You forgot your money at home and a colleague kindly agreed to buy you lunch. On the way back, you stopped by the store for a snack, and there they announced a super promotion for your favorite chocolates. You could not resist and took 5 pieces. You were so busy shopping that you forgot about your smartphone and did not calculate how much you owed a colleague in the end. The situation is not pretty. It would be much easier to just put everything together in your mind. But ... who needs it when every phone has a calculator for a long time!

Accounting in the mind can be as fast as on a calculator. Special when it comes to household issues. The main thing is to master the techniques of fast counting and practice them periodically. In the material we present the simplest of them.

Breaking the task into parts

Even the most complex arithmetic problems can be broken down into simple ones.

Example: how do you calculate a 15% discount if the full price of the item is known?

In this case, it makes sense to split 15 into 10% and 5%. 10% is easy enough to take away, and 5% is half of 10%.

Suppose we have a product for 900 rubles, 10% of it - 90 rubles, 5% - 45. Add up: 90 + 45 \u003d 135. The final cost of the goods with a 15% discount: 900 - 135 \u003d 765 rubles.

Rounding to the nearest integer

This technique involves the use of a complement - a number that fills the gap between the given number and the number that usually ends in 00.

For example, the complementary number for 87 would be 13, since their sum is 100.

Example 1234 - 678 seems complicated. Let's round 678 to 700. It will be much easier to calculate 1234 - 700, the result is 534.

Since we subtracted too big number, then the missing result must be returned: 700 - 678 = 22, add 22 to 534 and get the final result 556.

Multiply by 11

We know how easy it is to multiply any single-digit number by 11: just repeat it twice and you're done!

But few people have the skill of multiplying two-digit and even three-digit numbers by 11.

To multiply a two-digit number by 11, you need to spread its digits in different sides and write their sum in the middle. If the sum is more than 10, then in the middle we leave the second digit from the received number, and add ten, that is, one, to the first digit.

Example 1: 36×11 = 3 (3+6) 6 = 396

Example 2: 57×11 = 5 (5+7) 7 = 627

To multiply three digit numbers:

• Leave the first and last digit numbers.
• Add the penultimate digit with the last one and write down the result. If it is more than 10, remember the one.
• Add the second number to the first number and write down the result. If there is one left from the previous addition, add it to the result.
• If as a result of the last addition one remains, add it to the first digit of the original number.

Example 3 : 869×11

1. We remember 9 as a temporary result. Result: 8...9.
2. We add 6 and 9, we get 15. We write 5 before 9, 1 - remember. Result: 8...59 (1 in mind).
3. We add 8 and 6, we get 14, we add 1 from the previous result. Result: 8559 (1 in mind).
4. We add to 8 the unit from the previous result. Result: 9559.

Multiplication of numbers from 11 to 19

You can multiply such numbers using the following algorithm:

• Any number from the range from 11 to 19 is represented as tens and ones.
• We get the formula: (10+a)×(10+b).
• Expand the brackets: 100+10×b+10×a+a×b.
• We take the common factor out of brackets and get the final formula by which we can count and which makes sense to remember: 100+10×(a+b)+a×b.

Example: 13×17

1. Let's add the units - 3+7=10.
2. Multiply the result by 10: 10×10 = 100.
3. Let's add 100: 100+100=200.
4. Multiply units: 3 × 7 = 21.
5. Let's add to the result from step 3: 200+21 = 221.

mental arithmetic

You can learn to count in your mind by mastering the techniques of mental arithmetic. First, you learn how to perform arithmetic operations on Japanese abacus - soroban. Then you practice doing the same calculations by moving the knuckles in your mind. We have already written in more detail about. Mental arithmetic courses will fully help you master the technique!

“Mathematics should already be loved because it puts the mind in order,” said Mikhail Lomonosov. The ability to count mentally remains a useful skill for modern man, despite the fact that he owns all sorts of devices capable of counting for him. The ability to do without special devices and at the right time to quickly solve the set arithmetic problem is not the only application of this skill. In addition to the utilitarian purpose, mental counting techniques will allow you to learn how to organize yourself in various life situations. In addition, the ability to count in your mind will undoubtedly have a positive effect on the image of your intellectual abilities and distinguish you from the surrounding “humanities”.

mental counting training

There are people who can perform simple arithmetic operations in their minds. Multiply a two-digit number by a one-digit number, multiply within 20, multiply two small two-digit numbers, and so on. - all these actions they can perform in the mind and quickly enough, faster than the average person. Often this skill is justified by the need for constant practical use. As a rule, people who calculate well in their minds have a mathematical education or, according to at least, experience in solving numerous arithmetic problems.

Undoubtedly, experience and training plays a crucial role in the development of any ability. But the skill of mental counting is not based on experience alone. This is proved by people who, unlike those described above, are able to calculate in their minds much more complex examples. 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 do you need to know and be able to ordinary person to master such a phenomenal ability? Today, there are various techniques that help you learn how to quickly count in your mind. Having studied many approaches to teaching the skill of counting orally, we can distinguish 3 main components of this skill:

1. Ability. The ability to concentrate attention and the ability to keep 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 desired, most effective algorithm in each specific situation.

3. Training and experience, whose value for any skill has not been canceled. Constant training and the gradual complication of tasks and exercises will allow you to improve the speed and quality of mental arithmetic.

It should be noted that the third factor is of key importance. Without the necessary experience, you will not be able to surprise others quick score, even if you know the most convenient algorithm. However, do not underestimate the importance of the first two components, because having the ability and a set of necessary algorithms in your arsenal, you can outdo even the most experienced "bookkeeper", provided that you have been training for the same time.

Lessons on the site

The oral counting lessons presented on the site are aimed precisely at the development of these three components. The first lesson tells how to develop a predisposition for mathematics and arithmetic, as well as the basics of counting and logic. Then a number of lessons are given on special algorithms for performing various arithmetic operations in the mind. Finally, this training presents Additional materials, helping to train and develop the ability to count orally, in order to be able to apply your talent and your knowledge in life.

To multiply any two digit number by 11, just add these 2 numbers together and put their sum in the middle.

For example, if you want to multiply 53 by 11, add 5+3 to get an 8 and place it in the middle between 5 and 3 and this will give the correct answer 583.

If the sum of two digits is 10 or more, just add that number to the left digit. For example, if you want to multiply 97 by 11, add 9+7 = 16. Put 6 in the middle and add 1 to 9, which gives the correct answer - 1067.

Division by 5

When dividing by 5, multiply by 2 and remove the 0 at the end of the number.

For example, 480 divided by 5. Multiply by 2 (960) and remove 0. We get 96.

Now divide by 5 the following numbers yourself: 540, 290, 770, 1450. And check with a calculator!

It gives a moment of celebration.

When multiplied by 5 divide by 2 and assign 0.

Example. Multiply 480 by 5. Divide by 2, we get 240. Add 0. 2400.

Multiply by 5 yourself: 540, 290, 770, 1450

Multiply by 5, 50, 500

As you know, children love to multiply by 10, 100, 1000. You can also quickly and easily multiply by 5, 50, 500, especially even numbers.

68 x 5 = 34: 10 = 340

68 x 50 = (68:2) x 100 = 3400

It is possible and odd:

17 x 50 = (16 + 1) x 50 = 8 x 100 = 850

Division by 5, 50, 500

Everything happens in the reverse order: first, we double the dividend and discard 1, 2 or 3 zeros. For example:

135: 5 = (135 x 2): 10 = 27

2150: 50 = 2150 x 2: 100 = 4300: 100 = 43

Multiply by 25

24 x 25 = 24: 4 x 100 = 600 - easy when even. Odd ones are represented as a sum of terms (or a difference). For example:

37 x 25 = (36 + 1) x 25 = 36: 4 x 10 + 25 = 925

Multiply by 26 and by 24

We replace the terms 26 and 24 with the sum:

36 x 26 = 36 x (25 + 1) = 36: 4 x 100 + 36 = 936

36 x 24 = 36 x (25 - 1) = 900 - 36 = 864

When divided by 25 everything happens in reverse order:

360: 25 = (360 x 2) x 2 x 100 = 1440: 100 = 14.4

225: 25 = (225 x 2) x 2: 100 = 9.

Multiply by 125 is division by 8 and multiplication by 1000:

42 x 125 = 88: 8 x 1000 = 11,000

If the number is not divisible by 8, then we use one of the following methods:

42 x 125 = 40: 8 x 1000 + 2 x 125 = 5000 + 250 = 5250.

Multiply by 9, 99, 999

It is convenient to replace with 10 - 1, 100 - 1, 1000 - 1

Multiply even numbers by 15

We divide the number by 2 and add it to the desired number, then we multiply everything by 10. This technique only works for even numbers. For example:

14 x 15 = (14:2 + 14) x 10 = 21 x 10 = 210

26:15 = (26:2 + 26) x 10 = 39 x 10 = 390

The odd ones are presented as the sum of terms

23 x 15 = (22 + 1) x 15 = (22: 2 + 22) x 10 +15 = 330 +15 = 345

Using this technique, you can multiply by 16 and 14 - (15 +1) and (15 - 1):

66 x 16 = 66 x (15 + 1) = (66: 2 + 66) x 10 + 66 = 1156

Multiply numbers ending in 5 by themselves

35 x 35 \u003d 3 x 4 and we attribute 5 x 5, i.e. 35 x 35 = 1225

Multiply by 11 and by 111

a) 32 x 11 = 32 x 10 + 32 = 352

b) we push the numbers 3 and 2 and insert their sum between them: 3 5 2

c) when multiplied by 111, let's say 25:

Expanding the numbers of the multiplicand

Finding their sum

We enter it 2 times already:

25 x 111 = 2 7 7 5

If the sum of the digits of a two-digit number is greater than 10, then we do this:

The number of tens of the multiplier is increased by 1,

Expanding tens and ones

We enter the units of the sum of tens and units of the multiplicand:

78 x 11 = (7+1) (7+8) 8 = 8 15 8 = 858

d) to multiply a three-digit number by 11, you need:

Leave the hundreds and ones in place

Assign the sum of hundreds and tens of the multiplicand

Assign the sum of tens and ones

115 x 11 = 1 (1+1) (1+5) 5 = 1265

Addition of several consecutive natural numbers.

a) to add several consecutive numbers of the natural series (odd number), it is necessary to multiply the term in the middle by the number of terms:

6 + 7 + 8 + 9 + 10 = 8 x 5 = 40

b) if there are an even number of numbers, then we take two terms in the middle and multiply their sum by half the number of terms

6 + 7 + 8 + 9 + 10 + 11 = 8 + 9 x 3 = 51

INTRODUCTION

At all times, mathematics has been and remains one of the main subjects in school, because mathematical knowledge is necessary for all people. Not every student, studying at school, knows what profession he will choose in the future, but everyone understands that mathematics is necessary for solving many life problems: calculations in a store, payment for utilities, calculation family budget etc. In addition, all schoolchildren need to take exams in the 9th grade and in the 11th grade, and for this, starting from the 1st grade, it is necessary to master mathematics with high quality, and above all, you need to learn how to count.

Is it possible to imagine a world without numbers? Without numbers, you won’t make a purchase, you won’t know the time, you won’t dial a phone number. And what about spaceships, lasers and all other technical achievements?! They would simply be impossible if it were not for the science of numbers.

Two elements dominate mathematics - numbers and figures with their infinite variety of properties and relationships. In my work, preference is given to the elements of numbers and actions with them.

Now, at the stage of the rapid development of informatics and computer technology, modern schoolchildren do not want to bother themselves with mental arithmetic. So I decidedshow not only that the process of performing an action can be important, but also an interesting activity.

Target: to study the methods of fast counting, to show the need for their application to simplify calculations.

In accordance with the goal, the tasks:

1. Investigate whether students use quick counting techniques.
2. Learn quick counting techniques that you can use to make calculations easier.
3. Make a memo for students in grades 5-6 to use quick counting techniques.

Object of study:quick counting techniques.

Subject of study: calculation process.

Research hypothesis:if it is shown that the use of fast counting techniques facilitates calculations, then it can be achieved that the computational culture of students will increase, and it will be easier for them to solve practical problems.

The following were used in the work tricks and methods : survey (questionnaire), analysis (statistical data processing), work with information sources, practical work, observations.

This work refers toapplied research, because it shows the role of applying fast counting techniques for practical activities.

While working on a report, Iused the following methods:

1. search a method using scientific and educational literature, as well as searching for the necessary information on the Internet;
2. practical method of performing calculations using non-standard counting algorithms;
3. analysis data obtained during the study.

Relevance my research is that in our time more and more often calculators come to the aid of students, and an increasing number of students cannot count orally. But the study of mathematics develops logical thinking, memory, flexibility of the mind, accustoms a person to accuracy, to the ability to see the main thing, provides the necessary information for understanding challenging tasks arising in various fields of activity of modern man. Therefore, in my work, I want to show how you can count quickly and correctly and that the process of performing actions can be not only useful, but also interesting. It is the use of non-standard techniques in the formation of computational skills that enhances students' interest in mathematics and contributes to the development of mathematical abilities.

Behind the simple operations of addition, subtraction, multiplication and division lie the secrets of the history of mathematics. Accidentally heard the words "multiplication by a lattice", "chess way" intrigued. I wanted to know these and other methods of calculation, as well as compare them with today's ones.

Can you count? The question, perhaps even offensive for a person older than three years of age. Who can't count? Everyone will answer that for this, special art is not required. And he will be right. But the question is how to count? You can count on a calculator, you can count as a column in a notebook, or you can count verbally using quick counting techniques. I count very quickly verbally, I almost never solve in a column, in writing, all because I know and apply various methods of fast counting. Of my classmates, few people can count quickly orally, and I wanted to find out if they know the tricks of quick counting, if not, then help them master these tricks, for this purpose, compose a memo for them with quick counting tricks.

In order to find out whether modern schoolchildren know other ways to perform arithmetic operations, except for multiplication, addition, subtraction by a column and division by a "corner" and would like to learn new ways, a test survey was conducted.

To begin with, I conducted a survey in the 6th grade of our school. Asked the guys simple questions. Why do you need to know how to count? When studying what school subjects correct account required? Do they know how to count quickly? Would you like to learn how to count quickly orally? (Appendix I).

61 people took part in the survey. After analyzing the results, I concluded that the majority of students believe that the ability to count is useful in life and is necessary at school, especially when studying mathematics, physics, chemistry, computer science and technology. Several students know how to count quickly, and almost everyone would like to learn how to count quickly. (The results of the survey are reflected in the diagrams) (Appendix II).

After statistical processing of the data, I concluded that not all students know quick counting techniques, so it is necessary to make quick counting techniques for students in grades 5-6 in order to use them when performing calculations.

Survey results:

Question	5th grade	6 classes	Total
	Yes	No	don't know	Yes	No	don't know
Would you like to know?

Summary table of the survey:

Question	5, 6 grades
	Yes	No	don't know
Is it necessary for a modern person to be able to perform arithmetic operations with natural numbers?
Can you multiply, add, subtract numbers in a column, divide by a “corner”?
Do you know other ways to do arithmetic?
Would you like to know?

According to the results of the survey, it can be concluded that in most cases modern schoolchildren do not know other ways to perform actions other than multiplication, addition, subtraction by a column and division by a “corner”, since they rarely refer to material that is outside the school curriculum.

Chapter I. HISTORY OF THE ACCOUNT

1. HOW THE NUMBERS ARISED

People learned to count objects back in the ancient Stone Age - the Paleolithic, tens of thousands of years ago. How did it happen? At first, people only compared by eye different quantities identical items. They could determine which of the two piles had more fruit, which herd had more deer, and so on. If one tribe exchanged caught fish for stone knives made by people of another tribe, it was not necessary to count how many fish they brought and how many knives. It was enough to put a knife next to each fish for the exchange between the tribes to take place.

In order to be successful agriculture, arithmetic knowledge was required. Without counting days, it was difficult to determine when to sow the fields, when to start watering, when to expect offspring from animals. It was necessary to know how many sheep were in the flock, how many sacks of grain were put in the barns.
And more than eight thousand years ago, the ancient shepherds began to make mugs of clay - one for each sheep. To find out if at least one sheep was lost during the day, the shepherd put aside a mug each time the next animal entered the pen. And only after making sure that the same number of sheep returned as there were circles, he calmly went to sleep. But in his flock were not only sheep - he grazed cows, and goats, and donkeys. Therefore, other figures had to be made of clay. And farmers with the help of clay figurines kept records harvested crop, noting how many sacks of grain are put in the barn, how many jugs of oil are squeezed out of olives, how many pieces of linen are woven. If the sheep bore offspring, the shepherd added new mugs to the mugs, and if some of the sheep went for meat, several mugs had to be removed. So, still not knowing how to count, ancient people were engaged in arithmetic.

Then numerals appeared in the human language, and people were able to name the number of objects, animals, days. Usually there were few such numerals. For example, the Murray River tribe in Australia had two prime numbers: enea (1) and petcheval (2). They expressed other numbers with compound numerals: 3 = “petcheval-enea”, 4 “petcheval-petcheval”, etc. Another Australian tribe, the Camiloroi, had simple numerals mal (1), bulan (2), guliba (3). And here other numbers were obtained by adding smaller ones: 4="bulan-bulan", 5="bulan-guliba", 6="guliba-guliba", etc.

For many peoples, the name of the number depended on the items being counted. If the inhabitants of the Fiji Islands counted boats, then the number 10 was called "bolo"; if they counted coconuts, then the number 10 was called "karo". The Nivkhs living on Sakhalin near the banks of the Amur did the same. Back in the 19th century, they called the same number different words if they counted people, fish, boats, nets, stars, sticks.

We still use different indefinite numerals with the meaning "a lot": "crowd", "herd", "flock", "heap", "bundle" and others.

With the development of production and trade, people began to better understand what three boats and three axes, ten arrows and ten nuts have in common. The tribes often engaged in item-for-item exchanges; for example, they exchanged 5 edible roots for 5 fish. It became clear that 5 is the same for both roots and fish; so it can be called with one word.

Similar counting methods were used by other peoples. So there were numberings based on counting by fives, tens, twenties.

So far, I have talked about mental counting. How were the numbers written? At first, even before the advent of writing, they used notches on sticks, notches on bones, knots on ropes. The found wolf bone in Dolni-Vestonice (Czechoslovakia) had 55 cuts made more than 25,000 years ago.

When writing appeared, there were also numbers for writing numbers. At first, the numbers looked like notches on sticks: in Egypt and Babylon, in Etruria and Dates, in India and China, small numbers were written with sticks or dashes. For example, the number 5 was written with five sticks. The Aztecs and Mayans used dots instead of sticks. Then special signs appeared for some numbers, such as 5 and 10.

At that time, almost all numbering was not positional, but similar to Roman numbering. Only one Babylonian sexagesimal numbering was positional. But for a long time there was also no zero in it, as well as a comma separating the integer part from the fractional one. Therefore, the same figure could mean 1, 60, and 3600. One had to guess the meaning of the number according to the meaning of the problem.

Several centuries before new era invented new way writing numbers, in which the letters of the ordinary alphabet served as digits. The first 9 letters denoted the numbers tens 10, 20, ..., 90, and another 9 letters denoted hundreds. This alphabetical numbering was used until the 17th century. To distinguish “real” letters from numbers, a dash was placed above the letters-numbers (in Russia this dash was called “titlo”).

In all these numberings, it was very difficult to perform arithmetic operations. Therefore, the invention in the VI century by the Indians of decimal positional numbering is rightfully considered one of the greatest achievements of mankind. Indian numbering and Indian numerals became known in Europe from the Arabs and are usually referred to as Arabic.

When writing fractions for a long time, the whole part was recorded in the new decimal numbering, and the fractional part in sexagesimal. But at the beginning of the XV century. Samarkand mathematician and astronomer al-Kashi began to use decimal fractions in calculations.

The numbers we work with are positive and negative numbers. But it turns out that these are not all the numbers that are used in mathematics and other sciences. And you can find out about them without waiting high school, and much earlier, if you study the history of the emergence of numbers in mathematics.

Chapter II. OLD METHODS OF CALCULATION

2.1. RUSSIAN PEASANT METHOD OF MULTIPLICATION

In Russia, several centuries ago, among the peasants of some provinces, a method was spread that did not require knowledge of the entire multiplication table. It was only necessary to be able to multiply and divide by 2. This method was called PEASANT (there is an opinion that it originates from the Egyptian).

Example: multiply 47 by 35,

1. write the numbers on one line, draw a vertical line between them;
2. we will divide the left number by 2, multiply the right number by 2 (if a remainder occurs during division, then we discard the remainder);
3. the division ends when a unit appears on the left;
4. we cross out those lines in which there are even numbers on the left;35 + 70 + 140 + 280 + 1120 = 1645
5. then add the remaining numbers to the right - this is the result.

2.2. GRID METHOD

The outstanding Arab mathematician and astronomer Abu Abdalah Mohammed Ben Mussa al-Khwarizmi lived and worked in Baghdad. The scientist worked in the House of Wisdom, where there was a library and an observatory, almost all major Arab scientists worked here.

There is very little information about the life and work of Muhammad al-Khwarizmi. Only two of his works have survived - on algebra and on arithmetic. In the last of these books, four rules of arithmetic are given, almost the same as those used today.

	1	3
	0	1

In his "The Book of Indian Counting"the scientist described a method invented in ancient india, and later named"GRID METHOD". This method is even simpler than the one used today.

Example: multiply 25 and 63.

Let's draw a table in which two cells in length and two in width, we write one number in length and another in width. In the cells we write the result of multiplying these numbers, at their intersection we separate the tens and ones with a diagonal. We add the resulting numbers diagonally, and the result can be read along the arrow (down and to the right).

I have considered a simple example, however, any multi-digit numbers can be multiplied in this way.

Consider another example: multiply 987 and 12:

1. draw a 3 by 2 rectangle (according to the number of decimal places for each factor);
2. then we divide the square cells diagonally;
3. at the top of the table we write the number 987;
4. on the left of the table the number 12;
5. now in each square we enter the product of numbers located in the same line and in the same column with this square, tens below the diagonal, ones above;
6. after filling in all the triangles, the numbers in them are added along each diagonal on the right side;
7. the result is read by the arrow.

This algorithm for multiplying two natural numbers was common in the Middle Ages in the East and Italy.

I would like to note the inconvenience of this method in the laboriousness of preparing a rectangular table, although the calculation process itself is interesting and filling in the table resembles a game.

2.3. MULTIPLICATION ON FINGERS

The ancient Egyptians were very religious and believed that the soul of the deceased in afterlife subjected to a finger counting test. This already speaks of the importance that the ancients attached to this method of performing the multiplication of natural numbers (it was calledFINGER ACCOUNT).

They multiplied single-digit numbers from 6 to 9 on the fingers. To do this, they extended as many fingers on one hand as the first multiplier exceeded the number 5, and on the second they did the same for the second multiplier. The rest of the fingers were bent. After that, they took as many tens as the fingers extended on both hands, and added to this number the product of the bent fingers on the first and second hands.

Example: 8 ∙ 9 = 72

Later, the finger count was improved - they learned to show numbers up to 10,000 with the help of fingers.

finger movement - this is another way to help memory: with the help of fingers, remember the multiplication table for 9. Putting both hands side by side on the table, we number the fingers of both hands in order as follows: the first finger on the left will be denoted by 1, the second after it will be denoted by the number 2, then 3 , 4 ... up to the tenth finger, which means 10. If you need to multiply by 9 any of the first nine numbers, then for this, without moving your hands from the table, you need to lift up the finger whose number means the number by which nine is multiplied; then the number of fingers to the left of the raised finger determines the number of tens, and the number of fingers to the right of the raised finger indicates the number of units of the resulting product (see for yourself).

So, the old multiplication methods we have considered show that the algorithm for multiplying natural numbers used in school is not the only one and it was not always known.

However, it is quite fast and most convenient.

Chapter III. ORAL COUNTING - GYMNASTICS OF THE MIND

3.1. DIFFERENT WAYS OF ADDITION AND SUBTRACTION

ADDITION

The basic rule for doing mental addition is:

To add 9 to a number, add 10 to it and subtract 1; to add 8, add 10 and subtract 2; to add 7, add 10 and subtract 3, and so on. For example:

56+8=56+10-2=64;

65+9=65+10-1=74.

ADDITION IN THE MIND OF TWO-DIGITAL NUMBERS

If the number of units in the added number is greater than 5, then the number must be rounded up, and then subtract the rounding error from the resulting amount. If the number of units is less, then we add tens first, and then units. For example:

34+48=34+50-2=82;

27+31=27+30+1=58.

ADDITION OF THREE-DIGIT NUMBERS

We add from left to right, that is, first hundreds, then tens, and then ones. For example:

359+523= 300+500+50+20+9+3=882;

456+298=400+200+50+90+6+8=754.

SUBTRACTION

To subtract two numbers in your head, you need to round the subtracted, and then correct the resulting answer.

56-9=56-10+1=47;

436-87=436-100+13=349.

SUBTRACT A NUMBER LESS THAN 100 FROM A NUMBER OVER 100

If the subtrahend is less than 100 and the minuend is greater than 100 but less than 200, there is an easy way to calculate the difference in your mind. 134-76=58

76 is 24 less than 100. 134 is 34 more than 100. Add 24 to 34 and get the answer: 58.

152-88=64

88 is 12 less than 100, and 152 is more than 100 by 52, so

152-88=12+52=64

3.2. DIFFERENT WAYS OF MULTIPLICATION AND DIVISION

After studying the literature on this topic, I made a selection, from a variety of quick counting techniques, I chose multiplication and division techniques that are easy to understand and use for any student. I included these techniques in the memo (Appendix III), which will be useful for students in grades 5-6.

1. Multiplying and dividing a number by 4.

To multiply a number by 4, you need to multiply it by 2 twice.

For example:

26 4=(26 2) 2=52 2=104;

417 4=(417 2) 2=834 2=1668.

To divide a number by 4, you need to divide it twice by 2.

For example:

324:4=(324:2):2=162:2=81.

1. Multiplying and dividing a number by 5.

To multiply a number by 5, you need to multiply it by 10 and divide by 2.

For example:

236 5=(236 10):2=2360:2=1180.

To divide a number by 5, you need to multiply 2 and divide by 10, i.e. separate the last digit with a comma.

For example:

236:5=(236 2):10=472:10=47.2.

1. Multiplying a number by 1.5.

To multiply a number by 1.5, you need to add half of it to the original number.

For example: 34 1.5=34+17=51;

146 1.5=146+73=219.

1. Multiplying a number by 9.

To multiply a number by 9, add 0 to it and subtract the original number.

For example: 72 9=720-72=648.

1. Multiply by 25 a number divisible by 4.

To multiply by 25 a number that is divisible by 4, you need to divide it by 4 and multiply the resulting number by 100.

For example: 124 25=(124:4) 100=31 100=3100.

1. Multiplying a two-digit number by 11

When multiplying a two-digit number by 11, you need to enter the sum of these digits between the ones digit and the tens digit, and if the sum of the digits is more than 10, then one must be added to the highest digit (first digit).

For example:
23 11=253, because 2+3=5, so between 2 and 3 we put the number 5;
57 11=627, because 5+7=12, put the number 2 between 5 and 7, and add 1 to 5, write 6 instead of 5.

“Fold the edges, put them in the middle” - these words will help you remember easily this way multiplication by 11.

This method is only suitable for multiplying two-digit numbers.

1. Multiplying a two-digit number by 101.

In order to multiply a number by 101, you need to attribute given number to himself.

For example: 34 101 = 3434.

To clarify, 34 101 = 34 100+34 1=3400+34=3434.

1. Squaring a two-digit number ending in 5.

To square a two-digit number ending in 5, you need to multiply the tens digit by a digit greater than one, and add the number 25 to the right of the resulting product.
For example: 35 2 =1225, i.e. 3 4 \u003d 12 and we attribute 25 to 12, we get 1225.

1. Squaring a two-digit number starting with 5.

To square a two-digit number starting with five, you need to add the second digit of the number to 25 and assign the square of the second digit to the right, and if the square of the second digit is a single-digit number, then the number 0 must be assigned before it.

For example:
52 2 = 2704, because 25+2=28 and 2 2 =04;
58 2 = 3364, because 25+8=33 and 82=64.

3.3. GAMES

Guessing the received number.

1. Think of a number. Add 11 to it; multiply the amount received by 2; subtract 20 from this product; multiply the resulting difference by 5 and subtract a number from the new product that is 10 times the number you intended.I guess you got 10. Right?
2. Think of a number. Treat him. Subtract 1 from the result. Multiply the result by 5. Add 20 to the result. Divide the result by 15. Subtract the intended result from the result.You got 1.
3. Think of a number. Multiply it by 6. Subtract 3. Multiply by 2. Add 26. Subtract twice what you thought. Divide by 10. Subtract what you thought.You got 2.
4. Think of a number. Triple it. Subtract 2. Multiply by 5. Add 5. Divide by 5. Add 1. Divide by what you thought.You got 3.
5. Think of a number, double it. Add 3. Multiply by 4. Subtract 12. Divide by what you thought.You got 8.

Guessing the given numbers.

1. Invite your friends to think of any numbers. Let everyone add 5 to their intended number.
2. Let the resulting sum be multiplied by 3.
3. Let subtract 7 from the product.
4. Let's subtract 8 more from the result.
5. Let everyone give you a sheet with the final result. Looking at the sheet, you immediately tell everyone what number he has in mind.

(To guess the conceived number, the result, written on a piece of paper or told to you orally, is divided by 3).

CONCLUSION

We have entered the new millennium! Grand discoveries and achievements of mankind. We know a lot, we can do a lot. It seems something supernatural that with the help of numbers and formulas one can calculate the flight of a spaceship, the "economic situation" in the country, the weather for "tomorrow", describe the sound of notes in a melody. We know the statement of the ancient Greek mathematician, philosopher, who lived in the 4th century BC. - Pythagoras - "Everything is a number!".

Describing ancient ways of computing and modern techniques quick calculation, I tried to show that both in the past and in the future, one cannot do without mathematics, a science created by the human mind.

The study of ancient methods of calculation showed that these arithmetic operations were difficult and complex due to the variety of methods and their cumbersome execution.

Modern methods of computing are simple and accessible to everyone.

When getting acquainted with the scientific literature, I discovered faster and more reliable methods of calculation.

It is possible that the first time many will not be able to quickly, on the go, perform these or other calculations. Let at first fail to use the technique shown in the work. No problem. Constant computational training is needed. Lesson after lesson, year after year. It will help to acquire useful oral counting skills.

The German scientist Karl Gauss was called the king of mathematicians. His mathematical talent manifested itself already in childhood. Once at school (Gauss was 10 years old), the teacher asked the class to add up all the numbers from 1 to 100. While he was dictating the task, Gauss already had an answer ready. On his slate board it was written: 101 50=5050. How did he calculate? It's very simple - he applied the fast counting technique, he added the first number to the last, the second to the penultimate one, and so on. There are only 50 such sums and each is equal to 101, so he was able to give the correct answer almost instantly.

1+2+…+50+51+...+99+100=(1+100)+(2+99)+…+(50+51)=101 50=5050. This example shows best of all that it is possible to count quickly and correctly orally almost to all schoolchildren, for this you just need to know the methods of quick counting.

I designed the results of my work in a memo that I will offer to all my classmates, and I will also place it on the school thematic stand “It's interesting!”. It is possible that from the first time not everyone will be able to quickly, on the move, perform calculations using these techniques, even if at first you can’t use the technique shown in the memo, it’s okay, you just need constant computational training. It will help you to acquire useful skills of quick counting.

After statistical processing of the data, the following results were obtained. results:

1. You need to be able to count, because it will come in handy in life, 93% of students believe that in order to study well at school - 72%, to decide quickly - 61%, to be literate - 34% and it is not necessary to be able to count - only 3%.
2. Skills good score 100% of students believe that they are necessary in the study of mathematics, as well as in the study of physics - 90%, chemistry - 80%, informatics - 44%, technology - 36%.
3. 16% (many tricks), 25% (several tricks) know quick counting tricks, 59% of students do not know quick counting tricks.
4. 21% of students use fast counting techniques, sometimes they are used by 15%.
5. 93% of students would like to learn how to quickly count.

Conclusions:

1. Knowledge of fast counting techniques allows you to simplify calculations, save time, develop logical thinking and flexibility of mind.
2. There are practically no quick counting techniques in school textbooks, so the result of this work - a quick counting guide will be very useful for students in grades 5-6.

LIST OF USED LITERATURE

1. Vantsyan A.G. Mathematics: Textbook for grade 5. - Samara: Publishing House Fedorov, 1999
2. Kordemsky B.A., Akhadov A.A. amazing world numbers: Book of students, - M. Enlightenment, 1986.
3. Minskykh E.M. "From game to knowledge", M., "Enlightenment", 1982
4. Svechnikov A.A. Numbers, figures, tasks. M., Enlightenment, 1977. Yes No Don't know https://accounts.google.com

We are taught counting skills from childhood. These are the elementary operations of addition, subtraction, multiplication and division. In the case of small numbers, they are easily handled even junior schoolchildren, but the task becomes much more complicated when you need to perform an action with a two-digit or three-digit number. However, with the help of training, simple exercises and little tricks, it is quite possible to subordinate these operations to quick mental processing.

You may ask why this is necessary, because there is such a handy thing as a calculator, and in extreme cases, there is always paper at hand for making calculations. Quick mental arithmetic has many advantages:

Opportunity to address other aspects of the problem. Often, tasks contain at least two sides: purely arithmetic (operations with numbers) and intellectual and creative (choosing an appropriate solution for a specific task, a non-standard approach for a faster solution, etc.). If the student does not cope well and quickly with the first side, then the second side suffers from this: concentrating on the implementation of the arithmetic component, the child does not think about the meaning of the task, may not see the catch or more simple solution. If the counting operations are brought to automatism or simply do not require a large number time, then a detailed consideration of the meaning of the task “turns on”, it becomes possible to apply creativity To her.

Intelligence training. Accounting in the mind allows you to keep your intellect in good shape, constantly use thought processes. This is especially true for actions with big numbers when we select a method to simplify the operation as much as possible.

Table exercises

The exercises are designed for children of any age who have difficulty performing operations with prime numbers(single and double digits). Allows you to train the skills of oral counting, to bring simple arithmetic operations to automaticity.

Necessary materials: to complete the exercises, you will need a grid of one- and two-digit numbers. Example:

The first column contains the numbers with which you need to perform actions. In the second - the answers to these actions. Using a specially cut bookmark, you can check the correctness of the calculation. For example:

Exercise options:

Sequentially add in your mind the pairs of numbers in the grid. Say the answer out loud and check yourself with the second column and bookmark. The task can be performed at a free pace or for a while.

Sequentially subtract the numbers in your mind from the grid.

Sequentially add in your mind the pairs of numbers in the grid. Add the number 5 to each sum and say the answer aloud.

Sequentially put together in your mind the triplets of numbers in the grid.

Consistently with all the numbers in the grid, do the following: add the bottom number, subtract the next number in the column from the resulting amount.

On the basis of such tables, any tasks can be formed. Grids are compiled depending on the modification of the exercise.

IMPORTANT! For the exercise to give results, it must be performed regularly, until the skill is fully mastered.

Mastering multiplication

The exercise is intended for children who have mastered the multiplication table from 1 to 10. It trains the skill of multiplying a two-digit number by a one-digit number.

A column is made up of arbitrary two-digit numbers. Task for the child: successively multiply these numbers first by 1, then by 2, by 3, etc. The answer is spoken aloud. It is executed until the answers are remembered and will not be issued automatically.

The main thing is attention

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

Name the answer. Check yourself with a calculator.

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

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

Possible examples:

3000 – 700 — 60 – 500 — 40 – 300 -20 – 100 =

100:2:2*3*2 + 50 – 100 + 200 – 30 =

If most of the examples are solved with errors (BUT! not related to the ability to count in principle), then it makes sense to increase the concentration of attention. For this you can:

Minimize external stimuli. For example, if possible, go to another room, turn off the music, close the window, etc. If you need to focus on an example during a lesson, when there is no way to go out and achieve complete silence, you need to close your eyes and imagine the numbers with which the actions are carried out.

Add an element of contention. Knowing what is right and fast decision brings victory over the opponent and/or some kind of reward, the student is more willing to focus on the numbers and make the most effort in the calculation process.

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

Grouping. Sequentially adding / subtracting several numbers, you need to see which of them, when added / subtracted, will give an integer: 13 and 67, 98 and 32, 49 and 11, etc. First, perform actions with these numbers, and then move on to the rest. Example: 7+65+43+82+64+28=(7+43)+(82+28)+65+64=50+110+124=289

Decomposition into tens and units. When multiplying two two-digit numbers (for example, 24 and 57), it is advantageous to decompose one of them (ending in a smaller digit) into tens and ones: 24 as 20 and 4. The second number is multiplied first by tens (57 by 20), then by units ( 57 by 4). Then both values ​​are added. Example: 24×57=57×20+57×4=1140+228=1368

Multiply by 5. When multiplying any number by 5, it is more profitable to first multiply it by 10, and then divide by 2. Example: 45×5=45×10/2=450/2=225

Multiply by 4 and 8. When multiplying by 4, it is more profitable to multiply the number twice by 2; by 8 - three times by 2. Example: 63x4=63x2x2=126x2=252

Division by 4 and 8. Similar to multiplication: when dividing by 4, divide the number twice by 2, by 8 - three times by 2. Example: 192/8=192/2/2/2=96/2/2=48/2=24

Squaring numbers ending in 5. The following algorithm will facilitate this action: the number of tens, the squared number, is multiplied by the same plus one and is attributed at the end to 25. Example: 75^2=7x(7+1)=7×8=5625

Formula multiplication. In some cases, to facilitate the calculation, you can apply the difference of squares formula: (a+b)x(a-b)=a^2-b^2. Example: 52×48=(50+2)x(50-2)=50^2-2^2=2500-4=2496

P.S. These rules can greatly simplify mental counting, but regular training is necessary so that you can correctly use the rule at the right time. Therefore, it is recommended to solve such a number of examples for each of them, which will allow you to automate the skill. To begin with, you can write down the calculations on paper, gradually reducing the amount of writing and translating operations into a mental plan. At first, it is also recommended to check your answers with a calculator or standard calculations in a column.