The principle of operation is based on generating examples in mathematics of a suitable level of complexity for all classes, the solution of which contributes to the development of mental calculation skills.
The application has a beneficial 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 arithmetic simulator allows you to practice 4 well-known arithmetic operations at six difficulty levels.
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 different types of tasks: Example, Equation, Comparison.
This mode includes regular arithmetic examples in mathematics consisting of two or three numbers.
A mode in which the required number can be in any position.
A mode in which it is necessary to correctly place a comparison sign between the results of two examples.
All settings changes are immediately applied and you can immediately see what the new example will look like in the graph "For example". And when the selection of the desired characteristics is completed, click on the button GO.
A bonus is the ability to download and subsequently 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, user profile. It is also possible to enable/disable sound notifications or go to the Error and Hint Log.
You solve the given example, enter the answer using the on-screen keyboard, and click on 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 correct answer entered or about an error.
If for any reason you want to reset your results, click on the “Reset Result” icon.
Game form
The application also provides game animation “Fencer Battle”.
Depending on the correctness of the entered answer, one or another fencer strikes, pushing back his opponent. However, it is worth considering that every second of inaction the enemy crowds your player, and if you wait for a long time, he will pop up loss message.
This interface makes the process of solving mathematical examples more interesting, while also being a simple motivation for children.
If the animation mode bothers you, you can disable it on the settings page using the icon
Error log
At any time while 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 day 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 anyone to compete in solving arithmetic examples in mathematics, simply by passing them a link to the current mode.
Why is oral counting needed?
Mental arithmetic is an essential skill for people who work with numbers and money. At least that’s how it used to be; in the 21st century, everyone has small computing machines in their pockets called smartphones, and mental arithmetic has become a thing of the past.
But it can always happen that the smartphone sits down or lies in the car, in another room, in general, it will not be at hand. What to do in this case? Of course, you can run to get your phone, or you can just do the math in your head. Moreover, this can be done not only with single and double digit numbers, but even with three digit numbers.
With our advice you can add, subtract, multiply, divide, and also operate with percentages in your head.
The advantage of such calculations will be to charge the brain to keep it in good shape, and in some cases, you will be able to amaze others, especially the opposite sex. In general, get ready, now there will be a small warm-up for your gray matter!
Let's start with the simplest thing: mental addition.
The first thing you need to know to work with numbers in your head is accurately operate with numbers up to 10. When added up, it all comes down to manipulations with single digit numbers.
Common mistake:
Most people, when counting mentally, forget to move the unfortunate ten to the next place after addition. To prevent this from happening, we recommend using the “rely on ten” method. Its essence is that we mentally ask ourselves how much one of the terms is missing to 10 , and then add to 10 the difference remaining until the second term.
When the time comes to work with large numbers, then partitioning into the same categories mentioned above will come to your aid. Does everyone remember side-by-side addition? It's the same thing, only in your head.
What will this look like in practice? Let's say you have a task: add two numbers 1024 And 256 : Basically, what is 1024? 1000 + 20 +4. And 256 in turn: 200 + 50 + 6. Now we work by rank.
1024 + 256 = (1000 + 0) + (200 + 0) + (20 + 50) + (4 + 6) = 1000 + 200 + 70 + 10 = 1280 .
Subtraction in your head
With subtraction, the method is slightly different; you don’t need to split both numbers into digits; it will be enough to split the subtrahend. Suppose you decide to subtract 256 from 1024, what is the easiest way to do this? We break 128 into categories. 128=100 + 20 + 8. And now we produce subtraction.
1024 – 128 = 1024 – 100 – 20 – 8 = 924 – 20 – 8 = 904 – 8 = 796.
Multiplication in the mind
First, let's remember what multiplication is? This is the repetition of the addition operation a number of times. For example, if you want to find out the sum of five nines, then this means that you multiply 9 by 5.
9*5 = 9 + 9 + 9 + 9 + 9 = 45.
And in order to successfully multiply large numbers in your mind, you must first learn to accurately count the multiplication of single-digit ones, and here the good old multiplication table. You will not achieve any success in multiplying multi-digit numbers without it.
If you don’t remember the multiplication table by heart, we strongly recommend that you repeat it until it “bounces off your teeth.”
Multiplying multi-digit numbers by single-digit numbers
Have you memorized your multiplication tables? We move on to the next step, multiplying multi-digit numbers by single-digit numbers. Here, as with addition, division into categories comes to our aid. Let’s say we want to multiply 512 by 8. 512 is 500 + 10 + 2, and we multiply each of these elements by the eight we need:
512*8 = 500*8 + 10*8 + 2*8 = 4000 + 80 + 16 = 4096.
Multiplying a two-digit number by 11
Before learning how to multiply two-digit numbers by each other in your head, let's look at special cases. The first one will be multiply by 11.
Why is 11 such a special number, you ask. And the fact that when multiplying by it there is a trick: any two digit number, which you want to multiply by 11 will be calculated using the formula: ax*11 = a(a+x)x, where a is the first digit of a two-digit number, and x is the second digit. Difficult? Let's show by example.
- 11*11 = 1(1+1)1=121.
- 27*11 = 2(2+7)7=297.
- 37*11 = 3(3+7)7=407.
Multiplying by round numbers
Is multiplying by 11 easy? On round numbers Multiplying is even easier. It's like multiplying by a single-digit number with a zero added to the right. Examples:
- 373*300 = 373*3*100 = 111900.
- 172*80 = 172*8*10 = 13760.
Squaring a two-digit number with a 5 at the end
Have you rested on simple things? Let's make it more complicated. Squaring is multiplying a number by itself. Of course, multiplying 10 by 10 or 11 by 11 is not so difficult, but 45 by 45 will not work out right away. Fortunately, there is a trick here again.
The result of squaring will be equal to the product of the first digit of the number and the next one. The product ends with the square of the last digit. Again, we will show everything with examples.
- 75*75 = (7*8)(5*5) = 5625.
- 35*35 = (3*4)(5*5) = 1225.
- 45*45 = (4*5)(5*5) = 2025
Multiplying by two-digit numbers
The extravagant situations are over, now the most difficult part regarding multiplication. In fact, again, simple steps, of which there are just a little more.
Let's return to my favorite powers of two. And let's try to multiply 64 by 32. To do this, you need to reduce everything to multiplication using the methods described above, and then to addition.
64*32 = 64*30 + 64*2 = 1920 + 128 = 2048.
Tadam! Nothing complicated! Unfortunately, three-digit numbers are already more difficult to cope with mentally; here it is better to return to the achievements of technology.
Division in the mind
Division is the least favorite operation of almost all schoolchildren and students. Of course, when it comes to numbers up to one hundred, almost no one has any problems. The multiplication table will help, but what if we are talking about three or even four-digit numbers?
Dividing multi-digit numbers by single-digit numbers
In division there will always be our best friend, no, not a calculator, and the multiplication table. Let's say 6144 needs to be divided by 8. To do this, you need to imagine 6144 as the sum of the maximum convenient number for division and the remainder. 6144 = 5600 + 544. Now we perform the same operation with 544 = 480 + 64. And 64 is already conveniently divided by 8.
As a result, we get: 6144/8 = 5600/8 + 480/8 + 64/8 = 700 + 60 + 8 = 768.
Dividing multi-digit numbers into two-digit numbers
And here it is, the most difficult and complicated stage of this article. Usually, such things are rarely calculated in the mind; they resort to long division or a calculator. But if you don’t have a gadget at hand, or even a piece of paper with a pen, then your sharp mind is your last hope.
Let's immediately remember last digit rule. The rule states that the last digit when multiplying two multi-digit numbers is equal to the product of the last two digits of the factors. For example, let's hit the keyboard with our hand - 534153 and multiply this by another hand strike on the keyboard - 864324. In our minds, we count the product of the last digits: 3 * 4 = 12. That is, the last digit should be “2”. We check it on the calculator: 534153*864324 = 461681257572. Congratulations, everything came together! Let's remember this rule, it will be useful to us later.
Now let's move on to the task. Let's divide 4424 by 56.
The first thing to do is to decide within what framework our number will lie. Let's try to intuitively select the boundaries. Let it be 90. 90*56 = 5040. This is too much. Now 80. 56*80 = 4480. It’s already better, that is, our number will be less than 80, but more than 70. We will do the selection in this range!
And here the magnificent multiplication table and the same rule come to our aid. What number when multiplied by the last digit of 56, that is, by 6, gives the end result of 4? Two options suit us, it’s either 4 or 7. Let’s check both options
- 56*74 = 4144. Almost, but not quite.
- 56*79 = 4424. But this is the correct result. That is, 4424/56 = 79.
Unfortunately, all mental division methods are based on the fact that we know that we will get an integer answer, otherwise you will fail.
Working with percentages in your head
To work with percentages, you first need to understand what a “percentage” is.
Percentage is one hundredth of a number. From here you can draw convenient parallels that will simplify the calculation. 10% of the number is the original number divided by 10. And 50% of the number is half of the original number, that is, divided by 2. Based on this, you can make the following tricks for yourself:
- To find 5%, find 10% and divide by two.
- To find 15%, find 10% and then add 5%.
- To find 20%, find 10% and multiply by two.
- To find 25%, find 50% and divide by two.
- To find 60%, find 50% and add 10%.
- To find 75%, find 50% and then add 25%.
- To find 80%, find 20% and multiply by four.
We told you about the basic methods of working in your head with all the classical operations, now a few general tips so that they stick with you so that you can wake up in the middle of the night and ask: “What is 25% of 1024?”, and you immediately answer “ 256!”
- and went on to bed.
- Exercise every day.
- There are many apps for practicing mental counting, both on iOS and Android. Download and train with them.
If you liked our advice and want to get help from us in more serious matters, for example, do not hesitate to contact us. Our specialists are ready to help you by writing your coursework quickly and efficiently, so that in the end you receive an “excellent” grade.
Why do we need mental arithmetic if this is the 21st century, and all sorts of gadgets are capable of performing any arithmetic operations almost at lightning speed? You don’t even have to point your finger at your smartphone, but give a voice command and immediately receive the correct answer. Now this is successfully done even by elementary school students who are too lazy to divide, multiply, add and subtract on their own.
But this coin also has a flip side: scientists warn that if you don’t train, don’t overload with work and make tasks easier for him, he begins to be lazy and declines. In the same way, without physical training, our muscles weaken.
Mikhail Vasilyevich Lomonosov also spoke about the benefits of mathematics, calling it the most beautiful of sciences: “You have to love mathematics because it puts your mind in order.”
Oral arithmetic develops attention and reaction speed. It is not for nothing that more and more new methods of rapid mental calculation are appearing, intended for both children and adults. One of them is the Japanese mental counting system, which uses the ancient Japanese soroban abacus. The methodology itself was developed in Japan 25 years ago, and now it is successfully used in some of our mental counting schools. It uses visual images, each of which corresponds to a specific number. Such training develops the right hemisphere of the brain, which is responsible for spatial thinking, constructing analogies, etc.
It is curious that in just two years, students of such schools (they accept children aged 4–11 years) learn to perform arithmetic operations with 2-digit and even 3-digit numbers. Kids who don't know multiplication tables can multiply here. They add and subtract large numbers without writing them down. But, of course, the goal of training is the balanced development of the right and left.
You can also master mental arithmetic with the help of the problem book “1001 problems for mental arithmetic at school,” compiled back in the 19th century by a rural teacher and famous educator Sergei Aleksandrovich Rachinsky. This problem book is supported by the fact that it went through several editions. This book can be found and downloaded on the Internet.
People who practice quick counting recommend Yakov Trachtenberg’s book “The Quick Counting System.” The history of the creation of this system is very unusual. To survive the concentration camp where he was sent by the Nazis in 1941, and not lose his mental clarity, a Zurich mathematics professor began developing algorithms for mathematical operations that allow him to quickly count in his head. And after the war, he wrote a book in which the quick counting system is presented so clearly and accessiblely that it is still in demand.
There are also good reviews about Yakov Perelman’s book “Quick Counting. Thirty simple examples of mental counting." The chapters of this book are devoted to multiplying by single-digit and two-digit numbers, in particular multiplying by 4 and 8, 5 and 25, by 11/2, 11/4, *, dividing by 15, squaring, and formula calculations.
The simplest methods of mental counting
People who have certain abilities will master this skill faster, namely: the ability to think logically, the ability to concentrate and store several images in short-term memory at the same time.
No less important is knowledge of special action algorithms and some mathematical laws that allow, as well as the ability to choose the most effective one for a given situation.
And, of course, you can’t do without regular training!
Some of the most common quick counting techniques are:
1. Multiplying a two-digit number by a one-digit number
The easiest way to multiply a two-digit number by a single-digit number is to split it into two components. For example, 45 - by 40 and 5. Next, we multiply each component by the required number, for example, by 7, separately. We get: 40 × 7 = 280; 5 × 7 = 35. Then we add the resulting results: 280 + 35 = 315.
2. Multiplying a three-digit number
Multiplying a three-digit number in your head is also much easier if you break it down into its components, but present 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 we now have to 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. Next, we multiply by 5, first 130, and then 7, and add the results. That is, 137 × 5 = 130 × 5 + 7 × 5 = 650 + 35 = 685.
You can expand 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 multiply 470 by 3. Total 1410.
The same action can be done 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, by breaking down numbers into their components, you can perform addition, subtraction and division.
3. Multiplying by 10
Everyone knows how to multiply by 10: simply add zero to the multiplicand. For example, 15 × 10 = 150. Based on this, it is no less simple to multiply by 9. First, we add 0 to the multiplicand, that is, multiply it by 10, and then subtract the multiplicand from the resulting number: 150 × 9 = 150 × 10 = 1500 − 150 = 1,350.
4. Multiplication 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. Multiplying by 11
It’s interesting to multiply two-digit numbers by 11. Let’s take 18, for example. Let’s mentally expand 1 and 8, and between them write the sum of these numbers: 1 + 8. We get 1 (1 + 8) 8. Or 198.
6. Multiply by 1.5
If you need to multiply a 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 which we can train our brains in everyday life. For example, counting the cost of purchases while standing in line at the checkout. Or perform mathematical operations with numbers on the license plates of passing cars. Those who like to “play” with numbers and want to develop their thinking abilities can turn to the books of the above-mentioned authors.
Reading time: 11 minutes. Views 194 Published September 27, 2018
Many people ask how to learn to quickly count in their heads so that it looks unnoticeable and not stupid. After all, modern technologies allow us to use our memory and mental abilities less. But sometimes these technologies are not at hand and sometimes it is easier and faster to calculate something in your head. Many people have started counting even basic things on a calculator or phone, which is also not very good. The ability to count in your head remains a useful skill for modern man, despite the fact that he owns all kinds of devices that can count for him. The ability to do without special devices and quickly solve an arithmetic problem at the right time is not the only use of this skill. In addition to its utilitarian purpose, mental calculation techniques will allow you to learn how to organize yourself in various life situations. In addition, the ability to count in your head will undoubtedly have a positive impact on the image of your intellectual abilities and will distinguish you from the surrounding “humanists.”
Quick counting methods
There is a certain set of simple arithmetic rules and patterns that you not only need to know for mental calculation, but also constantly keep in mind in order to quickly apply the most effective algorithm at the right time. To do this, it is necessary to bring their use to automaticity, consolidate it in mechanical memory, so that from solving the simplest examples you can successfully move on to more complex arithmetic operations. Here are the basic algorithms that you need to know, remember and apply instantly, automatically:
Subtraction 7, 8, 9
To subtract 9 from any number, you need to subtract 10 from it and add 1. To subtract 8 from any number, you need to subtract 10 from it and add 2. To subtract 7 from any number, you need to subtract 10 from it and add 3. If usually If you think differently, then for a better result you need to get used to this new method.
Multiply by 9
You can quickly multiply any number by 9 using your fingers.
Dividing and multiplying by 4 and 8
Division (or multiplication) by 4 and 8 are double or triple division (or multiplication) by 2. It is convenient to perform these operations sequentially.
For example, 46*4=46*2*2 =92*2= 184.
Multiply by 5
Multiplying by 5 is very simple. Multiplying by 5 and dividing by 2 are practically the same thing. So 88*5=440, and 88/2=44, so always multiply by 5 by dividing the number by 2 and multiplying it by 10.
Multiply by 25
Multiplying by 25 is the same as dividing by 4 (followed by multiplying by 100). So 120*25 = 120/4*100=30*100=3000.
Multiplying by single digits
For example, let's multiply 83*7.
To do this, first multiply 8 by 7 (and add zero, since 8 is the tens place), and add to this number the product of 3 and 7. Thus, 83*7=80*7 +3*7= 560+21=581 .
Let's take a more complex example: 236*3.
So, we multiply the complex number by 3 bitwise: 200*3+30*3+6*3=600+90+18=708.
Defining Ranges
In order not to get confused in the algorithms and mistakenly give a completely wrong answer, it is important to be able to construct an approximate range of answers. Thus, multiplying single-digit numbers by each other can give a result of no more than 90 (9*9=81), two-digit numbers - no more than 10,000 (99*99=9801), three-digit numbers no more - 1,000,000 (999*999=998001).
Layout in tens and units
The method consists of dividing both factors into tens and ones and then multiplying the resulting four numbers. This method is quite simple, but requires the ability to hold up to three numbers in memory simultaneously and at the same time perform arithmetic operations in parallel.
For example:
63*85 = (60+3)*(80+5) = 60*80 + 60*5 +3*80 +3*5=4800+300+240+15=5355
Such examples can be easily solved in 3 steps:
1. First, tens are multiplied by each other.
2. Then add 2 products of units and tens.
3. Then the product of units is added.
This can be schematically described as follows:
— First action: 60*80 = 4800 — remember
— Second action: 60*5+3*80 = 540 – remember
— Third action: (4800+540)+3*5= 5355 – answer
For the fastest possible effect, you will need a good knowledge of the multiplication table for numbers up to 10, the ability to add numbers (up to three digits), as well as the ability to quickly switch attention from one action to another, keeping the previous result in mind. It is convenient to train the last skill by visualizing the arithmetic operations being performed, when you should imagine a picture of your solution, as well as intermediate results.
Mental visualization of columnar multiplication
56*67 – count in a column. Probably, counting in a column contains the maximum number of actions and requires constantly keeping auxiliary numbers in mind.
But it can be simplified:
First action: 56*7 = 350+42=392
Second action: 56*6=300+36=336 (or 392-56)
Third action: 336*10+392=3360+392=3,752
Private techniques for multiplying two-digit numbers up to 30
The advantage of the three methods of multiplying two-digit numbers for mental calculation is that they are universal for any numbers and, with good mental calculation skills, they can allow you to quickly come to the correct answer. However, the efficiency of multiplying some two-digit numbers in the head can be higher due to fewer steps when using special algorithms.
Multiplying by 11
To multiply any two-digit number by 11, you need to enter the sum of the first and second digits between the first and second digits of the number being multiplied.
For example: 23*11, write 2 and 3, and between them put the sum (2+3). Or in short, that 23*11= 2 (2+3) 3 = 253.
If the sum of the numbers in the center gives a result greater than 10, then add one to the first digit, and instead of the second digit we write the sum of the digits of the number being multiplied minus 10.
For example: 29*11 = 2 (2+9) 9 = 2 (11) 9 = 319.
You can quickly multiply by 11 orally not only two-digit numbers, but also any other numbers.
For example: 324 * 11=3(3+2)(2+4)4=3564
Squared sum, squared difference
To square a two-digit number, you can use the squared sum or squared difference formulas. For example:
23²= (20+3)2 = 202 + 2*3*20 + 32 = 400+120+9 = 529
69² = (70-1)2 = 702 – 70*2*1 + 12 = 4,900-140+1 = 4,761
Squaring numbers ending in 5. To square numbers ending in 5. The algorithm is simple. The number up to the last five, multiply by the same number plus one. Add 25 to the remaining number.
25² = (2*(2+1)) 25 = 625
85² = (8*(8+1)) 25 = 7,225
This is also true for more complex examples:
155² = (15*(15+1)) 25 = (15*16)25 = 24,025
The technique for multiplying numbers up to 20 is very simple:
16*18 = (16+8)*10+6*8 = 288
Proving the correctness of this method is simple: 16*18 = (10+6)*(10+8) = 10*10+10*6+10*8+6*8 = 10*(10+6+8) +6*8. The last expression is a demonstration of the method described above. Essentially, this method is a special way of using reference numbers. In this case, the reference number is 10. In the last expression of the proof, we can see that it is by 10 that we multiply the bracket. But any other numbers can be used as a reference number, the most convenient of which are 20, 25, 50, 100...
Reference number
Look at the essence of this method using the example of multiplying 15 and 18. Here it is convenient to use the reference number 10. 15 is greater than ten by 5, and 18 is greater than ten by 8.
In order to find out their product, you need to perform the following operations:
1. To any of the factors add the number by which the second factor is greater than the reference one. That is, add 8 to 15, or 5 to 18. In the first and second cases, the result is the same: 23.
2. Then we multiply 23 by the reference number, that is, by 10. Answer: 230
3. To 230 we add the product 5*8. Answer: 270.
The reference number when multiplying numbers up to 100. The most popular technique for multiplying large numbers in the mind is the technique of using the so-called reference number
Reference number for multiplication– this is the number to which both factors are close and by which it is convenient to multiply. When multiplying numbers up to 100 with reference numbers, it is convenient to use all numbers that are multiples of 10, and especially 10, 20, 50 and 100.
The technique for using the reference number depends on whether the factors are greater than or less than the reference number. There are three possible cases here. We will show all 3 methods with examples.
Both numbers are less than the reference (below the reference). Let's say we want to multiply 48 by 47.
These numbers are close enough to the number 50, and therefore it is convenient to use 50 as a reference number.
To multiply 48 by 47 using the reference number 50:
1. From 47, subtract as much as 48 is missing to 50, that is, 2. It turns out 45 (or
subtract 3 from 48 – it’s always the same)
2. Next we multiply 45 by 50 = 2250
3. Then add 2*3 to this result - 2,256
50 (reference number)
3(50-47) 2(50-48)
(47-2)*50+2*3=2250+6=2256
If the numbers are less than the reference number, then from the first factor we subtract the difference between the reference number and the second factor. If the numbers are greater than the reference number, then to the first factor we add the difference between the reference number and the second factor.
50(reference number)
(51+13)*50+(13*1)=3200+13=3213
One number is below the reference, and the other is above. The third case of using a reference number is when one number is greater than the reference number and the other is less. Such examples are no more difficult to solve than the previous ones. We increase the smaller factor by the difference between the second factor and the reference number, multiply the result by the reference number and subtract the product of the differences between the reference number and the factors. Or we reduce the larger factor by the difference between the second factor and the reference number, multiply the result by the reference number and subtract the product of the differences between the reference number and the factors.
50(reference number)
5(50-45) 2(52-50)
(52-5)*50-5*2=47*50-10=2340 or (45+2)*50-5*2=47*50-10=2340
When multiplying two-digit numbers from different tens, it is more convenient to use a reference number
take a round number that is greater than the larger factor.
90(reference number)
63 (90-27) 1 (90-89)
(89-63)*90+63*1=2340+63=2403
Thus, by using a single reference number, it is possible to multiply a large combination of two-digit numbers. The methods described above can be divided into universal (suitable for any numbers) and specific (convenient for specific cases).
As a last resort, you can use a “peasant” account. To multiply one number by another, say 21*75, we need to write the numbers in two columns. The first number in the left column is 21, the first number in the right column is 75. Then divide the numbers in the left column by 2 and discard the remainder until we get one, and multiply the numbers in the right column by 2. Cross out all lines with even numbers in the left column, and we add up the remaining numbers in the right column, we get the exact result.
Conclusion
Like all calculation methods, these fast calculation methods have their advantages and disadvantages:
PROS:
1.With the help of various methods of fast calculations, even the least educated person can count.
2. Quick counting methods can help get rid of a complex action by replacing it with several simpler ones.
3.Quick counting methods are useful in situations where columnar multiplication cannot be used.
4. Fast counting methods can reduce calculation time.
5. Mental arithmetic develops mental activity, which helps to quickly navigate difficult life situations.
6. The mental calculation technique makes the calculation process more fun and interesting.
MINUSES:
1. Often, solving an example using quick calculation methods turns out to be longer than simply multiplying by column, since you have to perform a larger number of actions, each of which is simpler than the original one.
2. There are situations when a person, out of excitement or something else, forgets the methods of quick counting or even gets confused in them; in such cases, the answer is incorrect, and the methods are actually useless.
3.Quick counting methods have not been developed for all cases.
4. When calculating using the quick counting technique, you need to keep many answers in your head, which can cause you to get confused and come to an erroneous result.
Undoubtedly, practice plays a vital role in the development of any ability. But the skill of mental calculation does not rely on experience alone. This is proven by people who are able to count complex examples in their heads. For example, such people can multiply and divide three-digit numbers, perform arithmetic operations that not every person can count in a column. What does an ordinary person need to know and be able to do in order to master such a phenomenal ability? Today, there are various techniques that help you learn how to quickly count in your head.
Having studied many approaches to teaching the skill of counting orally, we can highlight 3 main components of this skill:
1. Abilities. The ability to concentrate and the ability to hold several things in short-term memory at the same time. Predisposition to mathematics and logical thinking.
2. Algorithms. Knowledge of special algorithms and the ability to quickly select the necessary, most effective algorithm in each specific situation.
3. Training and experience, the importance of which for any skill has not been canceled. Constant training and gradual complication of solved problems and exercises will allow you to improve the speed and quality of mental calculation. It should be noted that the third factor is of key importance. Without the necessary experience, you will not be able to surprise others with a quick score, even if you know the most convenient algorithm. However, do not underestimate the importance of the first two components, since having in your arsenal the abilities and a set of the necessary algorithms, you can surprise even the most experienced “accountant”, provided that you have trained for the same amount of time.
Do you think well? What if you need to quickly add, subtract or divide three-digit numbers? What if it's four digits? Some children carry out these mental operations in a matter of seconds. Do you think they are child prodigies? Not at all. They are simply very familiar with mental arithmetic. Teacher Marina Brezovskaya told us what the secret of this system is.
Marina Brezovskaya
teacher of mental arithmetic at the children's development center "Lesenka",
Bereza
Children use imaginary abacus
Look how easy this girl is with numbers! How is this even possible?
— Marina, tell us what mental arithmetic is?
— This is a technique that trains the speed of perception and processing of information, the only technique in the world that simultaneously develops both hemispheres of the brain. This happens primarily due to the combination of visualization and computational calculations.
The beginning of the existence of mental arithmetic can be considered the invention of the counting board (suanpan) in China more than 5 thousand years ago. Those ancient abacus consisted of a tablet with special symbols and sand divided into lines.
A little later, similar devices for arithmetic calculations appeared in Egypt, Ancient Greece and Ancient Rome. They were more like modern abacus, since the counting was carried out on a board not with sand, but with stones or bones.
— Why is the girl fingering?
“She helps herself mentally move the pieces on the abacus. Now I will explain in more detail.
The main subject in mental arithmetic is an abacus called an abacus. First, we teach children to count on real abacuses that they can pick up, then instead we offer them a printed picture depicting these abacus.
At the final stage, students hold the imaginary abacus in their heads and simply imagine it. Mentally, the children move the bones on the rods in a certain way using the formulas they have learned. They use their fingers to help themselves so as not to get confused. A good teacher understands only by the movements of the students’ hands whether they are counting correctly or not.
The main thing is constant repetition
- Yes, definitely. With the help of mental arithmetic, not only counting speed is developed, but also concentration, analytical and creative thinking, observation, and memory. In addition, children gain self-confidence, determination, responsibility, and perceive and assimilate new information faster and easier.
Every child shows results. Mental arithmetic helps not only in mathematics. It promotes overall brain development. Therefore, some become successful in sports, some easily master foreign languages, some simply improve their performance at school and complete their homework faster.
— How long does one lesson last?
— Training usually takes place once a week, the lesson lasts 1.5 hours. Under the guidance of a teacher, children study and then work on a new topic, after which they consolidate it at home, honing their skills using an online simulator. Homework takes from 5 to 30 minutes. The time is selected individually for each child.
It is important to try not to skip short workouts at home. It is constant repetition that helps achieve the best results. This way, new interneuronal connections in the brain are strengthened faster.
Children count while reading poetry aloud
- With a good imagination - no. However, the problem of the modern generation is that most children find it difficult to hold a picture in their head for a long time, especially if it is constantly changing. That's why I say that, in addition to counting skills, we train imagination and the ability to retain information in our heads.
— Here the girl also recites a poem at the same time as counting? Is this even real?
- Yes, sure. Sometimes these are even poems or passages of prose in a foreign language. From the outside, this picture looks fantastic, but with regular training, anything is possible, believe me.
Sometimes the task becomes even more difficult. At the moment when the child is counting, the teacher asks him some questions. He must be able to add or subtract and answer these questions meaningfully. And everything works out!
Our brain is truly capable of performing several functions at the same time. A person is often simply lazy to develop these abilities.
— How large numbers can you manipulate in your head?
- Depends on how many rods and digits of abacus in your head you are able to mentally hold. Many students are good at counting four-digit numbers, but with great desire and perseverance, I think it is possible to work with even larger numbers. There is no limit to perfection.
At our center, children learn not only addition and subtraction. They also understand multiplication and division and can easily perform these operations on the abacus.
Adults find learning more difficult than children
- At what age should I start?
- Preferably from 5 years old.
— Aren’t such activities too much of a burden for the kids’ brains?
- No, our brain works constantly. But it needs to be developed. Mental arithmetic helps a lot with this.
In the modern world, where the flow of information is simply enormous, children just need to learn how to correctly analyze the data received. Just as when you exercise, your muscles are trained, so is your brain. The main thing is not to rush, increase the difficulty gradually.
— Isn’t it too late for adults to master mental arithmetic?
- Of course it’s not too late! I just warn you right away: it will be much more difficult for an adult. Children's thinking is more flexible. After all, it is easier for children to assimilate new information and imagine. But this does not mean that you don’t need to study. This is very useful for the brain, which has forgotten how to perform any functions other than the usual everyday ones.
A person will definitely notice positive changes: improved memory, concentration, sharpness of thinking, and so on. In general, I would highly recommend mental arithmetic for older people. This is excellent prevention.
— Does the skill last forever?
“Our memory is designed in such a way that without repetition, the acquired knowledge gradually fades away. The skill itself is unlikely to be completely forgotten, but in order to count accurately, you still need a certain regularity.
Let's talk about it
Oleg Smagin
psychologist, specialist in interpersonal communications and neuromarketing
— Is there any benefit from mental arithmetic? Undoubtedly! But - not for children.
For the elderly, fine motor skills of the 1st stage of mental arithmetic, the development of cognitive and thinking skills can really delay the onset of dementia. However, ordinary “Russian” abacus gives exactly the same effect. And learning foreign languages is even more important.
What are they promising us? They say that children will become more attentive, begin to concentrate better, systematize knowledge, adapt to new conditions and, thanks to all this, will be more successful in school.
How much of this is real? Psychologist David Barner conducted a study in India. Conclusions: thanks to this technique, some schoolchildren cope better with arithmetic operations, but the result depends on the child’s existing abilities, and not on “mental arithmetic” as a method.
American studies have shown that if there is a positive effect, it appears only in laboratory conditions or only in adults.
Targeted research on the “development of different areas of the brain” was carried out only in China and was funded, again, by centers promoting this project.
The child must develop. And his main task is to learn to interact with other people in society. Only after this can he gain knowledge that will help him be successful in a certain activity.
Studies conducted in different countries of the world have shown that children with emotional intelligence, who easily make contact and maintain it with other people, grow up to become prosperous and happy adults. Those who have not learned this are mostly outsiders. All tasks must be age appropriate.
Collective interaction and common play teach emotional intelligence. Knowledge acquired too early, especially to the detriment of games, extinguishes this intelligence.
Not all child prodigies necessarily become successful and happy... Maybe it’s worth thinking about how to develop a child in this regard, and not, following fashion, supporting the “mental arithmetic” business project?
Svetlana Leonova
mother of 7-year-old Sasha
— Sasha has been studying at the development center since she was 3 years old. When he was in the older group (4-5 years old), a new direction opened there - “mental arithmetic”. This technique was highly recommended to us by the teacher who taught Sashka’s school preparation classes. The son was inattentive, restless, quickly grasped, but it was impossible to maintain attention for a long time. I was afraid that we would have behavioral issues at school. This means that the child will not be comfortable in the classroom.
The teacher made an argument: mental arithmetic is the ability to concentrate: if you get distracted, miss one step out of 20, the example is not solved. With Sasha’s stubbornness and desire to win - just what you need!
At first, I somehow didn’t even delve into all these numbers (I myself am not of a mathematical mind at all). But when Sasha was recommended to go to the Olympiad based on the results of his studies and we began to prepare, I was very surprised: my son added two-digit numbers in his head within a hundred (he was 6)! Moreover, it was clear that he could do more. My child became the winner of the Republican Olympiad in one of the categories among preschoolers. And success is important for children.
During the first 3 months of training, a teacher at a music school and a behavior correction teacher-psychologist were pleased to report that Sasha began to concentrate, that the time he could spend on a task increased, and there were no questions at school.
I would recommend paying special attention to this direction for parents who endlessly hear about their baby: “How nimble he is, even too much…”, “He’s probably hyperactive...”. If you want your little one to have a calm and peaceful life in the school system, try focusing them with mental arithmetic. Explore for yourself. When I started helping my son study new topics, I noticed that I myself began to count better. I think I’ll start seriously in retirement so as not to let my brain dry out.
Maria Kamenetskaya
neuropsychologist, head of the Center for Practical Neuropsychology in Moscow
— Mental arithmetic (MA) is a popular area that allows you to automate the counting skill and increase its speed many times over. Some parents like it and rush to send their child to MA courses, others are suspicious, not understanding the principles and mechanisms, and are in no hurry to draw conclusions.
Let's try to figure it out.
The first advantage of MA is the automation of counting skills. Mental arithmetic is like learning to ride a bicycle again every time. Mental arithmetic is an automated counting, that is, the child will not waste energy on the counting operation, but will concentrate only on the condition of the task. At the same time, the brain’s counting mechanism itself shifts. If in the first case the child operates with symbols, then in MA he operates with visual images, shifting the localization of the process from the left to the right hemisphere.
The development of interhemispheric connections is also an absolute plus of the technique; the mechanics of working with fingers implies good reciprocal coordination.
The development of auditory-verbal and iconic memory in the MA method is achieved through working by ear and with flash cards.
If you decide to send your child to MA, know that classes must be regular, and it is very important to do homework, automating the skill. If this is not done, then the counting process will not be formed correctly either according to the classical scheme or according to the MA method, and it will be very difficult for the child.
Remember, counting in MA has a different brain basis than what we are used to, therefore, when sending a child to an MA club, you need to understand that this is not a replacement, but a complement to the process, learning to perform it in a different way, which requires long-term training.
There are certain restrictions when working with adults. The adult brain is not as plastic, so mental arithmetic is difficult, but counting on the abacus will be useful for brain development, as well as for maintaining its plasticity in later life.