How to learn to multiply large numbers in your head. How to quickly multiply two-digit numbers in your head? Arithmetic Mazes Game

Pure mathematics is, in its own way, the poetry of the logical idea. Albert Einstein

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 multiplication given percentage by this number, followed by discarding 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 simple signs 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.

3. Fast square root calculation

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 closest one to the given one square number, V in this case that's 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 deposit is made at 5% per annum, then it will take 14 s small years old so that it doubles.

Why exactly 72 (sometimes they take 70 or 69)? How it works? Wikipedia will answer these questions in detail.

5. Quick calculation of the time after which a cash deposit at a certain percentage will triple

In this case, the interest rate on the deposit 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, it is necessary to discard three from the stated amount last sign, then divide the resulting number by 2.

360,000 turns into 360 ÷ 2 = 180 rubles per hour. Other than that equal conditions It turns out that the second proposal 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.

In this way you can reproduce the entire column of multiplication by 9.

8. Multiply by 4 quickly

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.

9. Quickly determine the required minimum

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

To do this, we count how many points we have under/overtaken in the tests we have already passed, indicating a shortage negative numbers, and the results are more than 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. :(

10. Quickly represent the value of a fraction

Approximate value common fraction can be very quickly represented in the form decimal, if you first reduce it 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 David Blaine and surprise your 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 he will 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.

People rarely use the knowledge gained in algebra and geometry lessons in life. The most valuable and necessary skill related to 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?

In order to achieve good results, it is necessary to conduct training 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 in your head big numbers, 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. Learn how to quickly count in your head double figures, by addition. 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 one actual topic– 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.

The ability to quickly analyze a situation, calculate development options and create a single image of reality is one of the key skills of highly effective people. Personal development impossible without the intellectual, which is facilitated by quick calculations in the mind. In general, we will talk about the technique of increasing the speed of thinking in the article.

How our brain deceives us

Research in the field of brain function provides data that is difficult to believe. Most of of the population considers himself a curator of the brain. But this is an illusory idea. In fact, the brain has already made a decision for you and through nerve impulses transmitted it to consciousness.

Human thinking has practically not been studied; only a small picture of what is happening in the brain has been compiled. Roughly speaking, our actions are not determined by our own “I,” although this is a very vague formulation. And knowing this, you can begin to study the technique of quickly counting in your head.

How to study more effectively

Memory is differentiated into long-term and short-term; in the first case, knowledge is stored in the brain forever. And the second type is necessary for memorizing information and reading.

A modern young man is a multimedia personality with clip thinking. Postpone data to long-term memory It’s extremely difficult for him, because the constant flow of information clutters up his “hard drive.”

Therefore, learning how to quickly count in your head should take place in a calm state, when a person is not distracted by external stimuli. Otherwise, in a few hours he will forget everything.

Why should I learn this?

Yes, in currently There is no need to add numbers in your head. For this purpose special technical means, but not using the brain leads to personality degradation.

And the pursuit of knowledge is an eternity. Such people are confident in themselves, they only hope for own strength, and the acquired skills are used for their intended purpose, thereby enriching the individual spiritually and materially. Quick mental arithmetic develops a sense of control in a person and increases concentration.

Method one. For the lazy

Owners of devices on the Andorod and IOS platforms can download educational applications and games. Neuroscientists advise taking a comprehensive approach to quick counting in the mind. Training occurs in several stages, described below:

1. Applications are downloaded to develop attention, concentration, etc.
2. Then the user downloads memory developers.

In the first action, a person prepares his brain, so to speak, warms it up for intensive training. After which he begins to work on arithmetic in his head. Please note that applications should be easily regulated, both reducing or increasing the level of difficulty of tasks, and changing the time to work on it.

Method two. Basic knowledge

Tasks have been selected for a quick start entry level. Adding and subtracting small numbers, such as 3 and 10. The technique is called "Ten's Base."

Procedure:

1. Ask questions simple in nature, like how much is 3 + 8 or 9 + 1. Answer: 11 and 10.
2. How much does the number 10 need to become 14? Answer: 4.
3. Then take any number, for example 9, and find out how many 2s are in this number, and if there is a shortage, add the missing digits. Answer: four twos + 1.
4. Add the number from the second action (4) to the part that was missing to get (1) nine and add them up. Answer: 5.

Hone your skill to perfection and only then proceed to more difficult tests.

Method three. Multi-digit numbers

The skills acquired at school are used here. Column or row addition is the most popular among schoolchildren and students without computing resources. Let's look at two numbers as an example: 1345 and 6789. First, let's differentiate them:

• The number 1234 is made up of 1000, 200, 30 and 4.
• And 6789 is from 6000, 700, 80 and 9.

A quick mental calculation goes through the following steps:

1. Initially folded single digit values, this is 4 + 9 = 13.
2. Adds 30 + 80 = 110.
3. Let's move on to three-digit numbers, 700 + 200 = 900.
4. And then we count four digits: 1000 + 6000 = 7000.
5. Let's sum it up: 7000 + 900 + 110 + 13 = 8023 and check it on a calculator.

And a faster, but more imaginative way:

1. We imagine one number above another in our head.
2. Add the numbers starting from the end.
3. If 4 + 9 = 13, then put the unit in your head and add the following numbers to the final value.

In the screenshot this method is presented like this; in your thoughts it should have a similar structure.

Method four. Subtraction

As with addition, subtraction begins with introductory lesson. A person's attention should be focused exclusively on counting numerical values. You cannot be distracted by extraneous noises, otherwise nothing will come of it. This time we subtract 8 from 10 and see what comes out of it:

1. First, let's find out how much you need to subtract from ten to get eight. Answer: two.
2. From ten we subtract eight in parts - first this two, and then the remaining numbers. And let's calculate how many times you need to subtract to get zero. Answer: five.
3. Subtract five from ten. Answer: five.
4. And subtract the answer received from eight. Answer: three.

Method five. Combined

Appeared as a result of the interaction of addition and subtraction. The idea is simple, you need to take a number and start subtracting from it different numbers or add with some reformations. The number 9 is taken as the initial one, let's start:

1. Six is ​​subtracted from nine and four is added at the same time. Answer: seven.
2. Seven is broken down into its component parts, for example: 2 + 3 + 2.
3. And a random value is added to each, let’s take 2. It turns out, 2 + 2 = 4, 3 + 2 = 5 and 2 + 2 = 4.
4. Let's sum up the resulting numbers: 4 + 5 + 4 = 13.
5. We again arrange the value in parts and repeat the steps, using only subtraction.

And with subtraction of large numbers the situation is similar to addition. Say all actions out loud so that several types of memory work and quick mental calculations are accelerated.

How long does it take to become a superman?

There are four main mathematical operations:

1. Subtraction.
2. Addition.
3. Multiplication.
4. Division.

And everything will depend on how often a person engages in brain training. With fruitful work for 15-20 minutes a day, a noticeable result will come in two or three months. To maintain the effect of high-speed calculations, a superman will need to spend only 2-3 minutes a day repeating what he has learned. And in a few years it will become a habit, and the individual will not even notice, as he thinks in his mind.

Improving the computational skills of students in mathematics lessons using “quick” counting techniques.

Kudinova I.K., mathematics teacher

MKOU Limanovskaya Secondary School

Paninsky municipal district

Voronezh region

“Have you ever observed how people with a natural ability to count are receptive, one might say, to all sciences? Even all those who are slow to think, if they learn it and practice it, then even if they do not derive any benefit from it, they still become more receptive than they were before.”

Plato

The most important task education is the formation of universal educational activities that provide schoolchildren with the ability to learn, the ability for self-development and self-improvement. The quality of knowledge acquisition is determined by the diversity and nature of types universal actions. Forming the ability and readiness of students to implement universal learning activities makes it possible to increase the effectiveness of the learning process. All types of universal educational activities are considered in the context of the content of specific educational subjects.

Important role teaching schoolchildren skills plays a role in the formation of universal educational actions rational calculations. No one doubts that the development of the skills of rational calculations and transformations, as well as the development of skills in solving simple problems “in the mind” - essential element mathematical training students. INThere is no need to prove the importance and necessity of such exercises. Their significance is great in the formation of computational skills, and the improvement of knowledge of numbering, and in the development personal qualities child. Creating a specific system for consolidating and repeating the studied material gives students the opportunity to master knowledge at the level of automatic skill.

Knowledge of simplified methods of mental calculations remains necessary even with the complete mechanization of all the most labor-intensive computing processes. Mental calculations make it possible not only to quickly make mental calculations, but also to monitor, evaluate, find and correct errors. In addition, mastering computational skills develops memory and helps schoolchildren fully master physics and mathematics subjects.

It is obvious that rational calculation techniques are a necessary element of computational culture in the life of every person, primarily due to their practical significance, and students need it in almost every lesson.

Computational culture is the foundation for the study of mathematics and other academic disciplines, because in addition to the fact that calculations activate memory and attention, help rationally organize activities and significantly influence human development.

IN Everyday life, on training sessions When every minute is valuable, it is very important to quickly and rationally carry out oral and written calculations, without making mistakes and without using any additional computing tools.

Analysis of exam results in grades 9 and 11 shows that greatest number Students make mistakes when performing calculation tasks. Often, even highly motivated students reach final certification lose skills oral counting. They calculate poorly and irrationally, increasingly resorting to the help of technical calculators. the main task teachers - not only to maintain computational skills, but also to teach how to apply non-standard techniques oral calculation, which would significantly reduce the time spent working on the task.

Let's consider specific examples various techniques fast rational calculations.

DIFFERENT WAYS OF ADDING AND SUBTRACTING

ADDITION

The basic rule for doing addition in your head 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, etc. For example:

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

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

ADDING TWO-DIGITUAL NUMBERS IN THE MIND

If the units digit in the number being added is greater than 5, then the number must be rounded up, and then the rounding error must be subtracted 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.

ADDING 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 up the subtrahend, and then adjust the answer you get.

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

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

Multiplication multi-digit numbers by 9

1. Increase the number of tens by 1 and subtract it from the multiplicand

2. We attribute to the result the addition of the units digit of the multiplicand to 10

Example:

576 9 = 5184 379 9 = 3411

576 - (57 + 1) = 576 - 58 = 518 . 379 - (37 + 1) = 341 .

Multiply by 99

1. From a number, subtract the number of its hundreds, increased by 1

2. Find the complement of the number formed by the last two digits to 100

3. Attribute the addition to the previous result

Example:

27 99 = 2673 (hundreds - 0) 134 99 = 13266

27 - 1 = 26 134 - 2 = 132 (hundred - 1 + 1)

100 - 27 = 73 66

Multiplying any number by 999

1. From what is being multiplied, subtract the number of thousands increased by 1

2. Find the complement to 1000

23 999 = 22977 (thousands - 0 + 1 = 1)

23 - 1 = 22

1000 - 23 = 977

124 999 = 123876 (thousands - 0 + 1 = 1)

124 - 1 = 123

1000 - 124 = 876

1324 · 999 = 1322676 (thousand - 1 + 1 = 2)

1324 - 2 = 1322

1000 - 324 = 676

Multiply by 11, 22, 33, …99

To multiply a two-digit number, the sum of its 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 11 by a two-digit number, the sum of the digits of which is 10 or more than 10, you need to mentally move apart 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:

94 ×11 = 9 (9+4) 4 = 9 (13) 4 = (9+1) 34 = 1034;

59×11 = 5 (5+9) 9 = 5 (14) 9 = (5+1) 49 = 649.

To multiply a two-digit number by 22, 33...99, you need last number present as a product single digit number(from 1 to 9) by 11, i.e.

44= 4 × 11; 55 = 5×11, etc.

Then multiply the product of the first numbers by 11.

48 × 22 =48 × 2 × (22:2) = 96 × 11 =1056;

24 × 22 = 24 × 2 × 11 = 48 × 11 = 528;

23 × 33 = 23 × 3 × 11 = 69 × 11 = 759;

18 × 44 = 18 × 4 × 11 = 72 × 11 = 792;

16 × 55 = 16 × 5 × 11 = 80 × 11 = 880;

16 × 66 = 16 × 6 × 11 = 96 × 11 = 1056;

14 × 77 = 14 × 7 × 11 = 98 × 11 = 1078;

12 × 88 = 12 × 8 × 11 = 96 × 11 = 1056;

8 × 99 = 8 × 9 × 11 = 72 × 11 = 792.

In addition, you can apply the law of simultaneous increase in equal number times one factor and decreasing the other.

Multiplying by a number ending in 5

To multiply an even two-digit number by a number ending in 5, apply the following rule:if one of the factors is increased several times and the other is decreased by the same amount, the product will not change.

44 × 5 = (44:2) × 5 × 2 = 22 × 10 = 220;

28 × 15 = (28:2) × 15 × 2 = 14 × 30 = 420;

32 × 25 = (32:2) × 25 × 2 = 16 × 50 = 800;

26 × 35 = (26:2) × 35 × 2 = 13 × 70 = 910;

36 × 45 = (36:2) × 45 × 2 = 18 × 90 = 1625;

34 × 55 = (34:2) × 55 × 2 = 17 × 110 = 1870;

18 × 65 = (18:2) × 65 × 2 = 9 × 130 = 1170;

12 × 75 = (12:2) × 75 × 2 = 6 × 150 = 900;

14 × 85 = (14:2) × 85 × 2 = 7 × 170 = 1190;

12 × 95 = (12:2) × 95 × 2 = 6 × 190 = 1140.

When multiplying by 65, 75, 85, 95, numbers should be small, within the second ten. Otherwise, the calculations will become more complicated.

Multiplying and dividing by 25, 50, 75, 125, 250, 500

In order to verbally learn to multiply and divide by 25 and 75, you need to know well the divisibility sign and the multiplication table by 4.

Those and only those numbers that have two are divisible by 4. last digits numbers express a number divisible by 4.

For example:

124 is divisible by 4, since 24 is divisible by 4;

1716 is divisible by 4, since 16 is divisible by 4;

1800 is divisible by 4 since 00 is divisible by 4

Rule. To multiply a number by 25, you need to divide this number by 4 and multiply by 100.

Examples:

484 × 25 = (484:4) × 25 × 4 = 121 × 100 = 12100

124 × 25 = 124: 4 × 100 = 3100

Rule. To divide a number by 25, you need to divide this number by 100 and multiply by 4.

Examples:

12100: 25 = 12100: 100 × 4 = 484

31100: 25 = 31100:100 × 4 = 1244

Rule. To multiply a number by 75, you need to divide this number by 4 and multiply by 300.

Examples:

32 × 75 = (32:4) × 75 × 4 = 8 × 300 = 2400

48 × 75 = 48: 4 × 300 = 3600

Rule. To divide a number by 75, you need to divide this number by 300 and multiply by 4.

Examples:

2400: 75 = 2400: 300 × 4 = 32

3600: 75 = 3600: 300 × 4 = 48

Rule. To multiply a number by 50, you need to divide this number by 2 and multiply by 100.

Examples:

432×50 = 432:2×50×2 = 216×100 = 21600

848 × 50 = 848: 2 × 100 = 42400

Rule. To divide a number by 50, you need to divide that number by 100 and multiply by 2.

Examples:

21600: 50 = 21600: 100 × 2 = 432

42400: 50 = 42400: 100 × 2 = 848

Rule. To multiply a number by 500, you need to divide this number by 2 and multiply by 1000.

Examples:

428 × 500 = (428:2) × 500 × 2 = 214 × 1000 = 214000

2436 × 500 = 2436: 2 × 1000 = 1218000

Rule. To divide a number by 500, you need to divide that number by 1000 and multiply by 2.

Examples:

214000: 500 = 214000: 1000 × 2 = 428

1218000: 500 = 1218000: 1000 × 2 = 2436

Before you learn how to multiply and divide by 125, you need to know the 8 multiplication table and the divisibility test by 8 well.

Sign. Those and only those numbers whose last three digits express a number divisible by 8 are divisible by 8.

Examples:

3168 is divisible by 8, since 168 is divisible by 8;

5248 is divisible by 8 because 248 is divisible by 8;

12328 is divisible by 8, since 324 is divisible by 8.

To find out if it is dividing three digit number ending with the numbers 2, 4, 6. 8. by 8, you need to add half the digits of the ones to the number of tens. If the result is divisible by 8, then the original number is divisible by 8.

Examples:

632: 8, since i.e. 64:8;

712:8, since i.e. 72:8;

304:8, since i.e. 32:8;

376: 8, since i.e. 40:8;

208:8, since i.e. 24:8.

Rule. To multiply a number by 125, you need to divide this number by 8 and multiply by 1000. To divide a number by 125, you need to divide this number by 1000 and multiply

at 8.

Examples:

32 × 125 = (32:8) × 125 × 8 = 4 × 1000 = 4000;

72 × 125 = 72: 8 × 1000 = 9000;

4000: 125 = 4000: 1000 × 8 = 32;

9000: 125 = 9000: 1000 × 8 = 72.

Rule. To multiply a number by 250, you need to divide this number by 4 and multiply by 1000.

Examples:

36 × 250 = (36:4) × 250 × 4 = 9 × 1000 = 9000;

44 × 250 = 44: 4 × 1000 = 11000.

Rule. To divide a number by 250, you need to divide this number by 1000 and multiply by 4.

Examples:

9000: 250 = 9000: 1000 × 4 = 36;

11000: 250 = 11000: 1000 ×4 = 44

Multiplying and dividing by 37

Before learning how to verbally multiply and divide by 37, you need to have a good knowledge of the multiplication table by three and the sign of divisibility by three, which is studied in the school course.

Rule. To multiply a number by 37, you need to divide this number by 3 and multiply by 111.

Examples:

24 × 37 = (24:3) × 37 × 3 = 8 × 111 = 888;

27 × 37 = (27:3) × 111 = 999.

Rule. To divide a number by 37, you need to divide this number by 111 and multiply by 3

Examples:

999:37 = 999:111 × 3 = 27;

888:37 = 888:111 × 3 = 24.

Multiply by 111

Having learned to multiply by 11, it is easy to multiply by 111, 1111, etc. a number whose sum of digits is less than 10.

Examples:

24 × 111 = 2 (2+4) (2+4) 4 = 2664;

36 ×111 = 3 (3+6) (3+6) 6 = 3996;

17 × 1111 = 1 (1+7) (1+7) (1+7) 7 = 18887.

Conclusion. To multiply a number by 11, 111, etc., you need to mentally move the digits of this number into two, three, etc. steps, add the numbers and write them down between the spread digits.

Multiplying two side by side standing numbers

Examples:

Conclusion. When multiplying two adjacent numbers, you must first multiply the tens digits, then multiply the tens digits by the sum of the ones digits, and finally, you must multiply the ones digits. Let's get the answer (see examples)

Multiplying a pair of numbers whose tens digits are the same and the sum of their ones digits is 10

Example:

24 × 26 = (24 - 4) × (26 + 4) + 4 × 6 = 20 × 30 + 24 = 624.

We round the numbers 24 and 26 to tens to get the number of hundreds, and add the product of units to the number of hundreds.

18 × 12 = 2 × 1 cell. + 8 × 2 = 200 + 16 = 216;

16 × 14 = 2 × 1 × 100 + 6 × 4 = 200 + 24 = 224;

23 × 27 = 2 × 3 × 100 + 3 × 7 = 621;

34 × 36 = 3 × 4 cells. + 4 × 6 = 1224;

71 × 79 = 7 × 8 cells. + 1 × 9 = 5609;

82 × 88 = 8 × 9 cells. + 2 × 8 = 7216.

Can be resolved orally or more complex examples:

108 × 102 = 10 × 11 cells. + 8 × 2 = 11016;

204 × 206 = 20 × 21 cells. +4 × 6 = 42024;

802 × 808 = 80 × 81 cells. +2 × 8 = 648016.

Examination:

× 802

6416

6416__

648016

Multiplying two-digit numbers in which the sum of the tens digits is 10 and the ones digits are the same.

Rule. When multiplying two-digit numbers. for which the sum of the tens digits is 10, and the ones digits are the same, you need to multiply the tens digits. and add the units digit, we get the number of hundreds and add the product of units to the number of hundreds.

Examples:

72 × 32 = (7 × 3 + 2) cells. + 2 × 2 = 2304;

64 × 44 = (6 × 4 + 4) × 100 + 4 × 4 = 2816;

53 × 53 = (5 × 5 +3) × 100 + 3 × 3 = 2809;

18 × 98 = (1 × 9 + 8) × 100 + 8 × 8 = 1764;

24 × 84 = (2 × 8 + 4) ×100+ 4 × 4 = 2016;

63 × 43 = (6 × 4 +3) × 100 +3 × 3 = 2709;

35 × 75 = (3 × 7 + 5) × 100 +5 × 5 = 2625.

Multiplying numbers ending in 1

Rule. When multiplying numbers ending in 1, you must first multiply the tens digits and write the sum of the tens digits under this number to the right of the resulting product, and then multiply 1 by 1 and write it even further to the right. Adding it in a column, we get the answer.

Examples:

Multiplying two-digit numbers by 101, three-digit numbers by 1001

Rule. To multiply a two-digit number by 101, you need to add the same number to the right of this number.

648 1001 = 648648;

999 1001 = 999999.

Methods of oral rational calculations used in mathematics lessons help improve general level mathematical development;develop in students the skill of quickly identifying from the laws, formulas, and theorems known to them those that should be applied to solve the proposed problems, calculations and calculations;promote memory development, develop ability visual perception math facts, improve spatial imagination.

In addition, rational calculation in mathematics lessons plays an important role in increasing children’s cognitive interest to mathematics lessons, as one of the most important motives educational and cognitive activity, development of the child’s personal qualities.By developing the skills of oral rational calculations, the teacher thereby develops the skills of students conscious assimilation material being studied, teaches to value and save time, develops the desire to search rational ways solving the problem. In other words, cognitive, including logical, cognitive and sign-symbolic universal educational actions are formed.

The goals and objectives of the school are changing dramatically; a transition is taking place from the knowledge paradigm to personal-oriented learning. Therefore, it is important not just to teach how to solve problems in mathematics, but to show the operation of the basic mathematical laws in life, explain how the student can apply the acquired knowledge. And then children will have the main thing: the desire and meaning to learn.

Bibliography

Minskikh E.M. “From Game to Knowledge”, M., “Prosveshcheniye” 1982.

Kordemsky B.A., Akhadov A.A. Amazing world numbers: Book of students, - M. Education, 1986.

Sovaylenko VK. System of teaching mathematics in grades 5-6. From work experience. - M.: Education, 1991.

Cutler E. McShane R. “Quick counting system according to Trachtenberg” - M. Education, 1967.

Minaeva S.S. "Calculations in the classroom and extracurricular activities mathematics." - M.: Education, 1983.

Sorokin A.S. “Counting techniques (methods of rational calculations)”, M, Znani, 1976

http://razvivajka.ru/ Mental counting training

http://gzomrepus.ru/exercises/production/ Exercises for productivity and quick mental calculation

Every parent wants their child to grow up smart, well-developed and interested in learning. However, it is difficult to show a child’s interest in acquiring new knowledge. One of the first manifestations of interest in knowledge in children preschool age is the account.

It is at this moment that it is very important to create from math assignments a game that will captivate the baby.

This article will discuss how to quickly teach a child to add in his head. We will provide not only exercises, but also tell you where to start the exercises and how to transform them into a game form.

The basis of mathematics is mastering counting

The first step in educational process is the study of ordinal counting, in other words, the numbers of their location. Initial stage you can take everyday activities, i.e. introducing counting when you walk up the stairs with your baby, button his jacket or eat. The remaining stages of training also proceed smoothly one after another, so in such classes it is important to maintain consistency and systematicity.

The main tasks for primary stages are:

• teach your baby to distinguish multiple items from single ones, i.e. “many” and “one”;
• teach to separate concepts such as “equal”, “more” and “less”;
• ordinal and quantitative counting;
• teach an understanding of how the number of objects relates to a specific number;
• study the composition of numbers - first from one to ten, then from 10 to 20, etc.;
• simple arithmetic problems.

When you come to problems in mathematics, you should use not just one method of solving, but several. With this approach, it will be easier for the child to look for other solutions in the future, and his mind will become more flexible.

Answering the question, “how to learn to count in your head?”, We note that learning should begin systematically, when the child reaches the age of 3 or 4 years. Remember that the process should be playful. Otherwise, the baby’s desire to learn can be blocked.

Presentation: "Mental arithmetic in mathematics lessons"

Counting process

The mental process concerning counting always begins with simple actions. As a rule, they are divided into two components - speech and motor.

1. Speech action develops according to the scheme - first we talk about what we are doing, then we whisper, and then we count to ourselves. And only after this stage can you move on to a quick count. For example, when adding units 1+1, the next digit in the series is called, i.e. in his mind the child will immediately add 1,2,3,4...
2. The motor element develops from the usual shifting of objects from side to side. Thus, in game form objects will increase or decrease. At first, the child will follow the counting with his finger, then only with his eyes, performing mathematical operations in his mind.

When counting on fingers or sticks, kids do not try to remember the result. In view of this, when there are not enough fingers and sticks when counting, the child has difficulties.

If a parent wants to teach a child to count, then the subject should reduce their participation in the process as quickly as possible, but it will not be possible to remove them completely. How to learn to count quickly in your head? Read about this in the following sections.

The main component of learning is play

Each person develops individually. Making mistakes while learning the material is normal. However, many parents do not understand why smart child unable to understand simple things from an adult's point of view.

Note that the child’s brain is different in structure from the adult’s brain. Kids do not want and cannot remember what does not arouse their interest.

Children's memory is designed in such a way that it stores only what evokes an emotional response. It doesn’t matter whether the emotions are positive or negative.

So how do you teach a child to count mentally? The game will help you learn mathematical foundations you can start counting kittens on the street while, for example, you go to kindergarten. Having taught your child the numbers from 1 to 10, you can invite him to look for them on the way to the store, and when he comes home, count how many numbers were found and add them up in his head.

There are many methods, and we suggest you familiarize yourself with the most popular ones in the next section.

The ability to count is important not only when preparing for school, but also in later life any person. Counting to 10 is important, but a child is unlikely to be able to master it right away, so you need to start from 1 to 5, and then increase the complexity of the task.

In order to master counting quickly and successfully, we recommend using hints, but only at the beginning of training. Then they need to be gradually removed so that the baby learns to count in his head.

• fingers;
• educational TV programs;
• educational games and abacus;
• rhymes with numbers or counting rhymes;
• Count everything you see every day with your baby.

Quick counting techniques:

1. Cards. During the period of learning numbers, flashcards are very important. You can buy them, or make them yourself with your child. The latter will be more interesting for the child. At the beginning, show them to your baby sequentially, then change the order.
2. Shop. One of the most favorite games for kids. You should lay out “goods for sale” on the table, come up with a “currency” and assign a price tag to each item. Your child should be appointed as a cashier. When communicating with a store employee, you should not pay attention to the price tags; let the child tell the story himself and calculate how much the items cost.
3. Plasticine. A game in which you need to ask a child to make 4 legs for a bear, or two ears for a cat. Along the way, you should show him cards with these numbers.

How to teach a child to count in his head? Teaching a child to count is quite difficult, but all parents want him to do it without thinking. Daily exercise, fascinating shapes classes, coupled with your perseverance and patience, will help your child master the queen of sciences - mathematics.