Why do we need mental arithmetic if this is the 21st century, and all sorts of gadgets are capable of performing any arithmetic operations almost at lightning speed? You don’t even have to point your finger at your smartphone, but give a voice command and immediately receive the correct answer. Now this is successfully done even by elementary school students who are too lazy to divide, multiply, add and subtract on their own.

But this medal also has back side: scientists warn that if you don’t train, don’t load him with work and make his tasks easier, he begins to be lazy and his performance declines. In the same way, without physical training, our muscles weaken.

Mikhail Vasilyevich Lomonosov also spoke about the benefits of mathematics, calling it the most beautiful of sciences: “You have to love mathematics because it puts your mind in order.”

Oral arithmetic develops attention and reaction speed. It is not for nothing that more and more new methods of rapid oral counting, intended for both children and adults. One of them is the Japanese mental counting system, which uses the ancient Japanese soroban abacus. The methodology itself was developed in Japan 25 years ago, and now it is successfully used in some of our mental counting schools. It uses visual images, each of which corresponds to a specific number. Such training develops right hemisphere brain, responsible for spatial thinking, constructing analogies, etc.

It is curious that in just two years, students of such schools (they accept children aged 4–11 years) learn to perform arithmetic operations with 2-digit and even 3-digit numbers. Kids who don't know multiplication tables can multiply here. They add and subtract large numbers without writing them down. But, of course, the goal of training is the balanced development of the right and left.

Master verbal counting You can also use the problem book “1001 problems for mental arithmetic at school,” compiled back in the 19th century by a rural teacher and famous educator Sergei Aleksandrovich Rachinsky. This problem book is supported by the fact that it went through several editions. This book can be found and downloaded on the Internet.

People who practice quick counting recommend Yakov Trachtenberg’s book “The Quick Counting System.” The history of the creation of this system is very unusual. To survive the concentration camp where he was sent by the Nazis in 1941, and not lose his mental clarity, a Zurich mathematics professor began developing algorithms for mathematical operations that allow him to quickly count in his head. And after the war, he wrote a book in which the quick counting system is presented so clearly and accessiblely that it is still in demand.

There are also good reviews about Yakov Perelman’s book “Quick Counting. Thirty simple examples oral counting." The chapters of this book are devoted to multiplying by single-digit and two-digit numbers, in particular multiplying by 4 and 8, 5 and 25, by 11/2, 11/4, *, dividing by 15, squaring, and formula calculations.

The simplest methods of mental counting

People who have certain abilities will master this skill faster, namely: the ability to think logically, the ability to concentrate and store several images in short-term memory at the same time.

No less important is knowledge of special action algorithms and some mathematical laws that allow, as well as the ability to choose the most effective one for a given situation.

And, of course, you can’t do without regular training!

Some of the most common quick counting techniques are:

1. Multiplying a two-digit number by a one-digit number

The easiest way to multiply a two-digit number by a single-digit number is to split it into two components. For example, 45 - by 40 and 5. Next, we multiply each component by the required number, for example, by 7, separately. We get: 40 × 7 = 280; 5 × 7 = 35. Then we add the resulting results: 280 + 35 = 315.

2. Multiplying a three-digit number

Multiplying a three-digit number in your head is also much easier if you break it down into its components, but present the multiplicand in such a way that it is easier to perform mathematical operations with it. For example, we need to multiply 137 by 5.

We represent 137 as 140 − 3. That is, it turns out that we now have to multiply by 5 not 137, but 140 − 3. Or (140 − 3) x 5.

Knowing the multiplication table within 19 x 9, you can count even faster. We decompose the number 137 into 130 and 7. Next, we multiply by 5, first 130, and then 7, and add the results. That is, 137 × 5 = 130 × 5 + 7 × 5 = 650 + 35 = 685.

You can expand not only the multiplicand, but also the multiplier. For example, we need to multiply 235 by 6. We get six by multiplying 2 by 3. Thus, we first multiply 235 by 2 and get 470, and then multiply 470 by 3. Total 1410.

The same action can be done differently by representing 235 as 200 and 35. It turns out 235 × 6 = (200 + 35) × 6 = 200 × 6 + 35 × 6 = 1200 + 210 = 1410.

In the same way, by breaking down numbers into their components, you can perform addition, subtraction and division.

3. Multiplying by 10

Everyone knows how to multiply by 10: simply add zero to the multiplicand. For example, 15 × 10 = 150. Based on this, it is no less simple to multiply by 9. First, we add 0 to the multiplicand, that is, multiply it by 10, and then subtract the multiplicand from the resulting number: 150 × 9 = 150 × 10 = 1500 − 150 = 1,350.

4. Multiplication by 5

It is easy to multiply by 5. You just need to multiply the number by 10, and divide the resulting result by 2.

5. Multiplying by 11

It’s interesting to multiply two-digit numbers by 11. Let’s take 18, for example. Let’s mentally expand 1 and 8, and between them write the sum of these numbers: 1 + 8. We get 1 (1 + 8) 8. Or 198.

6. Multiply by 1.5

If you need to multiply a number by 1.5, divide it by two and add the resulting half to the whole: 24 × 1.5 = 24 / 2 + 24 = 36.

These are just the most simple ways mental calculations, with the help of which we can train our brain in everyday life. For example, counting the cost of purchases while standing in line at the checkout. Or perform mathematical operations with numbers on the license plates of passing cars. Those who like to “play” with numbers and want to develop their thinking abilities can turn to the books of the above-mentioned authors.

One of the main reasons for poor results in mathematics on the Unified State Exam or Unified State Exam is the inability to count. Many schoolchildren find it difficult to solve an example even on a piece of paper, not to mention quickly counting in their heads. But some parts of the brain atrophy if a person does not use mental skills. Therefore, it is important to develop mental abilities to their full potential.

The basis for developing mental arithmetic skills

Some parents believe that teaching a child to quickly count examples in his head is not necessary: ​​he will not need it in the future, because he can always use a calculator. But at the same time, they forget that such training is simply necessary for brain development: any learned method (technique) of counting is a new neural chain (connection), the more such chains there are, the smarter the student. Therefore, the main benefit of the quick counting skill is the development of the brain and intelligence.

It is impossible to learn to work with numbers in your head if you have a weak understanding of them and actions with them.

Counting skills develop gradually from a visual representation of numbers and actions with them to an abstract logical one:

1. First, the child learns to count forward and backward with the help of rhymes, nursery rhymes, practical exercises while walking, eating, playing games (counting how many objects are on the table, cars in the garage, birds in the tree). Gets acquainted with numbers, learns what they mean, learns to correlate numbers and quantities.
2. Then he masters the concepts of “more - less”, “equally”, learns to compare the number of objects, sizes.
3. After this, he gets acquainted with addition and subtraction and learns the meaning of these actions. All examples are illustrative (the child moves 2 more apples to two apples and counts how many they get).
4. Learns to count objects with his eyes, first pronounces out loud the actions and the result of the actions, and then in a whisper: if you add 2 more cars to 4, you get 6.
5. Repeated repetition of actions will lead to the fact that the baby will learn to recognize examples that he has already worked with and say the result out loud, bypassing the stage of pronunciation.

At the stage of learning to count, it is important to interest the child, support him in case of failure and rejoice with him in victories, even small ones. When, the skill will need to be developed by introducing the student to various techniques and techniques.

Development of mental arithmetic skills

• Improving the ability to work with numbers in your head.
• Acquaintance with new techniques and techniques.
• Training the ability to select the optimal solution algorithm in each specific case.

Ability to work with numbers

The following exercises will help you develop this skill:

• “Name the numbers in which...” - indicates the range and condition, for example, “Name the numbers from 5 to 50 that contain the digit 3” or “Name all two-digit numbers that contain the digit 0.” When performing this exercise, it is important to immediately work through all the mistakes made by the student. If he missed a number or said the wrong one, he starts over.
• “Maintaining progression” (range and arithmetic operations depend on age and development of counting skills). For example, “Go from 5 in steps of 3” or “Go backwards from 30 in steps of 4” - for children primary school. For those who have already learned the multiplication table, you can give tasks for multiplication and division: “Go from 2, multiplying all numbers by 3.”
• “Find the numbers from 1 to...” - children need to find and name in order all the numbers in the table.
• “Compare the numbers” - children determine which one is larger (smaller), by how much;
• “Examples” - schoolchildren are asked to solve examples in their minds, first the simplest ones (with small numbers), after working out the numbers are gradually increased. You should not introduce your child to two- or three-digit numbers if he does not know how to perform operations with numbers up to 5 perfectly.

Techniques for quickly counting numbers

Unfortunately, there is simply no single - universal - method that allows you to solve all examples equally quickly. Therefore, it is important to know and be able to put into practice several methods, from which you can then choose the most appropriate one.

Useful algorithms for solving some examples:

• To quickly subtract 7, 8 or 9 from a number, you must first subtract 10 and then add 3,2 or 1, respectively. For example: 45-9=45-10+1=36, or 36-8=36-10+2=28.
• You can also quickly multiply by 4, 8 and 16. To do this, you must first remember that 4=2*2, 8=2*2*2, 16=2*2*2*2. Then simply multiply the number by 2 several times: 6*16=6*2*2*2*2=96.
• To multiply a number by 9, it is first increased 10 times, and then the first factor is subtracted from the resulting one: 27*9=27*10-27=243. This technique will allow you to very quickly find the result of multiplying by 9, if you do not use a calculator.
• When multiplying by 2, it is more convenient to round non-round numbers, and then subtract or add (depending on which direction you rounded) the product of the remaining or missing number by 2: 132*2=130*2+2*2=264, or 138* 2=140*2-2*2=276.
• Similarly, numbers are divided by 2: 156/2=150/2+6/2=78, or 156/2=160/2-4/2=78.
• To multiply by 5, the number is divided by 2 and then increased by 10 times (the operation can be done the other way around): 27*5=27/2*10 or 27*10/2=135.
• Similar actions are performed when multiplying by 25: first divide by 4, and then increase by 100 times (simply add two zeros): 16*25=16/4*100=400. Of course, it is more convenient to use this method when the first factor is divisible by 4 without a remainder. Determining whether a number is divisible by 4 without a remainder is not difficult (non-tabular cases): a number consisting of its last two digits must be divisible by 4. For example, the number 124 is divisible by 4 (24/4=6), but 526 is not (26 is not divisible by 4 without a remainder).

And another way to multiply a multi-digit number by a single-digit number is to multiply the digit terms by the second factor and add the results. For example, 424*5=400*5+20*5+4*5=2000+100+20=2120.

In order not to make mistakes in calculations, it is important to be able to predict the future result, and several statements will help here:

• When multiplying single-digit numbers, the result does not exceed 81: 9*9=81.
• Likewise, 99*99=9801, so the result of multiplication is double digit numbers should not be more than this number, and when increasing three-digit numbers the maximum number is 998001.

Practicing mental arithmetic skills

The above algorithms are the basis for developing mental counting skills. Learn to count complex examples possible only with regular training, bringing the use of a skill to automaticity.

The effectiveness of work in this direction can be increased if during classes:

1. Create a game situation , turning the ordinary educational process into an interesting and unusual process.
2. Keep your child engaged interesting material constant change of activity.
3. Create a spirit of competition – the awareness that someone can do better will make you strive for new achievements; such classes will be more effective than memorizing “alone.”
4. Record personal achievements , set new goals to achieve new heights.

The ability to concentrate on solving a problem in any situation (even when others are in the way) also contributes to the development of counting skills (and not only). You can train this ability by solving examples with music on or while in a noisy company.

To prevent your child from becoming bored, it is important to learn how to deal with this feeling. Psychologists recommend using any action for this: for example, looking at what is happening outside the window, or observing the movement of the clock hands. If a child learns to cope with boredom and direct his energy in the right direction, then in class he will be able to absorb a greater amount of information, which will have a positive impact on his academic performance. .

IN Lately In Russia, a new method for developing intelligence is beginning to gain popularity in our country. Instead of the usual chess sections, parents send their children to mental arithmetic schools. How kids are taught to count in their heads, how much such classes cost and what experts say about them - in the material "AiF-Volgograd".

What is mental arithmetic?

Mental arithmetic is Japanese technique development of a child’s intellectual abilities through calculations on special abacus “soroban”, which is sometimes called “abacus”.

“When performing actions with numbers in their minds, children imagine these abacus and in a split second they mentally add, subtract, multiply and divide any numbers - even three-digit, even six-digit,” says Natalya Chaplieva, teacher of the Volga club, where children are taught using this method.

According to her, when children are just learning all these actions, they count the numbers directly on the soroban, fingering the bones. Then they gradually move from counting to a “mental map” - a picture depicting them. At this stage of learning, they stop touching the abacus and begin to imagine in their minds how they move the bones on it. Then, the children stop using the mental map and begin to completely visualize the soroban for themselves.

Abacus soroban. Photo: AiF/ Evgeniy Strokan

“We recruit children from 4 to 12 years old into groups. At this age, the brain is most plastic; the child absorbs information like a sponge, and therefore easily masters learning methods. It’s much more difficult for an adult to learn mental arithmetic,” says Ekaterina Grigorieva, teacher of the mental arithmetic club.

How much does it cost?

The abacus has a rectangular frame that contains 23-31 spokes, each of which has 5 bones strung on them, separated by a transverse crossbar. Above it there is one domino, which denotes “five”, and below it there are 4 dominoes, denoting ones.

You need to move the bones with only two fingers - the thumb and forefinger. The counting on the soroban starts from the very first knitting needle on the right. It stands for units. The knitting needle to the left of it is tens, the next one is hundreds, etc.

Soroban is not sold in regular stores. You can buy such accounts on the Internet. Depending on the number of knitting needles and material, the price of soroban can range from 170 to 1,000 rubles.

At the first stage, children work with abacus. Photo: AiF/ Evgeniy Strokan

If you don’t want to spend money on bills at all, you can download a free application for your phone - an online simulator that simulates an abacus.

Mental arithmetic classes for children in Volgograd cost about 500-600 rubles per hour. You can buy a subscription for 8 classes for 4,000 rubles and 16 classes for 7,200 rubles. Classes are held 2 times a week. The Volga school gives out abacus, mental maps and notebooks to children free of charge, and students can take them home. At the end of the course, the child can keep the soroban as a souvenir.

Children have to learn mental arithmetic for about 1-2 years, depending on their abilities.

Assignments for students. Photo: AiF/ Evgeniy Strokan

If you don't have money for classes at special school, then you can try to search for video tutorials on YouTube. True, some of them are posted on the website by organizations providing lessons for money for the purpose of self-promotion. Their videos are very short - 3 minutes long. With their help you can learn the basics of mental arithmetic, but nothing more.

What do experts say about this?

Teachers who conduct mental arithmetic classes are confident that the training is worth the money spent on it.

“Mental arithmetic develops well the child’s imagination, creativity, thinking, memory, fine motor skills, attentiveness, perseverance. Classes are aimed at ensuring that the child develops both hemispheres at the same time, which is very important, because the traditional preparation of a child for school develops only the right hemisphere of the brain,” believes teacher Natalya Chaplieva.

Psychologist Natalya Oreshkina believes that in the case of children 4-5 years old, mental arithmetic classes will be effective only if they take place in a playful way.

“Children of this age generally have difficulty concentrating for such a time, unless we're talking about not about watching a cartoon, says the expert. - But if the lesson is structured in a playful way, if children practice abacus and color something, then they will learn knowledge while being in their natural environment - in a game. In addition, it should not be difficult for children; they should not exceed the permissible load level. For example, for 4-year-olds, classes should last no more than 30 minutes. I can say that mental arithmetic for children it is very interesting. But if a child lags behind his peers in some way, then such activities will be too difficult for him. If a child does not have an internal resource for activities, then it will be a waste of time, effort and money.”

People rarely use the knowledge gained in algebra and geometry lessons in life. The most valuable and necessary skill associated with mathematics is the ability to do mental math quickly, so it's worth figuring out how to learn it. IN ordinary life this allows you to quickly count change, calculate time, etc.

It is best to develop it from childhood, when the brain absorbs information much faster. There are a few effective techniques which are used by many people.

How to learn to count very quickly in your head?

To achieve good results, you need to train regularly. After achieving certain goals, it is worth complicating the task. Great importance have human abilities, that is, the ability to retain several things in memory at once and concentrate attention. People with a mathematical mind can achieve the most. To quickly learn to count, you need to know the multiplication table well.

The most popular calculation methods:

1. Let's figure out how to quickly count two-digit numbers in your head if you need to multiply by 11. To understand the technique, consider one example: 13 multiplied by 11. The task is that between numbers 1 and 3 you need to insert their sum, that is, 4. As a result, it turns out that 13x11=143. When the sum of the digits gives a two-digit number, for example, if you multiply 69 by 11, then 6+9=15, then you only need to insert the second digit, that is, 5, and add 1 to the first digit of the multiplier. The result is 69x11=759. There is another way to multiply a number by 11. First, multiply by 10, and then add the original number to it. For example, 14x11=14x10+14=154.
2. Another way to quickly count large numbers in your head works for multiplying by 5. This rule is suitable for any number that first needs to be divided by 2. If the result is an integer, then you need to add a zero at the end. For example, to find out how much 504 will be multiplied by 5. To do this, 504/2 = 252 and add 0 at the end. The result is 504x5 = 2520. If, when dividing a number, the result is not an integer, then you simply need to remove the resulting comma. For example, to find out how much 173 is multiplied by 5, you need 173/2 = 86.5, and then simply remove the comma, and it turns out that 173x5 = 865.
3. Let's learn how to quickly count two-digit numbers in your head by adding. First you need to add tens, and then units. To get the final result, you should add the first two results. For example, let’s figure out how much 13+78 is. The first action: 10+70=80, and the second: 3+8=11. The final result will be: 80+11=91. This method can be used when you need to subtract another from one number.

Another hot topic is how to quickly calculate percentages in your head. Again, for a better understanding, let's look at an example of how to find 15% of a number. First, you should determine 10%, that is, divide by 10 and add half of the result -5%. Let's find 15% of 460: to find 10%, divide the number by 10, you get 46. The next step is to find half: 46/2=23. As a result, 46+23=69, which is 15% of 460.

There is another method for calculating interest. For example, if you need to determine how much 6% of 400 will be. First, you should find out 6% of 100 and it will be 6. To find out 6% of 400, then you need 6x4 = 24.

If you need to find 6% of 50, then you should use the following algorithm: 6% of 100 is 6, and for 50, it is half, that is, 6/2 = 3. As a result, it turns out that 6% of 50 is 3.

If the number from which you need to find a percentage is less than 100, then you should simply move the comma to the left. For example, to find 6% of 35. First, find 6% of 350 and it will be 21. The value of 6% for 35 is 2.1.

In this article we offer you a selection of simple mathematical techniques, many of which are quite relevant in life and allow you to count faster.

1. Quick interest calculation

Perhaps, in the era of loans and installment plans, the most relevant mathematical skill can be called masterly calculation of interest in the mind. The most in a fast way To calculate a certain percentage of a number is to multiply this percentage by this number and then discard the last two digits in the resulting result, because a percentage is nothing more than one hundredth.

How much is 20% of 70? 70 × 20 = 1400. We discard two digits and get 14. When rearranging the factors, the product does not change, and if you try to calculate 70% of 20, the answer will also be 14.

This method is very simple in the case of round numbers, but what if you need to calculate, for example, the percentage of the number 72 or 29? In such a situation, you will have to sacrifice accuracy for the sake of speed and round the number (in our example, 72 is rounded to 70, and 29 to 30), and then use the same technique with multiplication and discarding the last two digits.

2. Quick divisibility check

Is it possible to divide 408 candies equally among 12 children? It’s easy to answer this question without the help of a calculator, if you remember the simple signs of divisibility that we were taught at school.

· A number is divisible by 2 if its last digit is divisible by 2.

· A number is divisible by 3 if the sum of the digits that make up the number is divisible by 3. For example, take the number 501, imagine it as 5 + 0 + 1 = 6. 6 is divisible by 3, which means the number 501 itself is divisible by 3 .

· A number is divisible by 4 if the number formed by its last two digits is divisible by 4. For example, take 2,340. The last two digits form the number 40, which is divisible by 4.

· A number is divisible by 5 if its last digit is 0 or 5.

· A number is divisible by 6 if it is divisible by 2 and 3.

· A number is divisible by 9 if the sum of the digits that make up the number is divisible by 9. For example, take the number 6 390, imagine it as 6 + 3 + 9 + 0 = 18. 18 is divisible by 9, which means the number itself is 6 390 is divisible by 9.
·
A number is divisible by 12 if it is divisible by 3 and 4.

The square root of 4 is 2. Anyone can calculate this. What about the square root of 85?
For a quick approximate solution, we find the square number closest to the given one, in this case it is 81 = 9^2.

Now we find the next closest square. In this case it is 100 = 10^2.

The square root of 85 is somewhere between 9 and 10, and since 85 is closer to 81 than 100, then Square root this number will be 9-something.

4. Quick calculation of the time after which a cash deposit at a certain percentage will double

Do you want to quickly find out the time it will take for your money deposit at a certain interest rate to double? You don’t need a calculator here either, just know the “rule of 72.”

We divide the number 72 by our interest rate, after which we get the approximate period after which the deposit will double.

If the investment is made at 5% per annum, then it will take a little over 14 years for it to double.
Why exactly 72 (sometimes they take 70 or 69)? How it works? Wikipedia will answer these questions in detail.

In this case interest rate by contribution should become a divisor of the number 115.

If the investment is made at 5% per annum, it will take 23 years for it to triple.

6. Quickly calculate your hourly rate

Imagine that you are undergoing interviews with two employers who do not give salaries in the usual format of “rubles per month”, but talk about annual salaries and hourly wages. How to quickly calculate where they pay more?

Where the annual salary is 360,000 rubles, or where they pay 200 rubles per hour?

To calculate the payment for one hour of work when announcing the annual salary, you need to discard the last three digits from the stated amount, and then divide the resulting number by 2.

360,000 turns into 360 ÷ 2 = 180 rubles per hour. All other things being equal, it turns out that the second offer is better.

7. Advanced math on your fingers

Your fingers are capable of much more than simple operations addition and subtraction.
Using your fingers you can easily multiply by 9 if you suddenly forget the multiplication table.

Let's number the fingers from left to right from 1 to 10.

If we want to multiply 9 by 5, then we bend the fifth finger to the left.

Now let's look at the hands. It turns out four unbent fingers before the bent one. They represent tens. And five unbent fingers after the bent one. They represent units. Answer: 45.

If we want to multiply 9 by 6, then we bend the sixth finger to the left. We get five unbent fingers before the bent finger and four after. Answer: 54.

8. Fast multiplication by 4

There is extremely easy way lightning-fast multiplication of even large numbers by 4. To do this, it is enough to decompose the operation into two actions, multiplying the desired number by 2, and then again by 2.

See for yourself. Not everyone can multiply 1,223 by 4 in their head. Now we do 1223 × 2 = 2446 and then 2446 × 2 = 4892. This is much simpler.

Imagine that you are taking a series of five tests, for which you need a minimum score of 92 to pass. The last test remains, and the previous results are as follows: 81, 98, 90, 93. How to calculate the required minimum that you need to get in the last test?

To do this, we count how many points we have under/overtaken in the tests we have already passed, denoting the shortfall with negative numbers, and the results with a margin as positive.
So, 81 − 92 = −11; 98 − 92 = 6; 90 − 92 = −2; 93 − 92 = 1.

Adding these numbers, we get the adjustment for the required minimum: −11 + 6 − 2 + 1 = −6.

The result is a deficit of 6 points, which means that the required minimum increases: 92 + 6 = 98. Things are bad. :([But not your website:)]

10. Quick value representation common fraction

The approximate value of an ordinary fraction can be very quickly represented as a decimal fraction if it is first reduced to simple and understandable ratios: 1/4, 1/3, 1/2 and 3/4.

For example, we have a fraction 28/77, which is very close to 28/84 = 1/3, but since we increased the denominator, the original number will be slightly larger, that is, a little more than 0.33.

11. Number guessing trick

You can play a little like David Blaine [famous American illusionist - if anyone doesn’t know. For example, we didn’t know :) - website] and surprise our friends with an interesting, but very simple mathematical trick.

1. Ask a friend to guess any integer.

2. Let him multiply it by 2.

3. Then add 9 to the resulting number.

4. Now let him subtract 3 from the resulting number.

5. Now let him divide the resulting number in half (in any case, it will be divided without a remainder).

6. Finally, ask him to subtract from the resulting number the number he guessed at the beginning.

The answer will always be 3.

Yes, it’s very stupid, but often the effect exceeds all expectations.

Bonus

And, of course, we couldn’t help but insert into this post that same picture with a very cool method of multiplication.

Sources: wisebread.com, lifehacker.ru