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 the 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 for 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 there is no good teacher who 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.

• The ability to work with large numbers is beyond the scope of 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, at first you cannot know how to quickly count without prior preparation. 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 escape 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 regularly perform mathematical operations in their minds, such as 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 a very important point when learning the fast counting technique. 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.

• food

During the period of intense mental training, you should include in your diet foods for nourishing the brain, for example, walnuts, honey, and fruits are good.

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.

Mastering the addition of multi-digit numbers is also not difficult. 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 is suitable for calculating the product of two-digit numbers.

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 and double digit numbers in the mind is carried out according to a 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 the mind for a 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.

bart in Simple mathematics or how to learn to quickly count in your mind.

Can't imagine your life without a calculator anymore? Very in vain, scientists have proven that people who regularly count in their minds are insured against senile insanity and early dementia. So practice more often, and I will tell you some simple tricks for easy and fast mental counting.

1. Multiply by 11
We all know how to quickly multiply a number by 10, you just need to add a zero at the end, but did you know that there is a trick on how to easily multiply a two-digit number by 11?
Let's say we need to multiply 63 by 11. Take a two-digit number that needs to be multiplied by 11 and imagine a place between its two digits:
6_3
Now add the first and second digits of this number and place in this location:
6_(6+3)_3
And our multiplication result is ready:
63*11=693
If the result of adding the first and second digits is a two-digit number, insert only the second digit, and add one to the first digit of the original number:
79*11=
7_(7+9)_9
(7+1)_6_9
79*11=869

2. Fast squaring of a number ending in 5
If you need to frame a two-digit number ending in 5, then you can do it very simply in your mind. Multiply the first digit of the number by itself plus one and add 25 at the end and that's it:
45*45=4*(4+1)_25=2025

3. Multiply by 5
For most people, multiplying by 5 is easy for small numbers, but how do you quickly mentally count large numbers multiplied by 5?
You need to take this number and divide by 2. If the result is an integer, then add 0 at the end to it, if not, discard the remainder and add 5 at the end:
1248*5=(1248/2)_(0 or 5)=624_(0 or 5)=6240 (the result of dividing by 2 is an integer)
4469*5=(4469/2)_(0 or 5)=(2234.5)_(0 or 5)=22345 (result of dividing by 2 with remainder)

4. Multiply by 4
This is a very simple and, at first glance, obvious feature of multiplying any number by 4, but despite this, people do not know about it at the right time. To simply multiply any number by 4, you need to multiply it by 2, and then multiply by 2 again:
67*4=67*2*2=134*2=268

5. Calculate 15%
If you need to mentally calculate 15% of any number, then there is an easy way to do it. Take 10% of the number (dividing the number by 10) and add half of the resulting 10% to that number.
15% of 884 rubles \u003d (10% of 884 rubles) + ((10% of 884 rubles) / 2) \u003d 88.4 rubles + 44.2 rubles \u003d 132.6 rubles

6. Multiplication of large numbers
If you need to multiply large numbers in your mind and one of them is even, then you can use the method of simplifying the factors by reducing the even number by half, and the second by doubling:
32*125 is
16*250 is
8*500 is
4*1000=4000

7. Divide by 5
Dividing a large number by 5 in your head is very easy. All you need to do is multiply the number by 2 and move the decimal point back by one:
175/5
Multiply by 2: 175*2=350
Shift by one sign: 35.0 or 35
1244/5
Multiply by 2: 1244*2=2488
Shift by one sign: 248.8

8. Subtraction from 1000
To subtract a large number from a thousand, follow a simple technique, subtract all digits from 9 except the last, and subtract the last digit from 10:
1000-489=(9-4)_(9-8)_(10-9)=511
Of course, in order to learn how to quickly count in your mind, you need to practice using these techniques many times to bring them to automatism, a single reading will leave only zeros in your head.

“Mathematics should already be loved because it puts the mind in order,” said Mikhail Lomonosov. The ability to count in the mind remains a useful skill for a modern person, despite the fact that he owns all kinds of devices that can count for him. The ability to do without special devices and at the right time to quickly solve the set arithmetic problem is not the only application of this skill. In addition to the utilitarian purpose, mental counting techniques will allow you to learn how to organize yourself in various life situations. In addition, the ability to count in your mind will undoubtedly have a positive effect on the image of your intellectual abilities and distinguish you from the surrounding “humanists”.

mental counting training

There are people who can perform simple arithmetic operations in their minds. Multiply a two-digit number by a one-digit number, multiply within 20, multiply two small two-digit numbers, and so on. - all these actions they can perform in the mind and quickly enough, faster than the average person. Often this skill is justified by the need for constant practical use. As a rule, people who calculate well in their minds have a mathematical education or, according to at least, experience in solving numerous arithmetic problems.

Undoubtedly, experience and training plays a crucial role in the development of any ability. But the skill of mental counting is not based on experience alone. This is proved by people who, unlike those described above, are able to calculate in their minds much more complex examples. For example, such people can multiply and divide three-digit numbers, perform complex arithmetic operations that not every person can count in a column.

What does an ordinary person need to know and be able to master in order to master such a phenomenal ability? Today, there are various techniques that help you learn how to quickly count in your mind. Having studied many approaches to teaching the skill of counting orally, we can distinguish 3 main components of this skill:

1. Ability. The ability to concentrate attention and the ability to keep several things in short-term memory at the same time. Predisposition to mathematics and logical thinking.

2. Algorithms. Knowledge of special algorithms and the ability to quickly select the desired, most effective algorithm in each specific situation.

3. Training and experience, whose value for any skill has not been canceled. Constant training and the gradual complication of tasks and exercises will allow you to improve the speed and quality of mental arithmetic.

It should be noted that the third factor is of key importance. Without the necessary experience, you will not be able to surprise others with a fast score, even if you know the most convenient algorithm. However, do not underestimate the importance of the first two components, since having the abilities and a set of necessary algorithms in your arsenal, you can outdo even the most experienced "bookkeeper", provided that you have been training for the same time.

Lessons on the site

The oral counting lessons presented on the site are aimed precisely at the development of these three components. The first lesson tells how to develop a predisposition for mathematics and arithmetic, as well as the basics of counting and logic. Then a number of lessons are given on special algorithms for performing various arithmetic operations in the mind. And finally, this training provides additional materials to help train and develop the ability to count orally, in order to be able to apply your talent and your knowledge in life.

Practicing the computational skills of students in mathematics lessons using "quick" counting techniques.

Kudinova I.K., teacher of mathematics

MKOU Limanovskoy secondary school

Paninsky municipal district

Voronezh region

“Have you ever observed how people with natural counting abilities are susceptible, one might say, to all sciences? Even all those who are slow in thinking, if they learn and practice this, 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 of education is the formation of universal educational activities that provide students with the ability to learn, the ability for self-development and self-improvement. The quality of knowledge assimilation is determined by the variety and nature of the types of universal actions. Forming the ability and readiness of students to implement universal learning activities allows you to increase the effectiveness of the learning process. All types of universal educational activities are considered in the context of the content of specific academic subjects.

An important role in the formation of universal educational activities is played by teaching schoolchildren the skills of rational calculations.No one doubts that the development of the ability to rational calculations and transformations, as well as the development of skills for solving the simplest problems "in the mind" is the most important element in the mathematical preparation of students. ATThe importance and necessity of such exercises do not have to be proved. Their significance is great in the formation of computational skills, and the improvement of knowledge of numbering, and in the development of the child's personal qualities. The creation of a certain system of consolidation and repetition of the studied material gives students the opportunity to master knowledge at the level of automatic skill.

Knowledge of simplified methods of oral calculations remains necessary even with the complete mechanization of all the most labor-intensive computational processes. Oral calculations make it possible not only to quickly make calculations in the mind, but also to control, evaluate, find and correct errors. In addition, the development of computational skills develops memory and helps schoolchildren to fully master the subjects of the physical and mathematical cycle.

It is obvious that the methods of rational counting are a necessary element of the computational culture in the life of every person, primarily because of 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, since, in addition to the fact that calculations activate memory, attention, help rationally organize activities and significantly affect human development.

In everyday life, in training sessions, when every minute is valued, it is very important to quickly and rationally carry out oral and written calculations without making mistakes and without using any additional computing tools.

An analysis of the results of exams in grades 9 and 11 shows that students make the greatest number of mistakes when performing tasks for calculations. Often, even highly motivated students lose their oral counting skills by the time they enter the final assessment. They calculate badly and irrationally, increasingly resorting to the help of technical calculators. The main task of the teacher is not only to maintain computational skills, but also to teach how to use non-standard methods of oral counting, which would significantly reduce the time spent on the task.

Let's consider specific examples of various methods of fast rational computations.

DIFFERENT WAYS OF ADDITION AND SUBTRACTION

ADDITION

The basic rule for doing mental addition is:

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

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

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

ADDITION IN THE MIND OF TWO-DIGITAL NUMBERS

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

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

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

ADDITION OF THREE-DIGIT NUMBERS

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

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

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

SUBTRACTION

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

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

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

Multiplication of multi-digit numbers by 9

1. Increase the number of tens by 1 and subtract from the multiplier

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

Example:

576 9 = 5184 379 9 = 3411

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

Multiply by 99

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

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

3. We 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

Multiply by 999 any number

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

2. Find the complement of up to 1000

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

23 - 1 = 22

1000 - 23 = 977

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

124 - 1 = 123

1000 - 124 = 876

1324 999 = 1322676 (one 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 whose 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 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:

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, the last number must be represented as a product of a single-digit number (from 1 to 9) by 11, i.e.

44= 4 × 11; 55 = 5x11 etc.

Then multiply the product of the first numbers by 11.

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

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

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

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

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

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

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

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

8 x 99 = 8 x 9 x 11 = 72 x 11 = 792.

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

Multiply by a number ending in 5

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

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

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

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

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

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

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

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

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

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

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

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

Multiplication and division by 25, 50, 75, 125, 250, 500

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

Divisible by 4 are those, and only those, numbers in which the last two digits of the number 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 because 00 is divisible by 4

Rule. To multiply a number by 25, divide that number by 4 and multiply by 100.

Examples:

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

124 x 25 = 124: 4 x 100 = 3100

Rule. To divide a number by 25, divide that 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, divide that number by 4 and multiply by 300.

Examples:

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

48 x 75 = 48: 4 x 300 = 3600

Rule. To divide a number by 75, divide that 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, divide the number by 2 and multiply by 100.

Examples:

432 x 50 = 432:2 x 50 x 2 = 216 x 100 = 21600

848 x 50 = 848: 2 x 100 = 42400

Rule. To divide a number by 50, 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, divide that number by 2 and multiply by 1000.

Examples:

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

2436 × 500 = 2436: 2 × 1000 = 1218000

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

Examples:

214000: 500 = 214000: 1000 × 2 = 428

1218000: 500 = 1218000: 1000 × 2 = 2436

Before learning how to multiply and divide by 125, you need to have a good knowledge of the multiplication table by 8 and the sign of divisibility by 8.

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

Examples:

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

5248 is divisible by 8, since 248 is divisible by 8;

12328 is divisible by 8 because 324 is divisible by 8.

To find out if a three-digit number ending in 2, 4, 6. 8. is divisible by 8, you need to add half the units digits 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 x 125 = (32: 8) x 125 x 8 = 4 x 1000 = 4000;

72 x 125 = 72: 8 x 1000 = 9000;

4000: 125 = 4000: 1000 × 8 = 32;

9000: 125 = 9000: 1000 × 8 = 72.

Rule. To multiply a number by 250, divide that number by 4 and multiply by 1000.

Examples:

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

44 x 250 = 44: 4 x 1000 = 11000.

Rule. To divide a number by 250, divide that number by 1000 and multiply by 4.

Examples:

9000: 250 = 9000: 1000 × 4 = 36;

11000: 250 = 11000: 1000 × 4 = 44

Multiplication and division by 37

Before you learn how to verbally multiply and divide by 37, you need to know well 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, divide that number by 3 and multiply by 111.

Examples:

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

27 x 37 = (27:3) x 111 = 999.

Rule. To divide a number by 37, divide that 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 how 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. In order to multiply a number by 11, 111, etc., one must mentally expand the numbers of this number by two, three, etc. steps, add the numbers and write them down between the separated numbers.

Multiplying two adjacent numbers

Examples:

1) 12 × 13 = ? 1 x 1 = 1 1 × (2+3) = 5 2 x 3 = 6 2) 23 × 24 =? 2 x 2 = 4 2 × (3+4) = 14 3 x 4 = 12 3) 32 × 33 =? 3 x 3 = 9 3 × (2+3) = 15 2 x 3 = 6 1056 4) 75 × 76 =? 7 x 7 = 49 7 × (5+6) = 77 5 x 6 = 30 5700

Examination: × 12 Examination: × 23 Examination: × 32 1056 Examination: × 75 525_ 5700

Conclusion. When multiplying two adjacent numbers, you must first multiply the tens digits, then multiply the tens digit by the sum of the units digits, and finally, you need to multiply the units digits. Get an answer (see examples)

Multiplying a pair of numbers whose tens digits are the same and the unit digits add up to 10

Example:

24 x 26 = (24 - 4) x (26 + 4) + 4 x 6 = 20 x 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 x 12 = 2 x 1 cell. + 8 × 2 = 200 + 16 = 216;

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

23 x 27 = 2 x 3 x 100 + 3 x 7 = 621;

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

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

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

You can solve verbally and 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

Multiplication of two-digit numbers in which the sum of the tens digits is 10, and the units digits are the same.

Rule. When multiplying two-digit numbers. in which the sum of the tens digits is 10, and the units digits are the same, you need to multiply the tens digits. and add the number of units, 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 x 44 = (6 x 4 + 4) x 100 + 4 x 4 = 2816;

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

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

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

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

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

Multiply numbers ending in 1

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

Examples:

1) 81 × 31 =? 8 x 3 = 24 8 + 3 = 11 1 x 1 = 1 2511 81 × 31 = 2511

2) 21 × 31 =? 2 x 3 = 6 2 +3 = 5 1 x 1 = 1 21 x 31 = 651

3) 91 × 71 =? 9 x 7 = 63 9 + 7 = 16 1 x 1 = 1 6461 91 × ​​71 = 6461

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

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

648 1001 = 648648;

999 1001 = 999999.

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

In addition, rational counting in mathematics lessons plays an important role in increasing children's cognitive interest in mathematics lessons, as one of the most important motives for educational and cognitive activity, the development of a child's personal qualities.Forming the skills of oral rational calculations, the teacher thereby educates students in the skills of conscious assimilation of the material being studied, teaches them to appreciate and save time, develops a desire to find rational ways to solve a problem. In other words, cognitive, including logical, cognitive and sign-symbolic universal learning activities are formed.

The goals and objectives of the school are changing dramatically, a transition is being made from the knowledge paradigm to personally-oriented learning. Therefore, it is important not only to teach how to solve problems in mathematics, but to show the effect of basic mathematical laws in life, to explain how a student can apply the knowledge gained. And then the main thing will appear in children: the desire and meaning to learn.

Bibliography

Minskykh E.M. "From game to knowledge", M., "Enlightenment" 1982.

Kordemsky B.A., Akhadov A.A. The amazing world of numbers: A book of students, - M. Enlightenment, 1986.

Sovailenko VK. The system of teaching mathematics in grades 5-6. From experience.- M.: Education, 1991.

Cutler E. McShane R. "The Trachtenberg Quick Counting System" - M. Enlightenment, 1967.

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

Sorokin A.S. "Counting technique (methods of rational calculations)", M, Knowledge, 1976

http://razvivajka.ru/ Oral counting training

http://gzomrepus.ru/exercises/production/ Productivity exercises and quick mental counting