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. INThe 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 on numbering, and in the development personal qualities child. 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 the 9th and 11th grades shows that the largest number students make mistakes when performing assignments 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.

Consider concrete examples various tricks fast rational calculations.

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.

Can be solved orally 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 help to increase general level 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. amazing world numbers: Book of students, - M. Education, 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. "Calculation in the classroom and extracurricular activities 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

Do you want to speak and write Russian correctly, but make mistakes since school? Lost your literacy in the online communication race? The unique method of improving literacy under the program of Mikhail Shestov is now available to the general public!
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A lexeme is Greek for "word", and in the spelling section we will deal with the rules for writing words without their connection in a sentence. The various spellings of words will be of interest to us when we move on to the Morphology section.
So, spelling. To understand its basic laws, you need to make a short journey into the past.
How did our alphabet come about? It appeared like this: in the 9th century, two Bulgarians - Cyril and Methodius - who lived in Greece, carried the Word of God to the Slavic people. They carried him to Eastern Europe, to the Dnieper, then inhabited by the ancient Russian people. This community had already developed by the 9th century, and included the current Belarusians, Russians and Ukrainians. They spoke about the same. Cyril and Methodius, traveling through Ancient Rus', listened to the speech of the Eastern Slavs and, on the basis of the Greek alphabet known to them, compiled a new alphabet for the Russians. They composed it according to audible sounds: if there was a corresponding letter for them, they left the Greek one, if it was not found, they came up with a new, similar one. So we got many vowels: a, o, and (and). At that time, the pas did not have a letter and it appeared from the nasal sound ep (such a sound still exists in some languages, for example, in Polish), there was no y - there was a nasal op; but there were special sounds (letters): ь and ъ (by the way, these were once vowels denoting a very short sound that still sounds in many unstressed syllables, where o or a is written in place of stressed ones).

Introduction
Part 1. SPELLING AND MORPHOLOGY
Part 2. SYNTAX AND PUNCTUATION
Part 3. STYLISTICS OF THE RUSSIAN LANGUAGE. THE CONCEPT OF BUSINESS AND SCIENTIFIC STYLES.
Exercises
Applications
Answers

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Why count in the mind, if you can solve any arithmetic problem on a calculator. Modern medicine and psychology prove that mental counting is an exercise for gray cells. Performing such gymnastics is necessary for the development of memory and mathematical abilities.

There are many tricks to simplify mental calculations. Everyone who has seen the famous painting by Bogdanov-Belsky "Mental Account" is always surprised - how do peasant children solve such a difficult task as dividing the sum of five numbers that must first be squared?

It turns out that these children are students of the famous teacher-mathematician Sergei Alexandrovich Rachitsky (he is also depicted in the picture). These are not geeks - students primary school village school of the 19th century. But they all already know how to simplify arithmetic calculations and have learned the multiplication table! Therefore, it is quite possible for these kids to solve such a problem!

Secrets of mental counting

There are methods of oral counting - simple algorithms that it is desirable to bring to automatism. After mastering simple techniques, you can move on to mastering more complex ones.

We add the numbers 7,8,9

To simplify the calculations, the numbers 7,8,9 must first be rounded up to 10, and then subtract the increase. For example, to add 9 to a two-digit number, you must first add 10 and then subtract 1, and so on.

Examples :

Add two digit numbers quickly

If the last digit of a two-digit number is greater than five, round it up. We perform the addition, subtract the “additive” from the resulting amount.

Examples :

54+39=54+40-1=93

26+38=26+40-2=64

If the last digit of a two-digit number is less than five, then add up by digits: first add tens, then ones.

Example :

57+32=57+30+2=89

If the terms are reversed, then you can first round the number 57 to 60, and then subtract 3 from the total:

32+57=32+60-3=89

Adding three-digit numbers in your mind

Quick counting and addition of three-digit numbers - is it possible? Yes. To do this, you need to parse three-digit numbers into hundreds, tens, units and add them one by one.

Example :

249+533=(200+500)+(40+30)+(9+3)=782

Subtraction features: reduction to round numbers

Subtracted are rounded up to 10, up to 100. If you need to subtract a two-digit number, you need to round it up to 100, subtract, and then add an amendment to the remainder. This is true if the correction is small.

Examples :

576-88=576-100+12=488

Mind subtracting three-digit numbers

If at one time the composition of numbers from 1 to 10 was well mastered, then subtraction can be done in parts and in the indicated order: hundreds, tens, units.

Example :

843-596=843-500-90-6=343-90-6=253-6=247

Multiply and Divide

Instantly multiply and divide in your mind? It is possible, but one cannot do without knowledge of the multiplication table. is the golden key to quick mental counting! It applies to both multiplication and division. Recall that in the elementary grades of a village school in the pre-revolutionary Smolensk province (the painting "Mental Counting"), children knew the continuation of the multiplication table - from 11 to 19!

Although in my opinion it is enough to know the table from 1 to 10 in order to be able to multiply larger numbers. For example:

15*16=15*10+(10*6+5*6)=150+60+30=240

Multiply and divide by 4, 6, 8, 9

Having mastered the multiplication table for 2 and 3 to automatism, making the rest of the calculations will be as easy as shelling pears.

For multiplication and division of two- and three-digit numbers, we use simple tricks:

multiplying by 4 is twice multiplying by 2;

to multiply by 6 means to multiply by 2 and then by 3;

multiplying by 8 is three times multiplying by 2;

multiplying by 9 is twice multiplying by 3.

For example :

37*4=(37*2)*2=74*2=148;

412*6=(412*2) 3=824 3=2472

Similarly:

divided by 4 is twice divided by 2;

divide by 6 is first divide by 2 and then by 3;

divided by 8 is three times divided by 2;

Divide by 9 is twice divided by 3.

For example :

412:4=(412:2):2=206:2=103

312:6=(312:2):3=156:3=52

How to multiply and divide by 5

The number 5 is half of 10 (10:2). Therefore, we first multiply by 10, then we divide the result in half.

Example :

326*5=(326*10):2=3260:2=1630

More easier rule division by 5. First, multiply by 2, and then divide by 10.

326:5=(326 2):10=652:10=65.2.

Multiply by 9

To multiply a number by 9, it is not necessary to multiply it twice by 3. It is enough to multiply it by 10 and subtract the multiplied number from the resulting number. Compare which is faster:

37*9=(37*3)*3=111*3=333

37*9=37*10 - 37=370-37=333

Also, particular patterns have long been noticed that greatly simplify the multiplication of two-digit numbers by 11 or by 101. So, when multiplied by 11, a two-digit number seems to move apart. The numbers that make it up remain at the edges, and their sum is in the center. For example: 24*11=264. When multiplying by 101, it is enough to attribute the same to a two-digit number. 24*101= 2424. The simplicity and logic of such examples is admirable. Such tasks are very rare - these are entertaining examples, the so-called little tricks.

Counting on fingers

Today you can still meet many defenders " finger gymnastics”and methods of oral counting on the fingers. We are convinced that learning to add and subtract by bending and unbending fingers is very visual and convenient. The range of such calculations is very limited. As soon as the calculations go beyond one operation, difficulties arise: it is necessary to master the next technique. Yes, and bending your fingers in the era of iPhones is somehow undignified.

For example, in defense of the "finger" technique, the technique of multiplying by 9 is given. The trick of the technique is as follows:

• To multiply any number within the first ten by 9, you need to turn your palms towards you.
• Counting from left to right, bend the finger corresponding to the number being multiplied. For example, to multiply 5 by 9, you need to bend the little finger on your left hand.
• The remaining number of fingers on the left will correspond to tens, on the right - units. In our example - 4 fingers on the left and 5 on the right. Answer: 45.

Yes, indeed, the solution is quick and visual! But this is from the field of tricks. The rule only works when multiplying by 9. Isn't it easier to learn the multiplication table to multiply 5 by 9? This trick will be forgotten, and a well-learned multiplication table will remain forever.

There are also many more similar tricks using fingers for some single mathematical operations, but this is relevant while you use it and is immediately forgotten when you stop using it. Therefore, it is better to learn standard algorithms that will remain for life.

Oral account on the machine

First, you need to know the composition of the number and the multiplication table well.

Secondly, you need to remember the methods of simplifying calculations. As it turned out, there are not so many such mathematical algorithms.

Thirdly, in order for the technique to turn into a convenient skill, it is necessary to constantly conduct brief “brainstorming sessions” - to practice oral calculations using one or another algorithm.

Workouts should be short: mentally solve 3-4 examples using the same technique, then move on to the next one. We must strive to use every free minute - and useful, and not boring. Thanks to simple training, all calculations over time will be done at lightning speed and without errors. This is very useful in life and will help out in difficult situations.

There are a lot of methods for speed reading in our time. Only you can say which one is the best. Choose the one that suits you. You can find speed reading techniques in books, courses, download on the Internet. Many of the speed reading methods proposed by the author can be used online. Special exercises and trainings have been developed to expand the angle of view, increase concentration on the text, and develop memory.

Do I need to read and cram more?

Those who cram are not the most successful and quick-witted people. Have you heard anything about the life successes of high school students?

It never leaves me wanting to learn to read faster! The thing is, reading fast is a must. By the nature of my work, I have to read a lot of literature, documents, etc. But I just physically do not have time for all that I have to do. Ah, my progress in speed reading leaves a lot to be desired.

About speed reading techniques Dynamic reading- this is a set of tricks that allow you to significantly increase the speed of reading a speed reader without a big loss of reading comprehension. In particular, it should be borne in mind that there is no official separation between "slow" and "speed" reading methods, for the reason that many readers use suitable reading exercises for them. Basic Speed ​​Reading Techniques Elimination, regression, stops, recurrent eye movements. With the classical method of reading, rereading is common, and this significantly slows down the speed of reading and reduces the coefficient of assimilation of information. Practicing an instant selection habit main idea text, cutting off unnecessary information and reading useful and effective information. Suppression of internal articulation- development of a new reading strategy: . The average reader has the necessary visual reading skills. For example, logos are immediately decoded, ("Nike", "Pepsi", "Ford", "GM"). A lot of familiar phrases are perceived by the picture. At the same time, you should know that incomprehensible words need to be read by decoding words into sound images, that is, pronouncing the text. Review Reading. "Scanning" without carefully focusing on text of little importance. Field of view extensions. Special trainings are used (for example, the Schulte table), aimed at expanding the angle of view to two or three words, a page. Thanks to this skill, a reader who reads in one fixation of a glance can capture much more information than a reader who does not have this skill. There are many schools of directions, methods, courses that train the skill of speed reading. Most of them are based on the speed reading methods listed above.

Any exercises that develop your brain, your thinking is very useful! It is especially good when these are very different exercises. Then you can be sure that the various functions of your thinking will develop. And all this will lead to the fact that you will change qualitatively.

The exercise kills two birds with one stone - it expands the angle of view, which is one of the conditions for mastering the skill of fast reading. The second hare is the entry into a trance state during classes. Eyes unfocused, directed forward. All signs of trance.

Everyone can learn speed reading without courses and without wasting money.

And now let's use the visualization technology and write down the phrase "a small scientist, but a pedant". All links go to direct texts.

Speed ​​reading is not acceptable if our task is to deeply feel the attitude of the book. Any psychoanalyst will explain how important "insignificant" details are, how much information they carry.

Century information technologies came a long time ago and every day data and knowledge are beginning to have an increasing value, while the technical capacity of computers and data transmission facilities is constantly increasing, and the resources of the human brain are still used only to a small extent. Why do more and more people want to develop their abilities.

Write an article. Now squeeze it twice, then two more. Repeat until one or two words remain. In pair technology, there is such an algorithm for the work of a pair of students - it is called "paragraph reading". In it each has one text.

Before reading the literature, make a short review of it - find out what it is about and in what genre it is written. Review the abstract and content. In order not to load memory with unnecessary information, decide which parts of the book you will read. It is better to read in a calm environment, in the absence of distracting objects. The room should be light. Remember that information is absorbed worse if there is a TV, radio or children playing nearby. The back should be in a straight position.

Diagonal Reading Practice

What kind of people had the skill of speed reading?

He was one of the first to offer some speed reading techniques. They were mastered by Napoleon Bonaparte, Alexander Sergeevich Pushkin, John F. Kennedy. They had the best speed reading techniques

To which the leader replied that, of course, he reads otherwise he would not be able to get all the knowledge that he now has. One proven way to speed read is to move your eyes from left to right at a rate of one line per minute. To the greatest extent, the speed of reading reduces the mental pronunciation of the text. Speed ​​reading requires high concentration.

How Hitler read books

Adolf Hitler had his own speed reading technique. He took a book, a magazine, a scientific article in his hand and opened it to the last page. If I saw something worthwhile, I read it. From the memoirs of Adolf Hitler's secretary, we know that the leader was very sorry that he could not read a single fiction book, since his duties included only reading scientific literature. The famous choleric Alexander Sergeevich Pushkin had a phenomenal memory. He remembered the biographies of famous people up to the exact dates, the names of geographical objects. He read works very quickly.

Karl Marx and fast reading

He liked to bend the pages of books and put notes on their margins and Karl Marx. President Roosevelt was simply in love with speed reading. He could read the entire book in one sitting. Honore de Balzac told his contemporaries how easily he manages to read eight sentences at once, and at the same time single out one key one from them.

Passion even for self-testing hinders development. Testing is needed to understand what is happening. There were enough cases in practice when the absolute lack of improvement was associated precisely with the violation of the ban on frequent home self-testing.

Theodore Roosevelt on speed reading

All texts are different. Theodore Roosevelt read two sentences at a time, and then he could easily retell the text, sometimes even verbatim. Maxim Gorky had unique abilities. He perfectly mastered the technique of speed reading and when he took a fresh issue of the magazine, he cut the pages and read the text, as if “drawing” a zigzag. After reading one magazine, I took up new literature. This technique is called diagonal speed reading. To increase the speed of text perception, a set of speed reading techniques is used. Everyone works in tandem with a partner for a short time using the same algorithm. Students change partners many times until they have worked through their texts completely. As a result, everyone gets "very compressed and convex polyphonic" material of their text.

Is it possible to learn how to quickly read billiards?

Try to open the site with Arabic script or a Chinese letter, and you will experience the same feelings - "I look in a book - I see a fig."

Those who like to remember do not like to think. Is it possible to remember the history of the CPSU ( Communist Party Soviet Union)? Please! I got rid of pronunciation a long time ago and, unlike many of my friends without special efforts and there are almost no regressions, I always try to slide through the text with a vertical gaze. But the problem is that I constantly want to slide through the text faster and faster.

Those who retries gave up early. At first they gave up and proved to themselves that they were not capable. For example, they are unable to remember or pass the exam, and only then the teacher told them this phrase.

Do you want to speak and write Russian correctly, but make mistakes since school? Lost your literacy in the online communication race? The unique method of improving literacy under the program of Mikhail Shestov is now available to the general public! Mikhail Shestov is a linguist, international journalist, innovative teacher, author of his own methodology for teaching languages, recognized as one of the most successful in the world, adviser to Earth Day and a number of other organizations of the UN system, listed in the international Guinness Book (also holds the record for the Russian analogue Guinness Books - Books "Divo"), Interpol consultant and instructor in the field of improving the level of language proficiency, consultant to the Russian, American, Chinese and Israeli governments in the field of improving the effectiveness of teaching methods, head of the Department of Innovative Methods of Learning Foreign Languages ​​at the International University of Moscow. The book is intended for high school students, applicants, students, teachers and people who want to improve their literacy in the most effective way.

The work belongs to the genre Pedagogy. It was published in 2018 by Peter. On our website you can download the book "Russian without errors. A quick method for improving literacy" in fb2, rtf, epub, pdf, txt format or read online. Here, before reading, you can also refer to the reviews of readers who are already familiar with the book, and find out their opinion. In the online store of our partner you can buy and read the book in paper form.