Non-traditional methods of multiplying multivalued numbers. Methods of multiplication of numbers in different countries

Gassales Vasily

The theme of the work "Unusual Methods of Calculations" is interesting and relevant, as students constantly perform arithmetic actions on numbers, and the ability to quickly calculate, increases the success in learning and develops the flexibility of the mind.

Vasily managed to clearly state the reasons for his appeal to this topic, correctly formulated the goal and task of work. Having studied various sources of information, found interesting and unusual methods of multiplication and learned to apply them in practice. The student considered the pros and cons of each method and made the right conclusion. The reliability of the output confirms a new way of multiplication. At the same time, the student skillfully uses special terminology and knowledge outside the school curriculum of mathematics. The theme of the work corresponds to the content, the material is stated clearly and accessible.

The results of the work are practical and can be interesting to a wide range of people.

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MOU "Kurovskaya Secondary School No. 6"

Abstract for mathematics on the topic:

"Unusual methods of multiplication."

Fulfilled the student 6 "b" class

Cancer Vasily.

Leader:

Smirnova Tatiana Vladimirovna.

2011

Introduction .............................................................................. ....... 2
Main part. Unusual methods of multiplication ........................... ... 3

2.1. A little story ........................................................................ ..3

2.2. Multiplication on the fingers ............................................................... ... 4

2.3. Multiplication by 9 ........................................................................... 5

2.4. Indian method of multiplication ...................................................... .6

2.5. Multiplying by the way "Little Castle" ....................................... 7

2.6. Multiplication by the way "Jealousy" .................................................. ... 8

2.7. Peasant Method of Multiplication ....................................................... 9

2.8 New way .............................................................................10

Conclusion .............................................................................. ... 11
List of references ......................................................................12

I. Entry.

It's impossible to do without computing a person in everyday life. Therefore, in the lessons of mathematics, we are primarily taught to perform actions on numbers, that is, to count. We multiply, divide, fold and deduct we are familiar to all ways that are studied at school.

Once I accidentally came across the book S. N. Ololand, Yu. V. Nesterenko and M. K. Potapova "Ancient entertaining tasks". List through this book, my attention attracted a page called "Multiplication on the fingers". It turned out that you can multiply not only because they offer us in mathematics textbooks. It became interesting to me, and whether there are some other calculations. After all, the ability to quickly make calculations causes frank surprise.

The continuous use of modern computing equipment leads to the fact that students find it difficult to produce any calculations without having a table or counting machine at their disposal. Knowledge of simplified computing techniques makes it possible not only to quickly produce simple calculations in the mind, but also control, evaluate, find and correct errors as a result of mechanized calculations. In addition, the development of computing skills develops memory, increases the level of mathematical culture of thinking, helps to fully absorb objects of the physico-mathematical cycle.

Purpose of work:

Show unusual methods of multiplication.

Tasks:

Find as many unusual calculation methods as possible.
Learn to apply them.
Choose for yourself the most interesting or lighter than those are offered at school, and use them with the score.

II. Main part. Unusual methods of multiplication.

2.1. A little story.

Those methods of calculations we use now were not always so simple and comfortable. In the old days enjoyed more cumbersome and slow techniques. And if the 21st century schoolboy could be transferred to five centuries ago, he would have struck our ancestors to the speed and error of his calculations. The surrounding schools and monasteries would fly about it about him, eclipsed by the glory of the most scene counters of that era, and from all sides would come to learn from the New Great Master.

Especially difficult in the old days were the actions of multiplication and division. Then there was no one generated admission practice for each action. On the contrary, in the go was at the same time a little bit of a dozen of various ways of multiplication and division - the techniques of each other confusing, to remember who could not be a man of medium abilities. Each teacher of the Accounts was held by his favorite reception, each "Master of Denilation" (there were such specialists) praised his own way of doing this action.

In the book of V. Bellyustin "As people gradually reached the real arithmetic" set out 27 methods of multiplication, and the author notes: "It is very possible that there are still methods hidden in the caches of books, scattered in numerous, mainly handwritten collections."

And all these multiplication techniques are "chess or organizing", "bending", "cross", "lattice", "backward", "diamond" and others competed with each other and assimilated with great difficulty.

Let's consider the most interesting and simple methods of multiplication.

2.2. Multiplication on the fingers.

Ancient Russian method of multiplication on the fingers is one of the most common methods that Russian merchants have successfully used for many centuries. They learned to multiply on the fingers of unambiguous numbers from 6 to 9. At the same time, it was enough to own the initial skills of the finger account "units", "couples", "three", "fours", "fives" and "dozens". The fingers of the hands here served as auxiliary computing device.

For this, so many fingers pulled out on one hand, as far as the first factor exceeds the number 5, and on the second they did the same for the second factor. The remaining fingers were fucked. Then the number (total) elongated fingers was taken and was multiplied by 10, then multiplying the numbers showing how much fingers were hung on their hands, and the results were folded.

For example, multiply 7 on 8. In the considered example, 2 and 3 fingers will be replaced. If you fold the quantities of the bent fingers (2 + 3 \u003d 5) and multiply the amounts of non-bent (2 3 \u003d 6), then the number of tens and units of the desired work 56 is obtained. So you can calculate the product of any unambiguous numbers, more than 5.

2.3. Multiplication by 9.

Multiplication for number 9 - 9 · 1, 9 · 2 ... 9 · 10 - it is easier to eat out of memory and it is more difficult to manually by the method of addition, but it is for the number of 9 multiplication that "on the fingers" is easily reproduced. Pour your fingers on both hands and turn your hands with your palms from ourselves. Mentally presate the fingers sequentially numbers from 1 to 10, starting with the mother's maiden and ending with the little finger of the right hand (this is shown in the figure).

Suppose, we want to multiply 9 on 6. Beaging your finger with a number equal to the number that we will multiply nine. In our example, you need to bend a finger with number 6. The number of fingers to the left of the bent finger shows us the number of dozens in the answer, the number of fingers on the right is the number of units. On the left we have 5 fingers are not reducing, on the right - 4 fingers. Thus, 9 · 6 \u003d 54. Below in the figure, the entire principle of "calculations" is shown in detail.

Another example: need to calculate 9 · 8 \u003d?. In the course of the matter, let's say that the fingers of the hands may not necessarily act as a "counting machine". Take, for example, 10 cells in the notebook. Excrying the 8th cell. On the left there are 7 cells left, on the right - 2 cells. So 9 · 8 \u003d 72. Everything is very simple.

7 cells 2 cells.

2.4. Indian multiplication method.

The most valuable contribution to the treasury of mathematical knowledge was performed in India. Hindus offered the method of recording numbers used by us with ten signs: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.

The basis of this method is the idea that one and the same figure denotes units, dozens, hundreds or thousands, depending on what place this figure takes. The place occupied, in the absence of any discharges, is determined by zeros attributed to the numbers.

Hindus considered great. They came up with a very simple way of multiplication. They performed multiplied, starting with the older discharge, and recorded incomplete works just above the multiple, blessing. At the same time, the senior discharge of a complete work was immediately visible and, moreover, a pass of any number was excluded. The multiplication sign has not yet been known, so they left a small distance between the multipliers. For example, multiply in the way 537 to 6:

537 6

(5 ∙ 6 =30) 30

537 6

(300 + 3 ∙ 6 = 318) 318

537 6

(3180 +7 ∙ 6 = 3222) 3222

2.5. Multiplication by the method of "Little Castle".

Multiplication of numbers is now studying in the first class school. But in the Middle Ages, very few people owned the art of multiplication. A rare aristocrat could boast of knowledge of the multiplication table, even if he graduated from European University.

For the millennium, the development of mathematics was invented many ways to multiply numbers. Italian Mathematics of Luke Pachet in his treatise "The sum of knowledge of arithmetic, relationships and proportionality" (1494) leads eight different multiplication methods. The first of them is called "Little Castle", and the second no less romantic name "jealousy or lattice multiplication".

The top numbers, starting with the older discharge, alternately multiply on the lower number and are recorded in the column with the addition of the desired number of zeros. Then the results fold.

2.6. Multiplication of numbers by the "Jealousy" method.

The second method wears the romantic name "Jealousy", or "lattice multiplication".

First, the rectangle is drawn, separated into squares, and the sizes of the sides of the rectangle correspond to the number of decimal signs in the multiplier and multiplier. Then the square cells are divided according to the diagonal, and "... it turns out a picture similar to the lattice shutters-blinds," writes Pacheti. "Such shutters were hanging on the windows of Venetian houses, preventing street passers-by to see the windows sitting at the windows and nuns."

Multiply in this way 347 to 29. Note the table, write down the number 347 above it, and on the right number 29.

In each line, we write the work of numbers standing on this cell and to the right of it, with the number of tens of works, we write above the oblique feature, and the numbers are units under it. Now we add numbers in each oblique strip, performing this operation, right to left. If the amount is less than 10, then it is writing under the bottom of the strip. If it is more than 10, then we write only the number of units of the amount, and the figure of tens add to the next amount. As a result, we get the desired work 10063.

3 4 7

10 0 6 3

2.7. Peasant method of multiplication.

The most, in my opinion, the "native" and light way of multiplication is a way that Russian peasants consumed. This reception does not require knowledge of the multiplication table on the number 2. The essence of it is that the multiplication of any two numbers is reduced to a row of sequential divisions of one number in half while rejection of another number. The division in half is continued until 1, in parallel, doubles another number. The last tweed number and gives a desired result.

In the case of an odd number, it is necessary to learn a unit and divide the residue in half; But it will be necessary to add all those numbers of this column to the last number of the right column, which are against the odd numbers of the left column: the amount and will be the desired work

37……….32

74……….16

148……….8

296……….4

592……….2

1184……….1

The product of all pairs of corresponding numbers is the same, so

37 ∙ 32 = 1184 ∙ 1 = 1184

In the case when one of the numbers is odd or both odd, we do as follows:

24 ∙ 17

24 ∙ 16 =

48 ∙ 8 =

96 ∙ 4 =

192 ∙ 2 =

384 ∙ 1 = 384

24 ∙ 17 = 24∙(16+1)=24 ∙ 16 + 24 = 384 + 24 = 408

2.8. New way of multiplication.

Interesting a new way of multiplication, which recently appeared messages. The inventor of the new oral account system Candidate of Philosophical Sciences Vasily Okneshovnikov claims that a person is able to memorize a huge supply of information, the main thing - how to place this information. According to the scientist himself, the most advantageous in this regard is a nine-sized system - all data is simply placed in nine cells located like buttons on the calculator.

It is very simple to count on such a table. For example, multiply the number 15647 by 5. In terms of the table corresponding to the top selected, select the numbers corresponding to the numbers of the number in order: a unit, a five, six, fourth and seven. We get: 05 25 30 20 35

Left digit (in our example - zero), we leave unchanged, and the following numbers fold in pairs: a twin five, a top five, zero with a twos, zero with a triple. The last digit is also unchanged.

As a result, we get: 078235. The number 78235 and there is a result of multiplication.

If, when folding two digits, the number exceeding nine, its first digit is added to the previous figure of the result, and the second is written to "its" place.

III. Conclusion.

Of all the unusual ways found by me, the method of "lattice multiplication or jealousy" seemed more interesting. I showed it to my classmates, and he also really liked.

The simplest method of "doubling and split" seemed to me, which Russian peasants used. I use it when you multiply not too large numbers (it is very convenient to use it when multiplying two-digit numbers).

I was interested in a new way of multiplication, because it allows you to "turn" with huge numbers in the mind.

I think that our method of multiplication in the column is not perfect and you can come up with even faster and more reliable ways.

Literature.

Depima I. "Stories about mathematics." - Leningrad: Education, 1954. - 140 s.
Koreev A.A. The phenomenon of Russian multiplication. History. http://numbernautics.ru/
Olochnik S. N., Nesterenko Yu. V., Potapov M. K. "Ancient entertaining tasks". - M.: Science. The main editorial office of physico-mathematical literature, 1985. - 160 p.
Perelman Ya.I. Quick account. Thirty simple oral receptions. L., 1941 - 12 s.
Perelman Ya.I. Entertaining arithmetic. M. Russanova, 1994--205c.https://accounts.google.com.
Signatures for slides:
The work performed a student of 6 "b" class of the gods of Vasily. Leader: Smirnova Tatyana Vladimirovna Unusual methods of multiplication
Objective: show unusual methods of multiplication. Tasks: Find unusual methods of multiplication. Learn to apply them. Choose for yourself the most interesting or lighter and use them with the score.
Multiplication on the fingers.
Multiplication by 9.
The Italian Mathematics of Luke Pacioli was born in 1445.
Multiplying in the way "Little Castle"
Multiplication by the "Jealousy" method
Multiplying the grid meter. 3 4 7 2 9 6 8 1 4 3 6 6 3 7 2 3 6 0 10 347 29 \u003d 10063
Russian peasant method 37 32 37 ..........32 74 ..........16 148 ..........8 296 ..........4 592 ......... .2 1184 ......... 1 37 32 \u003d 1184
Thanks for attention

MBOU "SOSH S. Volnoe »Kharabalinsky district Astrakhan region

Project on:

« Unusual methods are multipliedand I»

Work performed:

students of grade 5 :

Tulesshev Amina,

Sultanov Samat,

Kujujugov Racita.

R project Card:

mathematic teacher

Fateeva T.V.

Volnoe 201. 6 year .

"All there is a number" Pythagora

Introduction

In the 21st century it is impossible to imagine the life of a person who does not produce calculations: these are sellers, and accountant, and ordinary schoolchildren.

The study of almost any subject in school involves good knowledge of mathematics, and without it you can not master these items. Two elements dominate mathematics - numbers and figures with their infinite variety of properties and actions with them.

We wanted to learn more about the history of mathematical action. Now, when computing techniques are rapidly developing, many do not want to bother themselves in mind. Therefore, we decided to show not only the fact that the process itself may be interesting, but even that, well, having learned the rapid account techniques, you can argue with a computer.

The relevance of this topic is that the use of non-standard techniques in the formation of computational skills increases the interest of students to mathematics and promotes the development of mathematical abilities.

Purpose of work:

ANDto heat some non-standard multiplication techniques and show that their application makes the calculation process rational and interesting And to calculate which, sufficiently oral account or the use of pencil, handles and paper.

Hypothesis:

E.if our ancestors were able to multiply by old ways, if having studied literature on this problem, whether a modern schoolboy can learn this, or some supernatural abilities are needed.

Tasks:

1. Find unusual methods of multiplication.

2. Learn to apply them.

3. Choose for yourself the most interesting or lighter than those are offered at school, and use them with the score.

4. Teach classmates to apply newe. methods Multiplication.

Object of study: mathematical action Multiplication

Subject of study: methods of multiplication

Research methods:

Search method using scientific and educational literature, the Internet;

Research method in determining multiplication methods;

Practical method when solving examples;

- - Questioning of respondents about knowledge of non-standard multiplication methods.

Historical reference

There are people with unusual abilities that, by the speed of oral computing, can compete with the computer. They are called "miracle - meters." And there are a lot of such people.

It is said that the Father Gauss, hoping with his workers at the end of the week, added payment for every day earnings for overtime hours. Once, after Gauss Father graduated from the calculations, which took place for the operations of the father of the child, who was 3 years, exclaimed: "Dad, counting is not true! This should be the amount! " Calculations were repeated and surprised were convinced that the boy indicated the correct amount.

In Russia, at the beginning of the 20th century, Roman Semenovich Levitan, famous for the pseudonym Arrago, shone his skills. Unique abilities began to appear in the boy already at an early age. For a few seconds, he built a square and a cube of ten-figure numbers, removed the roots of varying degrees. It seemed that all this was he made with extraordinary ease. But this lightness was deceptive and demanded great brain work.

In 2007, Mark Cherry, who then was 2.5 years old, struck the whole country with his intellectual abilities. The young participant of the show "Minute of Fame" was easily considered in the mind of multivalued numbers, ahead of the calculations of parents and the jury, which used calculators. Already in two years, he mastered the cosine and sinus table, as well as some logarithms.

The Cybernetics Institute of the Ukrainian Academy of Sciences held competitions of computer and man. The competition was attended by a young phenomenon phenomenon Igor Shelushkov and the "Peace". The car made many difficult operations in a few seconds, but Igor Shelushkov turned out to be the winner.

Sydney University in India also held a human and car competition. Shakuntala Devi also ahead of the computer.

Most of these people have excellent memory and have tasting. But some of them do not possess any special abilities to mathematics. They know the secret! And this secret is that they learned the rapid score, remembered several special formulas. It means that we can also use these techniques, quickly and accurately count.

Especially difficult in the old days were the actions of multiplication and division. Then there was no one generated admission practice for each action.

On the contrary, in the go was at the same time a little bit of a dozen of various ways of multiplication and division - the techniques of each other confusing, to remember who could not be a man of medium abilities. Each teacher of the Accounts was held by his favorite reception, each "Master of Denilation" (there were such specialists) praised his own way of doing this action.

And all these multiplication techniques are "chess or organizing", "bending", "cross", "lattice", "backward", "diamond" and others competed with each other and assimilated with great difficulty.

Let's consider the most interesting and simple methods of multiplication.

Ancient Russian method of multiplication on the fingers

This is one of the most common methods that Russian merchants have successfully used for many centuries.

The principle of this method: multiplication on the fingers of unambiguous numbers from 6 to 9. The fingers of the hands served here auxiliary computing device.

Very easily reproduced "on the fingers" multiplication for the number 9

R.starthosefingers on both hands and turn hands with palms from ourselves. Mentally assign fingers sequentially from 1 to 10, starting with the mother's hermit and ending with the little finger of the right hand. Suppose, we want to multiply 9 on 6. Beaging your finger with a number equal to the number that we will multiply nine. In our example, you need to bend a finger with number 6. The number of fingers to the left of the bent finger shows us the number of dozens in the answer, the number of fingers on the right is the number of units. On the left we have 5 fingers are not reducing, on the right - 4 fingers. Thus, 9 · 6 \u003d 54.

Multiplication by 9 using tetradi cells

Take, for example, 10 cells in the notebook. Excrying the 8th cell. On the left there are 7 cells left, on the right - 2 cells. So, 9 · 8 \u003d 72. Everything is very simple!

7 2

Method of multiplication "Little Castle"

The advantage of the method of multiplying the "Little Castle" is that from the very beginning the numbers of high-level digits are determined, and this is important if it is required to quickly appreciate the value.The top numbers, starting with the older discharge, alternately multiply on the lower number and are recorded in the column with the addition of the desired number of zeros. Then the results fold.

"Detergent multiplication"

First, the rectangle is drawn, separated into squares, and the sizes of the sides of the rectangle correspond to the number of decimal signs in the multiplier and multiplier.

Then square cells are divided diagonally, and "... It turns out a picture similar to the lattice shutters-blinds. Such shutters were hanging on the windows of Venetian houses ... "

"Russian peasant way"

In Russia among the peasants, a way was distributed, which did not require knowledge of the entire multiplication table. Here you only need to multiply the ability and divide the numbers by 2.

Watch one number on the left, and the other on the right on one line. The left number will be divided into 2, and the right - multiply by 2 and the results are recorded in the column.

If the balance has arisen, then it is discarded. Multiplication and division by 2 continue until the left remains 1.

Then, strike out those lines from the column in which even numbers are worth it. Now lay the remaining numbers in the right column.

This method of multiplication is much simpler previously discussed methods of multiplication. But he is also very cumbersome.

"Multiplication of Cross"

The ancient Greeks and Hindus in Starin called the reception of the cross multiplication "method of lightning" or "crossed multiplication".

24 and 32.

2 4

3 2

4x2 \u003d 8 - the last digit of the result;

2x2 \u003d 4; 4x3 \u003d 12; 4 + 12 \u003d 16; 6 - the penultimate figure of the result, the unit memorial;

2x3 \u003d 6 Yes, even retained in the mind of the digit, we have 7- this is the first digit of the result.

We get all the figures of the work: 7,6,8. Answer:768.

Indian method of multiplication

546 7

5 7=35 35

350+ 4 7=378 378

3780 + 6 7=3822 3822

546 7= 3822

W.the set is starting with the older discharge, and write incomplete works just above the multiple, is bonded. At the same time, the senior discharge of a complete work is immediately visible and, in addition, the pass of any number is excluded. The multiplication sign has not yet been known, so there was a small distance between the factors

Chinese (drawing) method of multiplication

Example №1: 12 × 321 = 3852
Drawfirst number from top to bottom, left to right: one green wand (1 ); Two orange sticks (2 ). 12 Drawn
Drawsecond number bottom up, to the left: Three blue wands (3 ); Two red (2 ); one lilac (1 ). 321 Drawn

Now we walk with a simple pencil with a drawing, the intersection points of numbers into parts split and proceed to the counting points. Moving to right left (clockwise):2 , 5 , 8 , 3 . Number-result We will "collect" from left to right (counterclockwise) received3852

Example number 2.: 24 × 34 = 816
In this example there are nuances ;-) When counting points in the first part it turned out16 . We send-add to the points of the second part (20 + 1 )…

Example number 3.: 215 × 741 = 159315

In the course of the project, we conducted a survey. Students responded to the following questions.

1. Is it necessary to a modern person an oral account?

Yes Not

2. Do you know other methods of multiplication other than multiplication in the column?

Yes Not

3. Do you use them?

Yes Not

4. Would you like to know other methods of multiplication?

Well no

We have surveyed students of grades 5-10.

This survey showed that modern schoolchildren do not know other ways to perform actions, as they rarely refer to the material outside the school curriculum.

Output:

In the history of mathematics there are many interesting events and discoveries, unfortunately not all this information comes to us, modern students.

This work, we wanted to at least slightly fill this space and convey information about the old methods of multiplication to our peers.

During the robots, we learned about the origin of the multiplication. In the old days, it was not easy to own this action, then there was no more, as now, one developed admission practice. On the contrary, in the go was at the same time a little bit of a dozen of various ways of multiplication - the receptions one of the other confusing, firmly, to remember which was unable to have a man of medium abilities. Each teacher of the Accounts held his favorite admission, each "Master" (there were such specialists) praised his own way of doing this action. It was even recognized that in order to master the art of the rapid and error-free multiplication of multivalued numbers, it is necessary to special natural dating, exceptional abilities; An ordinary people are inaccessible to ordinary people.

We have proven with your work that our hypothesis is true, you do not need to have supernatural abilities to be able to use the old methods of multiplication. And we have learned to pick up the material, process it, that is, allocate the main thing and systematize.

Having learned to consider all the presented ways, we came to the conclusion: that the most simple ways are those that we study at school, and maybe we just got used to them.

Modern method of multiplication is simple and accessible to everyone.

But we think that our method of multiplication in the column is not perfect and you can come up with even faster and more reliable ways.

It is possible that from the first time, many will not work quickly, with the go, perform these or other calculations.

No problem. Need a constant computing training. It will help to buy useful oral skills!

Bibliography

1. Glezer, G. I. History of mathematics at school / G. I. Glaser // History of mathematics at school: manual for teachers / edited by V. N. Young. - M.: Enlightenment, 1964. - P. 376.

Perelman Ya. I. Entertaining arithmetic: Riddles and wonder in the world numbers. - M.: Rusanova Publisher, 1994. - P. 142.

Encyclopedia for children. T. 11. Mathematics / Chapters. ed. M. D. Aksenova. - M.: Avat +, 2003. - P. 130.

Mathematics magazine №15 2011

Internet resources.

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"The score and calculations are the basis of order in the head."
Pestozzi

Purpose:

Get acquainted with antique multiplication techniques.
Expand knowledge on various multiplication techniques.
Learn to perform actions with natural numbers using vintage methods of multiplication.

An old way to multiply 9 on the fingers
Multiplying by Ferrela.
Japanese multiplication method.
Italian multiplication method ("grid")
Russian method of multiplication.
Indian multiplication method.

Structure occupation

The relevance of the use of rapid account techniques.

In modern life, each person often has to perform a huge amount of calculations and calculations. Therefore, the goal of my work is to show light, fast and accurate score methods that will not only help you during any calculations, but will cause a considerable surprise among familiar and comrades, because the free execution of countable operations can significantly testify to the uncommonness of your intelligence. The fundamental element of computational culture is conscious and durable computational skills. The problem of the formation of computational culture is relevant for the entire school course of mathematics, starting from the initial classes, and requires not a simple mastering of computational skills, but using them in various situations. The possession of computational skills and skills is of great importance for the assimilation of the material being studied, allows you to bring up valuable labor qualities: responsible attitude towards your work, the ability to detect and correct the errors allowed in the work, the accurate execution of the task, creative work towards work. However, recently, the level of computing skills, expression transformations has a pronounced tendency to decline, students allow a lot of errors in counting, are increasingly using a calculator, they do not think rationally, which adversely affects the quality of learning and the level of mathematical knowledge of students in general. One of the components of computational culture is verbal counting which is important. The ability to quickly and correctly produce simple calculations "in the mind" is necessary for each person.

Old ways of multiplication of numbers.

1. An old way to multiply 9 on the fingers

It's simple. To multiply any number from 1 to 9 to 9, look at your hands. Generate a finger that corresponds to a multiplying number (for example 9 x 3 - Generate the third finger), count your fingers to the bent finger (in the case of 9 x 3 it is 2), then count after the bent finger (in our case - 7). The answer is 27.

2. Multiplying by Ferrela.

For multiplication of units, the movement of multiplies varies multipliers to obtain dozens, dozens of one per unit of the other and the opposite are folded and the results are folded to produce hundreds. Ferrol method is easily multiplying orally two-digit numbers from 10 to 20.

For example: 12x14 \u003d 168.

a) 2x4 \u003d 8, we write 8

b) 1x4 + 2x1 \u003d 6, we write 6

c) 1x1 \u003d 1, we write 1.

3. Japanese Method of Multiplication

This technique resembles a multiplication of a column, but is carried out for quite a long time.

Use of reception. Suppose we need to multiply 13 to 24. Note the following drawing:

This drawing consists of 10 lines (the amount can be any)

These lines denote the number 24 (2 lines, indent, 4 lines)
And these lines indicate the number 13 (1 line, indent, 3 lines)

(intersections in the figure are indicated by points)

The number of intersections:

Upper left edge: 2
Lower left edge: 6
Upper right: 4
Lower right: 12

1) intersections in the upper left edge (2) - the first number of response

2) Amount of intersections of the lower left and upper right edges (6 + 4) - the second number of response

3) intersections in the lower right of the region (12) - the third number of response.

It turns out:2; 10; 12.

Because The last two numbers are double-digit and we cannot burn them, then write only units, and dozens add to the previous one.

4. Italian Method of Multiplication ("Grid")

In Italy, as well as in many countries of the East, this method has gained great fame.

Use of reception:

For example, multiply 6827 by 345.

1. Draw a square grid and write one of the numbers above the columns, and the second in height.

2. Multiply the number of each row sequentially in the number of each column.

6 * 3 \u003d 18. We write 1 and 8
8 * 3 \u003d 24. Record 2 and 4

If the multiplication turns out the unambiguous number, write up 0, and at the bottom of this number.

(As we have in the example when you multiply 2 on 3 it turned out 6. At the top we recorded 0, and below 6)

3. Fill the entire grid and fold the numbers by following diagonal stripes. We begin to fold on the right left. If the amount of one diagonal contains dozens, then add them to units of the following diagonal.

Answer: 2355315.

5. Russian method of multiplication.

This method of multiplication was used by Russian peasants about 2-4th century ago, and was still designed in ancient times. The essence of this method is: "For how much we divide the first factor, it is so many second." Here example: we need 32 multiplied by 13. That's how this example is 3-4th century ago, our ancestors:

32 * 13 (32 divide 2, and 13 multiply by 2)
16 * 26 (16 divide on 2, and 26 multiply by 2)
8 * 52 (etc)
4 * 104
2 * 208
1 * 416 =416

The division in half is continued until 1, in parallel, doubles another number. The last tweed number and gives a desired result. It is not difficult to understand what this method is based: the work does not change if one multiplier is doubled, and the other will double to increase. It is clear that as a result of the repeated repetition of this operation, a desired work is obtained

However, how to do if at the same time you have to share the odd number in half? People's way easily comes out of this difficulty. It is necessary - the rule says, - in the event of a different number, throw away the unit and divide the residue in half; But it will be necessary to add all those numbers of this column to the last number of the right column, which are against the odd numbers of the left column: the amount and will be the desired work. Almost this makes that all rows with even left numbers are burned; Only those that contain the left odd number remain. We give an example (asterisks indicate that this line must be shocked):

19*17
4 *68*
2 *136*
1 *272

Folding unlined numbers, we get quite the right result:

17 + 34 + 272 = 323.

Answer: 323.

6. Indian method of multiplication.

This method of multiplication was used in ancient India.

For multiplication, for example, 793 on 92, we will write a single number as multiplier and under it as a multiplier. To easier to navigate, you can use the mesh (a) as a sample.

Now we multiply the left digit of the multiplier on each digit of the multiple, that is, 9x7, 9x9 and 9x3. The obtained works we write to the grid (b), meaning the following rules:

Rule 1. The units of the first work should be written in the same column as the multiplier, that is, in this case, under 9.
Rule 2. The subsequent work must be written in such a way that the units are placed in the column directly to the right of the previous product.

We repeat the entire process with other figures of the multiplier, following the same rules (C).

Then we fold the numbers in the columns and get the answer: 72956.

As you can see, we get a large list of works. Indians who had a great practice wrote every figure not to the appropriate column, and from above, as far as it was possible. Then they folded the numbers in the columns and received the result.

Conclusion

We entered the new millennium! Grand discoveries and achievements of humanity. We know a lot, we know how much. It seems something supernatural that with the help of numbers and formulas, you can calculate the flight ship's flight, "economic - situation" in the country, weather on "tomorrow", describe the sound of notes in the melody. We know the statement of ancient Greek mathematics, the philosopher, who lived in the 4th century, D.N.- Pytagora - "Everything is there is a number!".

According to the philosophical view of this scientist and his followers, the numbers are managed not only by measure and weight, but also by all phenomena occurring in nature, and are the essence of harmony, reigning in the world, soul of space.

Describing the vintage methods of calculations and modern rapid account techniques, I tried to show that both in the past and in the future, without mathematics, science created by the mind of man, could not do.

"Who has been engaged in mathematics from ornamental years, he develops attention, trains the brain, his will, raises perseverance and perseverance in achieving the goal." (A.Markyshevich)

Literature.

Encyclopedia for children. "T.23". Universal Encyclopedic Dictionary \\ Ed. College: M. Aksyonova, E. Zhuravlyova, D. Riuri, and others. - M.: The world of the encyclopedia of Avanta +, Astrel, 2008. - 688 p.
Ozhegov S. I. Dictionary of Russian: OK. 57000 words / ed. CHL - Corr. Ansir N.Yu. Swedio. - 20th ed.- M.: Enlightenment, 2000. - 1012 p.
I want to know everything! Large illustrated intelligence encyclopedia / lane. from English A. Zykova, K. Malkova, O.Zhroeva. - M.: Publishing House ECMO, 2006. - 440 p.
Sheynina O.S., Solovyov G.M. Mathematics. Classes of school mug 5-6 cl. / O.S.Shinina, G.M. Solovyova - M.: Publishing House of NCNAS, 2007. - 208 p.
Cordemsky B. A., Ahadov A. A. Amazing World of Numbers: Student Book, - M. Enlightenment, 1986.
Minskie E. M. "From the game to Knowledge", M., "Education" in 1982.
Candles A. A. Numbers, figures, tasks M., Enlightenment, 1977.
http: // Matsievsky. newmail. RU / SYS-SCHI / File15.htm
http: //sch69.narod. RU / MOD / 1/6506 / HYSTORY. HTML.

In Ancient India, two methods of multiplication were used: grids and galleys.
At first glance, they seem very complicated, but if you follow step by step in the proposed exercises, you will see that it is quite simple.
Multiply, for example, numbers 6827 and 345:
1. Draw a square mesh and write one of the numbers above the columns, and the second height. In the proposed example, you can use one of these grids.

2. By selecting the grid, multiply the number of each row sequentially in the number of each column. In this case, consistently multiply 3 to 6, 8, 2 and by 7. Look at this scheme, as the product is written in the appropriate cell.

3. Look at how the grid looks like with all the baked cells.

4. In conclusion, we have a number, following diagonal strips. If the sum of one diagonal contains dozens, then add them to the next diagonal.

Look at how the results of the addition of digits in diagonals (they are highlighted with a yellow background) a number 2355315 is compiled, which is the product of Numbers 6827 and 345.

MOU "Kurovskaya Secondary School No. 6"

Abstract for mathematics on the topic:

« Unusual methods of multiplication».

Fulfilled the student 6 "b" class

Cancer Vasily.

Leader:

Smirnova Tatiana Vladimirovna.

Introduction…………………………………………………………………………2

Main part. Unusual methods of multiplication .............................. 3

2.1. A little story ........................................................................ ..3

2.2. Multiplication on your fingers .................................................................. 4

2.3. Multiplication by 9 ........................................................................... 5

2.4. Indian method of multiplication ...................................................... .6

2.5. Multiplying by the way "Little Castle" ....................................... 7

2.6. Multiplying in the way "Jealousy" ................................................... 8

2.7. Peasant Method of Multiplication ................................................... ..9

2.8 New way .............................................................................10

Conclusion ................................................................................. 11.

List of references ........................................................................12

I.. Introduction.

Purpose of work:

Show unusual Methods of multiplication.

Tasks:

Find as much as possible Unusual calculation methods.

Learn to apply them.

Choose for yourself the most interesting or lighter than thoseoffered At school, and use them with the score.

II.. Main part. Unusual methods of multiplication.

2.1. A little story.

And all these multiplication techniques are "chess or organizing", "bending", "cross", "lattice", "backward", "diamond" and others competed with each other and assimilated with great difficulty.

Let's consider the most interesting and simple methods of multiplication.

2.2. Multiplication on the fingers.

2.3. Multiplication by 9.

Multiplication for number 9 - 9 · 1, 9 · 2 ... 9 · 10 - it is easier to eat out of memory and it is more difficult to manually by the method of addition, but it is for the number of 9 multiplication that "on the fingers" is easily reproduced. Pour your fingers on both hands and turn your hands with your palms from ourselves. Mentally presate the fingers sequentially numbers from 1 to 10, starting with the mother's maiden and ending with the little finger of the right hand (this is shown in the figure).

7 cells 2 cells.

2.4. Indian method of multiplication.

(5 ∙ 6 =30) 30

(300 + 3 ∙ 6 = 318) 318

(3180 +7 ∙ 6 = 3222) 3222

2.5 . Multiplying method "Little Castle".

The top numbers, starting with the older discharge, alternately multiply on the lower number and are recorded in the column with the addition of the desired number of zeros. Then the results fold.

2.6. Multiplication of numbers The "Jealousy" method.

The second method wears the romantic name "Jealousy", or "lattice multiplication".

Multiply in this way 347 to 29. Note the table, write down the number 347 above it, and on the right number 29.

2.7. TOrestyansky way multiplication.

The product of all pairs of corresponding numbers is the same, so

37 ∙ 32 = 1184 ∙ 1 = 1184

In the case when one of the numbers is odd or both odd, we do as follows:

384 ∙ 1 = 384

24 ∙ 17 = 24∙(16+1)=24 ∙ 16 + 24 = 384 + 24 = 408

2.8 . New way of multiplication.

Interest A new way of multiplication, which has recently appeared messages. The inventor of the new oral account system Candidate of Philosophical Sciences Vasily Okneshovnikov claims that a person is able to memorize a huge supply of information, the main thing - how to place this information. According to the scientist himself, the most advantageous in this regard is a nine-sized system - all data is simply placed in nine cells located like buttons on the calculator.

Left digit (in our example - zero), we leave unchanged, and the following numbers fold in pairs: a twin five, a top five, zero with a twos, zero with a triple. The last digit is also unchanged.

As a result, we get: 078235. The number 78235 and there is a result of multiplication.

If, when folding two digits, the number exceeding nine, its first digit is added to the previous figure of the result, and the second is written to "its" place.

III. Conclusion.

Of all the unusual ways found by me, the method of "lattice multiplication or jealousy" seemed more interesting. I showed it to my classmates, and he also really liked.

I was interested in a new way of multiplication, because it allows you to "turn" with huge numbers in the mind.

I think that our method of multiplication in the column is not perfect and you can come up with even faster and more reliable ways.

Literature.

Depima I. "Stories about mathematics." - Leningrad: Education, 1954. - 140 s.

Koreev A.A. The phenomenon of Russian multiplication. History. http://numbernautics.ru/

Olochnik S. N., Nesterenko Yu. V., Potapov M. K. "Ancient entertaining tasks". - M.: Science. The main editorial office of physico-mathematical literature, 1985. - 160 p.

Perelman Ya.I. Quick account. Thirty simple oral receptions. L., 1941 - 12 s.

Perelman Ya.I. Entertaining arithmetic. M. Russanova, 1994-205c.

Encyclopedia "I will know the world. Mathematics". - M.: Astrel Ermak, 2004.

Encyclopedia for children. "Mathematics". - M.: Avanta +, 2003. - 688 p.

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