞ Multiplication and division example:

You deposited $36,000. e. at 11% and you need to calculate how much profit it will bring. The secret of the calculation is simple - the first and last digit will remain the same, and the middle will be the sum of the two extreme numbers. So 36 * 11 \u003d 3 (3 + 6) 6 \u003d 396, or in our case 396/100% \u003d 3,960 c.u. e.

In most mental methods of multiplication and division, a mandatory and uncontested condition is knowledge of the multiplication table up to ten. For elementary school children, the program for teaching oral counting will be different.

Children are faced with tasks of a different order. In addition to tedious memorization, they are also forced to multiply and divide apples and tomatoes, and if you ask why this is done, the teacher in best case will say “it is necessary”, and the child will lose interest in the whole process as a whole.

It is impossible to change the education system in a month, but helping a child develop oral counting skills is quite real.

Preparatory stage

Explain to the child in plain language why counting in your head is not only useful, but also interesting. If you decide to work with him on your own, select illustrated materials from various sources and make a schedule for joint classes. It is not necessary to practice daily and for many hours. It won't do any good. It is enough to devote twenty minutes to this three times a week, but at the same time, so that the child gets used to it.

Examples of exercises for children

Start with interesting tasks to "get in the game". Show how you can quickly get an answer to a difficult example and overtake all your classmates. Develop leadership qualities.

☞ Example:

Let's use the multiplication rule for two-digit numbers with the same first digits and the last digits that add up to "10" to solve the "44 * 46" example. We multiply the first digit by the one that follows it in order. We also multiply the last digits: 44 * 46 \u003d (4 * 5 \u003d 20; 4 * 6 \u003d 24) \u003d 2024.

At school similar examples solved in the old fashioned way, in a column. It takes a lot of time just to rewrite everything. Knowing the multiplication table for 4, this example can be solved mentally in a couple of seconds.

What is taught at school and is it possible to believe everything

The classical school as a whole is skeptical about the methods of accelerated calculation, citing as an example children who, having been taught the methods of mental mathematics, then do not seek to think logically in other subjects, they want to do everything quickly, as they are used to, and not qualitatively.

But this is due more to the rigidity of the educational program than to the real state of affairs.

Video information

Quick Counting Techniques: Magic Available to All

In order to understand the role that numbers play in our lives, set up a simple experiment. Try to do without them for a while. No numbers, no calculations, no measurements... You will find yourself in a strange world where you will feel absolutely helpless, bound hand and foot. How to get to a meeting on time? Distinguish one bus from another? Make a phone call? Buy bread, sausage, tea? Cook soup or potatoes? Without numbers, and therefore, without counting, life is impossible. But how hard this science is sometimes given! Try to quickly multiply 65 by 23? Does not work? The hand itself reaches for a mobile phone with a calculator. Meanwhile, semi-literate Russian peasants 200 years ago calmly did this, using only the first column of the multiplication table - multiplication by two. Don't believe? But in vain. This is reality.

stone age computer

Even without knowing the numbers, people have already tried to count. If our ancestors, who lived in caves and wore skins, needed to exchange something with a neighboring tribe, they acted simply: they cleared the site and laid out, for example, an arrowhead. Near lay a fish or a handful of nuts. And so on until one of the exchanged goods ran out, or the head of the "trading mission" decided that enough was enough. Primitive, but in its own way very convenient: you won’t get confused, and you won’t be deceived.

With the development of cattle breeding, the tasks became more complicated. A large herd had to be counted somehow in order to know whether all the goats or cows were in place. The "calculating machine" of illiterate but smart shepherds was a dugout pumpkin with pebbles. As soon as the animal left the pen, the shepherd put a pebble in the gourd. In the evening, the herd returned, and the shepherd took out a stone with each animal that entered the pen. If the gourd was empty, he knew the flock was all right. If there were pebbles, he went to look for the loss.

When the numbers appeared, things got more fun. Although for a long time our ancestors used only three numerals: "one", "pair" and "many".

Can you count faster than a computer?

Outrun a device that performs hundreds of millions of operations per second? Impossible... But the one who says this is cruelly disingenuous, or simply deliberately overlooks something. A computer is just a set of chips in plastic; it doesn't count by itself.

Let's put the question in another way: can a person, calculating in his mind, overtake someone who performs calculations on a computer? And here the answer is yes. Indeed, in order to receive an answer from the "black suitcase", the data must first be entered into it. This will be done by a person with the help of fingers or voice. And all these actions have time limits. Insurmountable restrictions. Nature itself supplied them to the human body. Everything except one organ. Brain!

The calculator can only perform two operations: addition and subtraction. Multiplication for him is multiple addition and division is multiple subtraction.

Our brain behaves differently.

The class where the future king of mathematics, Carl Gauss, studied, somehow received the task: add up all the numbers from 1 to 100. Carl wrote the absolutely correct answer on his board as soon as the teacher finished explaining the task. He did not diligently add numbers in order, as any self-respecting computer would do. He applied the formula he discovered himself: 101 x 50 = 5050. And this is far from the only trick that speeds up mental calculations.

The simplest tricks for quick counting

They are taught at school. The simplest: if you need to add 9 to any number, add 10 and subtract 1, if 8 (+ 10 - 2), 7 (+ 10 - 3), etc.

54 + 9 = 54 + 10 - 1 = 63. Fast and convenient.

Two-digit numbers add up just as easily. If the last digit in the second term is greater than five, the number is rounded up to the next ten, and then the "excess" is subtracted. 22 + 47 = 22 + 50 – 3 = 69

With three-digit numbers, there are no difficulties in the same way. We add them, as we read, from left to right: 321 + 543 \u003d 300 + 500 + 20 + 40 + 1 + 3 \u003d 864. Much easier than in a column. And much faster.

What about subtraction? The principle is the same: we round the subtracted to the nearest integer and add the missing one: 57 - 8 = 57 - 10 + 2 = 49; 43 - 27 \u003d 43 - 30 + 3 \u003d 16. Faster than on a calculator - and no complaints from the teacher even during the test!

Do I need to learn the multiplication table?

Children usually hate this. And they do it right. No need to teach her! But do not rush to be outraged. No one claims that the table does not need to be known.

Its invention is attributed to Pythagoras, but most likely great mathematician only gave a finished, concise form to what was already known. At the excavations of ancient Mesopotamia, archaeologists found clay tablets with the sacramental: "2 x 2". People have been using this to the highest degree for a long time. convenient system calculations and discovered many ways that help to comprehend the internal logic and beauty of the table, to understand - and not stupidly, mechanically memorize.

AT ancient China they began to learn the table by multiplying by 9. It’s easier this way, and not least because you can multiply by 9 “on your fingers”.

Place both hands on the table, palms down. The first finger from the left is 1, the second is 2, and so on. Let's say you need to solve a 6 x 9 problem. Raise your sixth finger. Fingers on the left will show tens, on the right - ones. Answer 54.

Example: 8 x 7. Left hand- the first multiplier, the right - the second. There are five fingers on the hand, and we need 8 and 7. We bend three fingers on the left hand (5 + 3 = 8), on the right 2 (5 + 2 = 7). We have five bent fingers, which means five dozen. Now multiply the rest: 2 x 3 = 6. These are units. Total 56.

This is just one of the simplest methods of "finger" multiplication. There are many of them. "On the fingers" you can operate with numbers up to 10,000!

The "finger" system has a bonus: the child perceives it as fun game. Engaged willingly, experiences a lot positive emotions and as a result, very soon begins to do all the operations in the mind, without the help of fingers.

You can also divide with your fingers, but it's a little more complicated. Programmers still use their hands to convert numbers from decimal to binary - it's more convenient and much faster than on a computer. But within the framework of the school curriculum, you can learn to quickly divide even without fingers, in your mind.

Let's say you need to solve example 91: 13. Column? No need to mess up paper. The dividend ends with one. And the divisor is three. What is the very first thing in the multiplication table where the triple is involved, and ends with one? 3 x 7 = 21. Seven! That's it, we got her. Need 84: 14. Remember the table: 6 x 4 = 24. The answer is 6. Simple? Still would!

number magic

Most of the quick counting tricks are similar to magic tricks. Take at least the most famous example of multiplying by 11. To, for example, 32 x 11, you need to write 3 and 2 along the edges, and put their sum in the middle: 352.

To multiply a two-digit number by 101, simply write the number twice. 34 x 101 = 3434.

To multiply a number by 4, multiply it by 2 twice. To divide, divide by 2 twice.

Many witty and, most importantly, quick tricks help to raise a number to a power, extract Square root. The famous "30 tricks of Perelman" for mathematical thinking people will be cooler than the Copperfield show, because they also UNDERSTAND what is happening and how it happens. Well, the rest can just enjoy the beautiful focus. For example, you need to multiply 45 by 37. Let's write the numbers on a sheet and separate them with a vertical line. We divide the left number by 2, discarding the remainder, until we get one. Right - multiply until the number of lines in the column is equal. Then we cross out from the RIGHT column all those numbers opposite which an even result is obtained in the LEFT column. We add the remaining numbers from the right column. It turns out 1665. Multiply the numbers in the usual way. The answer will fit.

"Charging" for the mind

Quick counting techniques can make life easier for a child at school, for mom in a store or kitchen, and for dad at work or in the office. But we prefer the calculator. Why? We don't like to stress. It's hard for us to keep numbers, even two-digit ones, in our heads. For some reason they don't hold up.

Try to go to the middle of the room and sit on the twine. For some reason "does not sit down", right? And the gymnast does it quite calmly, without straining. Need to train!

The easiest way to train and, at the same time, warm up the brain: verbal counting aloud (mandatory!) through the number to one hundred and back. In the morning, standing in the shower, or preparing breakfast, count: 2.. 4.. 6.. 100... 98.. 96. You can count in three, in eight - the main thing is to do it out loud. After just a couple of weeks of regular practice, you will be surprised how EASIER it becomes to deal with numbers.

You forgot your money at home and a colleague kindly agreed to buy you lunch. On the way back, you stopped by the store for a snack, and there they announced a super promotion for your favorite chocolates. You could not resist and took 5 pieces. You were so busy shopping that you forgot about your smartphone and did not calculate how much you owed a colleague in the end. The situation is not pretty. It would be much easier to just put everything together in your mind. But ... who needs it when every phone has a calculator for a long time!

Accounting in the mind can be as fast as on a calculator. Especially when it comes to domestic issues. The main thing is to master the techniques of fast counting and practice them periodically. In the material we present the simplest of them.

Breaking the task into parts

Even the most difficult arithmetic problems can be broken down into simple ones.

Example: how do you calculate a 15% discount if the full price of the item is known?

In this case, it makes sense to split 15 into 10% and 5%. 10% is easy enough to take away, and 5% is half of 10%.

Suppose we have a product for 900 rubles, 10% of it - 90 rubles, 5% - 45. Add up: 90 + 45 \u003d 135. The final cost of the goods with a 15% discount: 900 - 135 \u003d 765 rubles.

Rounding to the nearest integer

This technique involves the use of a complement - a number that fills the gap between the given number and the number that usually ends in 00.

For example, the complementary number for 87 would be 13, since their sum is 100.

Example 1234 - 678 seems complicated. Let's round 678 to 700. It will be much easier to calculate 1234 - 700, the result is 534.

Since we subtracted too big number, then the missing result must be returned: 700 - 678 = 22, add 22 to 534 and get the final result 556.

Multiply by 11

We know how easy it is to multiply any single-digit number by 11: just repeat it twice and you're done!

But few people have the skill of multiplying two-digit and even three-digit numbers by 11.

To multiply a two-digit number by 11, you need to spread its digits in different sides and write their sum in the middle. If the sum is more than 10, then in the middle we leave the second digit from the received number, and add ten, that is, one, to the first digit.

Example 1: 36×11 = 3 (3+6) 6 = 396

Example 2: 57×11 = 5 (5+7) 7 = 627

To multiply three digit numbers:

Leave the first and last digit of the number unchanged.
Add the penultimate digit with the last one and write down the result. If it is more than 10, remember the one.
Add the second number to the first number and write down the result. If there is one left from the previous addition, add it to the result.
If as a result of the last addition one remains, add it to the first digit of the original number.

Example 3 : 869×11

We remember 9 as a temporary result. Result: 8...9.
We add 6 and 9, we get 15. We write 5 before 9, 1 - remember. Result: 8...59 (1 in mind).
We add 8 and 6, we get 14, we add 1 from the previous result. Result: 8559 (1 in mind).
We add to 8 the unit from the previous result. Result: 9559.

Multiplication of numbers from 11 to 19

You can multiply such numbers using the following algorithm:

Any number from the range from 11 to 19 is represented as tens and ones.
We get the formula: (10+a)×(10+b).
Expand the brackets: 100+10×b+10×a+a×b.
We take the common factor out of brackets and get the final formula by which we can count and which makes sense to remember: 100+10×(a+b)+a×b.

Example: 13×17

Let's add the units - 3+7=10.
Multiply the result by 10: 10×10 = 100.
Let's add 100: 100+100=200.
Multiply units: 3 × 7 = 21.
Let's add to the result from step 3: 200+21 = 221.

mental arithmetic

You can learn to count in your mind by mastering the techniques of mental arithmetic. First, you learn how to perform arithmetic operations on Japanese abacus - soroban. Then you practice doing the same calculations by moving the knuckles in your mind. We have already written in more detail about. Mental arithmetic courses will fully help you master the technique!