How to count large numbers without a calculator. Game "Quick Count"

Why do I call my method easy and even surprisingly easy? Yes, simply because I have not yet come across a simpler and more reliable way of teaching kids to count. You will soon see this for yourself if you use it to educate your child. For a child, this will be just a game, and all that is required from parents is to devote a few minutes a day to this game, and if you follow my recommendations, sooner or later your child will definitely start counting in a race with you. But is this possible if the child is only three or four years old? It turns out that it is quite possible. In any case, I have been doing this successfully for over ten years.

I outline the entire learning process further in great detail, with a detailed description of each educational game, so that any mother can repeat it with her child. And, in addition, on the Internet on my website “Seven Steps to a Book,” I posted video recordings of fragments of my classes with children to make these lessons even more accessible for playback.

First, a few introductory words.

The first question that some parents have is: is it worth starting to teach your child arithmetic before school?

I believe that a child should be taught when he shows interest in the subject of study, and not after this interest has faded away. And children show interest in counting and counting early; it only needs to be slightly nourished and the games imperceptibly made more complex day by day. If for some reason your child is indifferent to counting objects, do not say to yourself: “He has no inclination for mathematics, I was also behind in mathematics at school.” Try to awaken this interest in him. Just include in his educational games what you have missed so far: counting toys, buttons on a shirt, steps when walking, etc.

The second question: what is the best way to teach a child?

You will get the answer to this question by reading here a complete description of my method of teaching mental arithmetic.

In the meantime, I want to warn you against using some teaching methods that do not benefit the child.

“To add 3 to 2, you must first add 1 to 2, you get 3, then add another 1 to 3, you get 4, and finally add another 1 to 4, the result is 5.” ; “- To subtract 3 from 5, you must first subtract 1, leaving 4, then subtract 1 more from 4, leaving 3, and finally subtract 1 more from 3, resulting in 2.”

This unfortunately common method develops and reinforces the habit of slow counting and does not stimulate the child’s mental development. After all, counting means adding and subtracting in whole numerical groups at once, and not adding and subtracting one by one, and even by counting fingers or sticks. Why is this method, which is not useful for a child, so widespread? I think because it’s easier for the teacher. I hope that some teachers, having become familiar with my methodology, will abandon it.

Don't start teaching your child to count with sticks or fingers and make sure that he doesn't start using them later on the advice of an older sister or brother. It's easy to learn to count on your fingers, but difficult to unlearn. While the child is counting on his fingers, the memory mechanism is not involved; the results of addition and subtraction in whole number groups are not stored in memory.

And finally, under no circumstances use the “ruler” method of counting that has appeared in recent years:

“To add 3 to 2, you need to take a ruler, find the number 2 on it, count from it to the right 3 times in centimeters and read the result 5 on the ruler”;

“To subtract 3 from 5, you need to take a ruler, find the number 5 on it, count from it to the left 3 times in centimeters and read the result 2 on the ruler.”

This method of counting, using such a primitive “calculator” as a ruler, seems to have been deliberately invented in order to wean a child from thinking and remembering. Instead of teaching how to count like this, it’s better not to teach at all, but to immediately show how to use a calculator. After all, this method, just like a calculator, eliminates memory training and inhibits the child’s mental development.

At the first stage of learning mental arithmetic, it is necessary to teach the child to count within ten. We need to help him firmly remember the results of all variants of adding and subtracting numbers within ten, just as we adults remember them.

At the second stage of education, preschoolers master the basic methods of adding and subtracting two-digit numbers in their heads. The main thing now is not the automatic retrieval of ready-made solutions from memory, but the understanding and memorization of addition and subtraction methods in subsequent tens.

Both at the first and second stages, learning mental arithmetic occurs using elements of play and competition. With the help of educational games built in a certain sequence, not formal memorization is achieved, but conscious memorization using the child’s visual and tactile memory, followed by consolidation in memory of each learned step.

Why do I teach mental arithmetic? Because only mental arithmetic develops the child’s memory, intelligence and what we call ingenuity. And this is exactly what he will need in his subsequent adult life. And writing “examples” with long thinking and calculating the answer on the fingers of a preschooler does nothing but harm, because discourages you from thinking quickly. He will solve examples later, at school, practicing the accuracy of the design. And intelligence must be developed at an early age, which is facilitated by mental calculation.

Even before starting to teach a child addition and subtraction, parents should teach him to count objects in pictures and in reality, count steps on a ladder, steps while walking. By the beginning of learning mental counting, a child should be able to count at least five toys, fish, birds, or ladybugs and at the same time master the concepts of “more” and “less.” But all these various objects and creatures should not be used in the future for teaching addition and subtraction. Learning mental arithmetic should begin with addition and subtraction of the same homogeneous objects, forming a certain configuration for each number. This will allow the child to use the visual and tactile memory when memorizing the results of addition and subtraction in whole number groups (see video file 056). As a tool for teaching mental counting, I used a set of small counting cubes in a counting box (detailed description below). And children will return to fish, birds, dolls, ladybugs and other objects and creatures later, when solving arithmetic problems. But by this time, adding and subtracting any numbers in the mind will no longer be difficult for them.

For ease of presentation, I divided the first stage of training (counting within the first ten) into 40 lessons, and the second stage of training (counting within the next tens) into another 10-15 lessons. Don't be intimidated by the large number of lessons. The breakdown of the entire training course into lessons is approximate; with prepared children, I sometimes go through 2-3 lessons in one lesson, and it is quite possible that your child will not need so many lessons. In addition, these classes can be called lessons only conditionally, because each lasts only 10-20 minutes. They can also be combined with reading lessons. It is advisable to study twice a week, and it is enough to spend 5-7 minutes on homework on other days. Not every child needs the very first lesson; it is designed only for children who do not yet know the number 1 and, looking at two objects, cannot say how many there are without first counting with their finger. Their training must begin practically “from scratch.” More prepared children can start immediately from the second, and some - from the third or fourth lesson.

I conduct classes with three children at a time, no more, in order to keep the attention of each of them and not let them get bored. When the level of preparation of children is slightly different, you have to work with them on different tasks one by one, all the time switching from one child to another. At the initial lessons, the presence of parents is desirable so that they understand the essence of the methodology and correctly perform simple and short daily homework with their children. But the parents must be placed so that the children forget about their presence. Parents should not interfere or discipline their children, even if they are naughty or distracted.

Classes with children in mental counting in a small group can begin from approximately the age of three, if they already know how to count objects with their fingers, at least up to five. And with their own child, parents can easily start elementary lessons using this method from the age of two.

Initial lessons of the first stage. Learning to count within five

To conduct initial lessons, you will need five cards with the numbers 1, 2, 3, 4, 5 and five cubes with an edge size of approximately 1.5-2 cm, installed in a box. For cubes, I use “knowledge cubes” or “learning bricks” sold in educational game stores, 36 cubes per box. For the entire training course you will need three such boxes, i.e. 108 cubes. For initial lessons I take five cubes, the rest will be needed later. If you are unable to find ready-made cubes, it will not be difficult to make them yourself. To do this, you just need to print out a drawing on thick paper, 200-250 g/m2, and then cut out cube blanks from it, glue them together in accordance with the instructions, fill them with any filler, for example, some kind of cereal, and cover the outside with tape. It is also necessary to make a box to place these five cubes in a row. Gluing it together is just as easy from a pattern printed on thick paper and cut out. At the bottom of the box, five cells are drawn according to the size of the cubes; the cubes should fit in it freely.

You have already understood that learning to count at the initial stage will be done with the help of five cubes and a box with five cells for them. In this regard, the question arises: why is the method of learning with the help of five counting cubes and a box with five cells better than learning with the help of five fingers? Mainly because the teacher can cover the box with his palm from time to time or remove it, due to which the cubes and empty cells located in it are very quickly imprinted in the child’s memory. But the child’s fingers always remain with him, he can see or feel them, and there is simply no need for memorization; the memory mechanism is not stimulated.

You should also not try to replace the box of cubes with counting sticks, other counting objects, or cubes that are not lined up in the box. Unlike cubes lined up in a box, these objects are arranged randomly, do not form a permanent configuration and therefore are not stored in memory as a memorable picture.

Lesson #1

Before the start of the lesson, find out how many cubes the child can identify at the same time, without counting them one by one with his finger. Usually, by the age of three, children can tell immediately, without counting, how many cubes are in a box, if their number does not exceed two or three, and only a few of them see four at once. But there are children who can only name one object so far. In order to say that they see two objects, they must count them by pointing with their finger. The first lesson is intended for such children. The others will join them later. To determine how many cubes the child sees at once, alternately place different numbers of cubes in the box and ask: “How many cubes are in the box? Don’t count, tell me right away. Well done! And now? And now? That’s right, well done!” Children can sit or stand at the table. Place the box with cubes on the table next to the child parallel to the edge of the table.

To complete the tasks of the first lesson, leave the children who can only identify one cube so far. Play with them one by one.

  1. Game "Putting numbers to dice" with two dice.
    Place a card with number 1 and a card with number 2 on the table. Place a box on the table and put one cube into it. Ask your child how many cubes are in the box. After he answers “one,” show and tell him the number 1 and ask him to put it next to the box. Add a second cube to the box and ask him to count how many cubes are in the box now. Let him, if he wants, count the cubes with his finger. After the child says that there are already two cubes in the box, show him and call the number 2 and ask him to remove the number 1 from the box and put the number 2 in its place. Repeat this game several times. Very soon the child will remember what two cubes look like and will begin to name this number immediately, without counting. At the same time, he will remember the numbers 1 and 2 and will move the number corresponding to the number of cubes in it towards the box.
  2. Game "Dwarves in a house" with two dice.
    Tell your child that you will now play the game “Gnomes in the House” with him. The box is a make-believe house, the cells in it are rooms, and the cubes are the gnomes who live in them. Place one cube on the first square to the left of the child and say: “One gnome came to the house.” Then ask: “And if another one comes to him, how many gnomes will be in the house?” If the child finds it difficult to answer, place the second cube on the table next to the house. After the child says that now there will be two gnomes in the house, allow him to place the second gnome next to the first on the second square. Then ask: “And if now one gnome leaves, how many gnomes will remain in the house?” This time your question will not cause difficulty and the child will answer: “One will remain.”

Then make the game more difficult. Say: “Now let’s put a roof on the house.” Cover the box with your palm and repeat the game. Every time the child says how many gnomes there are in the house after one came, or how many of them are left in it after one left, remove the palm roof and allow the child to add or remove the cube himself and make sure his answer is correct. . This helps connect not only the child’s visual, but also tactile memory. You always need to remove the last cube, i.e. second from the left.

Play games 1 and 2 alternately with all the children in the group. Tell the parents present at the lesson that they should play these games with their children once a day every day at home, unless the children themselves ask for more.

Comment on the article "A surprisingly easy way to teach your child mental arithmetic"

Doesn't understand math. How to teach a child not to be afraid of tests? Good afternoon. I am not an experienced mother, I have experience with Mathematics in How to teach a child mental arithmetic. Presentation "Mathematics for little ones, counting from 1 to 10 with adding one": methodological...

Discussion

My child was born with hypoxia, and there were some other diagnoses that were not critical for me at that time.
This resulted in speech therapy problems, but they were quickly resolved with a speech therapist.
Hyperactivity immediately became visible, but it was compensated for by the age of 11.
But concentration and Mathematics became a problem, and in the lower grades it was also 3-4-5, but in the fifth grade it was 2-3-4.
There was always a math tutor. I changed because I thought it was the tutor who didn’t explain it well!
But in November, in the 5th grade, I brought my child to Moscow to a neurologist, based on recommendations, and he told us, after examination and tests, that it was attention deficit.
The purpose was stratera (but this is only by prescription), pantogam. Also mandatory classes with a Neuropsychologist and a psychologist (cognitive techniques).
You know, I can’t believe it myself, but there is a result!
Now it’s February and she’s firmly in her 4th trimester.
And the math tutor praises me for paying attention!
And the math teacher herself (otherwise she called me in September to say that she had a 2 on a test and needed to study with her daughter! How else could she study if she studied all August and September!)

12.02.2019 20:19:40, Veronica-strawberry

Mental arithmetic - how to teach? Once you've mastered counting well within ten, you won't have any problems with counting when you start counting beyond ten. A surprisingly easy way to teach your child mental math. Initial lessons of the first stage.

Discussion

1. Work with him yourself in addition to school + other specialists.
2. Completely move away from the school methodology from the specific to the general; this “doesn’t work” for our children; they “can’t see the forest for the bushes.” The approach should be “from the general to the specific”, i.e. First you give a general vision, without going into details, then you disassemble one aspect and repeat it ad nauseam. For example:
We say - speech - parts of speech - independent (nominal) and service-independent: noun, adjective, numeral, adverb, verb, participle and gerund; auxiliary: preposition, conjunction, particle + special part of speech - interjection. Noun - proper, adverb. etc. We always start with the simplest: We speak - speech. Until you learn it, don’t move on to parts of speech. Then, when everything is mastered, go over the entire tree 100,500 times every day until the child’s teeth begin to bounce off. Next comes the complication of the task, we now rely on some familiar subsection and dance from it. But we regularly repeat the entire structure.
3. In mathematics, we count on our fingers for a long time and painfully. Then, when the counting becomes error-free and fast, we cover our fingers with a newspaper or towel, count by touch, then close our eyes and imagine the fingers in our minds, then we simply count in our minds.
4. We apply available types of differentiation (or selection). For example, number digits: ones are green, tens are yellow, hundreds are red. You can use tactile or sound - it depends on the child’s capabilities.
5. Work until you sweat, repeat until your tongue becomes calloused. No "hug and cry"! Our children have been given everything, the approach just needs to be DIFFERENT. And there the integrals with derivatives will also obey.

Where do you study?
Mine has the same thing, it’s also complicated by the fact that the beginning ends, there will be no continuation, I can’t imagine where to go(

Doesn't understand math. Education, development. Child from 7 to 10. I don’t understand what’s going on with math and how to help the child? My son is 11 years old and studies in the 6th grade. How to teach your child mental arithmetic. Print version.

Discussion

Hello, I would advise you to explain it more or less easily, let’s say the following example:
576-78=?
Please explain that I cannot subtract 78 from 76.
To 6 you need to add 10, that is, we take one ten.
Subtract 8 from 16 and get 8
So 8 is in the place of ones
Since we borrowed one ten from 70, it means not 70 but 60
Further:
From 560 I subtract 70 = 490, and we also remember that in place of units 8 we get 498.
I hope you improve your math!!!
Good luck.

26.12.2018 17:54:16, Kamilla Batrakanova

A tutor is needed if the child does NOT understand complex material, and the parents are NOT able to explain it. In your case, your daughter (having 3 explanations of the same thing) will be completely confused.
Try downloading flash games to your tablet or phone. Now there are many cool applications where you can improve mathematics, mental calculation, solve logic problems and generally practice spatial thinking in a playful way. Observe which tasks cause difficulties for your daughter, so you can highlight problem areas that are worth going over again.

08/14/2018 09:42:26, ​​Epsona

How to teach your child mental arithmetic. Presentation "Mathematics for little ones, counting from 1 to 10 with adding one": teaching material for educators. How to teach a child mental arithmetic and retain the skill of counting quickly for life?

Discussion

Peterson has successful translation schemes - look in the textbooks for grades 3 and 4. Or arrange it yourself - units of measurement in a row, from largest to smallest: 1t - 1c - 1kg - 1g. Between them at the bottom of the arc, under the arcs the ratio is (10, 100, 1000). And the arrows: to the right - we multiply (when converting to smaller ones), to the left - we divide (to large ones). Let's say, convert 35 tons into grams - 35 * 10 * 100 * 1000 = 35 * 1000000 = 35000000g.

I think the basic concept needs to be worked out very well. It is important for me not to go through the topic and forget, but for the child to understand and feel it.
I measured different things with the children using different MEASURES - for example, a room - with steps, rulers, briefcases, boa constrictors...
Then the area is also measured - a table, for example, with squares of paper: simply - how many of them will fit there, with notebooks. And if you take smaller squares, it will be more accurate, but longer.
Then we moved directly to calculations. But it turns out that you can not lay out the measurements by hand every time, but divide it arithmetically... The room is the length of 3 boa constrictors, and there is so much in the briefcases (because one boa constrictor can fit four briefcases in length), and so much in the pencil cases ( because the briefcase is equal in length to two pencil cases).
Then, as one of the types of measurements, they took meters, centimeters, hectares, square values

There, mental arithmetic is the basis of first grade. Sorry, Len, for intruding, but the problem is the same, we are also suffering, but my some I know that I am not a mathematician, and I wanted to make his “first-class” life easier - to understand (or learn) the composition of a number. As soon as you haven't played it, you can't remember it by heart...

Discussion

To do this, you need to memorize the composition of numbers up to 10 very well. This knowledge is vital when solving examples of addition and subtraction. In order to remember the composition of a number well, you just need to repeat the pairs that make up this number many times. There is an application for iPad and iPhone that makes this process easier for the child, turning it into a game with attractive features and sounds. The application has already been tested by many users for several years. This application, despite its simplicity, is very effective, experts in Singapore respond very well to it, and many educational institutions around the world use it in their practice. Especially for visitors to the site, we are giving 5 gift promotional codes for this application:
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You can download the Composition of Numbers to 10 application in the App Store:

Discussion

Example 3+4 will recalculate, and if you ask how much 3 candies and 4 more candies will be, the answer will immediately be seven.
By the way, in our schools we teach counting with fingers.

At the age of 4, my son counted using the composition of numbers. Now he is counting by counting units. I don’t understand what the connection is with future difficulties with algebra. In Mikulina’s notebook “Fairytale Numbers” (one of the authors of the textbook on mathematics ED), Mishenka solves all the examples with symbols in systems of linear equations at the speed of a pig squeal. What kind of tragedy is that? For a programmer, the idea of ​​moving along a number series is even preferable; many problems are solved this way. In exam problems that need to be solved in integers, this enumeration method is also convenient. In general, it’s more convenient for me to create an algorithm for solving a system of equations and put all this mess into a computer than to worry about numbers. I really don’t like the fact that huge abacus books have disappeared from school classrooms for first-graders; Perelman has written well about abacus; at the age of seven I figured it out myself from his book and enjoyed playing with the abacus. For centuries they counted on these knuckles, my mother was a virtuoso, the knuckles just flew, she didn’t need any adding machine. On fingers, knuckles, when counting in the mind, numbers are seen somehow differently, some patterns are noticed differently. Even though the children will try everything while they are small, they are still very, very far from real mathematics with proofs.

Number sense, minimal counting skills are the same element of human culture as speech and writing. And if you easily count in your mind, then you feel a different level of control over reality. In addition, this skill develops thinking abilities: concentration on objects and things, memory, attention to detail and switching between streams of knowledge. And if you are interested in how to learn to count quickly in your head, the secret is simple: you need to constantly practice.

Memory training: myth or reality?

In mathematics, everything is simple for those smart individuals who click equations like seeds. It is more difficult for other people to learn. But nothing is impossible, everything is possible if you practice a lot. There are the following mathematical operations: subtraction, addition, multiplication, division. Each of them has its own characteristics. To understand all the complexities, you need to understand them once, and then everything will be much simpler. If you practice for 10 minutes every day, in a few months you will reach a decent level and learn the truth of counting mathematical numbers.

Many people do not understand how they can vary numbers in their minds. How to become the master of numbers so that it does not look stupid and imperceptible from the outside? When you don’t have a calculator at hand, your brain begins to intensively process information, trying to calculate the necessary numbers in your head. But not all people are able to achieve the desired results, since each of us is an individual person with his own limits of capabilities. If you want to understand in your head, then you should study all the necessary information, armed with a pen, a notepad and patience.

The multiplication table will save the situation

We will not talk about those people who have an IQ level above 100; there are special requirements for such individuals. Let's talk about the average person who can learn many manipulations using the multiplication table. So, how to quickly count in your head without losing your health, energy and time? The answer is simple: memorize the multiplication table! In fact, there is nothing difficult here, the main thing is to have pressure and patience, and the numbers themselves will give in to your goal.

For such an amusing undertaking, you will need a smart partner who can test you and keep you company in this process that requires patience. The man who knows is in the mind of even the laziest student. Once you can multiply quickly, counting mentally will become routine. Unfortunately, there are no magic methods. How quickly you can learn a new skill is up to you. You can exercise your brain not only with the help of multiplication tables; there is a more exciting activity - reading books.

Books and no calculator train your brain

In order to learn how to perform computational activities verbally as quickly as possible, you need to constantly strengthen your brain with new information. But how can you learn to count quickly in umeza in a short time? You can train your memory only with useful books, thanks to which not only the work of your brain will be universal, but also, as a bonus, improving your memory and gaining useful knowledge. But reading books is not the end of training. Only when you can forget about the calculator will your brain begin to process information faster. Try to count in your head in any case, think through complex mathematical examples. But if it’s hard for you to do all this on your own, then enlist the help of a professional who will quickly teach you everything.

It may be difficult for you to understand how to learn to count quickly in your head when you are not familiar with mathematics and there is no good teacher who could make the task easier. But you shouldn’t give in to difficulties. Having studied all the necessary recommendations, you can easily quickly learn to count in your head and surprise your peers with new abilities.

  • The ability to work with large numbers goes beyond general development.
  • Knowing the “tricks” of counting will help you quickly overcome all obstacles.
  • Regularity is more important than intensity.
  • Don't rush, try to catch your rhythm.
  • Focus on correct answers, not on memorization speed.
  • Say your actions out loud.
  • Don't be discouraged if you don't succeed, because the main thing is to start.

Never give up in the face of difficulties

During your training, you may have many questions to which you do not know the answers. This shouldn't scare you. After all, you cannot know at first how to quickly count without prior preparation. The road can only be mastered by those who always move forward. Difficulties should only strengthen you, and not slow down your desire to join people with non-standard capabilities. Even if you are already at the finish line, return to the easiest thing, train your brain, do not give it the opportunity to relax. And remember, the more you speak information out loud, the faster you will remember it.

In the age of modern technology with many advanced gadgets, mental arithmetic 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 mental exercises, and counting, as you know, also includes this, increase a person’s intelligence and level of intelligence, which, in the future, affects his entire life. Such people navigate various situations much faster; at a minimum, they are more difficult to shortchange in a store or market, which is already a pleasant bonus of this ability.

It must be said that people who can count quickly in their heads are not necessarily some kind of genius or owner of special abilities, it’s all about years of practice, as well as knowledge of some tricky tricks, which we will talk about later. This question often arises when it is necessary to teach a schoolchild to count: as parents note, the child cannot count in his head, but on paper he can do it just fine.

If the age is very young, then problems may arise on paper, so how can you learn to quickly count in your head? It all depends on age: it is not without reason 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 appears in children at the age of 5-6 years, and sometimes earlier, the main thing is not to miss it and start developing it. 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 bed. You can count your favorite toys, be sure to get educational sets of cubes with numbers, rearrange them, 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 development game “gnomes in a house”. Think of a cardboard box - it will be a house. Take a few cubes and explain to your 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 gnome comes to visit the gnome, then how many gnomes will now be in the house?

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 mentally, but also visually sees the real result of the action. These are the first ways to develop mental math skills in a child.

How to learn to count in your head at an older age?

Of course, you can’t lure schoolchildren and adults with games anymore, and there’s no need for that. At an older age, the main thing is practice. The more a person practices, the easier it will be for him to give the correct answers. The second point is perfect knowledge of the multiplication tables by heart.

It may seem to you that this is stupid advice, who doesn’t 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 obtained.

It is impossible to learn how to quickly count in your head at the behest of a magic wand; you still have to work hard: at a minimum, remember special formulas that significantly simplify such counting. Secondly, learn to concentrate your attention: after all, when calculating, you will have to keep complex numbers in your mind, as well as their combinations.

Multiply by 11

There are several options for quickly and easily multiplying a number by 11. So, we’ll immediately show the first method with an example:

At the first stage, you need to add the numbers of the first factor, that is, 6+3=9. The next step is to place the resulting result between the first and last number of the multiplier, that is, 6(9)3. Here is the result!

Method No. 2. Let's look at other numbers:

At the first stage, we again add the components of the multiplier: 6+9=15. What to do if the result is two-digit? It’s simple: we move the unit to the left, (6+1)_in the center we leave 5_and add 9. The result of the formula is: 7_5_9=759.

Multiply by 5

The multiplication table “by 5” is easy to remember, but when it comes to complex numbers, counting is no longer so easy. And there is a trick here: any number that you want to multiply by five, simply divide it in half. Add a zero to the result obtained, but if the division results in a fractional number, then simply remove the comma. This always works, check this example:

Let's parse: 4568/2=2284

We add 0 to 2284 and get 22840. If you don’t believe me, check it yourself!

Multiplying two complex numbers

If you need to multiply two complex numbers in your head, 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 head

An interesting rule applies here: if one of the terms is increased by a certain number, then the same number must be subtracted from the resulting result. For example:

550+348=(550+348+2)-2=(550+350)-2=898

There are a lot of such techniques and interesting formulas that significantly simplify mental arithmetic; if this interests you, then many examples can always be found on the Internet. But to really achieve results, it is very important to practice a lot, so examples will help you!

The process of mental counting can be considered as a counting technology that combines human ideas and skills about numbers and mathematical arithmetic algorithms.

There are three types mental counting technologies, which use various physical capabilities of a person:

    audiomotor counting technology;

    visual counting technology.

Characteristic feature audiomotor mental counting is to accompany each action and each number with a verbal phrase like “twice two is four.” The traditional counting system is precisely an audiomotor technology. The disadvantages of the audiomotor method of calculations are:

    absence in the memorized phrase of relationships with neighboring results,

    the inability to separate tens and units of a product in phrases about the multiplication table without repeating the entire phrase;

    the inability to reverse the phrase from the answer to the factors, which is important for performing division with a remainder;

    slow speed of reproduction of a verbal phrase.

Supercomputers, demonstrating high speed of thinking, use their visual abilities and excellent visual memory. People who are good at speed calculations do not use words when solving an arithmetic example in their head. They demonstrate reality visual technology of mental counting, devoid of the main drawback - the slow speed of performing basic operations with numbers.

Perhaps our methods of multiplication are not perfect; Maybe an even faster and more reliable one will be invented.

Of course, it is impossible to know all the methods of quick counting, but the most accessible ones can be studied and applied.

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 quite quickly, 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 highlight3 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 necessary algorithms, you can “outdo” even the most experienced “accountant”, provided that you have trained for the same amount of time.

Several ways to count mentally:

1. Multiply by 5 It’s more convenient to do this: first multiply by 10, and then divide by 2

2. Multiply by 9. In order to multiply a number by 9, you need to add 0 to the multiplicand and subtract the multiplicand from the resulting number, for example 45 9 = 450-45 = 405.

3. Multiply by 10. Add a zero to the right: 48 10 = 480

4. Multiply by 11. two-digit number. Spread the numbers N and A, enter the amount in the middle (N+A).

for example, 43 11 = = = 473.

5. Multiply by 12. is done in approximately the same way as for 11. We double each digit of the number and add to the result the neighbor of the original digit on the right.

Examples.Let's multiplyon.

Let's start with the rightmost number - this is. Let's double itand add a neighbor (he is not present in this case). We get. Let's write it downand remember.

Let's move left to the next number. Let's double it, we get, add a neighbor,, we get, add. Let's write it downand remember.

Let's move left to the next number,. Let's double it, we get. Let's add a neighborand we get. Let's add, which we remembered, we get. Let's write it downand remember.

Let's move to the left to a non-existent number - zero. Let's double it, get and add a neighbor, which will give us . Finally, we add , which we remembered, and we get . Let's write it down. Answer: .

6. Multiplication and division by 5, 50, 500, etc.

Multiplication by 5, 50, 500, etc. is replaced by multiplication by 10, 100, 1000, etc., followed by division by 2 of the resulting product (or division by 2 and multiplication by 10, 100, 1000, etc. ). (50 = 100: 2, etc.)

54 5=(54 10):2=540:2=270 (54 5 = (54:2) 10= 270).

To divide a number by 5.50, 500, etc., you need to divide this number by 10,100,1000, etc. and multiply by 2.

10800: 50 = 10800:100 2 =216

10800: 50 = 10800 2:100 =216

7. Multiplication and division by 25, 250, 2500, etc.

Multiplication by 25, 250, 2500, etc. is replaced by multiplication by 100, 1000, 10000, etc. and the resulting result is divided by 4. (25 = 100: 4)

542 25=(542 100):4=13550 (248 25=248: 4 100 = 6200)

(if the number is divisible by 4, then multiplication does not take time; any student can do it).

To divide a number by 25, 25,250,2500, etc., this number must be divided by 100,1000,10000, etc. and multiply by 4: 31200: 25 = 31200:100 4 = 1248.

8. Multiplication and division by 125, 1250, 12500, etc.

Multiplication by 125, 1250, etc. is replaced by multiplication by 1000, 10000, etc. and the resulting product must be divided by 8. (125 = 1000 : 8)

72 125=72 1000: 8=9000

If the number is divisible by 8, then first divide by 8, and then multiply by 1000, 10000, etc.

48 125 = 48: 8 1000 = 6000

To divide a number by 125, 1250, etc., you need to divide this number by 1000, 10000, etc. and multiply by 8.

7000: 125 = 7000: 10008 = 56.

9. Multiplication and division by 75, 750, etc.

To multiply a number by 75, 750, etc., you need to divide this number by 4 and multiply by 300, 3000, etc. (75 = 300:4)

4875 = 48:4300 = 3600

To divide a number by 75,750, etc., you need to divide this number by 300, 3000, etc. and multiply by 4

7200: 75 = 7200: 3004 = 96.

10. Multiply by 15, 150.

When multiplying by 15, if the number is odd, multiply it by 10 and add half of the resulting product:

23 15=23 (10+5)=230+115=345;

if the number is even, then we proceed even simpler - we add half of it to the number and multiply the result by 10:

18 15=(18+9) 10=27 10=270.

When multiplying a number by 150, we use the same technique and multiply the result by 10, since 150 = 15 10:

24 150=((24+12) 10) 10=(36 10) 10=3600.

In the same way, quickly multiply a two-digit number (especially an even one) by a two-digit number ending in 5:

24 35 = 24 (30 +5) = 24 30+24:2 10 = 720+120=840.

11. Multiplying two-digit numbers less than 20.

To one of the numbers you need to add the number of units of the other, multiply this amount by 10 and add to it the product of the units of these numbers:

18 16=(18+6) 10+8 6= 240+48=288.

Using the described method, you can multiply two-digit numbers less than 20, as well as numbers that have the same number of tens: 23 24 = (23+4) 20+4 6=27 20+12=540+12=562.

Explanation:

(10+a) (10+b) = 100 + 10a + 10b + a b = 10 (10+a+b) + a b = 10 ((10+a)+b) + a b .

12. Multiplying a two-digit number by 101 .

Perhaps the simplest rule: assign your number to yourself. Multiplication is complete.
Example: 57 101 = 5757 57 --> 5757

Explanation: (10a+b) 101 = 1010a + 101b = 1000a + 100b + 10a + b
Similarly, three-digit numbers are multiplied by 1001, four-digit numbers by 10001, etc.

13. Multiplication by 22, 33, ..., 99.

To multiply a two-digit number 22.33, ...,99, you need to represent this factor as the product of a single-digit number by 11. Multiply first by a single-digit number, and then by 11:

15 33= 15 3 11=45 11=495.

14. Multiplying two-digit numbers by 111 .

First, let’s take as a multiplicand a two-digit number whose sum of digits is less than 10. Let’s explain with numerical examples:

Since 111=100+10+1, then 45 111=45 (100+10+1). When multiplying a two-digit number, the sum of the digits of which is less than 10, by 111, it is necessary to insert twice the sum of the digits (i.e., the numbers represented by them) of its tens and units 4+5=9 into the middle between the digits. 4500+450+45=4995. Therefore, 45,111=4995. When the sum of the digits of a two-digit multiplicand is greater than or equal to 10, for example 68 11, you need to add the digits of the multiplicand (6+8) and insert 2 units of the resulting sum into the middle between the digits 6 and 8. Finally, add 1100 to the composed number 6448. Therefore, 68 111 = 7548.

15. Squaring numbers consisting of only 1.

11 x 11 =121

111 x 111 = 12321

1111 x 1111 = 1234321

11111 x 11111 =123454321

111111 x 111111 = 12345654321

1111111 x 1111111 = 1234567654321

11111111 x 11111111 = 123456787654321

111111111 x 111111111 = 12345678987654321

Some non-standard multiplication techniques.

Multiplying a number by a single-digit factor.

To multiply a number by a single-digit factor (for example, 34 9) orally, you must perform actions starting from the highest digit, sequentially adding the results (30 9=270, 4 9=36, 270+36=306).

For effective mental counting, it is useful to know the multiplication table up to 19*9. In this case, multiplication is 147 8 is performed in the mind like this: 147 8=140 8+7 8= 1120 + 56= 1176 . However, without knowing the multiplication table up to 19 9, in practice it is more convenient to calculate all such examples by reducing the multiplier to the base number: 147 8=(150-3) 8=150 8-3 8=1200-24=1176, with 150 8=(150 2) 4=300 4=1200.

If one of the multiplied items is decomposed into single-digit factors, it is convenient to perform the action by sequentially multiplying by these factors, for example, 225 6=225 2 3=450 3=1350. Also, it may be easier to use 225 6=(200+25) 6=200 6+25 6=1200+150=1350.

Multiplying two-digit numbers.

1. Multiply by 37.

When multiplying a number by 37, if the given number is a multiple of 3, it is divided by 3 and multiplied by 111.

27 37=(27:3) (37 3)=9 111=999

If the given number is not a multiple of 3, then 37 is subtracted from the product or 37 is added to the product.

23 37=(24-1) 37=(24:3) (37 3)-37=888-37=851.

It is easy to remember the product of some of them:

3 x 37 = 111 33 x 3367 = 111111

6 x 37 = 222 66 x 3367 = 222222

9 x 37 = 333 99 x 3367 = 333333

12 x 37 = 444 132 x 3367 = 444444

15 x 37 = 555 165 x 3367 = 555555

18 x 37 = 666 198 x 3367 = 666666

21 x 37 = 777 231 x 3367 = 777777

24 x 37 = 888 264 x 3367 = 888888

27 x 37 = 999 297 x 3367 = 99999

2. If tens of two-digit numbers begin with the same digit, and the sum of the ones is 10 , then when multiplying them we find the product in this order:

1) multiply the ten of the first number by the ten of the second larger number by one;

2) multiply the units:

8 3x 8 7= 7221 ( 8x9=72 , 3x7=21)

5 6x 5 4=3024 ( 5x6=30 , 6x4=24)

  1. Algorithm for multiplying two-digit numbers close to 100

For example:97 x 96 = 9312

Here I use the following algorithm: if you want to multiply two

double-digit numbers close to 100, then do this:

1) find the disadvantages of factors up to a hundred;

2) subtract from one factor the deficiency of the second to a hundred;

3) add two digits to the result of the product of the shortcomings

factors up to hundreds.


The relevant literature mentions such methods of multiplication as “folding”, “lattice”, “back to front”, “diamond”, “triangle” and many others. I wanted to know what other non-standard multiplication techniques exist in mathematics? It turns out there are a lot of them. Here are some of these techniques.

Peasant method:

One of the multipliers is doubled, while the other is simultaneously decreased by the same amount. When the quotient becomes equal to one, the parallel product obtained is the desired answer.

If the quotient turns out to be an odd number, then one is removed from it and the remainder is divided. Then the products that stood opposite the odd quotients are added to the answer received

"Method of the Cross"

In this method, the factors are written one below the other and their numbers are multiplied in a straight line and crosswise.

3 1 = 3 – last digit.

2 1 + 3 3 = 11. The penultimate digit is 1, another 1 in the mind.

2 3 = 6; 6 + 1 = 7 is the first digit of the product

The required work is 713.

Sino-Japanese multiplication method.

It is no secret that teaching methods are different in different countries. It turns out that in Japan, first grade students can multiply three-digit numbers without knowing the multiplication table. For this it is used. The logic of the method is clear from the figure. After drawing, you just need to count the number of intersections in each area.

This method can be used to multiply even three-digit numbers. It is likely that when children later learn the multiplication tables, they will be able to multiply in a simpler and faster way, by columns. Moreover, the above method is too labor-intensive when multiplying numbers like 89 and 98, because you have to draw 34 stripes and count all the intersections. On the other hand, in such cases you can use a calculator. Many people will think that this method of Japanese or Chinese multiplication is too complicated and confusing, but this is only at first glance. It is visualization, that is, the image of all the points of intersection of lines (factors) on one plane, that gives us visual support, whereas the traditional method of multiplication involves a large number of arithmetic operations only in the mind. Chinese or Japanese multiplication not only helps you quickly and efficiently multiply two-digit and three-digit numbers by each other without a calculator, but also develops erudition. Agree, not everyone can boast that in practice they know the ancient Chinese method of multiplication (), which is relevant and works great in the modern world.



Multiplication can be done using a matrix table ts :

43219876=?

First we write the products of numbers.
2. Find the sums along the diagonal:
36, 59, 70, 70, 40, 19, 6
3. We get the answer from the end by adding the “extra” digits to the leading digit:2674196

Lattice method.

A rectangle is drawn, divided into squares. Next are square cells, divided diagonally. In each line we will write the product of the numbers above this cell and to the right of it, while we will write the tens digit of the product above the slash, and the units digit below it. Now we add the numbers in each oblique strip, performing this operation, from right to left. If it turns out to be greater than 10, then we write only the units digit of the sum, and add the tens digit to the next sum.

6

5

2

4

1 7

3

7

7

We write the answer numbers from left to right: 4, 5, 17, 20, 7, 5. Starting from the right, we write, adding “extra” numbers to the “neighbor”: 469075.

Got: 725 x 647 = 469075.

People rarely use the knowledge gained in algebra and geometry lessons in life. The most valuable and necessary skill associated with mathematics is the ability to do mental math quickly, so it's worth figuring out how to learn it. In everyday life, this allows you to quickly count change, calculate time, etc.

It is best to develop it from childhood, when the brain absorbs information much faster. There are several effective techniques that many people use.

How to learn to count very quickly in your head?

To achieve good results, you need to train regularly. After achieving certain goals, it is worth complicating the task. A person’s abilities are of great importance, that is, the ability to retain several things in memory at once and concentrate attention. People with a mathematical mind can achieve the most. To quickly learn to count, you need to know the multiplication table well.

The most popular calculation methods:

  1. Let's figure out how to quickly count two-digit numbers in your head if you need to multiply by 11. To understand the technique, consider one example: 13 multiplied by 11. The task is that between numbers 1 and 3 you need to insert their sum, that is, 4. As a result, it turns out that 13x11=143. When the sum of the digits gives a two-digit number, for example, if you multiply 69 by 11, then 6+9=15, then you only need to insert the second digit, that is, 5, and add 1 to the first digit of the multiplier. The result is 69x11=759. There is another way to multiply a number by 11. First, multiply by 10, and then add the original number to it. For example, 14x11=14x10+14=154.
  2. Another way to quickly count large numbers in your head works for multiplying by 5. This rule is suitable for any number that first needs to be divided by 2. If the result is an integer, then you need to add a zero at the end. For example, to find out how much 504 will be multiplied by 5. To do this, 504/2 = 252 and add 0 at the end. The result is 504x5 = 2520. If, when dividing a number, the result is not an integer, then you simply need to remove the resulting comma. For example, to find out how much 173 is multiplied by 5, you need 173/2 = 86.5, and then simply remove the comma, and it turns out that 173x5 = 865.
  3. Let's learn how to quickly count two-digit numbers in your head by adding. First you need to add tens, and then units. To get the final result, you should add the first two results. For example, let’s figure out how much 13+78 is. The first action: 10+70=80, and the second: 3+8=11. The final result will be: 80+11=91. This method can be used when you need to subtract another from one number.

Another hot topic is how to quickly calculate percentages in your head. Again, for a better understanding, let's look at an example of how to find 15% of a number. First, you should determine 10%, that is, divide by 10 and add half of the result -5%. Let's find 15% of 460: to find 10%, divide the number by 10, you get 46. The next step is to find half: 46/2=23. As a result, 46+23=69, which is 15% of 460.

There is another method for calculating interest. For example, if you need to determine how much 6% of 400 will be. First, you should find out 6% of 100 and it will be 6. To find out 6% of 400, then you need 6x4 = 24.

If you need to find 6% of 50, then you should use the following algorithm: 6% of 100 is 6, and for 50, it is half, that is, 6/2 = 3. As a result, it turns out that 6% of 50 is 3.

If the number from which you need to find a percentage is less than 100, then you should simply move the comma to the left. For example, to find 6% of 35. First, find 6% of 350 and it will be 21. The value of 6% for 35 is 2.1.