Examination
Mental arithmetic in mathematics lessons. Mental counting At the beginning of a word, oral counting

Mental arithmetic in mathematics lessons. Mental counting At the beginning of a word, oral counting

Department of Education of the Okhinsky Urban District

Municipal budget educational institution

average comprehensive school No. 1 Okha

Techniques

mental counting

The work was completed by:

Students of grade 5 "A"

Turboevskaya Eva

Bezinsky Stanislav

Project Manager:

mathematic teacher

Kravchuk Maria Arkadyevna

2017

CONTENT

INTRODUCTION………………………………………………………………………………...

Chapter 1. ACCOUNT HISTORY……………………………………………………………….....

Chapter 2. MULTIPLICATION TABLE ON YOUR FINGERS …………………………

2.1 Multiplication table by 9

2.2 Multiplying numbers from 6 to 9

Chapter 3. DIFFERENT METHODS OF MULTIPLICATION……………………….....

3.1 Multiplying a number by 9

3.2 Multiplying two-digit numbers by 11

3.3 Multiplying two-digit numbers by 111, 1111, etc.

3.4 Multiplying a two-digit number by 101, 1001, etc.

3.5 Multiplication by 5; 25; 125

3.7 Multiplying by 37

3.8 Multiplying a number by 1.5

Chapter 4.SQUAREING A TWO-DIGIT NUMBER …………...

4.1 Squaring a two-digit number ending in 5

4.2 Squaring a two-digit number starting with 5

CONCLUSION ……………………………………………………………….....

BIBLIOGRAPHY ………………………………………………………

ANNEX 1 ………………………………………………………………..

APPENDIX 2………………………………………………………………..

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, while 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, paying for utilities, calculating the family budget, etc. In addition, all schoolchildren must take exams in the 9th grade and in the 11th grade, and for this, studying from the 1st grade, it is necessary to master mathematics well and, above all, to learn to count.

The relevance of our project is that nowadays, calculators are increasingly coming 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 mind, accustoms a person to accuracy, to the ability to see the main thing, reports necessary information for understanding complex tasks arising in various fields of activity of modern man.

Objective of the project: study mental calculation techniques, show the need for their use to simplify calculations.

In accordance with the goal, we determinedtasks:

To investigate whether schoolchildren use mental counting techniques.

Learn mental counting techniques that can be used to simplify calculations.

Create a memo for students in grades 5-6 to use quick mental counting techniques.

Object of study: oral counting techniques.

Subject of study : calculation process.

Hypothesis: If you show that the use of fast mental calculation techniques makes calculations easier, then you can ensure that students’ computing culture improves and it will be easier for them to solve practical problems.

The following were used to carry out the work:techniques and methods : survey (questionnaire), analysis ( statistical processing data), work with information sources, practical work.

To begin with, we conducted a survey in the 5th and 6th grades of our school. We asked the guys simple questions.Why do you need to be able to count?When studying which school subjects will you need to count correctly?Do you know mental counting techniques?Would you like to learn fast mental counting techniques to count quickly?Annex 1

105 people took part in the survey. After analyzing the results, we concluded that the majority of studentsbelievethat the ability to count is useful in life and to be literate, especially when studying mathematics (100%), physics (68%), chemistry (50%), computer science (63%). A small number of students know mental counting techniques and almost all of them would like to learn how to quickly mental arithmetic (63%). Appendix 2

Having studied a number of articles, we discovered very interesting historical facts O in unusual ways mental calculation, as well as many patterns and unexpected results.Therefore, in our work we will show how you can count quickly and correctly and that the process of performing these actions can be not only useful, but also an interesting activity.

Chapter 1. ACCOUNT HISTORY

People learned to count objects back in the ancient Stone Age - Paleolithic, tens of thousands of years ago. How did this happen? At first, people only compared different quantities of identical objects by eye. They could determine which of two piles had more fruit, which herd had more deer, etc. If one tribe exchanged caught fish for stone knives made by people of another tribe, there was no need to count how many fish and how many knives they brought. It was enough to place a knife next to each fish for the exchange between the tribes to take place.

To practice successfully agriculture, arithmetic knowledge was needed. Without counting days, it was difficult to determine when to sow fields, when to start watering, when to expect offspring from animals. It was necessary to know how many sheep were in the herd, how many bags of grain were placed in the barns.
And more than eight thousand years ago, ancient shepherds began to make mugs out of clay - one for each sheep. To find out if at least one sheep had gone missing during the day, the shepherd put aside a mug each time another animal entered the pen. And only after making sure that as many sheep had returned as there were circles, he calmly went to bed. But in his herd there were not only sheep - he grazed cows, goats, and donkeys. Therefore, we had to make other figures from clay. And farmers, using clay figurines, kept records of the harvest, noting how many bags of grain were placed in the barn, how many jugs of oil were squeezed from olives, how many pieces of linen were woven. If the sheep gave birth, the shepherd added new ones to the circles, and if some of the sheep were used for meat, several circles had to be removed. So, not yet knowing how to count, the ancient people practiced 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 people of Australia had two prime numbers: enea (1) and petchewal (2). They expressed other numbers with compound numerals: 3 = “petcheval-enea”, 4 “petcheval-petcheval”, etc. Another Australian tribe, the Kamiloroi, 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, the number 10 was called "karo". The Nivkhs living on Sakhalin on the banks of the Amur did exactly the same thing. Also inXIXcentury they called the same number in different words, if you counted people, fish, boats, nets, stars, sticks.

We still use various indefinite numbers with the meaning “many”: “crowd”, “herd”, “flock”, “heap”, “bunch” and others.

With the development of production and trade exchange, people began to better understand what three boats and three axes, ten arrows and ten nuts have in common. Tribes often traded "item for item"; for example, they exchanged 5 edible roots for 5 fish. It became clear that 5 is the same for both roots and fish; This means that you can call it in one word.

Other peoples used similar methods of counting. This is how numberings based on counting in fives, tens, and twenties arose.

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

When writing appeared, numbers appeared to record numbers. At first, numbers resembled notches on sticks: in Egypt and Babylon, in Etruria and Phenice, in India and China, small numbers were written with sticks or lines. For example, the number 5 was written with five sticks. The Aztec and Mayan Indians used dots instead of sticks. Then special signs appeared for some numbers, such as 5 and 10.

At that time, almost all numberings were not positional, but similar to Roman numbering. Only one Babylonian sexagesimal numbering was positional. But for a long time there was no zero in it, as well as a comma separating the whole part from the fractional part. Therefore, the same number could mean 1, 60, or 3600. The meaning of the number had to be guessed according to the meaning of the problem.

Several centuries before new era invented a new way of writing numbers, in which the letters of the ordinary alphabet served as numbers. 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 Rus' this dash was called a “titlo”).

In all these numberings it was very difficult to perform arithmetic operations. Therefore, the invention inVIcentury by Indians, 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 called Arabic.

When writing fractions for a long time, the whole part was written in the new decimal numbering, and the fractional part in sexagesimal. But at the beginningXVV. 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 2. MULTIPLICATION TABLE ON YOUR FINGERS

2.1 Multiplication table by 9.

Finger movement - this is one way to help your memory: use your fingers to remember the multiplication table by 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 designated 1, the second one behind it will be designated 2, then 3, 4... to the tenth finger, which means 10. If you need to multiply any of the first nine numbers by 9, then to do this, without moving your hands from the table, you need to bend the finger whose number means the number by which nine is multiplied. The number of fingers lying to the left of the bent finger determines the number of tens, and the number of fingers lying to the right indicates the number of units of the resulting product.

3 9= 27

Try multiplying yourself using this method:6 · 9, 9 · 7.

2.2 Multiplying numbers from 6 to 9.

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

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

Example: 8 ∙ 9 = 72

Thus,7 7 = 49.

Chapter 3. DIFFERENT WAYS OF MULTIPLICATION

3.1 Multiplying a number by 9.

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

For example: 72 · 9 = 720 – 72 = 648.

3.2 Multiplying two-digit numbers by 11.

To multiply a number by 11, you need to mentally expand the digits of this number and put the sum of these digits between them.

45 ∙ 11 = 495

53 ∙ 11 = 583

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

To multiply by 11 a number whose sum of digits is 10 or more than 10, you need to mentally move apart the digits of this number, put the sum of these digits between them, and then add 1 to the first digit, leaving the second and third digits unchanged.

87 ∙ 11 = 957

94 ∙ 11 = 1024

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

3.3 Multiplying two-digit numbers by 111, 1111, etc., knowing the rules for multiplying a two-digit number by the number 11.

If the sum of the digits of the first factor is less than 10, you need to mentally expand the digits of this number by 2, 3, etc. step, add these numbers and write down their sum between the spread out numbers the appropriate number of times. Please note that the number of steps is always less than the number of units by 1.

Example:

24 111=2 (2+4) (2+4) 4 = 2664 (number of steps - 2)

24 1111=2 (2+4) (2+4) (2+4) 4 = 26664 (number of steps - 3)

42 · 111 111 = 4 (4+2) (4+2) (4+2) (4+2) (4+2) 2 = 4666662. (number of steps – 5)

If there are 6 units, then there will be 1 fewer steps, that is, 5.

If there are 7 units, then there will be 6 steps, etc.

It is a little more difficult to perform mental multiplication if the sum of the digits of the first factor is 10 or more than 10.

Examples:

86 · 111 = 8 (8+6) (8+6) 6 = 8 (14) (14) 6 = (8+1) (4+1) 46 = 9546.

In this case, you need to add 1 to the first digit 8, we get 9, then 4+1 = 5; and leave the last numbers 4 and 6 unchanged. We get the answer 9546.

3.4 Multiplying a two-digit number by 101, 1001, etc.

Perhaps the simplest rule: assign your number to yourself. Multiplication is complete. Example:

32 · 101 = 3232;

47 · 101 = 4747;

324 · 1001 = 324 324;

675 · 1001 = 675 675;

6478 · 10001 = 64786478;

846932 · 1000001 = 846932846932.

3.5 Multiplication by 5; 25; 125.

First multiply by 10, 100, 1000 and divide the result by 2, 4, 8

32 5 = 32 10: 2 = 320: 2 = 160

84 25 = 84 100: 4 = 8400: 4 = 2100

24 125 = 24 1000: 8 = 24000: 8 = 3000

Another way: 32 5 = 32: 2 10 = 160

3.6 Multiplication by 22, 33, …, 99

To multiply a two-digit number by 22.33,..., 99, this factor must be represented as the product of a single-digit number (from 2 to 9) by 11, that is, 33 = 3 x 11; 44 = 4 x 11, etc. Then multiply the product of the first numbers by 11.

Examples:

18 · 44 = 18 · 4 · 11 = 72 · 11 = 792;

42 · 22 = 42 · 2 · 11 = 84 · 11 = 924;

13 · 55 = 13 · 5 · 11 = 65 · 11 = 715;

24 · 99 = 24 · 9 · 11 = 216 · 11 = 2376.

3.7 Multiplying by 37

Before learning how to verbally multiply by 37, you need to know well the sign of divisibility and the multiplication table by 3. To verbally multiply a number by 37, you need to divide this number by 3 and multiply by 111.

Examples:

24 · 37 = (24: 3) · 37 · 3 = 8 · 111 = 888;

· 37 = (18: 3) · 111 = 6 · 111 = 666.

3.8 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.

Chapter 4.SQUAREING A TWO-DIGIT NUMBER

4.1 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 the digit greater than one, and add the number 25 to the right of the resulting product.

25 25 = 625

2 · (2 + 1) = 2 · 3 = 6, write 6; 5 5 = 25, write 25.

35 35 = 1225

3 · (3 + 1) = 3 · 4 = 12, write 12; 5 5 = 25, write 25.

4.2 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 add the square of the second digit to the right, and if the square of the second digit is single digit number, then the number 0 must be added in front of it.

For example:
52 2 = 2704, because 25 +2 = 27 and 2 2 = 04;
58 2 = 3364, because 25 + 8 = 33 and 8 2 = 64.

CONCLUSION

As we see, quick mental counting is no longer a sealed secret, but a scientifically developed system. Since there is a system, it means it can be studied, it can be followed, it can be mastered.

All the methods of oral multiplication that we have considered indicate the long-term interest of scientists, and ordinary people to the numbers game.

Using some of these methods in the classroom or at home, you can develop the speed of calculations, instill interest in mathematics, and achieve success in studying all school subjects. In addition, mastering these skills develops the student’s logic and memory.

Knowledge of quick counting techniques allows you to simplify calculations, save time, and develop logical thinking and mental flexibility.

There are practically no quick counting techniques in school textbooks, so the result of this work - a reminder for quick mental counting - will be very useful for students in grades 5-6.

We chose the topic “Tricks of mental calculation”because we love mathematics and would like to learn how to count quickly and correctly, without resorting to using a calculator.

LIST OF REFERENCES USED

Vantsyan A.G. Mathematics: Textbook for 5th grade. - Samara: Publishing house "Fedorov", 1999.

Kordemsky B.A., Akhadov A.A. Amazing world numbers: Book of students, - M. Education, 1986.

Oral counting, Kamaev P. M. 2007

“Mal arithmetic – mental gymnastics” G.A. Filippov

"Verbal counting". E.L.Strunnikov

Bill Handley “Count in your head like a computer”, Minsk, Potpourri, 2009.

Annex 1

QUESTIONNAIRE

1 . Why do you need to be able to count?

a) useful in life, for example, counting money;

b) to do well at school; c) to decide quickly;

d) to be literate; e) it is not necessary to be able to count.

2. List which school subjects you will need to count correctly when studying?

a) mathematics; b) physics; c) chemistry; d) technology; e) music; f) physical culture;

g) life safety; h) computer science; i) geography; j) Russian language; k) literature.

3. Do you know quick counting techniques?

a) yes, a lot; b) yes, several; c) no, I don’t know.

4. Would you like to learn fast counting tricks to count quickly?

a) yes; b) no.

Appendix 2

STATISTICAL DATA PROCESSING

1) Why do you need to be able to count?

Useful in life

To do well in school

To decide quickly

To be literate

You don't have to be able to count

Number of students

62%

30%

34%

57%

2) When studying which school subjects will you need to count correctly?

Mathematics

Physics

Chemistry

Technology

Music

Physical Culture

life safety fundamentals

Computer science

Geography

Russian language

Literature

Number of students

105

100%

68%

52%

35%

25%

63%

No,

Don't know

Number of students

17%

20%

63%

4) Would you like to learn quick counting techniques to solve quickly?

Yes

Number of students

91%

Municipal educational institution "Bryokhovskaya basic secondary school"

Mental arithmetic in mathematics lessons.

From V.’s work experience,

With. Brekhovo 2010

Come on, put the pencils aside!

No knuckles, no pens, no chalk.

Verbal counting! We're doing this thing

Only by the power of mind and soul.

The numbers converge somewhere in the darkness,

And the eyes begin to glow,

And there are only smart faces around.

Verbal counting! We count in our heads.

At the beginning of each mathematics lesson, I conduct mental calculations, during which I teach children to reason, think, analyze, compare, generalize, identify patterns, and teach fast and rational methods of mental calculations. I am working on the development of such mental qualities as perception, attention, imagination, memory, thinking. In addition, I am developing the ability to quickly switch from one type of activity to another.

I have the following requirements for organizing oral counting:

Entertaining

Originality

Diversity

Systematicity

Cognition

Subsequence.

When counting mentally I use entertaining tasks, rebuses, puzzles, games, magic squares, riddles, different types oral folk art. Using a wide variety of tasks, creating an atmosphere of interest, creativity, and cooperation, I cultivate in children independence, curiosity, a desire for creativity, and an interest in mathematics.

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

Intellectual workouts.

· You, me, and you and me. How many of us are there in total? (2)

· A merchant was driving along the sea, eating a cucumber with Alena. Did you eat half of it yourself, and give half of it to someone? (Alyona)

· My friend was walking and 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 you cook, but cannot eat? (lessons)

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

· The dog was tied to a 10-meter rope, but went 300 meters away. How come? (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 were flying? (3 birds)

Near Christmas trees and needles

A house was built on a summer day,

He is not visible behind the grass,

And there are a million residents there. (Anthill.)

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

Hello ten geese!

No, there are not ten of us. If you had been with us and two more geese, then it would have been

maybe ten.

How many geese are there in a flock of geese?

Find patterns.

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

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 and continue them.

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

Continue the columns:

36: 4 = 6 * 5 = □ : 6 = 3

32: 4 = 5 * 5 = □: 6 = 4

28: 4 = 4 * 5 = □: 6 = 5

……….. ………. ……….

………… ……….. ……….

Students are expected to identify a pattern in the composition of each column and continue with 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 kennels. Scarlet can't stand Polkan, so their booths are not nearby. Polkan cannot stand Rex - their houses are located apart. Rex doesn't like Mukhtar, so their houses are not next door. Rex's booth on the far left. What kind of booth does Mukhtar live in?

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A rebus is a riddle. Its peculiarity is that instead of words it contains signs, figures and even drawings - they must be solved.

Solve the following puzzles:

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Instead of question marks, replace the question marks with the names of the numbers to make nouns.

Formation of mental counting skills.

I develop mental counting skills in the games “Milchanka” and “Chain”, which can be played in all grades of elementary school, gradually increasing in complexity. 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 play a lot of games to develop table multiplication and division skills.

Pupils take turns standing up and reproducing 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 its answer sits down. And the one who made a mistake stands, that is, remains “in the sieve.”

Role-playing game.

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

In oral arithmetic I include tasks that promote the development of independence in the manifestation of variability.

What numbers can be inserted to make the equations true? (“The boxes” indicate the numbers that need to be substituted instead.)

700: 10 = □ + □

5 * □ = □ - 400

□ + 8 = □ : 50

630: □ = 70 - □

Build examples using diagrams where possible. Calculate. Where is it impossible to create an example? Explain why.

a) □□ + □ = □□□

b) □□ - □ = □□□

c) □□ - □ = □□

d) □□□ - □□ = □□

e) □ + □ + □ = □□□

f) □□□ - □ - □ = □

Children like to solve problems in verse.

Apple problem. L. Panteleev

I sent you 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 were counting

We are terribly tired

We are tired, we sat down

And they ate an apple.

And how many of them are left?

And there are so many of them left

What we thought so far -

Eight times we sat

Vacationed eight times

And they ate an apple.

And how many of them are left?

Oh, there are so many of them left

What, when in this box

We looked again

There at its clean bottom

Only the shavings turned white...

Only parsley shavings,

Only the shavings turned white.

So I ask you to guess

All the boys and girls:

How many of us brothers were there?

How many sisters were there?

We divided the apples

All without a trace.

But that was all there was

Fifty without ten.

Quick counting techniques.

From the first grade I teach children quick and rational methods of mental calculations. If one of the terms is 9, increase it by 1, while the second term must be decreased by 1. If one of the terms is 8, increase it by 2, while the second term must be decreased 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 meaning of expressions in which we need to add 9 to a two-digit number. To do this, we need to increase the number of tens by 1, and reduce the number of ones by 1.

13 + 9 =+ 9 =+ 9 = 98

How to quickly subtract 9 from a number? You need to 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 if the minuend or subtrahend is increased or decreased by the same number. You can easily solve these examples based on rounding 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 a two-digit, three-digit, or multi-digit number by 5?

For example: 2648 * 5

The trick is this: mentally divide 2648 by 2, and then add 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. This means that 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 a number is divided by 4 and then multiplied by 100, it will be multiplied by 25. If the multiplicand is not divisible by 4, then the remainder may be either 1, or 2, or 3. If the remainder is 1, then instead of two zeros We 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 help during oral counting

not only relieve tension, but also develop the child’s speech, enrich lexicon, train attention, memory, lay the foundations of creativity.

Children, do you know riddles with numbers? Make a guess and we will guess.

Now guess the following riddles:

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

· The sun ordered: “Stop,

The Seven Color Bridge is cool!” (rainbow)

· There are four legs under the roof,

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

His eyes are colored

Not eyes, but three lights.

He takes turns with them

Looks down at me. (traffic light)

What numbers were found in the riddles?

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

He who helped quickly helped twice.

One bee will make a little honey.

If you cut down one tree, you plant ten.

Better to see once than hear a hundred times.

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

It takes three years to learn hard work,

It only takes three days to learn laziness.

Try it on seven times, cut it once.

Seven do not wait for one.

Game "Transplants".

To consolidate theoretical knowledge in mathematics, I play the game “Transplants”. I'm asking a question. The student who answers this question correctly is seated in a separate chair. The student who answered the second question correctly takes the place of the first student, etc. At the end of the game I summarize. I ask: “Who transferred? Well done! Take your seats."

Questions could be:

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

What is perimeter?

How to find the perimeter of a rectangle? Square?

How to find the area of a rectangle?

What remainder can there be in division?

How to find unknown term? Subtrahend? Unknown multiplier?

What happens when you multiply a number by zero? And others.

Geometric material.

I include tasks of a geometric nature in oral calculation.

Which shapes are more numerous: triangles or quadrangles?

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Count how many triangles there are.

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How many segments?

644 " style="width:483.35pt;border-collapse:collapse;border:none">

Plus and minus.

Fairy-tale heroes.

Find the extra word.

Plus and minus.

Place 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 shouted: “Well, Hare, wait!”

Why was the Wolf angry? (The hare bought two servings each.)

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

20. Near the hut on chicken legs there are two barrels of water. One barrel contains 20 buckets of water, and the other contains 15 buckets. 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 took 3 minutes. Then he decided that 2 soft-boiled eggs would take twice as long, that is, 6 minutes. Is Dunno right? (No)

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

50. Donkey invited guests, including Piglet, to his birthday party at 9 o’clock. In order not to be late, Piglet left the house at 8 o'clock, taking a balloon as a gift. Piglet covered the first half of the journey in 10 minutes. For another 5 minutes he flew hot-air balloon, after which the balloon burst, cried bitterly for minutes and walked for 10 minutes to Donkey’s home. Was Piglet late for his birthday? (I wasn’t late, since he spent 45 minutes on the road.)

Find the odd one out.

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 weeks older

Sunday decision 14, 17, 20 days longer

https://pandia.ru/text/78/123/images/image020_7.gif" width="98" height="2 src=">20.

30. ses 3 tsy

na-tya-zero)

You can complete your mental counting with the following task: collect the words that are hidden under the following numbers.

Thank you so much!

beginning of mental counting

Alternative descriptions

One-time action

One (about quantity, when counting)

. "... in a year and the stick shoots"

. "... on... there is no need"

. "... on... there is no need" (pron)

. "... let's get to work - I wanted a drink"

. "..., two, they took it!" (loader's cry)

. "...-two, grief is not a problem!" (movie)

. "Here are those..."

. "One" into the microphone

. "Eh..., and also...!"

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

And forever

Two and done

Two three

. "do...!"

. "many, many more..."

. "first... to first class"

. "eh..., still..."

M. krata, reception, finally; unit, one. One, two, three, etc. Not once, not once, no matter 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 all at once. At once, at once or immediately, not to leave, in one mud, blow. You won’t guess right away, suddenly, soon. He was found at once, suddenly, instantly. Give him one! hit, give a punch. Grandma will give it to you once or twice! about an unpleasant accident. Count times, times, times. Take it at once! suddenly, together, amicably, swoop, at once, in full swing, whoop; strike from here. It’s better to sing at once (all together), but to speak separately. Once this way, once that way, it’s different. Ten times (ten) example, cut once (one again). For the first time, this time I forgive, but next time (nother times) don’t get caught. Once in a while, always, every time. If only you could visit them once again, sometimes. Time after time, in a row, time after time, every time. the king dines at once, song of the south. zap. together. once and a lot. For some it doesn’t take long, but for us it’s just that. It doesn't happen once at a time. One (first) time doesn't count. Once doesn't count. Not at once, but not too far ahead. Once I lost my mind, I was known as a fool forever; Once you steal, you become a thief forever. Born twice, never baptized, sang and sang and died. Born twice, never baptized, ordained as a sexton (rooster). Yes, not all at once (not all at once)! said the drunken Cossack, who climbed onto his horse, asking for help from the saints, and threw himself over the saddle to the ground. Once upon a time, once upon a time, somehow, someday. Once, on Epiphany evening, the girls were wondering, Zhukovsky. Once, once, once, once, once, once, once. Once, southern, pastenok, stennik, erroneous. insole, one layer of honeycomb. Each layer of honeycomb is called at once; disposable honey, cell phone. One-time, once, times related. One-time money, payment, according to the condition, actor or writer, for each time of the game, performance

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

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

Single action; one (about quantity, when counting)

Slap (colloquial)

An isolated case

First word into the microphone

Just like..., two, three

Ras, grew, once, is a combined preposition, meaning: a) the end of an action, like all prepositions in general: to make you laugh, to wake you up; b) division, singularity, difference: break, distribute, discern, disperse; into destruction, remaking again: to develop, to grow; to warm; d strong, highest degree of action or state: to decorate, to offend; subtle, beautiful, reasonable; run away, go wild. The spelling of this preposition, like others in z, is shaky. Once it changes into roses and grew when the stress is transferred to the preposition: but our surrounding population generally loves roses more: rozinya, to develop; unbend, etc. the cursing Little Russian says roses, cursing Belarus: once; Southern Great Russians, including Moscow, while northern and eastern ones are mostly roses, although literacy smooths out these pronunciations more. It will be enough to explain some of the words of this beginning with examples; but there cannot be completeness here: in meaning highest degree, since it can be attached to all verbs and to most names; eg It's a beaver hat, beaver! “Even if it’s a beaver, or even if it’s a beaver, I won’t buy it!” Razgrisha, razvanyushka, razdarushka, vm. Grisha, Vanya, Daria, humorously 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 woe is not a problem!"

Film "Do...!"

Yuzovsky's film "..., two - no trouble!"

. "... and forever"

. “here are those...”

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

. "first... to first class"

. "... on... there is no need"

. "many, many more..."

. “do...!”

. “stay where you are, ...-two”

. “eh..., and also...!”

. “eh..., still...”

The film "..., two sorrows are not a problem!"

Film "Do...!"

. "... let's get to work - I wanted a drink"

. "one" into the microphone

. “... on... there is no need” (pogov)

. "... in a year and the stick shoots"

. “..., two, they took it!” (loader's cry)

. "Eh..., and also...!"

. "... and forever" (express.)

. “... and forever” (express.)

And this is one of the main tasks of teaching mathematics at this stage. It is in the first years of education that the basic techniques of oral calculations are laid down, which activate the mental activity of students, develop memory, speech, and the ability to listen to what is said, increase attention and speed of reaction.

Phenomenal counters

The phenomenon of special abilities in mental calculation has been encountered for a long time. As you know, many scientists possessed them, in particular Andre Ampère and Carl 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 calculations 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, Moscow State University named after M.V. Lomonosov.

Among the famous Russian “super counters”:

Among foreign ones:

Although some experts insisted that it was a matter of innate abilities, others argued the opposite: “the matter is not only and not so much in some exceptional, “phenomenal” abilities, but in the knowledge of certain mathematical laws allowing for quick calculations” and willingly revealed these laws.

The truth, as usual, turned out to be on a certain “golden mean” of a combination of natural abilities and their competent, hardworking awakening, cultivation and use. Those who, following Trofim Lysenko, rely solely on will and assertiveness, with all the already well-known methods and techniques of mental calculation, usually, with all their efforts, do not rise above very, very average achievements. Moreover, persistent attempts to “properly load” the brain with such activities as mental arithmetic, blindfold 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, when using their talents indiscriminately in such an area as mental arithmetic, quickly “burn out” and cease to be able to show bright achievements for a long time and sustainably.

Mental counting competition

Trachtenberg method

Among those practicing mental calculation, the book “Quick Counting Systems” by Zurich mathematics professor 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 painting by the Russian artist Nikolai Bogdanov-Belsky “Oral Abacus. At the public school of S. A. Rachinsky,” written in 1895. The problem shown on the board, which the students are thinking about, requires fairly high mental arithmetic skills and ingenuity. Here is her condition:

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

Some mental counting techniques

To multiply a number by a single-digit factor (for example, 34*9) orally, you need to 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 of 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 as 147*8=(150-3)*8=150*8-3*8=1200-24=1176

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 simpler: 225*6=(200+25)*6=200*6+25*6=1200+150=1350.

There are several more ways of mental calculation, 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 special 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 from the resulting number, for example 45*9=450-45=405

It is more convenient to multiply by 5 this way: first multiply by 10, and then divide by 2

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

Notes

Literature

Bantova M. A. System for developing computing skills. //Start school - 1993.-No. 11.-s. 38-43.
Beloshistaya A.V. Technique for developing oral computational skills within 100 // Primary School. - 2001.- № 7
Berman G. N. Counting techniques, ed. 6th, M.: Fizmatgiz, 1959.
Borotbenko E I. Testing mental calculation skills. //Start school - 1972. - No. 7. - p. 32-34.
Vozdvizhensky A. Mental calculations. Rules and simplified examples of operations with numbers. - 1908.
Volkova S., Moro M. I. Addition and subtraction of multi-digit numbers. //Start school - 1998.-No. 8.-p.46-50
Voskresensky M. P. Techniques for shortened calculations. - M.D905.-148s.
Wroblewski. How to learn to count easily and quickly. - M.-1932.-132s.
Goldstein D. N. Course of simplified calculations. M.: State. educational pedagogical ed., 1931.
Goldstein D. N. Fast computing technique. M.: Uchpedgiz, 1948.
Gonchar D. R. Mental arithmetic and memory: riddles, development techniques, games // In collection. Mental arithmetic and memory. Donetsk: Stalker, 1997
Demidova T. E., Tonkikh A. P. Methods of rational calculations in initial course mathematics // Primary school. - 2002. - No. 2. - P. 94-103.
Cutler E. McShane R. Quick counting system according to Trachtenberg. - M.: Uchpedgiz. - 1967. −150 p.
Lipatnikova I. G. The role of oral exercises in mathematics lessons // Elementary school. - 1998. - No. 2.
Martel F. Fast counting techniques. - Pb. −1913. −34s.
Martynov I. I. Oral arithmetic is for a schoolchild what scales are for a musician. // Elementary School. - 2003. - No. 10. - P. 59-61.
Melentyev P.V."Fast and mental calculations." M.: Gostekhizdat, 1930.
Perelman Ya. I. Quick count. L.: Soyuzpechat, 1945.
Pekelis V.D.“Your possibilities, man!” M.: “Knowledge”, 1973.
Robert Toquet"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 mental calculations. //Start school - 1975.-No. 10.-p. 59-62.
Faddeicheva T. I. Teaching mental calculations // Primary school. - 2003. - No. 10.
Firemark D.S.“The task came from a painting.” M.: “Science”.

Links

V. Pekelis. Miracle counters // Technology for Youth, No. 7, 1974
S. Trankovsky. Oral counting // Science and Life, No. 7, 2006.
1001 tasks for mental calculation S.A. Rachinsky.

Wikimedia Foundation.

2010.
Ustinskaya

Environmental sustainability

See what “Oral counting” is in other dictionaries: oral - oral...

See what “Oral counting” is in other dictionaries: Russian spelling dictionary - pronounced, verbal, verbal, oral. Ant. written Dictionary of Russian synonyms. oral oral, verbal; verbal (special) Dictionary of synonyms of the Russian language. Practical guide. M.: Russian language. Z. E. Alexandrova. 2011…

Synonym dictionary ORAL - [sn], oral, oral. 1. Pronounced, not written down. Oral speech . Oral tradition. Oral test. Orally (adv.) convey an answer. 2. adj. to the mouth, oral (anat.). Oral muscles. ❖ Oral literature (philol.) is the same as folklore... ... Dictionary

Synonym dictionary Ushakova - ORAL, see mouth. Dahl's Explanatory Dictionary. IN AND. Dahl. 1863 1866 …

Phenomenal counters

Among the famous Russian “super counters”:

Among foreign ones:

Although some experts insisted that it was a matter of innate abilities, others argued the opposite: “the matter is not only and not so much in some exceptional, “phenomenal” abilities, but in the knowledge of certain mathematical laws that allow one to quickly make calculations” and willingly revealed these laws .

Mental counting competition

Trachtenberg method

Mental arithmetic in art

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

Some mental counting techniques

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

There are several more ways of mental calculation, 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

It is more convenient to multiply by 5 this way: first multiply by 10, and then divide by 2

Notes

Literature

Bantova M. A. System for developing computing skills. //Start school - 1993.-No. 11.-s. 38-43.
Beloshistaya A.V. Technique for developing oral computational skills within 100 // Elementary school. - 2001.- No. 7
Berman G. N. Counting techniques, ed. 6th, M.: Fizmatgiz, 1959.
Borotbenko E I. Testing mental calculation skills. //Start school - 1972. - No. 7. - p. 32-34.
Vozdvizhensky A. Mental calculations. Rules and simplified examples of operations with numbers. - 1908.
Volkova S., Moro M. I. Addition and subtraction of multi-digit numbers. //Start school - 1998.-No. 8.-p.46-50
Voskresensky M. P. Techniques for shortened calculations. - M.D905.-148s.
Wroblewski. How to learn to count easily and quickly. - M.-1932.-132s.
Goldstein D. N. Course of simplified calculations. M.: State. educational pedagogical ed., 1931.
Goldstein D. N. Fast computing technique. M.: Uchpedgiz, 1948.
Gonchar D. R. Mental arithmetic and memory: riddles, development techniques, games // In collection. Mental arithmetic and memory. Donetsk: Stalker, 1997
Demidova T. E., Tonkikh A. P. Techniques of rational calculations in the initial course of mathematics // Primary school. - 2002. - No. 2. - P. 94-103.
Cutler E. McShane R. Quick counting system according to Trachtenberg. - M.: Uchpedgiz. - 1967. −150 p.
Lipatnikova I. G. The role of oral exercises in mathematics lessons // Elementary school. - 1998. - No. 2.
Martel F. Fast counting techniques. - Pb. −1913. −34s.
Martynov I. I. Oral arithmetic is for a schoolchild what scales are for a musician. // Elementary School. - 2003. - No. 10. - P. 59-61.
Melentyev P.V."Fast and mental calculations." M.: Gostekhizdat, 1930.
Perelman Ya. I. Quick count. L.: Soyuzpechat, 1945.
Pekelis V.D.“Your possibilities, man!” M.: “Knowledge”, 1973.
Robert Toquet"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 mental calculations. //Start school - 1975.-No. 10.-p. 59-62.
Faddeicheva T. I. Teaching mental calculations // Primary school. - 2003. - No. 10.
Firemark D.S.“The task came from a painting.” M.: “Science”.

Links

V. Pekelis. Miracle counters // Technology for Youth, No. 7, 1974
S. Trankovsky. Oral counting // Science and Life, No. 7, 2006.
1001 tasks for mental calculation S.A. Rachinsky.

Wikimedia Foundation.

Environmental sustainability

See what “Oral counting” is in other dictionaries: oral - oral...

Pronounced, verbal, verbal, oral. Ant. written Dictionary of Russian synonyms. oral oral, verbal; verbal (special) Dictionary of synonyms of the Russian language. Practical guide. M.: Russian language. Z. E. Alexandrova. 2011… - pronounced, verbal, verbal, oral. Ant. written Dictionary of Russian synonyms. oral oral, verbal; verbal (special) Dictionary of synonyms of the Russian language. Practical guide. M.: Russian language. Z. E. Alexandrova. 2011…

- [sn], oral, oral. 1. Pronounced, not written down. Oral speech. Oral tradition. Oral test. Orally (adv.) convey an answer. 2. adj. to the mouth, oral (anat.). Oral muscles. ❖ Oral literature (philol.) is the same as folklore... ... Ushakov's Explanatory Dictionary

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

Mental arithmetic in mathematics lessons. Mental counting At the beginning of a word, oral counting

LIST OF REFERENCES USED

Phenomenal counters

Mental counting competition

Trachtenberg method

Mental arithmetic in art

Some mental counting techniques

see also

Notes

Literature

Links

Environmental sustainability

Phenomenal counters

Mental counting competition

Trachtenberg method

Mental arithmetic in art

Some mental counting techniques

see also

Notes

Literature

Links

Environmental sustainability