Fast multiplication techniques. Multiplying two-digit numbers

“You should love mathematics because it puts your mind in order,” said Mikhail Lomonosov. 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.”

Mental counting training

There are people who can perform simple arithmetic operations in their heads. Multiply a two-digit number by a single-digit number, multiply within 20, multiply two small two-digit numbers, etc. - they can perform all these actions in their minds and quickly enough, faster than the average person. Often this skill is justified by the need for constant practical use. Typically, people who are good at mental arithmetic have a background in mathematics or at least experience solving numerous arithmetic problems.

Undoubtedly, experience and training play 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, unlike those described above, are able to count much more complex examples in their minds. For example, such people can multiply and divide three-digit numbers, perform complex 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 to count quickly 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.

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 necessary algorithms, you can “outdo” even the most experienced “accountant”, provided that you have trained for the same amount of time.

Lessons on the site

The mental arithmetic lessons presented on the site are aimed specifically at developing these three components. The first lesson tells you how to develop a predisposition for mathematics and arithmetic, and also describes the basics of counting and logic. Then a series of lessons is given on special algorithms for performing various arithmetic operations in the mind. Finally, this training provides additional materials to help train and develop the ability to count orally, in order to be able to apply your talent and knowledge in life.

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 primary schoolchildren can easily cope with them, but the task becomes significantly more complicated when you need to perform an operation with a two- or three-digit number. However, with the help of training, simple exercises and little tricks, it is quite possible to subordinate these operations to rapid mental processing.

You may ask why this is necessary, because there is such a convenient thing as a calculator, and in case of emergency there is always paper at hand to carry out calculations. Quick mental arithmetic has many benefits:

Opportunity to address other aspects of the task. Often tasks contain at least two sides: purely arithmetic (operations with numbers) and intellectual and creative (choosing an appropriate solution for a specific problem, 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 suffers from this: concentrating on completing the arithmetic component, the child does not think about the meaning of the problem, and may not see a catch or a simpler solution. If counting operations are brought to automaticity or simply do not require a lot of time, then a detailed consideration of the meaning of the problem is “turned on”, and it becomes possible to apply a creative approach to it.

Intelligence training. Mental arithmetic allows you to keep your intellect in good shape and constantly engage your mental processes. This is especially true for operations with large numbers, when we select a method to simplify the operation as much as possible.

Exercises with tables

The exercises are designed for children of any age who have difficulty performing operations with prime numbers (one- and two-digit). Allows you to train mental calculation skills and bring simple arithmetic operations to automation.

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. The second contains responses to these actions. Using a specially cut bookmark, you can check the correctness of the calculation. For example:

Exercise options:

Consistently add pairs of numbers in a grid in your mind. Say the answer out loud and test yourself using the second column and bookmark. The task can be completed at a free pace or against time.

Consistently subtract numbers from the grid in your head.

Consistently add pairs of numbers in a grid in your mind. Add the number 5 to each sum and say the answer out loud.

Consistently add up triplets of numbers in a grid in your mind.

Perform the following actions sequentially with all the numbers in the grid: add the bottom number, subtract the next number in the column from the resulting amount.

Based on such tables, you can create any tasks. Grids are compiled depending on the modification of the exercise.

IMPORTANT! For the exercise to be effective, 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 single-digit number.

A column is made up of arbitrary two-digit numbers. Task for the child: sequentially multiply these numbers, first by 1, then by 2, by 3, etc. The answer is spoken out loud. This is carried out until the answers are remembered and given automatically.

The main thing is attention

Exercise: add the numbers in sequence: 3000 + 2000+ 30 + 2000 + 10 + 20 + 1000 + 10 + 1000 + 30 =

State the answer. Test yourself with a calculator.

If the answer is correct, you need to consolidate your success and solve several more similar examples (can be compiled arbitrarily). If there was an error in the answer, you need to go back to the sequence of numbers and correct it.

What's the idea: As a result of adding the numbers, the sum is 9100. But if you do this inattentively, the answer 10000 will automatically appear (the brain tries to round the sum, to make the answer more beautiful). Therefore, it is very important to maintain control over your actions when performing arithmetic problems in several steps.

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 concentration. To do this you can:

Minimize external stimuli. For example, if possible, go into another room, turn off the music, close the window, etc. If you need to concentrate on an example during a lesson, when it is not possible 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 competition. Knowing that a correct and quick solution will bring victory over the opponent and/or some kind of encouragement, the student will be more willing to focus on the numbers and make maximum effort in the calculation process.

Set personal records. You can visualize all the mistakes made by the student during the calculation process. For example, draw a flower with large petals (number of petals = number of solved examples). As many petals will be painted black as the number of examples that were solved with errors. The goal is to reduce the number of black petals as much as possible, setting personal records with each batch of examples.

Grouping. By 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 units: 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 together. 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 it by 2. Example: 45×5=45×10/2=450/2=225

Multiplying 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: 63×4=63x2x2=126×2=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 make this action easier: the number of tens squared is multiplied by the same number plus one and added at the end to 25. Example: 75^2=7x(7+1)=7×8=5625

Multiplication by formula. In some cases, to make calculations easier, you can use 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 significantly simplify mental counting, but regular training is necessary so that you can use the rule correctly at the right time. Therefore, it is recommended to solve as many examples for each of them as will allow you to automate the skill. To begin with, you can write down calculations on paper, gradually reducing the amount of writing and transferring the operations into a mental plan. At first, it is also recommended to check your answers using a calculator or standard column calculations.

It's no secret that there are some people who can perform moderately complex arithmetic operations in their heads with enviable speed. It is not difficult for them, for example, to multiply two two-digit numbers or divide several three-digit quantities by each other. They do this quickly and without the help of additional devices and do not even use notes, that is, they perform calculations in their heads! It is clear that for many it is not difficult to figure out how to learn to quickly count in your head - this is daily practice, forced work or occupation. But this does not mean that any of us who wants to learn how to learn to count in our heads is obliged to graduate from a mathematical university. So, today we will talk about how to learn to count. Count quickly!

Learning to count quickly, necessary preparation

Without a doubt, your experience and ability training will play an important role in developing such abilities. But this in no way means that the skill of fast counting is available only to people with experience. Mental arithmetic is a way of rationalization that relies on basic arithmetic. By following our tips on how to quickly learn to count, you will be able to surprise others with quick solutions to examples that not everyone can solve even with the help of a calculator.

What do you need to quickly master the technique of instant calculation “in your head”? The main components of success can be divided into three groups:

Predispositions and abilities. Your analytical mind will be a good help. The ability to retain several quantities in memory at a time is mandatory.
Directly the algorithms of your thinking. You can learn to count quickly only through strict algorithmization of your actions, their rationalization and the ability to choose the necessary method in a specific situation. We'll talk about situations and other things a little later.
Training and practice of skills. No one has denied the importance of these actions in any area of activity, especially in mental activity. The more you practice and perform various calculations, the better you will get at it.

You should pay attention to the third factor in the development of quick counting skills. Even if you are well versed in all existing algorithms, you are unlikely to be able to learn to count quickly if you do not have enough practice.

Tricks and basic algorithms for how to count quickly

Let's look at several generally accepted counting simplifications; with their help, you will be able to learn to count quickly. I would also like to draw your attention to the fact that no one forbids you to improvise - the remarkable thing about mathematics is that, with all its accuracy and rigor, it does not forbid you to act beautifully, like art. And the ability to count quickly is an art! So, some tricks on how to learn to count quickly.

Let's say you need to add multivalued terms. Easily! Add by digits: to a larger number, add the most significant digit of the smaller number, then add with the lower digits. Let's say you need to add 361 and 523. It won't be easy to remember right away, would you agree? Therefore, our course of action will be as follows:

The smaller number was determined - 361.
What is 361? This is 300+60+1. It is difficult to argue if you strive to be rational.
To 523 we first add 300. We get 823.
Then add 60 and we get 883.
And finally, our one, added to the amount obtained earlier, will give us the result 884.

You see, it was much easier to keep 3 numbers in your head than to add two three-digit numbers at once! We are starting to be able to count quickly in our heads!

Do the same with subtraction, but just by sequentially subtracting digits we will not achieve the required speed! We can cheat a little by adding another skill to our arsenal - increase/subtract to a round (convenient number).

For example, you need to subtract 93 from 250. Well, that’s inconvenient!

What is 93? That's right, it's 100-7!

250 – 100 = 150.

We make allowances for our “correction” of the number. If we added, we must add to the quotient, and vice versa. In our case, we “increased” the number 93 to 100 by adding 7. This means we add 7 to the quotient.

Check it on your calculator. Did you spend noticeably more time typing numbers than doing calculations? This is a sign that you are already quite good at counting quickly in your head!

Now with multiplication. You can speed up your counting in different ways. For example, when multiplying numbers, break down the factors into second-level factors.

For example:

Lots of ways to a solution! And here your algorithm may differ from the paths of other people - don’t be alarmed, that’s why we, the people, are geniuses and unique =)

You can do this: 12 = 3x4. Multiply 150 x 4 = 600, then 600 x 3 = 1800.

Without thinking, I began to count like this: 12 = 10 + 2. And now it’s elementary: (150 x 10) + (150 x2). All these are basic school rules that we, unfortunately, forget. It’s easy to see that in this case there’s practically no need to count - add zero to 150, getting one and a half thousand, and multiply 150 by 2, getting 300. The result is the same, 1800.

Based on the experience of fast multiplication, it is not difficult to guess how to quickly divide numbers in your head. You can again go in different ways, from parallel division by a simplified divisor of the dividend to rounding of the dividend up to the elementaryization of division with an amendment.

For example:

First, discard the same number of zeros. In this example it's simple - 39:4. Our brains are much more willing to operate with small numbers than with multi-digit values.

You've probably noticed that you just want to round the number 39 to 40. So, what's stopping us? (39+1):4 = 10.

But having changed the dividend, we need to adjust the answer. So, it is obvious that it will be less than 10, since we added a certain number 1 to the dividend. Now we need to subtract from 10 the result of dividing the corrector number by the divisor (4). If we took away, the procedure would be reversed, this goes without saying.

So 1:4 = 0.25

Answer: 9.75 (9 3 / 4)

It is much easier for our brain to perceive natural fractions, that is, we imagine 0.25 as 1/4 (one fourth, a quarter), and then it will be very easy to quickly calculate the result in our minds!

Remember, it's not that hard to figure out how to quickly learn to count. It is much more difficult to quickly select a method for a specific situation, but this can be solved with the help of enormous practice.

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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.

Division and multiplication 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 than 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 large quantity actions, each of which is simpler than the original.
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 to count quickly 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.

Every parent wants their child to grow up smart, well-developed and interested in learning. However, it is difficult to show a child’s interest in acquiring new knowledge. One of the first manifestations of interest in knowledge in preschool children is counting.

It is at this moment that it is very important to create a game out of mathematical tasks that will captivate the child.

This article will discuss how to quickly teach a child to add in his head. We will provide not only exercises, but also tell you where to start the exercises and how to transform them into a game form.

The basis of mathematics is mastering counting

The first step in the educational process is the study of ordinal counting, in other words, the numbers of their location. As an initial step, you can take everyday activities, i.e. introducing counting when you walk up the stairs with your baby, button his jacket or eat. The remaining stages of training also proceed smoothly one after another, so in such classes it is important to maintain consistency and systematicity.

The main tasks at the initial stages are:

teach the child to distinguish multiple objects from single ones, i.e. “many” and “one”;
teach to separate concepts such as “equal”, “more” and “less”;
ordinal and quantitative counting;
teach an understanding of how the number of objects relates to a specific number;
study the composition of numbers - first from one to ten, then from 10 to 20, etc.;
simple arithmetic problems.

When you come to problems in mathematics, you should use not just one method of solving, but several. With this approach, it will be easier for the child to look for other solutions in the future, and his mind will become more flexible.

Answering the question, “how to learn to count in your head?”, We note that learning should begin systematically, when the child reaches the age of 3 or 4 years. Remember that the process should be playful. Otherwise, the baby’s desire to learn can be blocked.

Presentation: "Mental arithmetic in mathematics lessons"

Counting process

The mental process regarding counting always begins with simple actions. As a rule, they are divided into two components - speech and motor.

Speech action develops according to the scheme - first we talk about what we are doing, then we whisper, and then we count to ourselves. And only after this stage can you move on to a quick count. For example, when adding units 1+1, the next digit in the series is called, i.e. in his mind the child will immediately add 1,2,3,4...
The motor element develops from the usual shifting of objects from side to side. Thus, in a playful way, objects will increase or decrease. At first, the child will follow the counting with his finger, then only with his eyes, performing mathematical operations in his mind.

When counting on fingers or sticks, kids do not try to remember the result. In view of this, when there are not enough fingers and sticks when counting, the child has difficulties.

If a parent wants to teach a child to count, then the subject should reduce their participation in the process as quickly as possible, but it will not be possible to remove them completely. How to learn to count quickly in your head? Read about this in the following sections.

The main component of learning is play

Each person develops individually. Making mistakes while learning the material is normal. However, many parents do not understand why a smart child is not able to understand simple things from an adult's point of view.

Note that the child’s brain is different in structure from the adult’s brain. Kids do not want and cannot remember what does not arouse their interest.

Children's memory is designed in such a way that it stores only what evokes an emotional response. It doesn’t matter whether the emotions are positive or negative.

So how do you teach a child to count mentally? The game will help you learn mathematical fundamentals; you can start counting kittens on the street while, for example, you are going to kindergarten. Having taught your child the numbers from 1 to 10, you can invite him to look for them on the way to the store, and when he comes home, count how many numbers were found and add them up in his head.

There are many methods, and we suggest you familiarize yourself with the most popular ones in the next section.

The ability to count is important not only when preparing for school, but also in the future life of any person. Counting to 10 is important, but a child is unlikely to be able to master it right away, so you need to start from 1 to 5, and then increase the complexity of the task.

In order to master counting quickly and successfully, we recommend using tips, but only at the beginning of training. Then they need to be gradually removed so that the baby learns to count in his head.

fingers;
educational TV programs;
educational games and abacus;
rhymes with numbers or counting rhymes;
Count everything you see every day with your baby.

Quick counting techniques:

Cards. During the period of learning numbers, flashcards are very important. You can buy them, or make them yourself with your child. The latter will be more interesting for the child. At the beginning, show them to your baby sequentially, then change the order.
Shop. One of the most favorite games for kids. You should lay out “goods for sale” on the table, come up with a “currency” and assign a price tag to each item. Your child should be appointed as a cashier. When communicating with a store employee, you should not pay attention to the price tags; let the child tell you and count how much the items cost.
Plasticine. A game in which you need to ask a child to make 4 legs for a bear, or two ears for a cat. Along the way, you should show him cards with these numbers.

How to teach a child to count in his head? Teaching a child to count is quite difficult, but all parents want him to do it without thinking. Daily exercises, exciting forms of study, coupled with your perseverance and patience will help your child master the queen of sciences - mathematics.