The program at the rate of quick counting. Forms of counting in elementary school

Bibliographic description: Vladimirov A. I., Mikhailova V. V., Shmeleva S. P. Interesting ways quick count // Young scientist. - 2016. - No. 6.1. - S. 15-17..03.2019).



Introduction

Mental counting is gymnastics for the mind. Mental counting is the oldest way of calculating. Mastering computational skills develops memory and helps to assimilate subjects of the natural and mathematical cycle.

There are many ways to simplify arithmetic operations. Knowledge of simplified calculation techniques is especially important in cases where the calculator does not have tables and a calculator at his disposal.

We want to dwell on the methods of addition, subtraction, multiplication, division, for the production of which it is enough to count or use a pen and paper.

The motivation for choosing the topic was the desire to continue the formation of computational skills, the ability to quickly and clearly find the result of mathematical operations.

The rules and techniques of calculations do not depend on whether they are performed in writing or orally. However, mastering the skills of oral calculations is of great value not because they are used more often in everyday life than written calculations. This is also important because they speed up written calculations, gain experience in rational calculations, and give a gain in computational work.

In mathematics lessons, we have to do a lot of oral calculations, and when the teacher showed us the method of fast multiplication by the numbers 11, we had an idea if there were still methods of fast calculation. We set ourselves the task of finding and testing other methods of fast calculation.

b) to do well in school; (sixteen%)

c) to decide quickly; (sixteen%)

d) to be literate; (52%)

2. List, when studying, which school subjects you will need to count correctly ?

a) mathematics; (80%)

b) physics; (fifteen%)

c) chemistry; (5%)

d) technology;

e) music;

3. Do you know how to count quickly?

a) yes, a lot;

b) yes, a few (85%);

c) no, I don't know (15%).

4. Do you use fast counting techniques in calculations?

b) no (85%)

5. Would you like to learn quick counting techniques to quickly count?

b) no (8%).

They say that if you want to learn how to swim, you must enter the water, and if you want to be able to solve problems, you must start solving them. But first you need to master the basics of arithmetic. Learning to count quickly, counting in the mind is possible only with great desire and systematic training in problem solving.

But the methods of fast mental counting have been known for a long time. The excellent mental arithmetic abilities of such brilliant mathematicians as Gauss, von Neumann, Euler or Wallis are a real delight. Much has been written about this. We want to tell and show some well-known computational secrets. And then a completely different math will open before you. Lively, useful and understandable.

1. Methods for fast multiplication

1. COUNTING ON FINGERS

A way to quickly multiply numbers within the first ten by 9.

Let's say we need to multiply 7 by 9.

Let's turn our hands with palms facing us and bend the seventh finger (starting counting from thumb left).

The number of fingers to the left of the bent one will be equal to tens, and to the right - units of the desired product.

Rice. 1. Finger counting

2. MULTIPLICATION OF NUMBERS FROM 10 TO 20

It is very easy to multiply such numbers.

To one of the numbers it is necessary to add the number of units of the other, multiply by 10 and add the product of units of numbers.

Example 1. 16∙18=(16+8) ∙ 10+6 ∙ 8=288, or

Example 2. 17 ∙ 17=(17+7) ∙ 10+7 ∙ 7=289.

Task: Multiply quickly 19 ∙ 13. Answer 19 ∙13=(19+3) ∙10 +9 ∙3=247.

3. MULTIPLY BY 11

To multiply a two-digit number whose sum of digits does not exceed 10 by 11, you need to move the digits of this number apart and put the sum of these digits between them.

72 ∙ 11 = 7 (7 + 2) 2 = 792;

35 ∙ 11 = 3 (3 + 5) 5 = 385.

To multiply by 11 a two-digit number whose sum of digits is 10 or more than 10, you must mentally push the digits of this number, put the sum of these digits between them, and then add one to the first digit, and leave the second and last (third) unchanged.

Example .

94 ∙ 11 = 9 (9 + 4) 4 = 9 (13) 4 = (9 + 1) 34 = 1034.

Task: Multiply quickly 54 ∙ 11 (594)

Task: Multiply quickly 67∙ 11 (737)

4. MULTIPLYING BY 22, 33, ..., 99

To multiply a two-digit number by 22, 33, ..., 99, this multiplier must be represented as a product of a single-digit number (from 2 to 9) by 11, that is, 44 \u003d 4 11; 55 = 5 ∙ 11 etc. Then multiply the product of the first numbers by 11.

Example 1. 24 ∙ 22 = 24 ∙ 2 ∙ 11 = 48 ∙ 11 = 528

Example 2. 23 ∙ 33 = 23 ∙ 3 ∙ 11= 69 ∙ 11 = 759

Task: Multiply 18∙44

5. MULTIPLY BY 5, BY 50, BY 25, BY 125

When multiplying by these numbers, you can use the following expressions:

a ∙ 5=a ∙ 10:2 a ∙ 50=a ∙ 100:2

a ∙ 25=a ∙ 100:4 a ∙ 125=a ∙ 1000:8

Example1. 17 ∙ 5=17 ∙ 10:2=170:2=85

Example 2. 43 ∙ 50=43 ∙ 100:2=4300:2=2150

Example 3. 27 ∙ 25=27 ∙ 100:4=2700:4=675

Example 4. 96 ∙ 125=96:8 ∙ 1000=12 ∙ 1000=12000

Task: multiply 824∙25

Task: multiply 348∙50

&2. Ways to quickly divide

1. DIVISION BY 5, BY 50, BY 25

When dividing by 5, by 50, by 25, you can use the following expressions:

a:5= a ∙ 2:10 a:50=a ∙ 2:100

a:25=a ∙ 4:100

35:5=35 ∙ 2:10=70:10=7

3750:50=3750 ∙ 2:100=7500:100=75

6400:25=6400 ∙ 4:100=25600:100=256

&3. Ways to quickly add and subtract natural numbers.

If one of the terms is increased by several units, then the same number of units must be subtracted from the resulting amount.

Example. 785+963=785+(963+7)-7=785+970-7= 1748

If one of the terms is increased by several units, and the second is reduced by the same number of units, then the sum will not change.

Example. 762+639=(762+8)+(639-8)=770 + 631=1401

If the subtrahend is reduced by several units and the minuend is increased by the same number of units, then the difference will not change.

Example. 529-435=(529-5)-(435+5)=524-440=84

Conclusion

There are ways to quickly add, subtract, multiply, divide, exponentiate. We have considered only a few ways to quickly count.

All the methods of mental calculation we have considered speak of the long-standing interest of scientists and ordinary people in playing with numbers. Using some of these methods in the classroom or at home, you can develop the speed of calculations, achieve success in the study of all school subjects.

Multiplication without a calculator is a training of memory and mathematical thinking. Computer technology is improving to this day, but any machine does what people put into it, and we have learned some tricks of mental counting that will help us in life.

We were interested in working on the project. So far, we have only studied and analyzed already known ways quick account.

But who knows, perhaps in the future we ourselves will be able to discover new ways of fast computing.

Literature:

1. Arutyunyan E., Levitas G. Entertaining Mathematics. - M .: AST - PRESS, 1999. - 368 p.
2. Gardner M. Mathematical miracles and secrets. - M., 1978.
3. Glazer G.I. History of mathematics at school. - M., 1981.
4. "First of September" Mathematics No. 3 (15), 2007.
5. Tatarchenko T.D. Methods for quick counting in the classroom, "Mathematics at School", 2008, No. 7, p.68.
6. Oral account / Comp. P.M. Kamaev. - M .: Chistye Prudy, 2007 - Library "First of September", series "Mathematics". Issue. 3(15).
7. http://portfolio.1september.ru/subject.php

Verbal counting- an occupation that in our time bothers less and less people. It is much easier to get a calculator on your phone and calculate any example.

But is it really so? In this article, we will present math hacks that will help you learn how to quickly add, subtract, multiply and divide numbers in your mind. Moreover, operating not in units and tens, but at least two-digit and three-digit numbers.

After mastering the methods in this article, the idea of ​​​​reaching the phone for a calculator no longer seems so good. After all, you can not waste time and calculate everything in your mind much faster, but at the same time stretch your brains and impress others (of the opposite sex).

We warn you! If you a common person, and not a child prodigy, then it will take training and practice, concentration and patience to develop mental numeracy. At first, everything can turn out slowly, but then things will go smoothly, and you can quickly count any numbers in your head.

Gauss and mental arithmetic

One of the mathematicians with a phenomenal rate of mental calculation was the famous Carl Friedrich Gauss (1777-1855). Yes, yes, the same Gauss who came up with the normal distribution.

According to him own words He learned to count before he could speak. When Gauss was 3 years old, the boy looked at payroll his father and declared: "The calculations are wrong." After the adults checked everything, it turned out that little Gauss was right.

In the future, this mathematician reached considerable heights, and his works are still actively used in theoretical and applied sciences. Until his death, Gauss did most of his calculations in his head.

Here we will not deal with complex calculations, but start with the simplest.

Adding numbers in your mind

To learn how to add large numbers in your mind, you need to be able to accurately add numbers up to 10 . Ultimately, any complex task comes down to performing a few trivial actions.

Most often, problems and errors occur when adding numbers with a "pass through 10 ". When adding (and even when subtracting), it is convenient to use the technique of “reliance on a dozen”. What is it? First, we mentally ask ourselves how much one of the terms is missing before 10 , and then add to 10 the difference remaining up to the second term.

For example, let's add the numbers 8 and 6 . To out 8 get 10 , lacks 2 . Then to 10 it remains to add 4=6-2 . As a result, we get: 8+6=(8+2)+4=10+4=14

The main trick with adding large numbers is to break them into bit parts, and then add these parts together.

Suppose we need to add two numbers: 356 and 728 . Number 356 can be imagined as 300+50+6 . Likewise, 728 will look like 700+20+8 . Now we add up:

356+728=(300+700)+(50+20)+(8+6)=1000+70+14=1084

Subtracting numbers in your mind

Subtracting numbers will also be easy. But unlike addition, where each number is divided into bit parts, when subtracting, you only need to “break” the number that we subtract.

For example, how much will 528-321 ? Breaking down the number 321 into bit parts and we get: 321=300+20+1 .

Now we consider: 528-300-20-1=228-20-1=208-1=207

Try to visualize the process of addition and subtraction. At school, everyone was taught to count in a column, that is, from top to bottom. One way to restructure thinking and speed up counting is not to count from top to bottom, but from left to right, breaking numbers into place parts.

Multiplying numbers in your mind

Multiplication is the repeated repetition of a number. If you need to multiply 8 on the 4 , which means that the number 8 need to repeat 4 times.

8*4=8+8+8+8=32

Since everything challenging tasks are reduced to simpler ones, you need to be able to multiply everything single digits. There is a great tool for this - multiplication table . If you do not know this table by heart, then we strongly recommend that you first learn it and only then take up the practice of mental counting. In addition, there is, in fact, nothing to learn there.

Multiplication of multi-digit numbers by single-digit

First, practice multiplying multi-digit numbers by single-digit numbers. Let's multiply 528 on the 6 . Breaking down the number 528 into ranks and go from oldest to youngest. We multiply first and then add the results.

528=500+20+8

528*6=500*6+20*6+8*6=3000+120+48=3168

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Multiplication of two-digit numbers

There is nothing complicated here either, only the load on short-term memory is a little more.

Multiply 28 and 32 . To do this, we reduce the whole operation to multiplication by single-digit numbers. Imagine 32 as 30+2

28*32=28*30+28*2=20*30+8*30+20*2+8*2=600+240+40+16=896

One more example. Let's multiply 79 on the 57 . This means that you need to take the number " 79 » 57 once. Let's break the whole operation into stages. Let's multiply first 79 on the 50 , and then - 79 on the 7 .

• 79*50=(70+9)*50=3500+450=3950
• 79*7=(70+9)*7=490+63=553
• 3950+553=4503

Multiply by 11

Here tricky trick a quick mental calculation that will help you multiply any two-digit number by 11 at phenomenal speed.

To multiply a two-digit number by 11 , we add two digits of the number with each other, and enter the resulting amount between the digits of the original number. The resulting three-digit number is the result of multiplying the original number by 11 .

Check and multiply 54 on the 11 .

• 5+4=9
• 54*11=594

Take any two digit number, multiply it by 11 and see for yourself - this trick works!

Squaring

With the help of another interesting method of mental counting, you can easily and quickly square two-digit numbers. It is especially easy to do this with numbers that end in 5 .

The result begins with the product of the first digit of the number by the one following it in the hierarchy. That is, if this figure is denoted by n , then the next digit in the hierarchy will be n+1 . The result ends with the square of the last digit, i.e. the square 5 .

Let's check! Let's square the number 75 .

• 7*8=56
• 5*5=25
• 75*75=5625

Division of numbers in the mind

It remains to deal with the division. In fact, this is the inverse operation of multiplication. With division up to 100 no problems should arise at all - after all, there is a multiplication table that you know by heart.

Division by a single number

When dividing multi-digit numbers by a single-digit one, it is necessary to select the largest possible part, which can be divided using the multiplication table.

For example, there is a number 6144 , to be divided by 8 . Remember the multiplication table and understand that on 8 will divide the number 5600 . Let's imagine an example in the form:

6144:8=(5600+544):8=700+544:8

544:8=(480+64):8=60+64:8

Left to divide 64 on the 8 and get the result by adding all the results of the division

64:8=8

6144:8=700+60+8=768

Division by two digits

When dividing by a two-digit number, you need to use the rule for the last digit of the result when multiplying two numbers.

When multiplying two multi-digit numbers, the last digit of the multiplication result always coincides with the last digit of the result of multiplying the last digits of these numbers.

For example, let's multiply 1325 on the 656 . As a rule, the last digit in the resulting number will be 0 , as 5*6=30 . Really, 1325*656=869200 .

Now, armed with this valuable information, consider dividing by a two-digit number.

How much will 4424:56 ?

Initially, we will use the “fitting” method and find the limits within which the result lies. We need to find the number that, when multiplied by 56 will give 4424 . Intuitively, let's try the number 80.

56*80=4480

So the required number is less than 80 and obviously more 70 . Let's determine its last digit. Her work on 6 must end with a number 4 . According to the multiplication table, the results are suitable for us 4 and 9 . It is logical to assume that the result of division can be either a number 74 , or 79 . We check:

79*56=4424

Done, solution found! If the number didn't fit 79 , the second option would certainly be correct.

In conclusion, we present a few useful tips, which will help you quickly learn oral counting:

• Don't forget to exercise every day;
• do not quit training if the result does not come as quickly as you would like;
• download mobile app for oral counting: so you don’t have to come up with examples for yourself;
• Read books on quick mental counting techniques. Exist different techniques arithmetic, and you will be able to master the one that best suits you.

The benefits of mental arithmetic are undeniable. Practice, and every day you will count faster and faster. And if you need help in solving more complex and multi-level tasks, contact the student service specialists for fast and qualified help!

A sense of number, minimal counting skills are the same element of human culture as speech and writing. And if you easily count in your mind, then you feel a different level of control over reality. In addition, such a skill develops mental abilities: concentration on objects and things, memory, attention to detail and switching between streams of knowledge. And if you are interested in how to learn how to quickly count in your mind, the secret is simple: you need to constantly train.

Memory training: myth or reality?

Math is easy for those smart people who pop equations like seeds. Other people find it harder to learn But nothing is impossible, everything is possible if you practice a lot. There are the following mathematical operations: subtraction, addition, multiplication, division. Each of them has its own characteristics. To understand all the difficulties, you need to understand them once, and then everything will be much easier. If you train for 10 minutes every day, then in a few months you will reach a decent level and learn the truth of counting mathematical numbers.

Many people do not understand how you can vary the numbers in your mind. How to become the master of numbers so that it does not look stupid and imperceptible from the outside? When there is no calculator at hand, the brain begins to intensively process information, trying to calculate necessary numbers in the mind. But not all people manage to achieve the desired results, since each of us is an individual person with his own limits. If you want to understand in your mind, then you should study all the necessary information, armed with a pen, notepad and patience.

Multiplication table will save the day

We will not talk about those people who have an IQ level above 100, there are special requirements for such individuals. Let's talk about the average person who, with the help of the multiplication table, can learn many manipulations. So, how to quickly count in the mind without losing health, strength and time? The answer is simple: memorize the multiplication table! In fact, there is nothing difficult here, the main thing is to have pressure and patience, and the numbers themselves will give up before your goal.

For such an interesting undertaking, you will need a smart partner who can check you out and keep you company in this patient process. A man who knows is in the mind of even the laziest student. Once you can multiply quickly, mental counting will be routine for you. Unfortunately, there are no magic methods. How quickly you can master a new skill is up to you. You can exercise your brain not only with the help of the multiplication table, there is a more exciting activity - reading books.

Books and no calculator train your brain

In order to learn how to conduct computational activities orally as quickly as possible, you need to constantly temper your brain with new information. But how to learn to count quickly in umeza a short time? You can train your memory only with useful books, thanks to which not only the work of your brain will be universal, but also, as a bonus, improving memory and gaining useful knowledge. But reading books is not the limit of training. Only when you can forget about the calculator will your brain begin to process information faster. Try to count in your mind in any case, think through complex mathematical examples. But if it’s hard for you to do all this on your own, then enlist the support of a professional who will quickly teach you everything.

It can be difficult for you to understand how to learn how to quickly count in your mind when you are not friends with mathematics and no good teacher which could make the task easier. But do not succumb to difficulties. Having studied all the necessary recommendations, you can easily quickly learn how to count in your head and surprise your peers with new abilities.

• Ability to work with big numbers- going beyond the general development.
• Knowing the "tricks" of counting will help you quickly overcome all obstacles.
• Regularity is more important than intensity.
• Do not rush, try to catch your rhythm.
• Focus on correct answers, not memorization speed.
• Speak actions out loud.
• Don't be discouraged if it doesn't work out for you, because the main thing is to start.

Never give up in the face of difficulties

During training, you may have many questions that you do not know the answers to. This shouldn't scare you. After all, you cannot at first know how to count quickly without pre-training. Only the one who always goes forward will master the road. Difficulties should only temper you, and not slow down the desire to join people with non-standard opportunities. Even if you are already at the finish line, go back to the easiest, train your brain, do not give it a chance to relax. And remember, the more you pronounce information out loud, the faster you will remember.

It is not difficult to learn how to quickly count in your mind, it only requires experience and training. The ability to operate with complex numbers increases the level of control over many life processes, makes a person more collected and organized. Also, a quick count in the mind allows you to distract from sad thoughts, improves memory, attention and a sense of self-confidence.

Features and Benefits of Quick Mental Counting

Practically every educated person can now operate in the mind with numbers up to 20. However, it is already difficult to make mental calculations with values ​​that have three numbers or more. This can only be done by those who mathematical operations in the mind regularly, these include mathematicians, scientists, accountants, etc.

How to master the same quick counting skills as these specialists? This is not something impossible. Each of us has a natural ability to do this. For some, they are developed to a greater extent, others need to be trained a little. Tasks for training can be found freely available on the Internet. You can develop your own methodology that will take into account all personal characteristics and help you quickly master the necessary skills.

In order to succeed in this business, the following basic rules must be observed:

• regular workouts

First you need to develop your own training regimen, and then, if you really want to achieve impressive results, strictly follow it. During the first month, training should be done once a day for 10-15 minutes. It is not recommended to do them longer, because you can get very tired and cool this activity.

If it is difficult, then you can take a break for one or two days. Take your time, learn the technique at your own pace. Learning to count quickly is like learning poetry. If something doesn’t work right away, then don’t back down, keep practicing and success will not keep you waiting.

• mindfulness and concentration

This is very important point when studying the method of fast counting. First of all, you need to remember the algorithm for working with complex numbers. Then, in the process of training, he will be remembered, and it will not be difficult to perform an action in the mind even with three- and four-digit numbers.

Try not to be distracted by extraneous matters so as not to overload the brain with unnecessary information and quickly master the necessary skills.

• compliance with the training regimen

This is one of the foundations of success. Only patience and regular work on yourself will allow you to get what you want. Make a schedule for what time you will practice. You can even mark there information about the exercise performed every day.

• motivation

It is also one of the keys to success, when a person sees a goal in front of him, he will strive to achieve it, even if this requires the acquisition of certain skills and abilities.

• patience

In any business, to achieve success, you need patience and perseverance, even if everything does not work out right away. All people are different, someone needs more time to acquire these skills, someone less. The main thing is not to give up after the first setbacks.

Also, before starting training, you must consider the following key points:

• natural ability

Not all people are naturally endowed with a mathematical mindset, so it will take them a little longer to master the speed counting algorithms. Just do not make this fact the main excuse not to learn the technique.

• knowledge and understanding of mathematical algorithms

This is necessary in order to further make quick calculations in the mind according to a previously learned scheme.

• nutrition

During the period of intense mental training, you should include in your diet products for nourishing the brain, for example, well suited walnuts, honey, fruits.

Using these skills, it will be very pleasant to carry out mental counting operations without resorting to the use of a calculator and other means of calculation.

Basic techniques

There are many ways to develop mental counting skills. Everyone can choose the most convenient for themselves. There are four operations with numbers: addition, multiplication, subtraction, division.

It is enough to understand the algorithm once in order to develop the necessary skills later. It will be enough to train 10-15 minutes a day, and then periodically maintain the acquired abilities with episodic training. The first results will be noticeable in half a month, and in two or three months you will be able to reach a decent account level.

• quick addition technique

This is the easiest level to start with when training. It's best to start with two-digit numbers. For example, you need to add the numbers 23 and 51. First, add the tens: 20+50 = 70, then add the remainder 3+1=4 to the resulting amount. As a result, we get the number 74.

Master the addition of multi-digit numbers, also will not be special work. For example, let's add 342 and 741. To do this, we divide these numbers into digits 300, 40, 2 and 700, 40 and 1, respectively. Then, by analogy with two-digit numbers, we begin to add in our minds: 300 + 700 = 1000, 40 + 40 = 80, 2 + 1 = 3, then add 1000 + 80 + 3 = 1083.

• technique for fast subtraction

Just as with addition, subtracting two values ​​is not difficult. Let's start with two-digit numbers, for example, we need to subtract the number 23 from 35. Let's also start with the digits: 30-20 \u003d 10, 5-3 \u003d 2, then add the resulting values ​​​​10 + 2 and get the desired number 12.

Subtracting multi-digit numbers is also easy, for example, subtract the number 154 from 377. To do this, we divide the digital values ​​into digits 300, 70, 7 and 100, 50 and 4, respectively.

Subtract 300-100 = 200, 70-50 = 20, 7-4 = 3, then add the resulting numbers: 200+20+3 = 223.

In the same way, you can subtract the numbers l in your mind with a higher bit depth.

• technique for fast multiplication

This procedure can be greatly facilitated by learning the multiplication table. We know that multiplication is a simplification of the operation of addition. For example, 3 * 6 = 18, but in fact this is the sum of three sixes. When multiplying, you can also use the bit depth technique, for example, you need to find the product of 42 * 3. First 2*3 = 6, 4*3 =12, then we combine these numbers, putting the last before the first, i.e. we get the number 126. This algorithm suitable for calculating the product of two-digit digits.

When multiplying a three-digit number in the mind, the technique will be slightly different. For example, we need to multiply 421 and 372. Here we have to apply addition. We multiply 421 in turn by each digit of the second number: 421 * 2 = 842, 421 * 7 = 2942, 421 * 3 = 1263, then add these numbers, observing the bit depth with an offset: 2000 + 1000 = 120000, 800 + 900 + 200 = 29800 , 40+40+60=6440, 2+7+3 = 372, as a result we get the number 156612.

When multiplying three-digit numbers, you need to be especially careful not to make a mistake with the addition of digits in your mind.

• rapid division technique

The division of single-digit and two-digit numbers in the mind is carried out according to simple principle using the multiplication table. For example, we need to divide 35 by 5, remembering the multiplication table, we know in advance that the result will be 7.

Dividing multi-digit numbers is a little more difficult. For example, we divide 345 by 5, we also do this taking into account the bit depth: 300/5 \u003d 60, 45/5 \u003d 9, then add 60 + 9 and get the desired number 69.

As far as you can see, the principle of making any calculations in the mind is based on the principle of bit depth.

Need to know

Acquiring the ability to quickly count in the mind is a significant advantage for the individual, since only a limited number of people have such skills. However, the following points must be taken into account:

• regularly maintain acquired skills;
• speak aloud mathematical operations during training;
• do not overdo it.

The road will be mastered by the walking one. Only with due patience and motivation, it is possible to keep the ability of quick mathematical calculation in mind for long time.

Learning to count quickly in your mind is not an impossible task. Everyone can master the technique of fast mathematical calculations, this requires perseverance, concentration and regular training. There are many ways to get this skill, everyone can choose for themselves the one that they like the most. The implementation of fast computational operations in the mind is based on the principle of bit depth.

This article was inspired by the topic “How and how fast do you calculate in your mind at an elementary level?” and is called upon 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 mental counting techniques 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 approach has special case: everything is greatly simplified when two factors have the same number 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 just submit voice command 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.