Mastering verbal arithmetic. Mental Counting: A Quick Mental Counting Technique What is Mental Counting

learning verbal arithmetic

This list of a few little-known math tricks will show you how to do mental arithmetic faster than 5 times 10, and your friends can use you as a calculator.

1. Multiply by 11
We all know how to quickly multiply a number by 10, you just need to add a zero at the end, but did you know that there is a trick on how to easily multiply a two-digit number by 11?
Let's say we need to multiply 63 by 11. Take a two-digit number that needs to be multiplied by 11 and imagine a place between its two digits:
6_3
Now add the first and second digits of this number and place in this location:
6_(6+3)_3
And our multiplication result is ready:
63*11=693
If the result of adding the first and second digits is a two-digit number, insert only the second digit, and add one to the first digit of the original number:
79*11=
7_(7+9)_9
(7+1)_6_9
79*11=869

2. Fast squaring a number ending in 5
If you need to square a two-digit number ending in 5, then you can do it very simply in your mind. Multiply the first digit of the number by itself plus one and add 25 at the end and that's it:
45*45=4*(4+1)_25=2025

3. Multiply by 5
For most people, multiplying by 5 is easy for small numbers, but how do you quickly mentally count large numbers multiplied by 5?
You need to take this number and divide by 2. If the result is an integer, then add 0 at the end to it, if not, discard the remainder and add 5 at the end:
1248*5=(1248/2)_(0 or 5)=624_(0 or 5)=6240 (the result of dividing by 2 is an integer)
4469*5=(4469/2)_(0 or 5)=(2234.5)_(0 or 5)=22345 (result of dividing by 2 with remainder)

4. Multiply by 4
This is a very simple and, at first glance, obvious feature of multiplying any number by 4, but despite this, people do not know about it at the right time. To simply multiply any number by 4, you need to multiply it by 2, and then multiply by 2 again:
67*4=67*2*2=134*2=268

5. Calculate 15%
If you need to mentally calculate 15% of any number, then there is an easy way to do it. Take 10% of the number (dividing the number by 10) and add half of the resulting 10% to that number.
15% of 884 rubles \u003d (10% of 884 rubles) + ((10% of 884 rubles) / 2) \u003d 88.4 rubles + 44.2 rubles \u003d 132.6 rubles

6. Multiplication of large numbers
If you need to multiply large numbers in your mind and one of them is even, then you can use the method of simplifying the factors, reducing the even number by half, and the second one by doubling:
32*125 is
16*250 is
8*500 is
4*1000=4000

7. Divide by 5
Dividing a large number by 5 in your head is very easy. All you need to do is multiply the number by 2 and move the decimal point back by one:
175/5
Multiply by 2: 175*2=350
Shift by one sign: 35.0 or 35
1244/5
Multiply by 2: 1244*2=2488
Shift by one sign: 248.8

8. Subtraction from 1000
To subtract a large number from a thousand, follow a simple technique, subtract all digits from 9 except the last, and subtract the last digit from 10:
1000-489=(9-4)_(9-8)_(10-9)=511

Of course, in order to learn how to quickly count in your mind, you need to practice using these techniques many times to bring them to automatism, a single reading will leave only zeros in your head.

MOU "Brekhovskaya basic comprehensive school"

Oral counting in mathematics lessons.

From the experience of V.,

With. Brekhovo 2010

Come on, pencils aside!

No knuckles, no pens, no chalk.

Verbal counting! We're doing this thing

Only by the power of the mind and soul.

Numbers converge somewhere in the darkness

And the eyes start to glow

And around only smart faces.

Verbal counting! We count in our minds.

At the beginning of each math lesson, I conduct an oral count, during which I teach children to reason, think, analyze, compare, generalize, identify patterns, teach quick and rational methods of oral calculations. I work on the development of such mental qualities as perception, attention, imagination, memory, thinking. In addition, I develop the ability to quickly switch from one type of activity to another.

I have the following requirements for the organization of the oral account:

amusement

Originality

Diversity

Systematic

Cognitiveness

Sequence.

During mental counting, I use entertaining tasks, rebuses, puzzles, games, magic squares, riddles, and various types of oral folk art. Applying a wide variety of tasks, creating an atmosphere of interest, creativity, cooperation, I educate children in independence, curiosity, the desire for creativity, and interest in mathematics.

I often start my lessons with an intellectual warm-up.

Smart workouts.

You, me, and we are with you. How many of us are there? (2)

· A merchant rode across the sea, ate a cucumber with Alena. He ate half himself, gave half to whom? (Alena)

· My friend was walking, he found a nickel. Let's go together, how much can we find? (You can't predict).

A man was walking into the city, and four of his acquaintances were walking towards him. How many people went to the city? (one)

What can be cooked but not eaten? (lessons)

· Seven candles burned, two went out. How many candles are left? (2)

· The dog was tied to a 10-meter rope, and went 300 meters away. How it is? (Gone with the rope)

· What has no length, width, depth, height and yet can be measured? (age)

· How to increase the number 86 by 12 without calculations? (Turn over.)

· A sparrow, a crow, a dragonfly, a swallow and a bumblebee flew across the sky. How many birds flew? (3 birds)

Near Christmas trees and needles

Building a house on a summer day

He is not visible behind the grass,

And it has a million residents. (Anthill.)

· A flock of geese was flying, and a gander was meeting them.

Hello ten geese!

No, we are not ten. If you were with us and two more geese, then it was

would be ten.

How many geese are in a flock?

Find patterns.

From the first grade, we include tasks to identify patterns in the oral account.

Continue the series of numbers using the identified pattern.

2, 4, 6, 8, …, …, … .

2, 5, 8, …, …, … .

Find the patterns by which the series of numbers are composed, continue them.

The numbers of the fourth column of the table are obtained as a result of performing operations on the numbers of the first two columns. Based on the results of the first rows, establish a rule by which the numbers of the fourth column are obtained. What numbers should be in the empty cells of the fourth column?

Continue columns:

36: 4 = 6 * 5 = □ : 6 = 3

32: 4 = 5 * 5 = □: 6 = 4

28: 4 = 4 * 5 = □: 6 = 5

……….. ………. ……….

………… ……….. ……….

It is expected that students will identify a pattern in the compilation of each column and continue it.

Tasks for the development of logical thinking.

Three boxes contain paper clips, buttons and matches. It is known that all three inscriptions are incorrect. Determine where everything is.

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· Guard dogs live in booths. Scarlet hates Polkan, so their booths are not nearby. Polkan can't stand Rex - their houses stand apart. Rex does not like Mukhtar, so their houses are not neighboring. Rex's booth on the far left. What booth does Mukhtar live in?

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Rebus is a mystery. Its peculiarity lies in the fact that instead of words it contains signs, figures and even drawings - they must be unraveled.

Solve the following puzzles:

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Replace the question marks with the names of the numbers so that you get nouns.

Formation of oral counting skills.

I form mental counting skills in the games "Silent", "Chain", which can be carried out in all grades of elementary school, gradually complicating. These games are good primarily because they are fast and entertaining.

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.gif" alt="8-pointed star: 8 +" width="104" height="114 src="> 9 7!}

I spend a lot of games to develop the skills of tabular multiplication and division.

Students take turns standing up and repeating the multiplication table. For example, on 2: the first student - 2 * 2 = 4, the second - 2 * 3 = 6, etc. The student who correctly named the example from the table and his answer sits down. And the one who made a mistake stands, that is, remains "in the sieve."

Role-playing game.

The first student of the first row stands up and names the dividend, the first student of the second row is the divisor, the first student of the third row is the quotient. Then the second students of each row get up and continue the game.

In the oral account I include tasks that contribute to the development of independence in the manifestation of variability.

What numbers can be inserted to make the equalities true? ("Boxes" denote numbers to be substituted for them.)

700: 10 = □ + □

5 * □ = □ - 400

□ + 8 = □ : 50

630: □ = 70 - □

Make examples according to diagrams where possible. Calculate. Where is it impossible to make an example? Explain why.

a) □□ + □ = □□□

b) □□ - □ = □□□

c) □□ - □ = □□

d) □□□ - □□ = □□

e) □ + □ + □ = □□□

f) □□□ - □ - □ = □

Children like to solve problems in verses.

Problem with apples. L. Panteleev

Sent a box of apples.

In this box of apples

There were, in general, a lot.

My sisters helped me

My brothers helped me.

And while we thought

We are terribly tired

We are tired, sit down

And they ate an apple.

And how many are left?

And there are so many left

What we thought so far

Eight times we sat

rested eight times

And they ate an apple.

And how many are left?

Oh, there are so many left

What when in this box

We looked again

There at the bottom of it clean

Only the shavings turned white ....

Only shavings, pied,

Only the shavings turned white.

Here I ask you to guess

All boys and girls:

How many of us brothers were there?

How many sisters were there?

We shared apples

All without a trace.

And all they were

Fifty without a dozen.

Quick counting tricks.

From the first grade, I teach children quick and rational methods of oral calculations. If one of the terms is 9, increase it by 1, while the second term must be reduced by 1. If one of the terms is 8, increase it by 2, while the second term must be reduced by 2.

9 + 5 = (9 + 1) + (5 – 1) = 10 + 4 = 14

8 + 4 = (8 + 2) + (4 – 2) = 10 + 2 = 12

In the second class, we find the value of expressions in which you need to add 9 to a two-digit number. To do this, you need to increase the number of tens by 1, and decrease the number of units by 1.

13 + 9 =+ 9 =+ 9 = 98

How to quickly subtract 9 from a number? Decrease the number of tens by 1 and increase the number of ones by 1.

34 – 9 =– 9 =– 9 = 33

How to quickly find the difference of multi-digit numbers? The difference does not change from an increase or decrease in the minuend and the subtracted by the same number. You can easily solve these examples based on rounding off the subtrahend.

572 - 395 = 572 - 400 +5 = 172 + 5 = 177 (Students will understand that if an extra five is subtracted from the minuend, then it must be added to the difference.)

25 406 – 4 991 =

How to quickly multiply by 5 a two-digit, three-digit, multi-digit number?

For example: 2648 * 5

And the trick is this: mentally divide 2648 by 2, and then assign 0 to the right.

13240 is the result.

What if the number is not divisible by 2?

When divided by 2, the remainder can only be 1. And if 1 is multiplied by 5, it will be 5. So, instead of zero at the end, you need to put 5.

For example, 125 * 5, 125: 5 = 62 (remaining 1), so 125 * 5 = 625

How to quickly multiply by 25?

48 * 25 = (48: 4) * 100 =1200

If the number is divided by 4, and then multiplied by 100, then it will be multiplied by 25. If the multiplicand is not divisible by 4, then the remainder can be either 1, or 2. or 3. If the remainder is 1, then instead of two zeros put 25, if the remainder is 2, then 50, if 3, then 75.

37 * 25, 37: 4 = 9 (remaining 1), so 37 * 25 = 925

38 * 25, 38: 4 = 9 (remaining 2), so 38 * 25 = 950

39 * 25, 39: 4 = 9 (remaining 3), so 39 * 25 = 975

Folklore.

Different types of oral folk art during oral counting help

not only relieve stress, but also develop the child's speech, enrich vocabulary, train attention, memory, lay the foundations of creativity.

Children, do you know riddles with numbers? Guess and we'll guess.

Now solve the following riddles:

Five steps - a ladder, on the steps - a song. (notes)

The sun ordered: “Stop,

The Seven Colored Bridge is cool!” (rainbow)

Four legs under the roof

And on the roof there is soup and spoons. (table)

He has colored eyes

Not eyes, but three lights.

He took turns by them

Looking up at me. (traffic light)

What numbers were found in riddles?

Do you know proverbs with numbers? You can play the game "Finish the proverb."

Who soon helped, he helped twice.

One bee will bring some honey.

You cut down one tree, plant ten.

It is better to see once than hear a hundred times.

A coward dies a hundred times, a hero only once.

It takes three years to learn hard work,

To learn laziness - only three days.

Try on seven times, cut once.

Seven do not wait for one.

Transplant game.

To consolidate theoretical knowledge in mathematics, I conduct the game "Transplants". I ask a question. The student who answered this question correctly is seated in a separate chair. The student who answered the second question correctly takes the place of the first student, and so on. At the end of the game, I summarize. I ask: “Who moved? Well done! Take your seats."

Questions may be:

What are numbers called when divided? When multiplying? When subtracting? When added?

What is a perimeter?

How to find the perimeter of a rectangle? Square?

How to find the area of a rectangle?

What is the remainder after division?

How to find the unknown term? Subtrahend? Unknown multiplier?

What happens when you multiply a number by zero? Other.

geometric material.

I include tasks of a geometric nature in the oral account.

Which shapes are more: triangles or quadrilaterals?

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Count how many triangles.

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How many cuts?

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Plus and minus.

Fairy-tale heroes.

Find the extra word.

Plus and minus.

Place the plus and minus signs in appropriate places.

Fairy-tale heroes.

10. The wolf and the hare went to buy ice cream. The wolf says: "I'm big and I'll buy three servings, and you're small, so ask for two." The hare agreed. The Wolf ate ice cream, looked at the Hare, and how he shouted: “Well, Hare, wait a minute!”

Why is the wolf angry? (The hare bought two servings twice.)

How many servings of ice cream did the Wolf and the Hare buy in total?

20. Near the hut on chicken legs there are two barrels of water. There are 20 buckets of water in one barrel and 15 buckets in the other. Baba Yaga took 5 buckets of water from one barrel. How many buckets of water are left in the barrels? (30 buckets)

30. Dunno noticed that the soft-boiled egg was cooked in 3 minutes. Then he decided that 2 eggs would boil soft-boiled twice as long, that is, 6 minutes. Is the stranger right? (No)

40. Dunno planted 50 pea seeds. Out of every ten, 2 seeds did not germinate. How many seeds didn't germinate? (10 seeds)

50. Donkey invited guests to his birthday party, including Piglet, by 9 o'clock. In order not to be late, Piglet left the house at 8 o'clock, taking a balloon as a gift. Piglet overcame the first half of the way in 10 minutes. For another 5 minutes he flew in a balloon, after which the balloon burst for minutes crying bitterly and for 10 minutes wandered to Donkey's home. Was Piglet late for his birthday? (He was not late, as he spent 45 minutes on the road.)

Find the extra.

Monday condition 3, 6, 9 year above

Wednesday answer 5, 8, 11 centimeter more expensive

February triangle 10, 13, 16 month thinner

Friday question 2, 4, 6 week older

Sunday decision 14, 17, 20 days longer

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30. ses 3 ts

na-ty-zeros)

You can finish the mental count with the following task: collect the words that lie under the following numbers.

With p a s and b o c e m!

And it is one of the main tasks of teaching mathematics at this stage. It is in the first years of training that the main methods of oral calculations are laid, which activate the mental activity of students, develop children's memory, speech, the ability to perceive what is said by ear, increase attention and speed of reaction.

Phenomenal Counters

The phenomenon of special abilities in mental counting has been around for a long time. As you know, many scientists possessed them, in particular, Andre Ampère and Karl Gauss. However, the ability to quickly count was also inherent in many people whose profession was far from mathematics and science in general.

Until the second half of the 20th century, performances by specialists in oral counting were popular on the stage. Sometimes they organized demonstration competitions among themselves, which were also held within the walls of respected educational institutions, including, for example, Lomonosov Moscow State University.

Among the well-known Russian "super counters":

Among foreign:

Although some experts assured that it was a matter of innate abilities, others argued the opposite with reason: “It’s not only and not so much about some exceptional,“ phenomenal ”abilities, but about the knowledge of some mathematical laws that allow you to quickly make calculations” and willingly disclosed these the laws .

The truth, as usual, turned out to be on a certain “golden mean” of a combination of natural abilities and their competent, industrious awakening, cultivation and use. Those who, following Trofim Lysenko, rely solely on will and assertiveness, with all the already well-known methods and methods of mental calculation, usually, with all their efforts, do not rise above very, very average achievements. Moreover, persistent attempts to "load" the brain well with such activities as mental counting, blind chess, etc. can easily lead to overstrain and a noticeable drop in mental performance, memory and well-being (and in the most severe cases, to schizophrenia). On the other hand, gifted people, with the indiscriminate use of their talents in such an area as mental arithmetic, quickly “burn out” and cease to be able to show bright achievements for a long time and steadily.

Oral counting competition

Trachtenberg method

Among those practicing mental arithmetic, the book "Quick Counting Systems" by the Zurich professor of mathematics Jacob Trachtenberg is popular. The history of its creation is unusual. In 1941, the Germans threw the future author into a concentration camp. To maintain clarity of mind and survive in these conditions, the scientist began to develop a system of accelerated counting. In four years, he managed to create a coherent system for adults and children, which he later outlined in a book. After the war, the scientist created and headed the Zurich Mathematical Institute.

Mental arithmetic in art

In Russia, the picture of the Russian artist Nikolai Bogdanov-Belsky “Mental Account. In the folk school of S. A. Rachinsky ”, written in 1895. The task given on the board, which the students are thinking about, requires fairly high mental counting skills and ingenuity. Here is her condition:

The phenomenon of rapid counting of an autistic patient is revealed in the film "Rain Man" by Barry Levinson and in the film "Pi" by Darren Aronofsky.

Some methods of oral counting

To multiply a number by a single-digit factor (for example, 34*9) orally, you need to perform actions, starting with the most significant 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, the multiplication 147*8 is mentally done 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 as 147*8=(150-3)*8=150*8-3*8=1200-24=1176

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

There are several other ways of mental counting, for example, when multiplying by 1.5, the multiplied must be divided in half and added to the multiplied, for example 48*1.5= 48/2+48=72

There are also features when multiplying by 9. in order to multiply a number by 9, you need to add 0 to the multiplicand and subtract the multiplicand to the resulting number, for example 45*9=450-45=405

Multiplying by 5 is more convenient like this: first multiply by 10, and then divide by 2

The squaring of a number of the form X5 (ending in five) is performed according to the scheme: we multiply X by X + 1 and assign 25 to the right, i.e. (X5)² = (X*(X+1))*100 + 25. For example, 65² = 6*7 and assign 25 = 4225 on the right or 95² = 9025 (9*10 and assign 25 on the right). Proof: (X*10+5)² = X²*100 + 2*X*10*5 + 25 = X*100*(X+1) + 25.

Notes

Literature

Bantova M. A. The system of formation of computational skills. //Begin. school - 1993.-№ 11.-p. 38-43.
Beloshistaya A.V. Reception of the formation of oral computing skills within 100 // Elementary School. - 2001.- No. 7
Berman G. N. Receptions of the account, ed. 6th, Moscow: Fizmatgiz, 1959.
Borotbenko E I. Control of skills of oral calculations. //Begin. school - 1972. - No. 7. - p. 32-34.
Vozdvizhensky A. Mental Computing. Rules and simplified examples of actions with numbers. - 1908.
Volkova S., Moro M. I. Addition and subtraction of multi-digit numbers. //Begin. school - 1998.-№ 8.-p.46-50
Voskresensky M.P. Abbreviated calculation methods. - M.D905.-148s.
Wroblewski. How to learn to count easily and quickly. - M.-1932.-132s.
Goldstein D.N. Simplified Computing Course. M.: State. educational-ped. ed., 1931.
Goldstein D.N. Technique of fast calculations. M.: Uchpedgiz, 1948.
Gonchar D. R. Oral Counting and Memory: Riddles, Developmental Techniques, Games // In Sat. Oral counting and memory. Donetsk: Stalker, 1997
Demidova T. E., Tonkikh A. P. Methods of rational calculations in the initial course of mathematics // Elementary school. - 2002. - No. 2. - S. 94-103.
Cutler E. McShane R. Trachtenberg fast counting system. - M.: Uchpedgiz. - 1967. -150s.
Lipatnikova I. G. The role of oral exercises in mathematics lessons // Primary school. - 1998. - No. 2.
Martel F. Quick counting tricks. - Pb. −1913. −34s.
Martynov I.I. Mental arithmetic is for a schoolboy what scales are for a musician. // Elementary School. - 2003. - No. 10. - S. 59-61.
Melentiev P.V."Fast and verbal calculations." Moscow: Gostekhizdat, 1930.
Perelman Ya. I. Quick account. L .: Soyuzpechat, 1945.
Pekelis V.D."Your opportunities, man!" M.: "Knowledge", 1973.
Robert Toque"2 + 2 = 4" (1957) (English edition: The Magic of Numbers (1960)).
Sorokin A. S. Counting technique. M.: "Knowledge", 1976.
Sukhorukova A. F. More emphasis on verbal calculations. //Begin. school - 1975.-No. 10.-p. 59-62.
Faddeycheva T. I. Teaching Oral Computing // Elementary School. - 2003. - No. 10.
Faermark D.S."The task came from the picture." M.: "Science".

Links

V. Pekelis. Miracle counters // Technique-youth, No. 7, 1974
S. Trankovsky. Oral account // Science and life, No. 7, 2006.
1001 mental arithmetic tasks by S.A. Rachinsky.

Wikimedia Foundation. 2010 .

See what "Mental Counting" is in other dictionaries:

oral- oral... Russian spelling dictionary

Spoken, verbal, verbal, oral. Ant. Written dictionary of Russian synonyms. oral verbal, verbal; verbal (special) Dictionary of synonyms of the Russian language. Practical guide. M.: Russian language. Z. E. Alexandrova. 2011 ... Synonym dictionary

- [sn], oral, oral. 1. Pronounced, not fixed in writing. Oral speech. oral tradition. Oral report. Orally (adv.) convey the answer. 2. adj. to the mouth, oral (anat.). oral muscles. ❖ Oral literature (philol.) is the same as folklore. ... ... Explanatory Dictionary of Ushakov

ORAL, see mouth. Dahl's Explanatory Dictionary. IN AND. Dal. 1863 1866 ... Dahl's Explanatory Dictionary

beginning of verbal counting

Alternative descriptions

Single Action

One (about the number, when counting)

. "... a year and a stick shoots"

. "... on... not necessary"

. "... on ... not necessary" (Pogov)

. "... let's get down to business - I wanted to drink"

. "..., two, they took!" (call of the loader)

. "...-two, grief - it doesn't matter!" (movie)

. "Here are those..."

. "One" into the microphone

. "Eh... and more...!"

. "Stay where you are, ...-two"

And forever

Two and done

Two three

. "do...!"

. "many, many more..."

. "first... first class"

. "uh... still..."

M. krata, reception, nakon; unit, one. One, two, three, etc. Not once, not once, how many times it was ordered. I see him for the first time, for the first time or for the first time. You can't do it all at once, or you can't do it all at once. At once, not to leave at once or immediately, with one tinge, blow. You won’t guess right away, suddenly, soon. He was found at once, suddenly, instantly. Give it time! hit, give a cuff. Here's one time, another, grandmother will give! about an unfortunate accident. Count times, times, times. Take times! suddenly, together, together, swish, at once, impudently, uhni; blow out from here. It is better to sing at once (all together), but to speak separately. Once so, once that way, different. Ten times (ten) example, once (one) cut. For the first time, this time I forgive, but the next time (other times) don't get caught. Once in a while, always, every time. If only you could visit them another time, sometimes. Once upon a time, in a row, over and over again, every time. dine with the king at once, the song of the south. app. together. times yes much. who is not long, but we have just. Once in a while it doesn't have to. One (first) time doesn't count. One time doesn't count. Once not at once, but not much ahead. At times the mind was gone, until forever he was known as a fool; Once he stole, he became a thief forever. Born twice, never baptized, sang, sang and died. He was born twice, never baptized, ordained as a sacristan (rooster). Yes, not all at once (not all of a sudden)! said the drunken Cossack, who mounted his horse, asking for help from the saints, and threw himself over the saddle to the ground. Once, sometime, somehow, sometime. Once, on Epiphany evening, the girls wondered, Zhukovsky. One time, one time, one time, one time, one time, one time, one time. Once, southern, shepherd, stennik, erroneous. bedding, one layer of honeycombs. Each layer of honeycombs is called at once; single honey, honeycomb. One-time, to once, times related. One-time money, payment, by condition, to an actor or writer, for each time a game, performance

adv. more than once, more than once, repeatedly, many times, many times, often

Designation of a single action (when counting, indicating the quantity)

Single action; one (about the number, when counting)

Slap (colloquial)

individual case

First word into the microphone

Just like..., two, three

Ras, grew up, once, a continuous preposition, meaning: a) the end of the action, like all prepositions in general: to make laugh, wake up; b) division, separation, difference: break, distribute, bite through, disperse; into destruction, alteration again: develop, grow; to warm; d strong, highest degree of action or state: decorate, offend; thin, beautiful, reasonable; run away, get angry. The spelling of this preposition, like the others in z, is shaky. Once it changes into roses and grew when the emphasis was shifted to the preposition: but our surrounding population generally loves roses more: rose, to develop; unbend, etc. the small-haired boy says roses, the Belarusian one says: one; southern Great Russian, including Moscow, once, northern and eastern, mostly roses, although literacy smoothes these pronunciations more. Some of the words of this beginning will suffice to be explained by examples; but completeness cannot be here: in the meaning of the highest degree, since it can be attached to all verbs and to most of the names; e.g. Why, beaver hat, beaver! "Though razbobrovaya, even razbober, so I will not buy!" Razgrisha, razvanyushka, razdaryushka, vm. Grisha, Vanya, Daria, jokingly and affectionately, sometimes reproachfully

Seven... measure

The case of phenomena in a series of single-row actions, manifestations of something

Oral start of counting

The film "..., two grief is not a problem!"

Film "Do...!"

Yuzovsky's film "..., two - grief is not a problem!"

. "... and forever"

. "Here are those..."

Yuzovsky's film "..., two - grief is not a problem!"

. "first... first class"

. "... on ... not necessary"

. "many, many more..."

. "do it..."

. "Stay where you are, ...-two"

. "Eh... and even more...!"

. "uh... still..."

The film "..., two grief is not a problem!"

The film "Do ...!"

. "... let's get down to business - I wanted to drink"

. "one" into the microphone

. "... on ... not necessary" (Pogov)

. "... a year and a stick shoots"

. “..., two, they took!” (call of the loader)

. "Eh... and more...!"

. "...and forever" (ex.)

This article was written by me several years ago for a tutoring site. When posting, the site administrator misrepresented not only my last name, but also the purpose of my article. I intended it for schoolchildren, and the administrator of that site redirected it to .... beginner tutors, with the title "What calculations does a math tutor do in her head?" At the same time, the ceiling of the mental calculation indicated by him in his article on this topic is reduced only to calculating in the mind the multiplication of a two-digit number by a single-digit one. He writes: "Let's say it's 29x7. The "soundtrack" from the tutor could be the following:" 29 is twenty and 9. Twenty by 7 will be .... (student answers 14), and 9 by 7 will be .... (student answers 63) One hundred and forty and sixty-three will be ... "" Not only is there an error in this text (Twenty by seven will be 140, not 14) - you need to check, read what is written (!!!), not only is thirty much more convenient multiply by seven and subtract seven, so this technique in the article of that tutor is the only one (????) in the matter of mental counting.
What happens? Are fast mental counting skills superfluous for schoolchildren and only tutors can use them? But no! In my classes, I always welcome when a student strives to count in his mind. Yes, this is usually not taught in school. But as experience shows, every schoolchild can use the skills of quick oral counting if desired. And this in itself is useful, because it allows you to "feel" the numbers and understand how much can be obtained by multiplication, and how much cannot. It is only important to learn to think a little differently than they teach in school. And after all, these techniques can be useful to a student throughout the entire school curriculum, and at exams, where, as you know, it is not allowed to use a calculator.
For example, you need to subtract 9487 from 11531. How do they teach at school? It is necessary to write a column, while constantly occupying, counting the difference. Meanwhile, if you borrow several times, you can easily make a mistake where you borrowed and where you didn’t. And you can calculate it in your mind in a completely different way, without even thinking in a column. It can be seen that in the minuend, the numbers are mostly small, and in the subtrahend, mostly large. Then we consider in this way: How much more is 11531 than 11000? - By 531. How much is 9487 less than 10000? - At 513. Between 11,000 and 10,000 is one thousand.

11531 – 9487 = 11000 + 531 – (10000 – 513) = 11000 – 10000 + 531 + 513 = 2044
This technique is most conveniently remembered with the help of a picture:

Now let's look at a more complicated example - multiplication. How much will 64 * 15 be? What is 15? 15 is 1.5 * 10. How is a number multiplied by 1.5, i.e. for one and a half? To do this, you need to add half of itself to this number. If the example does not include 1.5, but 15, or 150, then a certain number of zeros must also be added to the right. Thus, 64 plus half of this number, that is, we attribute 32 and zero.
That is, 64 + 32 = 96; 96 * 10 = 960.

64 * 15 = 64 * 1,5 * 10 = (64 + 32) * 10 = 960

Now let's multiply 84 by 25. A similar example, but in this case it can be calculated in different ways. You can think of 25 as 2.5 * 10. In other words, take 84 twice and add 42 to the result, and then multiply by 10.

84 * 25 = (84 + 84 + 42) * 10 = 2100
And assign zero. And it is possible in another way. 84 * 0.25 * 100. That is, we break 25 into 0.25 and 100. Why do we need this? The fact is that 0.25 is ¼ (one fourth). In other words, we divide 84 by 4, we get 21, and we assign two zeros. It turns out the same 2100:

84 * 25 = 84 * 0,25 * 100 = 84: 4 * 100 = 2100
It may seem that such techniques can hardly be needed at school, that only examples of the 29x7 type are found in the school curriculum. Meanwhile, some textbooks are full of examples that involve the use of fast counting methods, it is only important to be able to recognize these methods. It is important to note in this connection that in the textbooks of the 6th grade there are often tasks "Calculate in the most rational way", and in the textbooks of the next grades such tasks are usually absent. This does not mean that such methods should be forgotten in high school. Here is an example from a real class with an 8th grade student. He met in one task
375 * 48. It would seem that you can multiply three-digit numbers by two-digit numbers only in a column. But the result of multiplying these two numbers is easier to get mentally. What is 375?
- That's 125 * 3. The number 125 is 0.125 * 1000 (one-eighth times a thousand). Therefore, we turn 375 into 0.375 (three eighths) * 1000. We get

48 * 375 = 48 * 0,375 * 1000 = 48 * 3: 8 * 1000 = 48: 8 * 3 * 1000 = 18000
Knowing this technique, all actions are obtained automatically in the mind and the student can be sure that he did not make a mistake anywhere. Whereas when counting in a column, where in fact it is necessary to perform several actions, the probability of error is much greater.
For a quick mental calculation, it’s good to know by heart not only the multiplication table, but also the table of squares, at least up to thirty. Practice shows that this is relatively easy, and there are schoolchildren with such knowledge. In addition, this knowledge sometimes allows not only to square, but also to count in the mind examples like 39 * 26, using the decomposition technique into "known" factors. It is easy to see that 39 is 13 * 3,
and 26 is 13 * 2. Knowing by heart that 13 * 13 = 169, only 169 * 6 remains. 170 * 6 will be 170 * 3 * 2 = 1020 and minus 6, it turns out 1014.

39 * 26 = 3 * 13 * 2 * 13 = 169 * 6 = 170 * 6 – 6 = 1014

By the way, about the table of squares. Yes, the table of squares is published on the flyleaf of textbooks, it is published in collections for preparing for exams, it is allowed to be used in the exam. It turns out that it is not necessary to know the table of squares by heart. However, before the revolution, when there were no calculators and computers, schoolchildren, at least in the Rachinsky school (the artist N.P. Bogdanov-Belsky has a painting "Mental Counting", reminiscent of this), were able to square numbers up to 100 in mind. Not in a column, but in the mind. How did they do it? It would seem that the process is rather time-consuming, even if, for example, the abbreviated multiplication formulas are used. Indeed, let's take, for example, the number 96 and square it using the formula for the square of the sum (90 + 6) 2 . Three terms are obtained, which are sometimes inconvenient to add up. It is even less convenient if we take the formula for the square of the difference (100 - 4) 2 . However, there is a simpler trick, but for now it’s worth making a digression and talking about abbreviated multiplication formulas. Curiously, in the school curriculum, these formulas are used in various sections of mathematics - from algebraic fractions to trigonometric transformations, but not for quick multiplication of numbers. Only with a direct study of the topic are several examples given with the help of these formulas, and such tasks are found at entrance exams to lyceums. Why? Yes, because it is not very convenient to make calculations in the mind using these formulas, and the methods are not universal. Of course, in some cases, these formulas can be used for a quick calculation. This is especially true for the difference of squares formula. Indeed, if you need to multiply 37 by 43, 26 by 32, 35 by 25, etc. (if the difference between the numbers is even), then the difference of squares formula can achieve a quick result, although this again requires knowing the table of squares (37 * 43 \u003d (40 - 3) * (40 + 3) \u003d 1600 - 9 \u003d 1591; 26 * 32 \u003d (29 - 3) * (29 + 3) \u003d 841 - 9 \u003d 832;
35 * 25 = (30 + 5) * (30 - 5) = 900 - 25 = 875). Another way of squaring is more convenient than using abbreviated multiplication formulas. For example, let's take the same number 96 squared.
First, let's look at the rule for quickly squaring numbers ending in 5. For example, 25 squared, 35 squared, 45 squared, 95 squared. The rule is. To do this, multiply the number of tens of the squared number (for example, 9 in 95) by a number that is one more (that is, 10 in the case of 95) and assign 25. It turns out 9025. Let's calculate in this way, for example 85 2:

85 2 = 8 * 9 * 100 + 25 = 7225
(we multiply by 100 because the product 8 * 9 gives us the first two digits of the final result).
I won’t comment on why this happens within the framework of this article, I will only note that this rule also applies to three-digit numbers, which has become common, for example, at the OGE, and in the opposite direction - in the form of extracting the arithmetic square root of a five-digit number ending in ...25. In all likelihood, the compilers of the assignments began to take into account that the table of squares published everywhere includes squaring only two-digit numbers, and it is necessary to check schoolchildren with something that goes beyond this table. In fairness, it must be said that in schools, some teachers introduce students to this technique. Although it is usually not said that with its help you can easily get the result of squaring any number from the table. How it's done? Among the numbers that are squared, there is a so-called. "base" numbers. These are, firstly, 10, 20, 30, 40, .... 90 and, secondly, 15, 25, 35 ... 95. These are the numbers that are very easy to square. Now we take the number 96 and square it. To do this, add 95 and 96 to 9025. Add 200 and subtract (5 + 4 are numbers that complement 95 and 96 to 100). We write the result - 9216. Why is that?

96 * 96 = (95 + 1) * 96 = 95 * 96 + 1 * 96 = 95 * (95 + 1) + 1 * 96 = 95 * 95 + 95 + 96 = 9216.
In a similar way, with appropriate training, you can square any number from the table of squares, up to showing tricks of fast counting or phenomenal memory in front of classmates. For those who are still afraid of such large numbers, the principle of operation can be explained with a simple example. 4 squared. This will be 16. Now let's square 5. This will be 25. Knowing 4 squared, the result of the next squared number is obtained by adding the sum of the squared numbers to the previous one. For example, 5 squared is 4 squared + 5 + 4 (i.e. 16 + 9).
A student who has become proficient in applying these methods of rapid mental counting may well come up with his own methods, carefully peering into the numbers and find their own patterns in them. As experience shows, this desire teaches him not to make mistakes in counting, and the search for his own methods instills in him an interest in the subject, allows him to be creative in his study and find something of his own in it. Some schoolchildren tend to show off such skills in front of their classmates, or even completely pro-demon-stri-ro-vat "trick" by counting large numbers in their minds. This is to be welcomed, although not in all schools teachers believe that students can calculate something in their minds, and not on a calculator. In my memory, there is a completely anecdotal case from the series “you can’t think of it on purpose”, when a student in the 5th grade wrote: 22 + 33 = 55. It would seem that what is wrong here? But the teacher crossed it out for him, offering to rewrite the same thing ... in a column. Instead of teaching children to count in their heads, sometimes there are "incredulous" teachers who believe that if the column is not written, it means that the student counted with a calculator.
In individual lessons with a math tutor, it can be useful to pay attention to the study of fast mental counting techniques.