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 × 88 = 8 × 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
Under the game there is a description, instructions and rules, as well as thematic links to similar materials - we recommend that you read it.
There is definitely something sporty in this game. The emotional surge increases with the growth of the rate of presentation of examples. The process looks simpler than a steamed turnip. You see an example on the screen, say "8 - 5 =", enter the answer "3" on the keyboard and move on to the next one. However, the faster you manage to solve these simple problems, the faster the next examples begin to appear, as the speed increases, so does the complexity, operations with multiplication and division begin to appear. A great game for those who want to test their mental arithmetic skills as well as practice basic math.
Can download game SPEED COUNT on your computer, it will not take up much space, but think about whether it makes sense to do this, because here it is always available, you just need to open this page.
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The game in categories Logic, Sports is available is free, around the clock and without registering with a description in Russian on Min2Win. If the capabilities of the electronic desktop allow, you can expand the ORAL ACCOUNT IN SPEED plot to full screen and enhance the effect of the passage of scenarios. Many things really make sense to consider in more detail.
The principle of operation is based on the generation of examples in mathematics of a suitable level of complexity for all classes, the solution of which contributes to the development of mental counting skills.
The application has a positive effect on the mental activity of both children and adults.
Variety of modes
On the mode settings page, you can set the necessary parameters for generating examples in mathematics for any class.
The mental counting simulator allows you to work out 4 well-known arithmetic operations at six levels of difficulty.
At this stage of development, modes were thought out and implemented that allow you to work with two sets of numbers: positive and negative. In each of them, you can practice in different types of tasks: "Example", "Equation", "Comparison".
This mode includes the usual arithmetic math examples consisting of two or three numbers.
The mode in which the desired number can be in any position.
The mode in which it is necessary to place the comparison sign correctly between the results of two examples.
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Counting Process
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The application also provides game animation "Battle of the swordsmen".
Depending on the correctness of the entered answer, one or another fencer strikes, pushing his opponent back. However, it should be borne in mind that every second of inactivity, the enemy crowds your player, and with a long wait, he jumps out loss message.
Such an interface makes the process of solving mathematical examples more interesting, and is also a simple motivation for children.
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Thus, you can easily invite any person to compete in solving arithmetic examples in mathematics, simply by passing him a link to the current mode.
Why do we need a mental account, if it is the 21st century in the yard, and all kinds of gadgets are capable of almost instantly performing any arithmetic operations? You can even not poke your finger at the smartphone, but give a voice command - and immediately get the right answer. Now even elementary school students who are too lazy to independently divide, multiply, add and subtract are doing this successfully.
But this medal also has a downside: scientists warn that if you don’t train, don’t load it with work and make it easier for him, he starts to be lazy, he is reduced. In the same way, without physical training, our muscles also weaken.
Mikhail Vasilyevich Lomonosov spoke about the benefits of mathematics, calling it the most beautiful of sciences: “Mathematics is already worth loving because it puts the mind in order.”
The oral account develops attention, speed of reaction. No wonder there are more and more new methods of quick oral counting, designed for both children and adults. One of them is the Japanese oral counting system, which uses the ancient Japanese soroban abacus. The technique itself was developed in Japan 25 years ago, and now it is successfully used in some of our schools of oral counting. It uses visual images, each of which corresponds to a certain number. Such training develops the right hemisphere of the brain, which is responsible for spatial thinking, building analogies, etc.
It is curious that in just two years, students of such schools (children aged 4–11 years old are accepted here) learn to perform arithmetic operations with 2-digit, or even 3-digit numbers. Kids who do not know multiplication tables here know how to multiply. They add and subtract large numbers without writing down their column. But, of course, the goal of training is the balanced development of the right and.
You can also master mental arithmetic with the help of the problem book “1001 tasks for mental arithmetic at school”, compiled back in the 19th century by a village teacher and well-known educator Sergey Alexandrovich Rachinsky. This problem book is supported by the fact that it went through several editions. This book can be found and downloaded online.
People who practice quick counting recommend Yakov Trakhtenberg's book "Quick Counting System". The history of this system is very unusual. In order to survive in the concentration camp where he was sent by the Nazis in 1941, and not to lose his mental clarity, the Zurich professor of mathematics began to develop algorithms for mathematical operations that allow him to quickly calculate in his head. And after the war, he wrote a book in which the quick counting system is presented in such a clear and accessible way that it is still in demand.
Good reviews about the book by Yakov Perelman “Quick Count. Thirty Simple Examples of Oral Counting. The chapters in this book are devoted to multiplication by single and double digits, in particular, multiplying by 4 and 8, 5 and 25, by 11/2, 11/4, *, dividing by 15, squaring, calculating by formula.
The simplest ways of oral counting
People with certain abilities will quickly master this skill, namely: the ability to think logically, the ability to concentrate and store several images in short-term memory at the same time.
Equally important is the knowledge of special action algorithms and some mathematical laws that allow, as well as the ability to choose the most effective for a given situation.
And, of course, you can not do without regular training!
The most common quick counting methods are as follows:
1. Multiplying a two-digit number by a one-digit number
Multiplying a two-digit number by a one-digit number is easiest by decomposing it into two components. For example, 45 - by 40 and 5. Next, we multiply each component by the desired number, for example, by 7, separately. We get: 40 × 7 = 280; 5 × 7 = 35. Then add the results: 280 + 35 = 315.
2. Multiply a three-digit number
Multiplying a three-digit number in your mind is also much easier if you decompose it into its components, but presenting the multiplicand in such a way that it is easier to perform mathematical operations with it. For example, we need to multiply 137 by 5.
We represent 137 as 140 - 3. That is, it turns out that now we must multiply by 5 not 137, but 140 - 3. Or (140 - 3) x 5.
Knowing the multiplication table within 19 x 9, you can count even faster. We decompose the number 137 into 130 and 7. Then we multiply by 5, first 130, and then 7, and add the results. So 137 x 5 = 130 x 5 + 7 x 5 = 650 + 35 = 685.
You can decompose not only the multiplicand, but also the multiplier. For example, we need to multiply 235 by 6. We get six by multiplying 2 by 3. Thus, we first multiply 235 by 2 and get 470, and then we multiply 470 by 3. Total 1410.
The same operation can be performed differently by representing 235 as 200 and 35. It turns out 235 × 6 = (200 + 35) × 6 = 200 × 6 + 35 × 6 = 1200 + 210 = 1410.
In the same way, decomposing numbers into components, you can perform addition, subtraction and division.
3. Multiply by 10
Everyone knows how to multiply by 10: just add zero to the multiplicand. For example, 15 × 10 = 150. Based on this, it is no less easy to multiply by 9. First, add 0 to the multiplicand, that is, multiply it by 10, and then subtract the multiplier from the resulting number: 150 × 9 = 150 × 10 = 1500 − 150 = 1350.
4. Multiply by 5
It is easy to multiply by 5. You just need to multiply the number by 10, and divide the resulting result by 2.
5. Multiply by 11
It is interesting to multiply two-digit numbers by 11. Let's take, for example, 18. Let's mentally expand 1 and 8, and write the sum of these numbers between them: 1 + 8. We get 1 (1 + 8) 8. Or 198.
6. Multiply by 1.5
If you need to multiply some number by 1.5, divide it by two and add the resulting half to the whole: 24 × 1.5 = 24 / 2 + 24 = 36.
These are just the simplest ways of mental counting, with the help of which we can train our brain in everyday life. For example, counting the cost of purchases while standing in line at the checkout. Or perform mathematical operations with the numbers on the numbers of cars passing by. Those who like to "play" with numbers and want to develop their mental abilities can refer to the books of the above-mentioned authors.
Calculation trainer- easily and significantly increases the intellectual potential of a person.
The result of acquiring skills and completing the standard qualification will be the assignment of a sports category (I category, II category, III category, candidate master of sports, master of sports and grandmaster).
- People from the group are distinguished both by the ability to speak beautifully and correctly, and by the ability to quickly count in the mind, and, as a rule, they are classified as smart. The ability to quickly count in the mind allows a student to study more successfully, and an engineer and a scientist to reduce the time for obtaining the result of their activities.
- CS is needed not only for schoolchildren, but also for engineers, teachers, medical workers, scientists and managers of various levels. Who quickly considers, it is easier for him to study and work. US is not a toy, although it entertains. It allows the student to return to those "rails" from which he once fell; increases the speed and quality of perception of information; disciplines and produces accuracy in everything; teaches to notice details and trifles; teaches to save; creates images of objects and phenomena; allows you to foresee the future and develops human intelligence.
- "Renovation" in the head should begin with simple arithmetic operations that allow you to structure the brain.
- The ability to quickly count in the mind gives the student self-confidence. As a rule, those who do well at school or at a university are the fastest to count in their minds. If a lagging student is taught to quickly count in his mind, then this will certainly have a beneficial effect on his academic performance, and not only in natural, but also in all other subjects. This has been proven by practice.
- Arbitrary attention and interest during oral counting changes the wandering gaze of a lagging student to a fixed one, and the concentration of attention reaches several floors of the depth of the subject or process that is being studied.
- “The study of mathematics disciplines thinking, accustoms to the correct verbal expression of thoughts, to the accuracy, conciseness and clarity of speech, cultivates perseverance, the ability to achieve the intended goal, develops working capacity, and contributes to the correct self-assessment of mastering the subject that is being studied.” (Kudryavtsev L.D. - corresponding member of the Russian Academy of Sciences. 2006.).
- A student who has learned to quickly count in his mind, as a rule, begins to think faster.
- He who by his nature counts well will naturally discover the mind in any other science, and the one who thinks slowly, learning this art and mastering it, will be able to improve his mind, make it sharper (Plato).
- The acquired skills of oral counting will be enough for some for 5-10 years, and for others for life.
- It will be easier for our descendants to learn and gain knowledge. However, the culture of oral counting will always be an integral part of human culture.
- Those who quickly calculate in their minds tend to think clearly, perceive quickly, and see deeper.
- Mastering the CS develops figurative, diagrammatic and systemic thinking, expands the working memory, the range of perception, accustoms to thinking a few steps ahead, improves the quality of thinking, operating with the quantitative characteristics of objects.
- SS increases clarity of thinking, self-confidence, as well as strong-willed qualities (patience, perseverance, endurance, diligence). Accustoms to a deep and stable concentration of attention, conjecture and finishing the started phrases (especially for preschoolers and primary school students).
