Calculus 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.
- RS is needed not only for schoolchildren, but also for engineers, teachers, medical workers, scientists and leaders different 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 find intelligence in any other science, and he who calculates 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 several moves 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 students primary school).
A convenient and multifunctional application for android that will help users learn how to quickly make calculations. This free program has a wide range of various tests and tasks that will improve your skills. In each type of exercise, you can choose the difficulty, which will allow you to gain experience gradually. Doing these exercises daily will greatly improve your skills, and soon you will be able to quickly count in your mind.
Functional:
- This android program has a variety of parameters and settings for difficulty, time and reminders. You can create the necessary schedule to stick to it, and the software will automatically remind you to complete the task. It is very convenient and you won't miss your workouts. If you wish, you can always view the statistics, which will indicate the number of examples already solved, their percentage, the number of visits, and much more.
Control:
- Management in the android program is very simple, intuitive. First you need to choose the complexity of the examples, the duration of the training, as well as the direction of mathematical operations of interest. Thus, exercises will be selected as close as possible to the required ones.
Relevance:
- a useful application for students, and not only. Indeed, at any age there are gaps in the calculations. Even if you do not have them, this application will increase the speed of making calculations. A trifle, but nice, and very useful in Everyday life.
Decor:
- The application has a light design, with a large font. All menu items are medium in size, which makes them comfortable to use. The tasks will be displayed at the top of the screen, and you will need to quickly enter the correct answer. At the end of the task, a report will be displayed with detailed information.
Peculiarities:
Simple control
Common Math Functions
Convenient interface
Detailed information on the session
Conclusion:
- a convenient simulator of mathematical calculations for android, in which each user can increase the speed of calculations in the mind and get detailed information about their progress.
The principle of operation is based on the generation of examples in mathematics of the level of complexity that suits you for all classes, the solution of which contributes to the development of mental counting skills.
The application has a positive effect on mental activity 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.
All settings changes are immediately applied and you can immediately see how the new example will look in the column "For example". And when the selection of the desired characteristics is over, click on the button GO.
A bonus is the ability to download and later print " independent work» in PDF format, consisting of 26 examples of the corresponding mode, click on the icon Printer.
Counting Process
At the top there are 4 quick access buttons: to the main page of the site, to the user profile. It is also possible to enable/disable sound notifications or go to the Error and Tip Log.
You solve the given example, enter the answer using the on-screen keyboard, and press the CHECK button. If you find it difficult to answer, use the hint. After checking the result, you will see a message either about the correctly entered answer, or about an error.
If for any reason you want to reset your results, click on the "Reset result" icon on the right.
game form
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.
If the animation mode bothers you, you can turn it off on the settings page using the icon
Error log
At any moment of working with the simulator, you can go to the "Error log" section of the application by clicking on the corresponding icon at the top, or by scrolling down the page.
Here you can see your statistics (number of examples by category) for the last 24 hours and for the last mode.
And also see a list of errors and hints (maximum 6 pieces), or go to detailed statistics.
Additional Information
site domain + application section + encoding of this mode
For example: website/app/#12301
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.
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
“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 learning activities providing students with the ability to learn, the ability for self-development and self-improvement. The quality of knowledge assimilation is determined by the diversity and nature of species universal action. 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 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 of the study of mathematics and other academic disciplines, because in addition to the fact that calculations activate memory, attention, help rationally organize activities and significantly affect human development.
In everyday life, on 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 methods of 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, one can apply the law of simultaneous increase in equal number times one factor and a decrease in 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.
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 visual perception mathematical facts, improve spatial imagination.
In addition, rational counting in mathematics lessons plays an important role in increasing children's cognitive interest to the lessons of mathematics, as one of the most important motives for educational and cognitive activity, the development of the 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. "System quick count according to Trachtenberg "- 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
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 of the primary classes of the village schools XIX 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, which 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 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. Let us remember that in primary school village school in the pre-revolutionary Smolensk province (picture "Mental count"), the 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
The rule of division by 5 is even simpler. First, we multiply by 2, and then we divide the result 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 of "finger gymnastics" and the method of mental 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.
