Research work in mathematics in elementary school

Brief Abstract Research
Each schoolboy can multiply multivalued numbers "Stumpy." In this paper, the author draws attention to the existence of alternative methods of multiplication, affordable to younger schoolchildren who can "tedious" calculations to turn into a merry game.
The paper discusses six non-traditional methods of multiplying multivalued numbers used in various historical era: Russian peasant, lattice, small castle, Chinese, Japanese, according to Table V.Okonheshnikova.
The project is intended for the development of cognitive interest in the subject studied, to deepen knowledge in the field of mathematics.
Table of contents
Introduction 3.
Chapter 1. Alternative Methods of Multiplication 4
1.1. A little story 4.
1.2. Russian peasant method of multiplication 4
1.3. Multiplying in the way "Little Castle" 5
1.4. Multiplication of numbers by the "jealousy" or "lattice multiplication" 5
1.5. Chinese Method of Multiplication 5
1.6. Japanese multiplication method 6
1.7. Table Okneshikov 6.
1.8.Motion by the Stage. 7.
Chapter 2. Practical Part 7
2.1. Peasant Method 7.
2.2. Little Castle 7.
2.3. Multiplication of numbers by the "jealousy" or "lattice multiplication" 7
2.4. Chinese method 8.
2.5. Japanese method 8.
2.6. Table Okneshikov 8.
2.7. Questioning 8.
Conclusion 9.
Appendix 10.

"The subject of mathematics is so serious that it is useful not to lose cases of doing it a little entertaining."
B. Pascal
Introduction
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. The question arose: Are there any other alternative methods of calculations? I wanted to explore them in more detail. In search of a response to the questions, this study was carried out.
The purpose of the study: identification of non-traditional multiplication methods to explore the possibility of their use.
In accordance with the purpose of the goal, we have formulated the following tasks:
- Find as many unusual multiplication methods as possible.
- Learn to apply them.
- Choose for yourself the most interesting or lighter than those offered at school, and use them with the score.
- Check in practice multiplication of multivalued numbers.
- Conduct the survey of students in 4th grades
Object of study: Various non-standard multiplication algorithms multiplying numbers
Subject: Mathematical action "Multiplication"
Hypothesis: If there are standard methods for multiplying multi-valued numbers, there may be alternative ways.
Relevance: Dissemination of knowledge about alternative multiplication methods.
Practical significance. In the course of the work, many examples were solved and the album was created, which includes examples with different algorithms multiplying multi-valued numbers by several alternative methods. It may be interested in classmates to expand the mathematical outlook and will serve as the beginning of new experiments.

Chapter 1. Alternative Methods of Multiplication

1.1. A bit of history
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 a modern schoolboy could go for five hundred years ago, he would have struck all 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.
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 techniques of multiplication competed with each other and digested with great difficulty.
Consider the most interesting and simple methods of multiplication.
1.2. Russian peasant method of multiplication
In Russia, 2-3 centuries ago, a method was distributed among the peasants of some provinces that did not require knowledge of the entire multiplication table. It was necessary only to be able to multiply and divide on 2. This method was called the peasant.
To multiply two numbers, they were recorded near, and then the left number was divided into 2, and the right was multiplied by 2. The results are recorded in the column until the left will remain 1. The residue is discarded. We highlight the lines in which there are even numbers. The remaining numbers in the right column are folded.
1.3. Multiplication of the way "Little Castle"
The Italian Mathematics of Luke Pachet in his treatise "The amount of knowledge of arithmetic, relationships and proportionality" (1494) leads eight different multiplication methods. The first of them is called "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.
1.4. Multiplication of numbers by the "jealousy" or "lattice multiplication"
The second method of Luke Pachet is called "Jealousy" or "Detergent Multiplication".
First draws a rectangle, separated into squares. Then the square cells are divided diagonally and "... It turns out a picture similar to the lattice shutters," Pachet writes. "Such shutters were hanging on the windows of Venetian houses, preventing street passers-by to see the windows sitting at the windows and nuns."
Multipling each figure of the first factor with each number of the second, the works are written to the corresponding cells, there are tens of diagonal, and units under it. The figures of the works are obtained by adding numbers in oblique bands. The results of the additions are recorded under the table, as well as to the right.
1.5. Chinese way multiplication
Now imagine the multiplication method, the rapidly discussed on the Internet, which is called Chinese. When multiplying numbers, the points of intersection of direct, which correspond to the number of numbers of each discharge of both multipliers are considered.
1.6. Japanese way multiplication
A Japanese multiplication method is a graphic method using circles and lines. No less fun and interesting than Chinese. Even something like him.
1.7. Table Okoneshikov
Candidate of Philosophical Sciences Vasily Okneshnikov, part-time inventor of a new oral account system, believes that schoolchildren will be able to learn to master and multiply millions, billions and even sextillion with quadrillion. 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.
According to the thoughts, before becoming a computing "computer", you need to send the table created by it.
The table is divided into 9 parts. They are located on the principle of mini calculator: on the left in the lower corner "1", on the right in the upper corner of "9". Each part is the multiplication table of numbers from 1 to 9 (along the same "key" system). In order to multiply any number, for example, on 8, we find a large square corresponding to the number 8 and write out of this square of the number corresponding to the numbers of a multi-valued multi-alarm. The numbers obtained are specifically: the first digit remains unchanged, and all the rest are folded pairwise. The resulting number will be the result of multiplication.
If when two digits are addition, it turns out the number superior to nine, then its first digit is added to the previous figure of the result, and the second is written to "its" place.
The new technique was tested in several Russian schools and universities. The Ministry of Education of the Russian Federation allowed publishing in notebooks into the cells along with the usual Pythagore table a new multiplication table - so far just for dating.
1.8. Multiplication of a column.
Not many know that the author of our usual way of multiplying a multi-valued number to multi-equity should be considered Adam Riza (Appendix 7). This algorithm is considered the most convenient.
Chapter 2. Practical Part
Mastering the listed methods of multiplication, a variety of examples were solved, an album was decorated with samples of various calculation algorithms. (Application). Consider the calculation algorithm on the examples.
2.1. Peasant fashion
Multiply 47 on 35 (Appendix 1),
- Purchased numbers on one line, carry out a vertical line between them;
- by 2, we will divide 2, right - multiplied by 2 (if the residue occurs during the division, then the residue is discarding);
- ending when one appears on the left;
-The strings in which there are left numbers;
-The appropriate numbers on the right - this is the result.
35 + 70 + 140 + 280 + 1120 = 1645.
Output. The method is convenient because it is enough to know the table only on 2. However, when working with large numbers it is very cumbersome. It is convenient for working with double-digit numbers.
2.2. Little castle
(Appendix 2). Output. The method is very similar to our modern "column". Yes, and immediately define the numbers of senior discharges. This is important if you need to quickly appreciate the value.
2.3. Multiplication of numbers by the "jealousy" or "lattice multiplication"
Multiply, for example, numbers 6827 and 345 (Appendix 3):
1. Draw a square grid and write one of the multipliers over the columns, and the second is height.
2. Multiply the number of each row sequentially in the number of each column. Consistently multiply 3 by 6, by 8, 2 and 7, etc.
4. We fold the numbers by following diagonal stripes. If the sum of one diagonal contains dozens, then add them to the next diagonal.
From the results of the addition of figures on the diagonals, the number 2355315 is composed, which is the product of Numbers 6827 and 345, that is, 6827 ∙ 345 \u003d 2355315.
Output. The "lattice multiplication" method is not worse than the generally accepted. It is even simpler because there are numbers directly from the multiplication table without simultaneous addition, which is present in the standard method.
2.4. Chinese fashion
Suppose you need to multiply 12 to 321 (Appendix 4). On a sheet of paper, alternately draw lines, the number of which is determined from this example.
We draw the first number - 12. To do this, from top to bottom, to the left, we draw:
one green wand (1)
and two orange (2).
We draw the second number - 321, from the bottom up, to the left to the right:
Three blue sticks (3);
two red (2);
one lilac (1).
Now a simple pencil separating the intersection points and proceed to their calculation. Moving right left (clockwise): 2, 5, 8, 3.
Received result Read from left to right - 3852
Output. An interesting way, but spend 9 direct when multiplying 9 somehow for a long time and uninteresting, and then another point of intersection count. Without skill it is difficult to understand the division of the number on the discharge. In general, no multiplication table do not do!
2.5. Japanese fashion
Multiply 12 to 34 (Appendix 5). Since the second multiplier is a two-digit number, and the first figure of the first factor 1, we build two single circles in the upper line and two binary circles in the bottom line, since the second figure of the first factor is 2.
Since the first digit of the second multiplier 3, and the second 4, divide the circles of the first column into three parts, the second column into four parts.
The number of parts on which circles were divided and is the answer, that is, 12 x 34 \u003d 408.
Output. The method is very similar to Chinese graphic. Only direct are replaced with circles. It is easier to define discharges in the number, however draw circles less convenient.
2.6. Table Okoneshikov
It is required to multiply 15647 x 5. Immediately remember the large "button" 5 (it is in the middle) and we mentally find small buttons 1, 5, 6, 4, 7 (they are also located, as on the calculator). They correspond to numbers 05, 25, 30, 20, 35. The obtained numbers fold: the first digit 0 (remains unchanged), 5 mentally add from 2, we get 7 - this is the second digit of the result, 5 fold with 3, we get the third digit - 8 0 + 2 \u003d 2, 0 + 3 \u003d 3 and the last digit of the work remains - 5. As a result, it turned out 78,235.
Output. The method is very convenient, but you need to learn by heart or always have a table at hand.
2.7. Questioning of students
Quart term books were conducted. 26 people took part (Appendix 8). On the basis of the survey, it was revealed that all respondents can multiply in a traditional way. But about the unconventional methods of multiplication, most guys do not know. And there are wishing to meet them.
After the primary questionnaire, an extracurricular occupation "Multiplication with passion" was carried out, at which the guys got acquainted with alternative multiplication algorithms. After that, a survey was conducted to identify the most likely ways. The undisputed leader was the most modern method of Vasily Okneshikov. (Appendix 9)
Conclusion
Having learned to count by all the presented ways, I believe that the most convenient multiplication method is the "Little Castle" method - because it looks like this now!
From all those found by me of unusual ways of account, the Japanese method seemed more interesting. The simplest method of "doubling and split" seemed to me, which Russian peasants used. I use it when multiplying is not too large numbers. It is very convenient to use it when multiplying two-digit numbers.
Thus, I reached my research goals - I studied and learned to apply non-traditional methods for multiplying multivalued numbers. My hypothesis was confirmed - I took possession of six alternative ways and found out that this is not all possible algorithms.
Non-traditional multiplication methods studied by me are very interesting and have the right to exist. And in some cases they even easier to use. I believe that the existence of these methods can be told at school, at home and surprise your friends and acquaintances.
While we just studied and analyzed the already known methods of multiplication. But who knows, perhaps, in the future, we will be able to open new ways of multiplication. I also do not want to stop at reached and continue the study of non-traditional multiplication methods.
List of sources of information
1. List of references
1.1. Harutyunyan E., Levitas. Entertaining mathematics. - M.: AST - Press, 1999. - 368 p.
1.2. Bellyustina V. How gradually reached people to real arithmetic. - LKI, 2012.-208 p.
1.3. Depman I. Stories about mathematics. - Leningrad: Education, 1954. - 140 s.
1.4. Likum A. All about everything. T. 2. - M.: Philological Society "Word", 1993. - 512 p.
1.5. Olochnik S. N., Nesterenko Yu. V., Potapov M. K .. Vintage entertaining tasks. - M.: Science. The main editorial office of physico-mathematical literature, 1985. - 160 p.
1.6. Perelman Ya.I. Entertaining arithmetic. - M.: Rusanova, 1994 - 205c.
1.7. Perelman Ya.I. Quick account. Thirty simple oral receptions. L.: Lenzdat, 1941 - 12 p.
1.8. Savin A.P. Mathematical miniatures. Entertaining mathematics for children. - M.: Children's literature, 1998 - 175 p.
1.9. Encyclopedia for children. Mathematics. - M.: Avanta +, 2003. - 688 p.
1.10. I will know the world: Children's Encyclopedia: Mathematics / Sost. Savin A.P., Stozo V.V., Kotova A.Yu. - M.: LLC "Publisher AST", 2000. - 480 p.
2. Other sources of information
Internet resources:
2.1. Koreev A.A. The phenomenon of Russian multiplication. History. [Electronic resource]

Indian method of multiplication

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:

Multiplication of the way "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 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.

Municipal General Education

Staromaximkinskaya Main Composition School

District Scientific - Practical Conference on Mathematics

"Step into science"

Research work

"Non-standard account algorithms or a quick expense without a calculator"

Leader:,

mathematic teacher

from. Art. Maximkino, 2010.

Introduction ................................................................................................3

Chapter 1. Account History

1.2. Miracle - counters .............................................................................. ... 9

Chapter 2. Vintage Methods Multiplication

2.1. Russian peasant way of multiplication ... .. ................................................................. ......................................... …….………..13

2.3. Indian method of multiplication ......................................................................... ..15

2.4. Egyptian method of multiplication .......................................................................... .16

2.5. Multiplication on your fingers .......................................................................17

Chapter 3. Oral Account - Mind Gymnastics

3.1. Multiplication and division at 4 ..............................................................................19

3.2. Multiplication and division by 5 ..............................................................................19

3.3. Multiplication by 25 .............................................................................. 19

3.4. Multiplication by 1.5 ............................................................................... 20

3.5. Multiplication by 9 ................................................................................................20

3.6. Multiplication by 11 ........................................................................................

3.7. Multiplying three-digit number per 101 ................................................................. 21

3.7. Erend into the square of the number ending the digit 5 \u200b\u200b............................. 21

3.8. The construction of a number close to 50 ............................................... 22

3.9. Games ............................................................................................................22

Conclusion ........................................................................................ ... 24

List of used literature ......................................................... ... 25

Introduction

Is it possible to imagine the world without numbers? Without numbers, no purchase will not do nor the time you do not know, no phone number will do. And space ships, lasers and all other technical achievements?! They would simply be impossible if it were not for the science of numbers.

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

Now, at the stage of the rapid development of computer science and computing equipment, modern schoolchildren do not want to bother themselves in mind. Therefore, we considered it is important to show not only that the process itself can be interesting, but even that, well, having learned the rapid account techniques, you can argue with a computer.

Objectstudies are account algorithms.

Subject Studies acts as the process of calculation.

Purpose:examine non-standard techniques for computing and experimentally to identify the reason for the refusal to use these methods when teaching mathematics of modern schoolchildren.

Tasks:

Reveal the history of the occurrence of the account and the phenomenon of "Miracle - Counters";

Describe the vintage methods of multiplication and experimentally experimentally to identify difficulties in their use;

Consider some instruments of oral multiplication and on specific examples show the advantages of using them.

Hypothesis:in the old days they said: "Multiplication is my torment." So, it used to be difficult and difficult to multiply. Is our modern way of multiplication?

When working on the report I enjoyed the following methods :

Ø search method using scientific and educational literature, as well as the search for the necessary information on the Internet;

Ø practical method of performing calculations using non-standard account algorithms;

Ø analysis obtained during the study of data.

Relevance This topic is that the use of non-standard techniques in the formation of computing skills enhances students' interest in mathematics and promotes the development of mathematical abilities.

Behind the simple action of multiplications hid the secrets of the history of mathematics. Accidentally heard the words "multiplication with grid", "chess method" intrigued. I wanted to learn these and other methods of multiplication, to compare them with our today's multiplication.

In order to find out if modern schoolchildren know other methods of performing arithmetic action, except for the multiplication of a column and division "corner" and would like to learn new ways, an oral survey was conducted. 20 students of grades 5-7 were interviewed. 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.

Results of survey:

(In diagrams are presented as a percentage of students' affirmative responses).

1) Do you need to be able to perform arithmetic action with natural numbers to modern person?

2) a) Do you know how to multiply, fold,

b) Do you know other ways to perform arithmetic action?

3) And would you like to know?

Chapter 1. Account History

1.1. How to have numbers

Calculate items People learned even in the ancient stone age - Paleolitis, tens of thousands of years ago. How did it happen? First, people only compared different number of identical objects. They could determine which of the two heaps more fruits, in which herd more deer, etc. If one tribe changed the stone knives made by people of another tribe, it was not necessary to consider how much fish brought and how many knives. It was enough to put next to each fish on the knife so that the exchange between tribes took place.

In order to successfully engage in agriculture, arithmetic knowledge needed. Without counting days it was difficult to determine when the fields should be seized when to start watering when to wait for the offspring from animals. It was necessary to know how many sheep in the herd, how many grain bags were put in barns.
And now more than eight thousand years ago, the ancient shepherds began to make a mug from clay - one for each sheep. To find out if there was no one sheep per day, the shepherd was postponed in the direction of the mug every time the next animal went into the pen. And only making sure that sheep returned as much as there were circles, he calmly went to sleep. But in his herd there were not only sheep - he pass and cows, and goats, and donkeys. Therefore, I had to do from clay and other figures. And the farmers with the help of clay figures were taking into account the collected crop, noting how many grain bags are put in the barn, how many jugs are squeezed out of olives, how much wear linen pieces. If the sheep brought the rats, the shepherd added new ones, and if part of the sheep went on meat, several circles had to be removed. So, not knowing how to count, the ancient people were engaged in arithmetic.

Then, in the human language, numerals appeared, and people were able to call the number of objects, animals, days. Usually there were few such numerical. For example, at the Murray River tribe in Australia, there were two simple numerals: ENEA (1) and Petcheval (2). Other numbers they expressed composite numerical: 3 \u003d "Petcheval-Enea", 4 "Petcheval-Petcheval", etc. Another Australian tribe - Camiloroev had simple numeral small (1), Bulan (2), Guliba (3). And here, other numbers were additioned by adding less: 4 \u003d "BULON - BULLAN", 5 \u003d "BULON - GULIBA", 6 \u003d "GULIBA - GULIBA", and so on.

In many nations, the name of the number depended on the objects calculated. If residents of Fiji Islands considered boats, then the number 10 was called "Bolo"; If they considered coconuts, the number 10 was called "Caro". In the same way, Amur Nivhi, living on the Sakhalin and the shores of Amur. Even in the last century, the same number was called different words if people, fish, boats, networks, stars, sticks thought.

We now use different indefinite numeral with the meaning of "many": "crowd", "flock", "flock", "bunch", "beam" and others.

With the development of production and trade exchange, people began to better understand that in common three boats and three axes, ten arrows and ten nuts. The tribes often conducted an exchange of "subject for the subject"; For example, they exchanged 5 edible roots for 5 fish. It became clear that 5 is the same for roots, and for fish; It means that you can call it in one word.

Similar accounts used other nations. This is how the numbering based on the score of the top five, tens, twenty.

Until now, we told about the oral account. And how did the numbers recorded? At first, even before writing, we used scubons on sticks, notches on the bones, nodules on the rods. Found wolf bone in Dolny - Westonice (Czechoslovakia), had 55 notches made more than 25,000 years ago.

When writing appeared, and numbers appeared to record numbers. At first, the numbers were reminded of scubons on sticks: in Egypt and Babylon, in Etruria and dates, in India and China, small numbers were recorded with chopsticks or screenshots. For example, the number 5 was recorded by five chopsticks. Axtie and Maya Indians instead of chopsticks used points. Then special signs appeared for some numbers, such as 5 and 10.

At that time, almost all numbering were not positional, but similar to Roman numbering. Only one Babylonian six-month numbering was positional. But there was no zero in it for a long time, as well as a comma separating the whole part of the fractional. Therefore, one and the same figure could mean 1, and 60, and 3600. Guess the value of the number accounted for the meaning of the problem.

For a few centuries, a new method of recording numbers invented a new era, in which the letters of the usual alphabet served the numbers. The first 9 letters denoted the numbers tens of 10, 20, ..., 90, and another 9 letters denoted hundreds. Such an alphabetical numbering was used to 17 V. To distinguish the "real" letters from the numbers, they put the screenshots above the letters (this Chestochka was called "Titlo").

In all these numbering it was very difficult to perform arithmetic action. Therefore, the invention is 6 V. Indians of decimal position is considered to be one of the largest achievements of humanity. Indian numbering and Indian figures became known in Europe from Arabs, and they are usually called Arabic.

When writing fractions for a long time, a whole part was recorded in a new, decimal numbering, and fractional - in a sixteen. But at the beginning of the 15th century. Samarkand Mathematics and Astronomer Al - Kashi began to use decimal fractions in the calculations.

The numbers with which we work with positive and negative numbers. But it turns out that this is not all the numbers that are used in mathematics and other sciences. And you can learn about them without waiting for the eldest school, but much earlier, if you study the history of the emergence of numbers in mathematics.

1.2 "Miracle - Counters"

He understands everything from a half-clow and immediately formulates the conclusion to which an ordinary person may be, will come through long and painful thinking. Books it absorbs with an incredible speed, and in the first place in his short-list of bestsellers - a textbook on entertaining mathematics. At the time of solving the most difficult and unusual tasks in his eyes, fire inspiration is burning. Requests to go to the store or wash the dishes remain ignore either performed with great discontent. The best reward is a campaign in a lecture, and the most valuable gift is a book. It is most practical and in his actions mainly obeys the mind and logic. He refers to the people around him and prefers roller skating a chess game with a computer. As a child, he does not realize its own shortcomings, it is characterized by increased emotional stability and adaptability to external circumstances.

This portrait is not written with the analyst of the CIA.
So, according to psychologists, a calculator person looks like, an individual with unique mathematical abilities, allowing it in the blink of an eye to produce the most complex calculations in the mind.

Behind the threshold of consciousness Miracle - accounts capable of a calculator to make unimaginable complex arithmetic actions, possess unique features of memory that distinguishes them from other people. As a rule, besides huge lines, formulas and computing, these people (scientists call them mnemonics - from the Greek word Mnemonika, meaning "the art of memorization") keep the list of addresses not only friends, but also random acquaintances, as well as numerous organizations, where once had to be.

In the laboratory of the Research Institute of Psychotechnology, where they decided to explore the phenomenon, conducted such an experiment. Invited the Unicum - an employee of the central state archive of St. Petersburg offered to him for memorizing various words and numbers. He had to repeat them. For some couple of minutes, he could fix in memory to seventy elements. Tens of words and numbers literally "downloaded" in memory of Alexander. When the number of elements exceeded two hundred, they decided to check its capabilities. To the surprise of participants in the experiment, it did not give a single failure to megapaund. From a second, he moved her lips, he with astounding accuracy, as if reading, began to reproduce the entire number of elements.

For example, for example, one scientist - the researcher conducted an experiment with Madmoiselle Osaka. The subject asked to build a square 97, get the tenth of that number. She did it instantly.

In the Vansky district of Western Georgia, Aron Chikashvili lives. He quickly and accurately produces the most complicated calculations in the mind. Somehow friends decided to check the possibilities of the "Miracle Counter". The task was difficult: how many words and letters will say the announcer, commenting on the second half of the Spartak football match (Moscow) - Dynamo (Tbilisi). At the same time, the tape recorder was turned on. The answer followed as soon as the announcer said the last word: 17427 letters, 1835 words. On check gone ... .5 hours. The answer was correct.

It is said that Gauss's father usually paid the work at the end of the week, adding to every day earnings for overtime hours. Once after Gauss Father finished the calculations, which took place for the operations of the father of the child, who was three years, exclaimed: "Dad, counting is not true! This should be the amount. " The calculations were repeated and surprised were convinced that the baby pointed out the correct amount.

Interestingly, many "miracle counters" do not have the concepts in general, as they consider. "We believe, that's all! And as we believe, God knows him. " Some "counters" were completely uneducated people. Berkston Englishman, "Virtuoso Counter", never learned to read; The American Neg-Counter Thomas Falleler died illiterate at the age of 80.

Competitions at the Cybernetics Institute of the Ukrainian Academy of Sciences were held. The competition was attended by a young "counter-phenomenon" Igor Shelushkov and EUM "Peace". The car has made many complex mathematical operations in a few seconds. Igor Shelushkov came out the winner in this competition.

The majority of such people has excellent memory and have dating. But some of them do not possess any abilities to mathematics. They know the secret! And this secret is that they well learned the rapid score techniques, remembered several special formulas. But the Belgian employee, which in 30 seconds to the multi-valued number proposed to him, obtained from multiplying a number of itself 47 times, calls this number (extracts the root of the 47th

the degree from a multi-valued number), achieved such amazing success in the score as a result of many years of workout.

So, many "phenomena counters" use special rapid score techniques and special formulas. So we can also use some of these techniques.

ChapterII. . Vintage methods of multiplication.

2.1. Russian peasant method of multiplication.

In Russia, 2-3 centuries ago, a method was distributed among the peasants of some provinces that did not require knowledge of the entire multiplication table. It was necessary only to be able to multiply and divide on 2. This method was called peasantic (There is an opinion that he originates from Egyptian).

Example: Multiply 47 on 35,

We write the number on one line, carry out a vertical line between them;

The left number will be divided by 2, right - multiplied by 2 (if the residue occurs during the division, then the residue is discarding);

The division ends when the unit will appear on the left;

We highlight those lines in which the left numbers stand;

35 + 70 + 140 + 280 + 1120 = 1645.

2.2. The "lattice" method.

one). An outstanding Arab mathematician and astronomer Abu Mussia Al-Khorezmi lived and worked in Baghdad. Al-Khorezmi literally means "from Khorezmi", i.e., born in the city of Khorezma (now included in Uzbekistan). The scientist worked in the wisdom house, where there were a library and observatory, almost all major Arab scientists worked here.

Information about the life and activities of Mohammed Al - Khorezmi is very small. Only two works have been preserved - according to algebra and arithmetic. The last of these books give four arithmetic action rules, almost the same as used in our time.

2). In Ov "The book about the Indian account" Scientist described the method invented in ancient India, and later called "Method of lattice" (He is "jealousy"). This method is even easier than applied today.

Let you need to multiply 25 and 63.

Draw a table in which two cells in length and two in the width of writing one number along the length of the other in width. In the cells, write the result of multiplying data numbers, on their intersection there are tens and units of diagonal. The figures obtained are diagonally, and the result obtained can be read by arrow (down and right).

We considered a simple example, however, in this way you can multiply any multivalued numbers.

Consider another example: varnish 987 and 12:

We draw a rectangle 3 to 2 (by the number of decimal signs of each multiplier);

Then square cells divide diagonally;

At the top of the table write the number 987;

On the left table number 12 (see Figure);

Now in each square we will enter the product of numbers - factors located in one line and in one column with this square, dozens above the diagonal, units below;

After filling all the triangles, the numbers in them are folded along each diagonal;

The result is written to the right and at the bottom of the table (see Figure);

987 ∙ 12=11844

This algorithm multiplying two natural numbers was distributed to the Middle Ages in the East and Italy.

The inconvenience of this method, we noted in the complexity of the preparation of the rectangular table, although the process of calculation itself is interesting and filling the table resembles the game.

2.3 Indian Mode Multiplication

Some experienced teachers in the last century believed that this method should replace in our school a generally accepted method of multiplication.

He liked the Americans so much that they even called him "American Method". However, they used the inhabitants of India in the VI century. n. er, and it is more correct to call it "Indian way." Multiply two of any two-digit numbers, say 23 to 12. I immediately write what happens.

You see: very quickly received the answer. But how is it received?

First step: x23 I say: "2 x 3 \u003d 6"

Second step: x23 I say: "2 x 2 + 1 x 3 \u003d 7"

Third step: x23 I say: "1 x 2 \u003d 2".

12 I write 2 left numbers 7

276 Get 276.

We got acquainted with this way on a very simple example without transition through a category. However, our studies have shown that they can be used and when multiplying numbers with a transition through a category, as well as with multiplying multivalued numbers. We give examples:

x528 x24 x15 x18 x317

123 30 13 19 12

In Russia, this method was known as a way to multiply a cross.

In this "cross" and is the inconvenience of multiplication, it is easy to get confused, besides, it is difficult to keep all intermediate works in mind, the results of which then need to be folded.

2.4. Egyptian method of multiplication

The designations of the numbers that were used in antiquity were more or less suitable for recording the result of the account. But it was very difficult to perform arithmetic actions with their help, especially this concerned the action of multiplication (try, to multiply: ξφß * τδ). The exit from this situation found the Egyptians, so the method was called egyptian. They replaced multiplication by any number - doubling, that is, the addition of the number with itself.

Example: 34 ∙ 5 \u003d 34 ∙ (1 + 4) \u003d 34 ∙ (1 + 2 ∙ 2) \u003d 34 ∙ 1+ 34 ∙ 4.

T. to. 5 \u003d 4 + 1, then the numbers standing in the right column against numbers 4 and 1, i.e. 136 + 34 \u003d 170 remained to obtain the answer.

2.5. Multiplication on the fingers

The ancient Egyptians were very religious and believed that the soul of the deceased in the afterlife was subjected to an exam on the fingers. This is already talking about the meaning that the ancient method was attached to perform multiplication of natural numbers (he received a name finger account).

Moved on the fingers unambiguous numbers from 6 to 9. For this, so many fingers pulled out on one hand, as far as the first factor exceeded the number 5, and on the second they did the same for the second factor. The remaining fingers were fucked. After that, they took so many dozens of how much fingers stretched on both hands, and added to this number the work of the curved fingers on the first and second hand.

Example: 8 ∙ 9 \u003d 72

Later, the result was improved - learned to show the number of fingers to 10,000

Finger move

But one of the ways to help the memory: with the help of fingers to remember the multiplication table by 9. Putting both hands next to the table, in order to do the fingers of both hands as follows: the first finger on the left will designate 1, the second for it we denote the number 2, then 3 , 4 ... to the tenth finger, which means 10. If you need to multiply by 9 any of the first nine numbers, then for this, without moving hands from the table, you need to lift up the finger whose number means the number to which nine is multiplied; Then the number of fingers lying left from the raised finger determines the number of tens, and the number of fingers lying to the right of the raised finger denotes the number of units of the resulting product.

Example. Let it be necessary to find a product 4x9.

Putting both hands on the table, raise the fourth finger, counting from left to right. Then, to the raised finger there are three fingers (tens), and after raised - 6 fingers (units). The result of the work 4 to 9, which means equal to 36.

Another example:

Let you need to multiply 3 * 9.

From left to right, find the third finger, that finger straightened will be 2 fingers, they will mean 2 dozen.

On the right of the bent finger straightened by 7 fingers, they mean 7 units. Fold, 2 dozen and 7 units will be 27.

The fingers themselves showed this number.

// // /////

So, the vintage methods considered by us show that the algorithm of multiplication of natural numbers used in the school is not the only one and is not always known.

However, it is fast enough and most convenient.

Chapter 3. Oral Account - Mind Gymnastics

3.1. Multiplication and division by 4.

To multiply the number 4, it doubled it twice.

For example,

214 * 4 = (214 * 2) * 2 = 428 * 2 = 856

537 * 4 = (537 * 2) * 2 = 1074 * 2 = 2148

To divide the number to 4, it is divided twice to 2.

For example,

124: 4 = (124: 2) : 2 = 62: 2 = 31

2648: 4 = (2648: 2) : 2 = 1324: 2 = 662

3.2. Multiplication and division by 5.

To multiply the number 5, you need to multiply it by 10/2, that is, multiply by 10 and divided into 2.

For example,

138 * 5 = (138 * 10) : 2 = 1380: 2 = 690

548 * 5 (548 * 10) : 2 = 5480: 2 = 2740

In order for the number to divide by 5, you need to multiply it by 0.2, that is, in a doubled source number, it is possible to separate the firing of the last digit.

For example,

345: 5 = 345 * 0,2 = 69,0

51: 5 = 51 * 0,2 = 10,2

3.3. Multiplication by 25.

To multiply a number 25, you need to multiply it by 100/4, that is, multiply by 100 and divided by 4.

For example,

348 * 25 = (348 * 100) : 4 = (34800: 2) : 2 = 17400: 2 = 8700

3.4. Multiplication by 1.5.

To multiply a number of 1.5, it is necessary to add it half to the initial number.

For example,

26 * 1,5 = 26 + 13 = 39

228 * 1,5 = 228 + 114 = 342

127 * 1,5 = 127 + 63,5 = 190,5

3.5. Multiplication by 9.

To multiply a number of 9, it is attributed to it 0 and take the initial number. For example,

241 * 9 = 2410 – 241 = 2169

847 * 9 = 8470 – 847 = 7623

3.6. Multiplication by 11.

1 way. In order for the number to multiply by 11, it is attributed to it 0 and add the initial number. For example:

47 * 11 = 470 + 47 = 517

243 * 11 = 2430 + 243 = 2673

2 way. If you want to multiply a number of 11, then do this: I write down the number that needs to be multiplied by 11, and insert the amount of these numbers between the numbers of the original number. If the sum is obtained by a two-digit number, then 1 add to the first digit of the initial number. For example:

45 * 11 = * 11 = 967

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

3.7. Multiplying three-digit number per 101.

For example 125 * 101 \u003d 12625

(Increase the first factor on the number of its hundreds and attribute to it to the right two last figures of the first factor)

125 + 1 = 126 12625

This technique is easily assimilated when writing a calculation in a column

x x125. 101 + 125 125 _ 12625

x x348. 101 +348 348 _ 35148

Another example: 527 * 101 = (527+5)27 = 53227

3.8. The erection of the number ending in the square 5.

In order to build a number in a square, the number 5 (for example, 65), the number of its tens (6) is multiplied by the number of dozens, increased by 1 (by 6 + 1 \u003d 7), and the obtained number is attributed to 25

(6 * 7 \u003d 42 Answer: 4225)

For example:

3.8. Erend into a square of a number close to 50.

If you want to build a number close to 50, but more than 50, then do this:

1) deduction from this number 25;

2) Eashes to the result by two digits square excess of a given number over 50.

Explanation: 58 - 25 \u003d 33, 82 \u003d 64, 582 \u003d 3364.

Explanation: 67 - 25 \u003d 42, 67 - 50 \u003d 17, 172 \u003d 289,

672 = 4200 + 289 = 4489.

If you want to build a number close to 50, but less than 50, then do this:

1) deduction from this number 25;

2) Eashes to the result by two digits square lack of this number up to 50.

Explanation: 48 - 25 \u003d 23, 50 - 48 \u003d 2, 22 \u003d 4, 482 \u003d 2304.

Explanation: 37 - 25 \u003d 12, \u003d 13, 132 \u003d 169,

372 = 1200 + 169 = 1369.

3.9. Games

Gaying out the resulting number.

1. Think some number. Add 11 to it; multiply the resulting amount by 2; From this product, take 20; Multiply the resulting difference at 5 and from the new product take the number 10 times more than the number you have.

I guess: you got 10. Right?

2. Think the number. In the morning of it. Dedule from the resulting 1. The resulting multiply to 5. To the resulting add 20. Divide the obtained by 15. From the resulting deduction.

You got 1.

3. Think the number. Multiply it to 6. Delete 3. Multiply 2. Add 26. Delete double-minded. Divide on 10. Remove the planned.

You got 2.

4. Think the number. Watch it. Delete 2. Multiply 5. Add 5. Divide on 5. Add 1. Divide on the planned one. You have 3.

5. Think about the number, double it. Add 3. Multiply 4. Delete 12. Divide on the planned one.

You got 8.

Guessing conceived numbers.

Offer your comrades to think any numbers. Let everyone add to his intended number 5.

Let the amount allow for 3 let it be multiplied by 3.

From the work let us take 7.

From the result obtained, let it be deducted 8 more.

Lesson with the final result Let everyone give you. Looking at the sheet, you immediately tell everyone, what number he thought.

(To guess the intended number, the result written on a piece of paper or spent orally, divided by 3)

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 of the spacecraft, the "economic - situation" in the country, the weather on "tomorrow", describe the sound of notes in the melody. We know the statement of ancient Greek mathematics, a 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, we tried to show that both in the past and in the future, without mathematics, science created by the mind of man, could not do.

The study of the old multiplication methods showed that this arithmetic effect was difficult and difficult due to the variety of methods and their bulkness of execution.

Modern method of multiplication is simple and accessible to everyone.

When finding out scientific literature, they discovered faster and reliable methods of multiplication. Therefore, the study of the multiplication is the topic is promising.

It is possible that from the first time, many will not work quickly, to perform these or other counts from the go. Suppose first it will not be possible to use the reception shown in the work. No problem. Need a constant computing training. From the lesson in the lesson, from year to year. It will help to purchase useful oral account skills.

List of used literature

1. Vanzian: textbook for grade 5. - Samara: Publishing House

Fedorov, 1999.

2., Akhadov World of Numbers: Student Book, - M. Enlightenment, 1986.

3. "From the game to Knowledge", M., "Enlightenment" in 1982.

4. Candles, figures, tasks M., Enlightenment, 1977.

5. http: // Matsievsky. ***** / SYS-SCHI / File15.htm

6. http: // ***** / MOD / 1/6506 / HYSTIRY. HTML

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.

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.

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.

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.

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

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.

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.

Objective: Explore and show unusual ways to multiply the task: find unusual methods of multiplication. Learn to apply them. Choose for yourself the most interesting or lighter than those are offered at school, and use them with the score. Teach classmates to apply a new method of multiplication

Methods: search method using scientific and educational literature, as well as search for the necessary information on the Internet; The practical method of performing computing using non-standard account algorithms; Analysis of the data obtained during the study is the relevance of this topic lies in the fact 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

In mathematics lessons, we studied an unusual way to multiply a column. We liked it and we decided to learn other ways of multiplying natural numbers. We asked our classmates if they know other expenses of the account? Everyone spoke only about those methods that are studied at school. It turned out that all our friends do not know anything about other ways. In the history of mathematics, about 30 methods of multiplication, characterized by the recording scheme or the stroke of the calculation itself are known. Multiplication method "In the Column", which we study at school - one of the ways. But is this the most effective way? Let's see! Introduction

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. 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 (23 \u003d 6), then the number of tens and units of the desired work 56, respectively. So you can calculate the product of any unambiguous numbers more than 5.

It is very easy to reproduce "on the fingers" multiplication for the number 9 pop up your fingers on both hands and turn your hands with your 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.

The method of multiplying the "Little Castle" The advantage of the method of multiplication "Little Castle" is that from the very beginning the numbers of senior discharges are determined, and this is important if it is necessary 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.

"Jealousy" or "lattice multiplication" first draws a rectangle, separated into squares, and the sizes of the sides of the rectangle correspond to the number of decimal signs in the multiplier and the multiplier then square cells are divided according to the diagonal, and "... it turns out a picture similar to the lattice shutters, - Writes Pacheti. - Such shutters were hanging on the windows of Venetian houses ... "

Multiplying with lattice \u003d +1 +2

The peasant method is the way of Velikorvsky peasants the essence of it lies in the fact that multiplication of any numbers is reduced to a number of consecutive divisions of one number in half, while reworking the other number ..........32 74 ......... ..........4 592 ......... ......... 1 3732 \u003d 1184

Peasant method (odd numbers) 47 x \u003d 1645

Step 1. First number 15: We draw the first digit - one line. We draw a second digit - five lines. Step 2. Second number 23: We draw the first digit - two lines. We draw the second digit - three lines. Step 3. Count the number of points in groups. Step 4. Result - 345. Move two two digits: 15 * 23

Indian method of multiplication (cross) 24 and x 3 2 1) 4x2 \u003d 8 - the last digit of the result; 2) 2x2 \u003d 4; 4x3 \u003d 12; 4 + 12 \u003d 16; 6- the penultimate digit of the result, the unit remember; 3) 2x3 \u003d 6 Yes, the number is still retained in the mind, we have 7 is the first digit of the result. We get all the figures of the work: 7,6,8. Answer: 768.

Indian method of multiplication \u003d \u003d \u003d \u003d 3822 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. Multiplication is starting with the older discharge, and write incomplete works just above the multiple, blessing. 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

Reference number multiply 18 * 19 20 (reference number) * 2 1 (18-1) * 20 \u003d answer: 342 short record: 18 * 19 \u003d 20 * 17 + 2 \u003d 342

New method of multiplication x \u003d, 5 + 2, 5 + 3, 0 + 2, 0 + 3, 5

Conclusion: I have 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 from all the considered unusual ways of the account, a way of graphic multiplication seemed to be more interesting. We showed it to your classmates, and he also really liked it. The simplest method of "doubling and splitting", which used Russian peasants working with literature and materials on the Internet, we realized that we considered a very small number of multiplication methods, which means that there is a lot of interesting things in front

Conclusion Describing the vintage methods of calculations and modern rapid account techniques, we tried to show that, both in the past and in the future, without mathematics, science created by the mind of a person, not to do the study of old methods of multiplication showed that this arithmetic effect was difficult and Complex due to the manifold of the methods and their bulkness of execution, the modern method of multiplication is simple and accessible to everyone. But, we think that our method of multiplication in a column is not perfect and you can come up with even faster and more reliable ways possible that many people will not work quickly, with the go, do these or other calculations do not matter. Need a constant computing training. It will help to buy useful oral skills!

Used materials: HTML encyclopedia for children. "Mathematics". - M.: Avanta +, - 688 p. Encyclopedia "I will know the world. Mathematics". - M.: Astrel Ermak, Perelman Ya.I. Quick account. Thirty simple oral receptions. L., s.