We are taught counting skills from childhood. These are the elementary operations of addition, subtraction, multiplication and division. In the case of small numbers, even younger students can easily cope with them, but the task becomes much more complicated when you need to perform an action with a two-digit or three-digit number. However, with the help of training, simple exercises and little tricks, it is quite possible to subordinate these operations to quick mental processing.
You may ask why this is necessary, because there is such a handy thing as a calculator, and in extreme cases, there is always paper at hand for making calculations. Quick mental arithmetic has many advantages:
Opportunity to address other aspects of the problem. Often, tasks contain at least two sides: purely arithmetic (operations with numbers) and intellectual and creative (choosing an appropriate solution for a specific task, a non-standard approach for a faster solution, etc.). If a student does not cope well and quickly with the first side, then the second side suffers from this: concentrating on the implementation of the arithmetic component, the child does not think about the meaning of the task, may not see a catch or a simpler solution. If the counting operations are brought to automatism or simply do not require a lot of time, then a detailed consideration of the meaning of the task “turns on”, it becomes possible to apply a creative approach to it.
Intelligence training. Accounting in the mind allows you to keep your intellect in good shape, constantly engage thought processes. This is especially true for operations with large numbers, when we select a method to simplify the operation as much as possible.
Table exercises
The exercises are designed for children of any age who have difficulty performing operations with prime numbers (single and double digits). Allows you to train the skills of oral counting, to bring simple arithmetic operations to automaticity.
Necessary materials: to complete the exercises, you will need a grid of one- and two-digit numbers. Example:
The first column contains the numbers with which you need to perform actions. In the second - the answers to these actions. Using a specially cut bookmark, you can check the correctness of the calculation. For example:
Exercise options:
Sequentially add in your mind the pairs of numbers in the grid. Say the answer out loud and check yourself with the second column and bookmark. The task can be performed at a free pace or for a while.
Sequentially subtract the numbers in your mind from the grid.
Sequentially add in your mind the pairs of numbers in the grid. Add the number 5 to each sum and say the answer aloud.
Sequentially put together in your mind the triplets of numbers in the grid.
Consistently with all the numbers in the grid, do the following: add the bottom number, subtract the next number in the column from the resulting amount.
On the basis of such tables, any tasks can be formed. Grids are compiled depending on the modification of the exercise.
IMPORTANT! For the exercise to give results, it must be performed regularly, until the skill is fully mastered.
Mastering multiplication
The exercise is intended for children who have mastered the multiplication table from 1 to 10. It trains the skill of multiplying a two-digit number by a one-digit number.
A column is made up of arbitrary two-digit numbers. Task for the child: successively multiply these numbers first by 1, then by 2, by 3, etc. The answer is spoken aloud. It is executed until the answers are remembered and will not be issued automatically.
The main thing is attention
Exercise: add the numbers in sequence: 3000 + 2000+ 30 + 2000 + 10 + 20 + 1000 + 10 + 1000 + 30 =
Name the answer. Check yourself with a calculator.
If the answer turned out to be correct, it is necessary to consolidate the success and solve several more similar examples (they can be compiled arbitrarily). If there was an error in the answer, you need to return to the sequence of numbers and correct it.
What is the idea: As a result of adding numbers, the sum is 9100. But if you do this inattentively, the answer 10000 will automatically come up (the brain tends to round the amount, to make the answer more beautiful). Therefore, it is very important to maintain control over your actions when performing arithmetic problems in several actions.
Possible examples:
3000 – 700 — 60 – 500 — 40 – 300 -20 – 100 =
100:2:2*3*2 + 50 – 100 + 200 – 30 =
If most of the examples are solved with errors (BUT! not related to the ability to count in principle), then it makes sense to increase the concentration of attention. For this you can:
Minimize external stimuli. For example, if possible, go to another room, turn off the music, close the window, etc. If you need to focus on an example during a lesson, when there is no way to go out and achieve complete silence, you need to close your eyes and imagine the numbers with which the actions are carried out.
Add an element of contention. Knowing that a correct and quick decision will bring victory over the opponent and / or some kind of encouragement, the student is more willing to focus on the numbers and make maximum efforts in the calculation process.
Set personal records. You can visualize all the mistakes made by the student in the calculation process. For example, draw a flower with large petals (the number of petals = the number of solved examples). As many petals will be painted black as the number of examples was solved with errors. The task is to reduce the number of black petals as much as possible, setting personal records with each set of examples.
Grouping. Sequentially adding / subtracting several numbers, you need to see which of them, when added / subtracted, will give an integer: 13 and 67, 98 and 32, 49 and 11, etc. First, perform actions with these numbers, and then move on to the rest. Example: 7+65+43+82+64+28=(7+43)+(82+28)+65+64=50+110+124=289
Decomposition into tens and ones. When multiplying two two-digit numbers (for example, 24 and 57), it is advantageous to decompose one of them (ending in a smaller digit) into tens and ones: 24 as 20 and 4. The second number is multiplied first by tens (57 by 20), then by units ( 57 by 4). Then both values are added. Example: 24×57=57×20+57×4=1140+228=1368
Multiply by 5. When multiplying any number by 5, it is more profitable to first multiply it by 10, and then divide by 2. Example: 45×5=45×10/2=450/2=225
Multiply by 4 and 8. When multiplying by 4, it is more profitable to multiply the number twice by 2; by 8 - three times by 2. Example: 63x4=63x2x2=126x2=252
Division by 4 and 8. Similar to multiplication: when dividing by 4, divide the number twice by 2, by 8 - three times by 2. Example: 192/8=192/2/2/2=96/2/2=48/2=24
Squaring numbers ending in 5. The following algorithm will facilitate this action: the number of tens, the squared number, is multiplied by the same plus one and is attributed at the end to 25. Example: 75^2=7x(7+1)=7×8=5625
Formula multiplication. In some cases, to facilitate the calculation, you can apply the difference of squares formula: (a+b)x(a-b)=a^2-b^2. Example: 52×48=(50+2)x(50-2)=50^2-2^2=2500-4=2496
P.S. These rules can greatly simplify mental counting, but regular training is necessary so that you can correctly use the rule at the right time. Therefore, it is recommended to solve such a number of examples for each of them, which will allow you to automate the skill. To begin with, you can write down the calculations on paper, gradually reducing the amount of writing and translating operations into a mental plan. At first, it is also recommended to check your answers with a calculator or standard calculations in a column.
In the age of cash registers and calculators, people are less and less counting in their heads. They have almost completely switched to computer technology, but it often fails, or it simply will not be there when it is needed. We imperceptibly lose the skills of accurate and fast counting, and sometimes belatedly realize that we are no longer so good at this business. But, quickly counting in the mind is an undeniable advantage and advantage. A person who easily operates with numbers will almost never be deceived in calculations. But the important thing is that it will develop and keep in shape mental abilities, which is important for children and young people.
How to learn to quickly count in the mind of a child
All skills are best developed and reinforced in childhood. You can learn to count, as well as to read, from 1.5-2 years old. The peculiarities of this age are that the child will first accumulate passive knowledge - he will understand, know, but because of the small vocabulary, he will speak little. Up to five years old, a baby can learn to perform simple actions in his mind - subtraction and addition within twenty. If at two or three and a half years old you use visual methods in teaching, then later the baby will be able to operate only with numbers, without reinforcement with visual material.
If you want your child to have more chances that the process of operating with large values and mathematical operations will be easier and faster, then you need to teach him to count as early as possible.
It is better to teach children under four years of age with visual materials. You can count whatever you want. Fire trucks rushing to a fire, motorcyclists roaring past you, cats basking in the sun, flocks of birds - everything around you can be counted. With counting skills, observation and attention will simultaneously develop. Gradually increase the load. In the morning you saw 2 cats, and when you returned home, 3 more. Ask your child: “Did he notice that there are so many cats today! How much did he notice? Praise him for his accuracy and observation, because these qualities will be useful to him in life.
In elementary school, the kid needs to quickly and freely make any calculations within the limits defined by the school curriculum. To learn how to count quickly, constant training is needed. Therefore, the task of parents is to encourage the baby to count and make it interesting. The more often your child trains, the easier it will be for him to make accurate and quick calculations in his mind.
How to learn to count quickly as an adult
If a child has been trained in quick counting since childhood, then over time he will operate with large values without much effort. But if a person of a more mature age or a student decides to master a quick account, then it is necessary to apply a simple technique that will undoubtedly bring positive results.
Every learning starts small. If you know the multiplication table, that's great. If you forgot, or never knew, you should use this method of counting. For example, you need to find out how much 8x6 will be. We write the example like this:
What happens when a dog licks its face
How to behave if you are surrounded by boors
Ten habits that make people chronically unhappy
2 4
—-=48
8x6
Answer 48. We got it by writing an 8x6 example, drew a straight line over it and wrote down over each digit how much is missing to 10. Over 8 we write 2, on 6 we write 4. The first digit of the answer is the difference between the numbers in the lower and upper rows diagonally. 8-4=4, 6-2=4 - you can take any pair for calculation - the answer will always be the same. So we realized that the first digit is 4. Now let's find the second one. To do this, multiply the numbers of the top row 2x4 = 8. Our example is solved: 8x6=48.
Larger numbers are considered slightly different. For example, you need to calculate 11x13.
1 3
——=140+3=143
11x13
In the bottom line we write an example 11x13. At the top we write how much these numbers exceed 10. We get 1 and 3. Add the numbers diagonally. We get 11+3=14, 13+1=14. We got 14 tens, since the original numbers exceed 10. Therefore, we multiply 14 by 10. 14x10 \u003d 140. It remains only to multiply the upper numbers 1x3 \u003d 3 and add the resulting figure to the answer.
Such calculation methods are difficult to carry out only at first. Therefore, start with simple examples and gradually complicate. But in order to learn to count in your mind, you must completely get rid of the notes, and do everything in your head.
Children can also be taught in this way, but only when they fully know the school curriculum. Otherwise, you will not achieve positive results, but only harm the assimilation of school knowledge.
When you master the manipulation of two-digit numbers, you can move on to calculating multi-digit numbers - hundreds and even thousands.
Video lessons
December 23, 2013 at 03:10 pm
Effective counting in the mind or warm-up for the brain
- Maths
This article was inspired by the topic and is intended to spread the techniques of S.A. Rachinsky for oral counting.
Rachinsky was a wonderful teacher who taught in rural schools in the 19th century and showed from his own experience that it is possible to develop the skill of fast mental counting. It wasn't much of a problem for his students to calculate a similar example in their minds:
Using round numbers
One of the most common methods of mental counting is that any number can be represented as the sum or difference of numbers, one or more of which is “round”:Because on the 10
, 100
, 1000
and other round numbers to multiply faster, in the mind you need to reduce everything to such simple operations as 18x100 or 36x10. Accordingly, it is easier to add by “splitting off” a round number, and then adding a “tail”: 1800 + 200 + 190
.
Another example:
31 x 29 = (30 + 1) x (30 - 1) = 30 x 30 - 1 x 1 = 900 - 1 = 899.
Simplify multiplication by division
When calculating mentally, it is more convenient to operate with a dividend and a divisor than with an integer (for example, 5 present in the form 10:2 , a 50 as 100:2 ):68 x 50 = (68 x 100) : 2 = 6800: 2 = 3400; 3400: 50 = (3400 x 2) : 100 = 6800: 100 = 68.
Similarly, multiplication or division by 25 , after all 25 = 100:4 . For example,
600: 25 = (600: 100) x 4 = 6 x 4 = 24; 24 x 25 = (24 x 100) : 4 = 2400: 4 = 600.
Now it doesn't seem impossible to multiply in the mind 625 on the 53 :
625 x 53 = 625 x 50 + 625 x 3 = (625 x 100) : 2 + 600 x 3 + 25 x 3 = (625 x 100) : 2 + 1800 + (20 + 5) x 3 = = (60000 + 2500): 2 + 1800 + 60 + 15 = 30000 + 1250 + 1800 + 50 + 25 = 33000 + 50 + 50 + 25 = 33125.
Squaring a two-digit number
It turns out that to simply square any two-digit number, it is enough to remember the squares of all numbers from 1 before 25 . Good, squares up 10 we already know from the multiplication table. The remaining squares can be seen in the table below:Reception Rachinsky is as follows. In order to find the square of any two-digit number, you need the difference between this number and 25
multiply by 100
and to the resulting product add the square of the complement of the given number to 50
or the square of its excess over 50
-Yu. For example,
37^2 = 12 x 100 + 13^2 = 1200 + 169 = 1369; 84^2 = 59 x 100 + 34^2 = 5900 + 9 x 100 + 16^2 = 6800 + 256 = 7056;
In general ( M- two-digit number):
Let's try to apply this trick when squaring a three-digit number, first breaking it into smaller terms:
195^2 = (100 + 95)^2 = 10000 + 2 x 100 x 95 + 95^2 = 10000 + 9500 x 2 + 70 x 100 + 45^2 = 10000 + (90+5) x 2 x 100 + + 7000 + 20 x 100 + 5^2 = 17000 + 19000 + 2000 + 25 = 38025.
Hmm, I wouldn't say it's much easier than stacking, but maybe you can get used to it with time.
And, of course, you should start training with squaring two-digit numbers, and there you can already reach disassembly in your mind.
Multiplication of two-digit numbers
This interesting technique was invented by a 12-year-old student of Rachinsky and is one of the options for adding up to a round number.Let two two-digit numbers be given, in which the sum of units is equal to 10:
M = 10m + n, K = 10a + 10 - n.
Compiling their product, we get:
For example, let's calculate 77x13. The sum of the units of these numbers is equal to 10
, because 7 + 3 = 10
. First put the smaller number in front of the larger one: 77 x 13 = 13 x 77.
To get round numbers, we take three units from 13
and add them to 77
. Now let's multiply the new numbers 80x10, and to the result we add the product of the selected 3
units to the difference of the old number 77
and a new number 10
:
13 x 77 = 10 x 80 + 3 x (77 - 10) = 800 + 3 x 67 = 800 + 3 x (60 + 7) = 800 + 3 x 60 + 3 x 7 = 800 + 180 + 21 = 800 + 201 = 1001.
This technique has a special case: everything is greatly simplified when two factors have the same number of tens. In this case, the number of tens is multiplied by the number following it, and the product of the units of these numbers is attributed to the result. Let's see how elegant this technique is with an example.
48x42. Number of tens 4
, the next number: 5
; 4 x 5 = 20
. Product of units: 8x2= 16
. So 48 x 42 = 2016.
99x91. Number of tens: 9
, the next number: 10
; 9 x 10 = 90
. Product of units: 9 x 1 = 09
. So 99 x 91 = 9009.
Yeah, that is, to multiply 95x95, it is enough to calculate 9 x 10 = 90 and 5 x 5 = 25 and the answer is ready:
95 x 95 = 9025.
Then the previous example can be calculated a little easier:
195^2 = (100 + 95)^2 = 10000 + 2 x 100 x 95 + 95^2 = 10000 + 9500 x 2 + 9025 = 10000 + (90+5) x 2 x 100 + 9000 + 25 = = 10000 + 19000 + 1000 + 8000 + 25 = 38025.
Instead of a conclusion
It would seem, why be able to count in the mind in the 21st century, when you can simply give a voice command to your smartphone? But if you think about it, what will happen to humanity if it puts on machines not only physical work, but also any mental work? Is it degrading? Even if you do not consider mental counting as an end in itself, it is quite suitable for tempering the mind.References:
“1001 tasks for mental arithmetic at the school of S.A. Rachinsky.
Can't imagine your life without a calculator anymore? Very in vain, scientists have proven that people who regularly count in their minds are insured against senile insanity and early dementia. So practice more often, and I will tell you some simple tricks for easy and fast mental counting.
1. Multiply by 11
We all know how to quickly multiply a number by 10, you just need to add a zero at the end, but did you know that there is a trick on how to easily multiply a two-digit number by 11?
Let's say we need to multiply 63 by 11. Take a two-digit number that needs to be multiplied by 11 and imagine a place between its two digits:
6_3
Now add the first and second digits of this number and place in this location:
6_(6+3)_3
And our multiplication result is ready:
63*11=693
If the result of adding the first and second digits is a two-digit number, insert only the second digit, and add one to the first digit of the original number:
79*11=
7_(7+9)_9
(7+1)_6_9
79*11=869
2. Fast squaring of a number ending in 5
If you need to frame a two-digit number ending in 5, then you can do it very simply in your mind. Multiply the first digit of the number by itself plus one and add 25 at the end and that's it:
45*45=4*(4+1)_25=2025
3. Multiply by 5
For most people, multiplying by 5 is easy for small numbers, but how do you quickly mentally count large numbers multiplied by 5?
You need to take this number and divide by 2. If the result is an integer, then add 0 at the end to it, if not, discard the remainder and add 5 at the end:
1248*5=(1248/2)_(0 or 5)=624_(0 or 5)=6240 (the result of dividing by 2 is an integer)
4469*5=(4469/2)_(0 or 5)=(2234.5)_(0 or 5)=22345 (result of dividing by 2 with remainder)
4. Multiply by 4
This is a very simple and, at first glance, obvious feature of multiplying any number by 4, but despite this, people do not know about it at the right time. To simply multiply any number by 4, you need to multiply it by 2, and then multiply by 2 again:
67*4=67*2*2=134*2=268
5. Calculate 15%
If you need to mentally calculate 15% of any number, then there is an easy way to do it. Take 10% of the number (dividing the number by 10) and add half of the resulting 10% to that number.
15% of 884 rubles \u003d (10% of 884 rubles) + ((10% of 884 rubles) / 2) \u003d 88.4 rubles + 44.2 rubles \u003d 132.6 rubles
6. Multiplication of large numbers
If you need to multiply large numbers in your mind and one of them is even, then you can use the method of simplifying the factors by reducing the even number by half, and the second by doubling:
32*125 is
16*250 is
8*500 is
4*1000=4000
7. Divide by 5
Dividing a large number by 5 in your head is very easy. All you need to do is multiply the number by 2 and move the decimal point back by one:
175/5
Multiply by 2: 175*2=350
Shift by one sign: 35.0 or 35
1244/5
Multiply by 2: 1244*2=2488
Shift by one sign: 248.8
8. Subtraction from 1000
To subtract a large number from a thousand, follow a simple technique, subtract all digits from 9 except the last, and subtract the last digit from 10:
1000-489=(9-4)_(9-8)_(10-9)=511
Of course, in order to learn how to quickly count in your mind, you need to practice using these techniques many times in order to bring them to automatism, a single reading will leave only zeros in your head.
In the age of modern technology with many progressive gadgets, counting in the mind still has not lost its relevance. Today, it is far from uncommon when, in order to add or multiply the simplest numbers, a person reaches for a phone or a calculator so as not to strain too much. And this is completely wrong!
Regular exercises of the mind, and, as you know, counting is also included there, increase a person’s quick wit and level of intelligence, which, in the future, affects his whole life. Such people navigate much faster in various situations, at least they are more difficult to cheat in a store or in the market, which is already a nice bonus of such an ability.
I must say that people who can count quickly in their minds are not necessarily some kind of genius or possessing special abilities, it's all about years of practice, as well as the knowledge of some tricky tricks, which we will talk about later. Often and acutely, such a question arises when it is necessary to teach a schoolchild to count: as parents notice, the child does not know how to count in his mind, but on paper - quite, please.
If the age is very young, then problems may arise on paper, as how to learn how to quickly count in your mind? It all depends on age: it is not for nothing that they say that everything has its time, it is in childhood that it is very important to develop the skills of correct and quick counting.
How to teach a child?
Many parents wonder at what age should they start teaching counting? The earlier the better! Usually the first interest is shown in children at the age of 5-6 years, and sometimes even earlier, the main thing is not to miss and start developing. Count everything that comes to your mind - birds on a branch, cars in a parking lot, people on a bench or flowers in a garden. You can count your favorite toys, be sure to get developing sets of cubes with numbers, rearrange, carry out the first addition and subtraction operations using a visual example.
In general, in childhood, everything should resemble a game: for example, there is a wonderful developmental “gnomes in the house”. Think of a cardboard box - it will be a house. Take a few cubes - explain to the child that these are gnomes. Place one gnome in the house and say - "one gnome came to the house." Now you need to ask the child, if another one comes to visit the gnome, then how many gnomes will be in the house now?
Do not expect the correct answers right away, but as soon as you hear the correct one, place the required number of cubes in the box so that the child not only in his mind, but also visually sees the real result of the action. These are the first ways to develop in a child the ability to count in the mind.
How to learn to count in the mind at an older age?
Of course, you can’t lure schoolchildren and adults with games, and there is no need for this either. At an older age, the main thing is practice. The more a person exercises, the easier it will be for him to give the correct answers. The second point is the perfect knowledge of the multiplication table by heart.
It may seem to you that this is stupid advice, who does not know the simplest table? Believe me, anything can happen. And third - forget about the existence of auxiliary gadgets, they can only be used to check the results.
It is impossible to learn how to quickly count in your mind at the behest of a magic wand, you still have to work hard: at least remember special formulas that greatly simplify such a calculation. Secondly, learn to concentrate your attention: after all, when calculating, you will have to keep in mind complex numbers, as well as their combinations.
Multiply by 11
There are several options for how to quickly and simply multiply a number by 11. So, we will immediately show the first method with an example:
At the first stage, you need to add the numbers of the first multiplier, that is, 6 + 3 = 9. The next step is to place the result obtained between the first and last number of the multiplier, that is, 6(9)3. Here is the result!
Method number 2. Let's look at other numbers:
At the first stage, we again add the components of the multiplier: 6+9=15. What if the result is double digit? It's simple: move the unit to the left, (6 + 1) _ leave 5_ in the center and add 9. As a result of the formula, it turns out: 7_5_9 = 759.
Multiply by 5
The multiplication table “by 5” is easy to remember, but when it comes to complex numbers, it’s not so easy to count. And here there is a trick: any number that you want to multiply by five, just divide in half. Add zero to the result, but if the result of division is a fractional number, then simply remove the comma. It always works, check with an example:
Parse: 4568/2=2284
We add 0 to 2284 and get 22840. If you don't believe me, check it out for yourself!
Multiplying two complex numbers
If you need to mentally multiply two complex numbers, one of which is even, then you can also use an interesting formula:
48x125 is the same as:
24x250 is the same as:
12x500 is the same as:
Adding complex natural numbers in your mind
One interesting rule applies here: if one of the terms is increased by some number, then the same number must be subtracted from the result. For example:
550+348=(550+348+2)-2=(550+350)-2=898
There are a lot of such tricks and interesting formulas that greatly simplify mental counting, if you are interested, then many examples can always be found on the Internet. But in order to really get results, it is very important to practice a lot, so examples will help you!
