Indian way of multiplication

The most valuable contribution to the treasury of mathematical knowledge was made in India. The Hindus proposed the way we use to write numbers using ten signs: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.

The basis of this method is the idea that the same digit stands for units, tens, hundreds or thousands, depending on where this figure occupies. The place occupied, in the absence of any digits, is determined by zeros assigned to the numbers.

The Indians thought well. They came up with a very simple way to multiply. They performed multiplication, starting with the highest order, and wrote down incomplete products just above the multiplicand, bit by bit. At the same time, the senior digit of the complete product was immediately visible and, in addition, the omission of any digit was excluded. The multiplication sign was not yet known, so they left a small distance between the factors. For example, let's multiply them in the way 537 by 6:

Multiplication using the "LITTLE CASTLE" method

Multiplication of numbers is now studied in the first grade of the school. But in the Middle Ages, very few mastered the art of multiplication. A rare aristocrat could boast of knowing the multiplication table, even if he graduated from a European university.

Over the millennia of the development of mathematics, many ways to multiply numbers have been invented. The Italian mathematician Luca Pacioli, in his treatise The Sum of Knowledge in Arithmetic, Relations and Proportionality (1494), lists eight different methods of multiplication. The first of them is called "Little Castle", and the second is no less romantic called "Jealousy or Lattice Multiplication".

The advantage of the “Little Castle” multiplication method is that the digits of the highest digits are determined from the very beginning, and this can be important if you need to quickly estimate the value.

The digits of the upper number, starting from the most significant digit, are alternately multiplied by the lower number and written in a column with the addition of the required number of zeros. Then the results are added up.

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

2. Having chosen the grid, we multiply the number of each row sequentially by the numbers of each column. In this case, we sequentially multiply 3 by 6, by 8, by 2 and by 7. Look at this diagram how the product is written in the corresponding cell.

3. See what the grid looks like with all the filled cells.

4. Finally, add up the numbers, following the diagonal stripes. If the sum of one diagonal contains tens, then we add them to the next diagonal.

See how the results of adding the numbers along the diagonals (they are highlighted in yellow) form the number 2355315, which is the product of the numbers 6827 and 345.

The purpose of the work: To explore and show unusual ways of multiplication. Tasks: To find unusual ways of multiplication. Learn to apply them. Choose for yourself the most interesting or easier ones than those offered at school, and use them when counting. Teach classmates to use a new way of multiplication

Methods: search method using scientific and educational literature, as well as searching for the necessary information on the Internet; a practical method for performing calculations using non-standard counting algorithms; analysis of the data obtained during the study The relevance of this topic lies in the fact that the use of non-standard methods in the formation of computational skills enhances students' interest in mathematics and contributes to the development of mathematical abilities

In math class, we learned an unusual way of multiplication by a column. We liked it and decided to learn other ways to multiply natural numbers. We asked our classmates if they knew other ways of counting? Everyone spoke only about those methods that are studied at school. It turned out that all our friends do not know anything about other methods. In the history of mathematics, about 30 methods of multiplication are known, differing in the recording scheme or in the very course of the calculation. The multiplication method "in a column", which we study at school, is one of the ways. But is it the most efficient 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 single-digit numbers from 6 to 9. The fingers here served as an auxiliary computing device. To do this, on one hand they extended as many fingers as the first factor exceeds the number 5, and on the second they did the same for the second factor. The rest of the fingers were bent. Then the number (total) of outstretched fingers was taken and multiplied by 10, then the numbers were multiplied showing how many fingers were bent on the hands, and the results were added up. For example, let's multiply 7 by 8. In the considered example, 2 and 3 fingers will be bent. If we add the number of bent fingers (2 + 3 = 5) and multiply the number of not bent fingers (23 = 6), then we get the numbers of tens and units of the desired product 56, respectively. So you can calculate the product of any single-digit numbers greater than 5.

The multiplication for the number 9 is very easy to reproduce "on the fingers". Spread the fingers on both hands and turn the palms away from you. Mentally assign the numbers from 1 to 10 to the fingers in sequence, starting with the little finger of the left hand and ending with the little finger of the right hand. Let's say we want to multiply 9 by 6. We bend a finger with a number equal to the number by which we will multiply the nine. In our example, you need to bend the finger with number 6. The number of fingers to the left of the bent finger shows us the number of tens in the answer, the number of fingers to the right - the number of ones. On the left, we have 5 fingers not bent, on the right - 4 fingers. Thus, 9 6=54.

The "Little Castle" multiplication method The advantage of the "Little Castle" multiplication method is that the high-order digits are determined from the very beginning, which is important if you need to quickly estimate the value. The digits of the upper number, starting from the most significant digit, are alternately multiplied by the lower number and written in a column with the addition of the required number of zeros. Then the results are added up.

“Jealousy” or “lattice multiplication” First, a rectangle is drawn, divided into squares, and the dimensions of the sides of the rectangle correspond to the number of decimal places for the multiplier and multiplier. Then the square cells are divided diagonally, and “... a picture is obtained that looks like lattice shutters-blinds, - writes Pacioli. - Such shutters were hung on the windows of Venetian houses ... "

Lattice multiplication = +1 +2

Peasant method This is the method of the Great Russian peasants. Its essence lies in the fact that the multiplication of any numbers is reduced to a series of successive divisions of one number in half, while doubling another number ……….32 74……… ……….8 296……….4 592……… ………1 3732=1184

Peasant way (odd numbers) 47 x =1645

Step 1. first number 15: Draw the first number - in one line. We draw the second figure - five lines. Step 2. second number 23: Draw the first number - two lines. We draw the second figure - three lines. Step 3. Count the number of points in groups. Step 4. The result is 345. Let's multiply two two-digit numbers: 15 * 23

Indian multiplication method (cross) 24 and X 3 2 1)4x2=8 - the last digit of the result; 2)2x2=4; 4x3=12; 4+12=16 ; 6 - the penultimate digit of the result, remember the unit; 3) 2x3=6 and even the number kept in mind, we have 7 - this is the first digit of the result. We get all the digits of the product: 7,6,8. Answer: 768.

Indian method of multiplication = = = = 3822 The basis of this method is the idea that the same digit stands for units, tens, hundreds or thousands, depending on where this figure occupies. The place occupied, in the absence of any digits, is determined by zeros assigned to the numbers. we start the multiplication from the highest order, and write down the incomplete products just above the multiplicand, bit by bit. In this case, the most significant digit of the complete product is immediately visible and, in addition, the omission of any digit is excluded. The multiplication sign was not yet known, so a small distance was left between the factors

Base Number Multiply 18*19 20 (base number) * 2 1 (18-1)*20 = Answer: 342 Short Note: 18*19 = 20*17+2 = 342

New multiplication method X = , 5+2, 5+3, 0+2, 0+3, 5

Conclusion: Having learned to count in all the presented ways, we came to the conclusion that the simplest methods are those that we study at school, or maybe we just got used to them Of all the considered unusual counting methods, the method of graphic multiplication seemed more interesting. We showed it to our classmates and they also liked it very much. The “doubling and doubling” method used by Russian peasants seemed to be the simplest.

Conclusion Describing the ancient methods of calculations and modern methods of fast counting, we tried to show that, both in the past and in the future, one cannot do without mathematics, a science created by the human mind. Studying the ancient methods of multiplication showed that this arithmetic operation was difficult and complex due to the variety of methods and their cumbersome implementation 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 methods. It is possible that the first time many will not be able to quickly, on the go, perform these or other calculations. It does not matter. Constant computational training is needed. It will help you develop useful mental counting skills!

Materials used: html Encyclopedia for children. "Maths". – M.: Avanta +, – 688 p. Encyclopedia “I know the world. Maths". - M .: Astrel Ermak, Perelman Ya.I. Quick account. Thirty simple methods of mental counting. L., s.

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“Counting and calculations are the basis of order in the head.”
Pestalozzi

Target:

• Familiarize yourself with the old methods of multiplication.
• Expand your knowledge of various multiplication techniques.
• Learn to perform operations with natural numbers using the old methods of multiplication.

1. The old way to multiply by 9 on your fingers
2. Multiplication by the Ferrol method.
3. Japanese way of multiplication.
4. Italian way of multiplication (“Grid”)
5. Russian way of multiplication.
6. Indian way of multiplication.

Lesson progress

The relevance of using quick counting techniques.

In modern life, each person often has to perform a huge amount of calculations and calculations. Therefore, the purpose of my work is to show easy, fast and accurate counting methods that will not only help you during any calculations, but will cause considerable surprise among friends and comrades, because the free performance of counting operations can largely indicate the extraordinaryness of your intellect. A fundamental element of a computing culture is conscious and strong computing skills. The problem of forming a computational culture is relevant for the entire school course in mathematics, starting from the primary grades, and requires not just mastering computational skills, but using them in various situations. The possession of computational skills and abilities is of great importance for the assimilation of the material being studied, it allows one to cultivate valuable labor qualities: a responsible attitude to one's work, the ability to detect and correct mistakes made in the work, accurate execution of the task, and a creative attitude to work. However, recently the level of computational skills, expression transformations has a pronounced downward trend, students make a lot of mistakes when calculating, they increasingly use a calculator, do not think rationally, which negatively affects the quality of education and the level of mathematical knowledge of students in general. One of the components of computing culture is verbal counting which is of great importance. The ability to quickly and correctly make simple calculations “in the mind” is necessary for every person.

Ancient ways of multiplying numbers.

1. The old way of multiplying by 9 on your fingers

It's simple. To multiply any number between 1 and 9 by 9, look at the hands. Bend the finger that corresponds to the number being multiplied (for example 9 x 3 - bend the third finger), count the fingers up to the crooked finger (in the case of 9 x 3 it is 2), then count after the crooked finger (in our case 7). The answer is 27.

2. Multiplication by the Ferrol method.

To multiply the units of the multiplication product, multiply the units of factors, to get tens, multiply the tens of one by the units of the other and vice versa and add the results, to get hundreds, multiply the tens. Using the Ferrol method, it is easy to verbally multiply two-digit numbers from 10 to 20.

For example: 12x14=168

a) 2x4=8, write 8

b) 1x4+2x1=6, write 6

c) 1x1=1, write 1.

3. Japanese multiplication method

This technique resembles multiplication by a column, but it takes quite a long time.

Use of reception. Let's say we need to multiply 13 by 24. Let's draw the following picture:

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

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

(intersections in the figure are indicated by dots)

Number of crossings:

• Top left edge: 2
• Bottom left edge: 6
• Top right: 4
• Bottom right: 12

1) Crossings in the upper left edge (2) - the first number of the answer

2) The sum of the intersections of the lower left and upper right edges (6 + 4) - the second number of the answer

3) Intersections in the lower right edge (12) - the third number of the answer.

It turns out: 2; 10; 12.

Because the last two numbers are two-digit and we cannot write them down, then we write down only units, and add tens to the previous one.

4. Italian way of multiplication (“Grid”)

In Italy, as well as in many countries of the East, this method has become very famous.

Reception usage:

For example, let's multiply 6827 by 345.

1. We 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 by the numbers of each column.

• 6*3 = 18. Write down 1 and 8
• 8*3 = 24. Write down 2 and 4

If multiplication produces a single-digit number, we write 0 at the top, and this number at the bottom.

(As in our example, when multiplying 2 by 3, we got 6. At the top, we wrote 0, and at the bottom 6)

3. Fill in the entire grid and add up the numbers following the diagonal stripes. We begin to fold from right to left. If the sum of one diagonal contains tens, then we add them to the units of the next diagonal.

Answer: 2355315.

5. Russian way of multiplication.

This multiplication technique was used by Russian peasants about 2-4 centuries ago, and was developed in ancient times. The essence of this method is: “By how much we divide the first factor, we multiply the second by so much.” Here is an example: We need to multiply 32 by 13. This is how our ancestors would have solved this example 3-4 centuries ago:

• 32 * 13 (32 divided by 2, and 13 multiplied by 2)
• 16 * 26 (16 divided by 2, and 26 multiplied by 2)
• 8 * 52 (etc.)
• 4 * 104
• 2 * 208
• 1 * 416 =416

Bisection continues until the quotient is 1, while doubling another number in parallel. The last doubled number gives the desired result. It is not difficult to understand what this method is based on: the product does not change if one factor is halved, and the other is doubled. It is clear, therefore, that as a result of repeated repetition of this operation, the desired product is obtained

However, what to do if you have to divide an odd number in half? The popular way easily gets out of this difficulty. It is necessary, - says the rule, - in the case of an odd number, discard the unit and divide the remainder in half; but on the other hand, to the last number of the right column it will be necessary to add all those numbers of this column that stand against the odd numbers of the left column: the sum will be the desired product. In practice, this is done in such a way that all lines with even left numbers are crossed out; only those that contain an odd number to the left remain. Here is an example (asterisks indicate that this line should be crossed out):

• 19*17
• 4 *68*
• 2 *136*
• 1 *272

Adding the uncrossed numbers, we get a completely correct result:

• 17 + 34 + 272 = 323.

Answer: 323.

6. Indian way of multiplication.

This method of multiplication was used in ancient India.

To multiply, for example, 793 by 92, we write one number as a multiplier and under it another as a factor. To make it easier to navigate, you can use the grid (A) as a reference.

Now we multiply the left digit of the multiplier by each digit of the multiplicand, that is, 9x7, 9x9 and 9x3. We write the resulting products in the grid (B), bearing in mind the following rules:

• Rule 1. The units of the first product 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 immediately to the right of the previous work.

Repeat the whole process with other multiplier numbers, following the same rules (C).

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

As you can see, we get a large list of works. The Indians, who had great practice, wrote each figure not in the corresponding column, but on top, as far as possible. Then they added up the numbers in the columns and got the result.

Conclusion

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

According to the philosophical view of this scientist and his followers, numbers govern not only measure and weight, but also all phenomena occurring in nature, and are the essence of harmony that reigns in the world, the soul of the cosmos.

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

“Whoever has been involved in mathematics since childhood develops attention, trains the brain, his will, cultivates perseverance and perseverance in achieving the goal.”(A.Markushevich)

Literature.

1. Encyclopedia for children. "T.23". Universal Encyclopedic Dictionary \ ed. collegium: M. Aksyonova, E. Zhuravleva, D. Lury and others - M .: World of encyclopedias Avanta +, Astrel, 2008. - 688 p.
2. Ozhegov S.I. Dictionary of the Russian language: approx. 57000 words / Ed. member - corr. ANSIR N.Yu. Shvedova. - 20th ed. - M .: Education, 2000. - 1012 p.
3. I want to know everything! The Great Illustrated Encyclopedia of Intelligence / Per. from English. A. Zykova, K. Malkov, O. Ozerova. – M.: Publishing House of EKMO, 2006. – 440 p.
4. Sheinina O.S., Solovieva G.M. Maths. Classes of the school circle 5-6 cells / O.S. Sheinina, G.M. Solovieva - M .: Publishing House of NTsENAS, 2007. - 208 p.
5. Kordemsky B. A., Akhadov A. A. The Amazing World of Numbers: A Book of Students, - M. Education, 1986.
6. Minskykh E. M. “From the game to knowledge”, M., “Enlightenment”, 1982
7. Svechnikov A. A. Numbers, figures, tasks M., Enlightenment, 1977.
8. http://matsievsky.ru newmail. ru/sys-schi/file15.htm
9. http://sch69.narod. en/mod/1/6506/history. html

Research work in mathematics in elementary school

Brief abstract of the research work
Each student knows how to multiply multi-digit numbers with a “column”. In this paper, the author draws attention to the existence of alternative methods of multiplication available to younger students, which can turn "tedious" calculations into a fun game.
The paper discusses six non-traditional ways of multiplication of multi-digit numbers used in different historical eras: Russian peasant, lattice, small castle, Chinese, Japanese, according to the table of V. Okoneshnikov.
The project is designed to develop cognitive interest in the subject being studied, to deepen knowledge in the field of mathematics.
Table of contents
Introduction 3
Chapter 1. Alternative Ways of Multiplication 4
1.1. A bit of history 4
1.2. Russian peasant way of multiplying 4
1.3. Multiplication using the "Little Castle" method 5
1.4. Multiplication of numbers by the method of "jealousy" or "lattice multiplication" 5
1.5. Chinese multiplication method 5
1.6. Japanese multiplication method 6
1.7. Table Okoneshnikov 6
1.8. Multiplication by a column. 7
Chapter 2. Practical part 7
2.1. Peasant way 7
2.2. Little Castle 7
2.3. Multiplication of numbers by the method of "jealousy" or "lattice multiplication" 7
2.4. Chinese way 8
2.5. Japanese way 8
2.6. Table Okoneshnikov 8
2.7. Questionnaire 8
Conclusion 9
Annex 10

"The subject of mathematics is so serious that it is useful to take opportunities to make it a little entertaining."
B. Pascal
Introduction
It is impossible for a person to do without calculations in everyday life. Therefore, in mathematics lessons, we are first of all taught to perform operations on numbers, that is, to count. We multiply, divide, add and subtract in the usual ways for everyone that are studied at school. The question arose: are there any other alternative ways of computing? I wanted to study them in more detail. In order to answer these questions, this study was carried out.
The purpose of the study: to identify non-traditional methods of multiplication in order to study the possibility of their application.
In accordance with the goal, we formulated the following tasks:
- Find as many unusual ways of multiplication as possible.
- Learn to apply them.
- Choose for yourself the most interesting or easier ones than those offered at school, and use them when counting.
- Check in practice the multiplication of multi-digit numbers.
- Conduct a survey of 4th grade students
Object of study: various non-standard multi-digit multiplication algorithms
Subject of research: mathematical action "multiplication"
Hypothesis: If there are standard ways to multiply multi-digit numbers, perhaps there are alternative ways.
Relevance: dissemination of knowledge about alternative methods of multiplication.
Practical significance. In the course of the work, many examples were solved and an album was created, which includes examples with various algorithms for multiplying multi-valued numbers in several alternative ways. This may interest classmates to expand their mathematical horizons and serve as the beginning of new experiments.

Chapter 1

1.1. A bit of history
The methods of calculation that we use now were not always so simple and convenient. In the old days, more cumbersome and slower methods were used. And if a modern schoolboy could go back five hundred years, he would amaze everyone with the speed and accuracy of his calculations. The rumor about him would have spread around the surrounding schools and monasteries, eclipsing the glory of the most skillful counters of that era, and people would come from all over to study with the new great master.
The operations of multiplication and division were especially difficult in the old days.
In the book by V. Bellyustin “How people gradually came to real arithmetic”, 27 methods of multiplication are outlined, and the author notes: “it is quite possible that there are more methods hidden in the recesses of book depositories, scattered in numerous, mainly handwritten collections.” And all these methods of multiplication competed with each other and were assimilated with great difficulty.
Consider the most interesting and simple ways of multiplication.
1.2. Russian peasant way of multiplication
In Russia, 2-3 centuries ago, a method was spread among the peasants of some provinces that did not require knowledge of the entire multiplication table. It was only necessary to be able to multiply and divide by 2. This method was called the peasant method.
To multiply two numbers, they were written side by side, and then the left number was divided by 2, and the right one was multiplied by 2. Record the results in a column until 1 remains on the left. The remainder is discarded. We cross out those lines in which there are even numbers on the left. The remaining numbers in the right column are added.
1.3. Multiplication using the "Little Castle" method
The Italian mathematician Luca Pacioli in his treatise "The sum of knowledge in arithmetic, ratios and proportionality" (1494) gives eight different methods of multiplication. The first of them is called "Little Castle".
The advantage of the “Little Castle” multiplication method is that the digits of the highest digits are determined from the very beginning, and this can be important if you need to quickly estimate the value.
The digits of the upper number, starting from the most significant digit, are alternately multiplied by the lower number and written in a column with the addition of the required number of zeros. Then the results are added up.
1.4. Multiplying numbers using the "jealousy" or "lattice multiplication" method
Luca Pacioli's second method is called "jealousy" or "lattice multiplication".
First, a rectangle is drawn, divided into squares. Then the square cells are divided diagonally and “... it turns out a picture that looks like lattice shutters, blinds,” writes Pacioli. “Such shutters were hung on the windows of Venetian houses, preventing passers-by from seeing the ladies and nuns sitting at the windows.”
Multiplying each digit of the first factor with each digit of the second, the products are written in the corresponding cells, placing tens above the diagonal, and units below it. The digits of the product are obtained by adding the digits in oblique stripes. The results of additions are recorded under the table, as well as to the right of it.
1.5. Chinese multiplication method
Now let's imagine a method of multiplication, heatedly discussed on the Internet, which is called Chinese. When multiplying numbers, the points of intersection of lines are considered, which correspond to the number of digits of each digit of both factors.
1.6. Japanese multiplication method
The Japanese multiplication method is a graphical method using circles and lines. No less funny and interesting than Chinese. Even something like him.
1.7. Okoneshnikov's table
PhD in Philosophy Vasily Okoneshnikov, who is also the inventor of a new system of mental counting, believes that schoolchildren will be able to learn to add and multiply millions, billions and even sextillions with quadrillions orally. According to the scientist himself, the nine-decimal system is the most advantageous in this regard - all data is simply placed in nine cells arranged like buttons on a calculator.
According to the scientist, before becoming a computing "computer", it is necessary to memorize the table he created.
The table is divided into 9 parts. They are arranged according to the principle of a mini calculator: on the left in the lower corner "1", on the right in the upper corner "9". Each part is a multiplication table of numbers from 1 to 9 (according to the same "push-button" system). In order to multiply any number, for example, by 8, we find a large square corresponding to the number 8 and write out from this square the numbers corresponding to the digits of the multi-valued multiplier. We add the resulting numbers especially: the first digit remains unchanged, and all the rest are added in pairs. The resulting number will be the result of the multiplication.
If the addition of two digits results in a number greater than nine, then its first digit is added to the previous digit of the result, and the second is written in its “own” place.
The new methodology was tested in several Russian schools and universities. The Ministry of Education of the Russian Federation allowed the publication of a new multiplication table in squared notebooks along with the usual Pythagorean table - so far just for acquaintance.
1.8. Column multiplication.
Not many people know that the author of our usual method of multiplying a multi-digit number by a multi-digit number by a column should be considered Adam Rize (Appendix 7). This algorithm is considered the most convenient.
Chapter 2. Practical part
Mastering the listed methods of multiplication, a lot of examples were solved, an album with samples of various calculation algorithms was designed. (Application). Let's consider the calculation algorithm with examples.
2.1. peasant way
Multiply 47 by 35 (Appendix 1),
-write the numbers on one line, draw a vertical line between them;
-we will divide the left number by 2, multiply the right number by 2 (if a remainder occurs during division, then we discard the remainder);
- division ends when a unit appears on the left;
- cross out those lines in which there are even numbers on the left;
We add the numbers remaining on the right - this is the result.
35 + 70 + 140 + 280 + 1120 = 1645.
Conclusion. The method is convenient because it is enough to know the table only by 2. However, when working with large numbers, it is very cumbersome. Convenient for working with two-digit numbers.
2.2. small castle
(Appendix 2). Conclusion. The method is very similar to our modern "column". Moreover, the numbers of the highest ranks are immediately determined. This is important if you need to quickly estimate the value.
2.3. Multiplying numbers using the "jealousy" or "lattice multiplication" method
Let's multiply, for example, the numbers 6827 and 345 (Appendix 3):
1. We draw a square grid and write one of the multipliers above the columns, and the second - in height.
2. Multiply the number of each row sequentially by the numbers of each column. We successively multiply 3 by 6, by 8, by 2 and by 7, etc.
4. Add up the numbers following the diagonal stripes. If the sum of one diagonal contains tens, then we add them to the next diagonal.
From the results of adding the numbers along the diagonals, the number 2355315 is compiled, which is the product of the numbers 6827 and 345, that is, 6827 ∙ 345 = 2355315.
Conclusion. The "lattice multiplication" method is no worse than the conventional one. It is even simpler, since numbers are entered into the cells of the table directly from the multiplication table without simultaneous addition, which is present in the standard method.
2.4. Chinese way
Suppose you need to multiply 12 by 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, from left to right, we draw:
one green stick (1)
and two orange (2).
We draw the second number - 321, from bottom to top, from left to right:
three blue sticks (3);
two red (2);
one lilac (1).
Now we separate the intersection points with a simple pencil and proceed to count them. We move from right to left (clockwise): 2, 5, 8, 3.
Read the result from left to right - 3852
Conclusion. An interesting way, but to draw 9 straight lines when multiplied by 9 is somehow long and uninteresting, and then count more intersection points. Without skill, it is difficult to understand the division of a number into digits. In general, you can’t do without a multiplication table!
2.5. Japanese way
Multiply 12 by 34 (Appendix 5). Since the second factor is a two-digit number, and the first digit of the first factor is 1, we build two single circles in the top row and two binary circles in the bottom row, since the second digit of the first factor is 2.
Since the first digit of the second factor is 3, and the second is 4, we divide the circles of the first column into three parts, the second column into four parts.
The number of parts into which the circles are divided is the answer, that is, 12 x 34 = 408.
Conclusion. The method is very similar to Chinese graphic. Only straight lines are replaced by circles. It is easier to determine the digits of a number, but drawing circles is less convenient.
2.6. Okoneshnikov's table
It is required to multiply 15647 x 5. We immediately recall the large “button” 5 (it is in the middle) and on it we mentally find small buttons 1, 5, 6, 4, 7 (they are also located, as on a calculator). They correspond to the numbers 05, 25, 30, 20, 35. We add the resulting numbers: the first digit is 0 (remains unchanged), 5 is mentally added to 2, we get 7 - this is the second digit of the result, 5 is added to 3, we get the third digit - 8 , 0+2=2, 0+3=3 and the last digit of the product remains - 5. The result is 78,235.
Conclusion. The method is very convenient, but you need to learn by heart or always have a table at hand.
2.7. Student survey
A survey was conducted among fourth-graders. 26 people took part (Appendix 8). Based on the survey, it was revealed that all respondents know how to multiply in the traditional way. But most guys do not know about non-traditional methods of multiplication. And there are those who want to get to know them.
After the initial survey, an extra-curricular activity "Multiplication with passion" was held, where the children got acquainted with alternative multiplication algorithms. After that, a survey was conducted in order to identify the most liked methods. The undisputed leader was the most modern method of Vasily Okoneshnikov. (Annex 9)
Conclusion
Having learned to count in all the ways presented, I believe that the most convenient method of multiplication is the "Little Castle" method - because it is so similar to our current one!
Of all the unusual counting methods I found, the “Japanese” method seemed more interesting. The simplest method seemed to me to be the “doubling and splitting” method used by Russian peasants. I use it when multiplying not too large numbers. It is very convenient to use it when multiplying two-digit numbers.
Thus, I achieved the goal of my research - I studied and learned how to apply non-traditional ways of multiplying multi-digit numbers. My hypothesis was confirmed - I mastered six alternative methods and found out that these are not all possible algorithms.
The unconventional multiplication methods I studied are very interesting and have the right to exist. And in some cases, they are even easier to use. I think that you can talk about the existence of these methods at school, at home and surprise your friends and acquaintances.
So far, we have only studied and analyzed the already known methods of multiplication. But who knows, perhaps in the future we ourselves will be able to discover new ways of multiplying. Also, I do not want to stop there and continue to study non-traditional methods of multiplication.
List of information sources
1. List of references
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1.7. Perelman Ya.I. Quick account. Thirty simple methods of mental counting. L.: Lenizdat, 1941 - 12 p.
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2. Other sources of information
Internet resources:
2.1. Korneev A.A. The phenomenon of Russian multiplication. Story. [Electronic resource]