published 20.04.2012
Dedicated to Elena Petrovna Karinskaya ,
my school math teacher and class teacher
Alma-Ata, ROFMSh , 1984-1987
"Science achieves perfection only when it manages to use mathematics"... Karl Heinrich Marx
these words were inscribed above the blackboard in our math classroom ;-)
Informatics lessons (lecture materials and workshops)

What is multiplication?
This is an addition action.
But not too pleasant
Because many times ...
Tim Sobakin

Let's try to do this action
pleasant and fun ;-)

WAYS OF MULTIPLICATION WITHOUT MULTIPLICATION TABLE (gymnastics for the mind)

I offer the readers of the green pages two methods of multiplication, which do not use the multiplication table ;-) I hope that this material will appeal to teachers of computer science, which they can use when conducting optional classes.

This method was used in the everyday life of Russian peasants and inherited from ancient times. Its essence is that the multiplication of any two numbers is reduced to a series of successive divisions of one number in half while doubling another number, multiplication table in this case unnecessarily :-)

The division in half is continued until the quotient is 1, while another number is doubled in parallel. The last doubled number gives the desired result (picture 1). It is easy 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 therefore clear that as a result of repeated repetition of this operation, the desired product is obtained.

However, what to do if you have to halve an odd number? In this case, we discard one from the odd number and divide the remainder in half, while all those numbers of this column that are opposite the odd numbers of the left column will need to be added to the last number of the right column - the sum will be the desired product (Figures: 2, 3).
In other words, cross out all lines with even left numbers; leave and then summarize not strikethrough numbers right column.

For Figure 2: 192 + 48 + 12 = 252
The correctness of the reception will become clear if you take into account that:
5 × 48 \u003d (4 + 1) × 48 \u003d 4 × 48 + 48
21 × 12 \u003d (20 + 1) × 12 \u003d 20 × 12 + 12
It is clear that the numbers 48 , 12 , lost when dividing an odd number in half, must be added to the result of the last multiplication to get the product.
The Russian way of multiplication is both elegant and extravagant at the same time ;-)

§ Logic puzzle about Serpent Gorynyche and the famous Russian heroes on the green page "Which of the heroes defeated the Serpent Gorynych?"
solving logic problems by means of logic algebra
For those who love to learn! For those who are happy gymnastics for the mind ;-)
§ Solving logical problems in a tabular way

We continue the conversation :-)

Chinese??? The drawing way of multiplication

My son introduced me to this method of multiplication, having provided me with several pieces of paper from a notebook with ready-made solutions in the form of intricate drawings. The process of decrypting the algorithm has boiled drawing way of multiplication :-) For clarity, I decided to use colored pencils, and ... gentlemen of the jury broke the ice :-)
I bring to your attention three examples in color pictures (in the upper right corner check post).

Example # 1: 12 × 321 = 3852
Draw first number top to bottom, left to right: one green stick ( 1 ); two orange sticks ( 2 ). 12 drew :-)
Draw second number bottom to top, left to right: three blue sticks ( 3 ); two reds ( 2 ); one lilac ( 1 ). 321 drew :-)

Now, with a simple pencil, we will walk through the drawing, divide the intersection points of the numbers-sticks into parts and start counting the points. Moving from right to left (clockwise): 2 , 5 , 8 , 3 . Result number we will "collect" from left to right (counterclockwise) and ... voila, we got 3852 :-)

Example # 2: 24 × 34 = 816
There are some nuances in this example ;-) When counting points in the first part, it turned out 16 ... We send the unit-add to the dots of the second part ( 20 + 1 )…

Example # 3: 215 × 741 = 159315
No comment:-)

At first it seemed to me somewhat pretentious, but at the same time intriguing and surprisingly harmonious. On the fifth example, I caught myself thinking that multiplication goes into flight :-) and works in autopilot mode: draw, count points, we don't remember the multiplication table, it seems like we don't know it at all :-)))

To be honest, by checking drawing way of multiplication and turning to multiplication by a column, and more than once, and not twice, to my shame, I noticed some slowdowns, indicating that my multiplication table rusted in some places :-( and you shouldn't forget it. When working with more "serious" numbers drawing way of multiplication has become too cumbersome, and column multiplication went into joy.

Multiplication table (sketch of the back of the notebook)

P.S.: Glory and praise to the native Soviet column!
In terms of construction, the method is unpretentious and compact, very fast, memory trains - the multiplication table does not allow forgetting :-) And therefore, I strongly recommend that you and yourself and you, if possible, forget about calculators in phones and computers ;-) and periodically indulge yourself with multiplication by a column. Otherwise, it’s not even an hour and the plot from the movie “Rise of the Machines” will unfold not on the cinema screen, but in our kitchen or the lawn next to our house ...
Three times over the left shoulder ... knocking on wood ... :-))) ... and most importantly do not forget about gymnastics for the mind!

For the curious: Multiplication denoted by [×] or [·]
The [×] sign was introduced by an English mathematician William Outread in 1631.
The [·] sign was introduced by a German scientist Gottfried Wilhelm Leibniz in 1698.
In the letter designation, these signs are omitted and instead of a × b or a · b write ab.

In the piggy bank of the webmaster: Some math symbols in HTML

°	° or °	degree
±	± or ±	plus or minus
¼	¼ or ¼	fraction - one quarter
½	½ or ½	fraction - one second
¾	¾ or ¾	fraction - three quarters
×	× or ×	multiplication sign
÷	÷ or ÷	division sign
ƒ	ƒ or ƒ	function sign
′	' or '	single stroke - minutes and feet
″	" or "	double prime - seconds and inches
≈	≈ or ≈	roughly equal sign
≠	≠ or ≠	not equal
≡	≡ or ≡	identically
>	\u003e or\u003e	more
<	< или	smaller
≥	≥ or ≥	more or equal
≤	≤ or ≤	less or equal
∑	∑ or ∑	summation sign
√	√ or √	square root (radical)
∞	∞ or ∞	infinity
Ø	Ø or Ø	diameter
∠	∠ or ∠	angle
⊥	⊥ or ⊥	perpendicular

Municipal budgetary educational institution

Secondary school with. Hoses

Municipal District Aurgazinsky District RB

Research work

"UNUSUAL WAYS OF MULTIPLICATION"

Vasiliev Nikolay

Leader -

2013-2014 account g.

1. Introduction……………………………………………………………......

2. Unusual ways of multiplication ……………………………………… ...

1) A little history ……… .. ……… .. ………………………………… ..

2) Multiplication by 9 ………………………………………… ..............

3) Multiplication on fingers ………………………………………………

4) Pythagoras table …………………………………………………

5) Okoneshnikov's table …………………………………………….

6) Peasant way of multiplication ………………………. ……… ....

7) Multiplication by the “Little Castle” method …………. ……………….

8) Multiplication by the "Jealousy" method …………………………………….

9) The Chinese way of multiplication …………………………………………

10) The Japanese way of multiplication …………………………………………

3. Conclusion ………………………… .. ………………………………… ...

4. References ……………………………………………………….

Introduction

To a person in everyday life it is impossible to do without calculations. Therefore, in mathematics lessons, we are primarily taught to perform actions on numbers, that is, to count. We multiply, divide, add and subtract in the usual ways that are taught in school.

Once I accidentally came across a page on the Internet with an unusual multiplication method used by children in China (as it is written there). I read, studied and liked this method. It turned out that it is possible to multiply not only as they suggest to us in mathematics textbooks. I wondered if there were any other ways of calculating. After all, the ability to quickly perform calculations is frankly surprising.

The constant use of modern computing technology leads to the fact that students find it difficult to make any calculations without having tables or a calculating machine at their disposal. Knowledge of simplified calculation techniques makes it possible not only to quickly perform simple calculations in the mind, but also to control, evaluate, find and correct errors as a result of mechanized computation. In addition, mastering computational skills develops memory, raises the level of mathematical culture of thinking, and helps to fully master the subjects of the physics and mathematics cycle.

Objective:

Show unusual ways to multiply.

Tasks:

Ø Find as many unusual ways of computing as possible.

Ø Learn to apply them.

Ø Choose for yourself the most interesting or lighter ones than those offered at the school, and use them when counting.

I wondered if modern schoolchildren, my classmates and others know other ways of performing arithmetic operations, except for multiplication by a column and division by a "corner" and would like to learn new ways? I conducted an oral survey. 20 students of grades 5-7 were interviewed. This survey showed that modern schoolchildren do not know other ways of performing actions, since they rarely turn to material outside the school curriculum.

Results of the survey:

https://pandia.ru/text/80/266/images/image002_6.png "align \u003d" left "width \u003d" 267 "height \u003d" 178 src \u003d "\u003e

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

https://pandia.ru/text/80/266/images/image004_2.png "align \u003d" left "width \u003d" 264 height \u003d 176 "height \u003d" 176 "\u003e

3) would you like to know?

Unusual ways to multiply.

A bit of history

The methods of computing that we use now have not always been so simple and convenient. In the old days, they used more cumbersome and slow methods. And if a schoolboy of the 21st century could travel back five centuries, he would amaze our ancestors with the speed and accuracy of his calculations. Rumors about him would have spread around the surrounding schools and monasteries, eclipsing the glory of the most skillful enumerators of that era, and people would come from all sides to learn from the new great master.

The actions of multiplication and division were especially difficult in the old days. At that time, there was no one method developed by practice for each action. On the contrary, almost a dozen different methods of multiplication and division were in use at the same time - the methods of each other are more intricate, which a person of average abilities could not remember. Each teacher of counting adhered to his favorite technique, each “master of division” (there were such specialists) praised his own way of doing this.

In the book by V. Bellustin "How people gradually got to real arithmetic" 27 methods of multiplication are set forth, and the author notes: "it is quite possible that there are still methods hidden in the caches of book depositories, scattered in numerous, mainly manuscript collections."

And all these methods of multiplication - "chess or organ", "bending", "cross", "lattice", "backward", "diamond" and others competed with each other and were absorbed with great difficulty.

Let's look at the most interesting and simple ways to multiply.

Multiplication by 9

Multiplication for the number 9 - 9 · 1, 9 · 2 ... 9 · 10 - more easily disappears from memory and is more difficult to recalculate manually by the method of addition, however, it is for the number 9 that multiplication is easily reproduced "on the fingers." Spread your fingers on both hands and turn your palms away from you. Mentally assign the numbers from 1 to 10 to your fingers in sequence, starting with the little finger of your left hand and ending with the little finger of your right hand (this is shown in the figure).

computing ".

counting machine "fingers can not necessarily protrude. Take, for example, 10 cells in a notebook. Cross out the 8th cell. On the left there are 7 cells, on the right - 2 cells. So 9 · 8 \u003d 72. Everything is very simple.

7 cells 2 cells.

Multiplication on fingers

The Old Russian method of multiplication on the fingers is one of the most common methods that Russian merchants have successfully used for many centuries. They learned to multiply single-digit numbers from 6 to 9 on their fingers. At the same time, it was enough to master the initial skills of finger counting “ones”, “pairs”, “threes”, “fours”, “fives” and “tens”. The fingers here served as an auxiliary computing device.

To do this, on one hand, they stretched out 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 the extended 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.

For example, multiply 7 by 8. In this example, 2 and 3 fingers will be bent. If you add up the number of bent fingers (2 + 3 \u003d 5) and multiply the number of unbent fingers (2 3 \u003d 6), you get the number of tens and units of the desired product 56, respectively. This way you can calculate the product of any single digit numbers greater than 5.

Pythagoras table

Let us recall the main rule of ancient Egyptian mathematics, which says that multiplication is performed by doubling and adding the results obtained; that is, each doubling is the addition of a number to itself. Therefore, it is interesting to look at the result of a similar doubling of numbers and numbers, but obtained by the modern method of "column folding", known even in the elementary grades of school.

Okoneshnikov table

Students will be able to learn orally to add and multiply millions, billions, and even sextillions and quadrillions. And the candidate of philosophical sciences Vasily Okoneshnikov, who is also the inventor, will help them in this. new system oral counting. The scientist claims that a person is able to memorize a huge store of information, the main thing is how to arrange this information.

According to the scientist himself, the most advantageous in this respect is the ninefold system - all the data are simply placed in nine cells, located like buttons on a calculator.

According to the scientist, before becoming a computing "computer", it is necessary to memorize the table he created. The numbers in it are not easily distributed in nine cells. According to Okoneshnikov, the human eye and his memory are so cunningly arranged that the information located according to his method is remembered, firstly, faster, and secondly, firmly.

The table is divided into 9 parts. They are located according to the principle of a mini calculator: in the lower left corner "1", in the upper right corner "9". Each part is a multiplication table for numbers from 1 to 9 (again in the lower left corner by 1, next to the right by 2, etc., according to the same "button" system). How to use them?
for example, you need to multiply 9 on the 842 ... We immediately recall the big "button" 9 (it is at the top right and on it we mentally find small buttons 8,4,2 (they are also located like on a calculator). They correspond to the numbers 72, 36, 18. We add the resulting numbers separately: the first digit 7 ( remains unchanged), mentally add 2 to 3, we get 5 - this is the second digit of the result, 6 we add to 1, we get the third digit -7, and the last digit of the required number remains - 8. As a result, we get 7578.
If the addition of two digits results in a number exceeding nine, then its first digit is added to the previous digit of the result, and the second is written in its "proper" place.

With the help of the Okoneshnikov matrix table, according to the author himself, it is possible to study foreign languages, and even the periodic table. The new technique has been tested in several Russian schools and universities. The Ministry of Education of the Russian Federation allowed to publish a new multiplication table in notebooks in a box along with the usual Pythagorean table - so far just for acquaintance.

Example : 15647 x 5

https://pandia.ru/text/80/266/images/image015_0.jpg "alt \u003d" (! LANG: Figure5" width="220 height=264" height="264"> 35 + 70 + 140 + 280 + 1120 = 1645.!}

LITTLE CASTLE multiplication

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

The advantage of the "Little Castle" multiplication method is that the most significant 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 with 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. The results are then added up.

Multiplication of numbers by the "jealousy" method.

https://pandia.ru/text/80/266/images/image018.jpg "width \u003d" 303 "height \u003d" 192 id \u003d "\u003e. jpg" width \u003d "424 height \u003d 129" height \u003d "129"\u003e

3. This is how the grid looks like with all filled cells.

Mesh 1

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

Mesh1

From the results of adding the digits along the diagonals (they are highlighted in yellow), a number is compiled 2355315 , which is the product of numbers 6827 and 345,i.e 6827 x 345 \u003d 2355315.

Chinese way of multiplication

Now let's imagine the multiplication method, which is widely discussed on the Internet, which is called Chinese. When multiplying numbers, the points of intersection of straight lines are considered, which correspond to the number of digits of each digit of both factors.

https://pandia.ru/text/80/266/images/image024_0.png "width \u003d" 92 "height \u003d" 46 "\u003e Example : multiply 21 on the 13 ... In the first factor there are 2 tens and 1 unit, so we build 2 parallel straight lines and 1 straight line at some distance.

The lines intersect at points, the number of which is the answer, that is 21 x 13 \u003d 273

It's funny and interesting, but drawing 9 straight lines when multiplying by 9 is somehow long and uninteresting, and then counting the intersection points ... In general, you can't do without a multiplication table!

Japanese way of multiplication

The Japanese way of multiplying is a graphical way using circles and lines. No less funny and interesting than Chinese. Even something like him.

Example: multiply 12 on the 34. Since the second factor is a two-digit number, and the first digit of the first factor 1 , we build two single circles in the top line and two binary circles in the bottom line, since the second digit of the first factor is 2 .

12 x 34

The number of parts into which the circles are divided is the answer, that is 12 x 34 \u003d 408.

Of all the unusual counting methods I found, the more interesting was the "lattice multiplication or jealousy" method. I showed it to my classmates, and they also really liked it.

The simplest method seemed to me to be the “doubling and doubling” method used by the Russian peasants. I use it when multiplying not too large numbers (it is very convenient to use it when multiplying two-digit numbers).

I think that our method of long multiplication is not perfect and we can come up with even faster and more reliable methods.

Literature

1. "Stories about mathematics". - Leningrad .: Education, 1954 .-- 140 p.

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

3., "Ancient entertaining tasks". - M .: Science. Main edition of physical and mathematical literature, 1985. - 160 p.

4. Perelman account. Thirty Easy Verbal Counting Techniques. L., 1941 - 12 p.

5. Perelman arithmetic. M. Rusanova, 1994-205s.

6. Encyclopedia “I get to know the world. Maths". - M .: Astrel Ermak, 2004.

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

Tretyakova Anastasia, Temkina Alina

Purpose and objectives of the project:

Purpose: familiarization with various methods of multiplying natural numbers that are not used in the lessons, and their application in calculating numerical expressions.

Tasks:

1. Find and parse different ways of multiplying.
2. Learn to demonstrate some ways of multiplying.
3. Explain new ways to multiply and teach students to use them.
4. To develop skills of independent work: search for information, selection and design of the found material.

Hypothesis: “Knowledge is revealed only by that.

Who knows with different numbers !!! "

Download:

Preview:

Municipal budgetary educational institution

secondary school No. 35 of the city district of Samara

Project on:

"Multiplication methods

Natural numbers "

The work was completed by: pupils of 5 "A" class

Tretyakova Anastasia,

Temkina Alina.

Supervisor:

mathematic teacher

Ruzanova I.M.

Samara, 2014

Purpose and objectives of the project:

Purpose: familiarization with various methods of multiplying natural numbers that are not used in the lessons, and their application in calculating numerical expressions.

Tasks:

1. Find and parse different ways of multiplying.
2. Learn to demonstrate some ways to multiply.
3. Explain new ways to multiply and teach students to use them.
4. To develop skills of independent work: search for information, selection and design of the found material.

Hypothesis: “Knowledge is revealed only by that.

Who knows with different numbers !!! "

Pythagoras.

1. Introduction. 4 p.
2. Main part. 5 - 13 pages

1. Russian-peasant way of multiplication. 5 - 6 pages
2. Pythagoras square. 6 - 7 pages
3. Okoneshnikov's table. 7 - 9 pages
4. The Indian way of multiplying. 9 - 11 pp.
5. Egyptian multiplication method. 11 - 12 p.
6. Chinese way of multiplication. 12 p.
7. The Japanese way of multiplying. 13 pp.

1. Conclusion. 14 pp.
2. Literature. 14 pp.

1. Introduction.

….. You cannot multiply multi-digit numbers - even two-digit numbers - unless you remember by heart all the results of multiplying single-digit numbers, that is, what is called the multiplication table. In the ancient "Arithmetic" of Magnitsky, the need for a solid knowledge of the multiplication table is glorified in such - it must be confessed, alien to the modern ear - verses:

If anyone does not repeat

tables and proud

Can't cognize

the number to multiply

And in all science, not free from torment,

Colico will not be considered

And in favor it will not if you forget.

Magnitsky himself, the author of these verses, obviously did not know or overlooked that there are ways to multiply numbers without knowing the multiplication table. These methods are not similar to our school methods, some were used in everyday life of the Great Russian peasants and were inherited by them from ancient times, some are still used in our time.

At school, they study the multiplication table, and then teach children to multiply numbers in a column. Of course, this is not the only way to multiply. In fact, there are several dozen ways to multiply multidigit numbers. In this work, we will give several methods of multiplication, perhaps they will seem simpler and you will use them.

1. Main part.

1. Russian-peasant way of multiplication.

Its essence is that the multiplication of any two numbers is reduced to a series of consecutive divisions of one number in half while simultaneously doubling another number. Example:32 x 13

Multiplier \u003d 32	Multiplier \u003d 13

Table 1.

The division in half (see the left half of Table 1) is continued until the quotient is 1, in parallel doubling another number (the right side of Table 1). 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 therefore clear that as a result of repeated repetition of this operation, the desired product is obtained:(32 x 13) \u003d (1 x 416)

Those who are especially attentive will notice "What about odd numbers that are not divisible by 2?"

So, let's say we need to multiply two numbers: 987 and 1998. We write one on the left, and the second on the right on one line. The left number will be divided by 2, and the right number will be multiplied by 2 and the results are written in a column. If there is a remainder during division, then it is discarded.

We continue the operation until there is 1 on the left. Then cross out those lines in which there are even numbers on the left and add the remaining numbers in the right column. This is the desired work. A graphic illustration is given for this description. (see Table 2.)

Table 2.

1. Pythagoras square.

1 2 3

4 5 6

7 8 9

This is the well-known Pythagorean Square, which reflects the world number system, consisting of nine digits: from 1 to 9. In modern terms, it is a nine-bit numerical matrix in which the numbers that are the basis for further calculations of any complexity are arranged in ascending order. The square of Pythagoras is also called the Ennead, and the three numbers are called the triad. You can consider triples of numbers located horizontally (123, 456, 789) and vertically (147, 258, 369). Moreover, written in this way, the triplets of numbers begin to denote special numbers that obey the laws of mathematical proportion and harmony.

Let us recall the main rule of ancient Egyptian mathematics, which says that multiplication is performed by doubling and adding the results obtained; that is, each doubling is the addition of a number to itself. Therefore, it is interesting to look at the result of such a doubling of numbers and numbers, but obtained by the modern method of "column folding", known even in the elementary grades of school. This will resemble the Egyptian number system, in fact, with the difference that all numbers or numbers are written in one column (without specifying one or another action in the next column - like the Egyptians).

Let's start with the numbers that make up the Pythagorean Square: from 1 to 9.

1 2 3 4 5 6 7 8 9

2 4 6 8 10 12 14 16 18

3 6 9 12 15 18 21 24 27

4 8 12 16 20 24 28 32 36

5 10 15 20 25 30 35 40 45

6 12 18 24 30 36 42 48 54

7 14 21 28 35 42 49 56 63

8 16 24 32 40 48 56 64 72

9 18 27 36 45 54 63 72 81

10 20 30 40 50 60 70 80 90

Digit 1: Normal sequential row of digits.

Number 9: the left column is a clear ascending row ("flow").

the right column is a clear descending row of consecutive numbers. Let us agree to call an ascending series in which the values \u200b\u200bof numbers increase from top to bottom; in the downward direction, on the contrary: the values \u200b\u200bof the numbers decrease from top to bottom.

Number 2: the even numbers 2,4,6,8 are repeated in the right column ("in the period").

Number 8: the same repeat - only in reverse order - 8,6,4,2.

Digits 4 and 6: even digits "in the period" 4,8,2,6 and 6,2,8,4.

Digit 5: obeys the rule of addition of digit 5 \u200b\u200b- alternation of 5 and 0.

Number 3: the right column is a descending row of no longer numbers, but numbers that form triplets of vertical rows in the Pythagorean square - 369, 258, 147. Moreover, the counting goes "from the right corner of the square" or from right to left. The rule of the ascending - descending series adopted above also applies here. But the ascending series is a movement from three of 147 to three of 369; descending - from 369 to 147.

Number 7: Ascending series of numbers 147,258,369 from the "left corner" or from left to right. However, it all depends on how the nine-digit numerical matrix itself is depicted - where to put the number 1.

1. Okoneshnikov's table.

Students will be able to learn orally to add and multiply millions, billions, and even sextillions and quadrillions. And they will be helped in this by Vasily Okoneshnikov, Candidate of Philosophy, who is also the inventor of a new system of oral counting. The scientist claims that a person is able to memorize a huge store of information, the main thing is how to arrange this information.
According to the scientist himself, the most advantageous in this respect is the ninefold system - all the data are simply placed in nine cells, located like buttons on a calculator.

According to the scientist, before becoming a computing "computer", it is necessary to memorize the table he created. The numbers in it are not easily distributed in nine cells. According to Okoneshnikov, the human eye and his memory are so cunningly arranged that the information located according to his method is remembered, firstly, faster, and secondly, firmly.
The table is divided into 9 parts. They are located according to the principle of a mini calculator: in the lower left corner "1", in the upper right corner "9". Each part is a multiplication table for numbers from 1 to 9 (again in the lower left corner by 1, next to the right by 2, etc., according to the same "button" system). How to use them?
for example , you need to multiply9 at 842 ... We immediately recall the big "button" 9 (it is at the top right and on it we mentally find small buttons 8,4,2 (they are also located like on a calculator). They correspond to the numbers 72, 36, 18. We add the resulting numbers separately: the first digit 7 ( remains unchanged), mentally add 2 to 3, we get 5 - this is the second digit of the result, 6 we add to 1, we get the third digit -7, and the last digit of the required number remains - 8. As a result, we get 7578.
If the addition of two digits results in a number exceeding nine, then its first digit is added to the previous digit of the result, and the second is written in its "proper" place.
With the help of the Okoneshnikov matrix table, according to the author himself, one can study foreign languages, and even the periodic table. The new technique has been tested in several Russian schools and universities. The Ministry of Education of the Russian Federation allowed to publish a new multiplication table in notebooks in a box along with the usual Pythagorean table - so far just for acquaintance.

Example: 15647 x 5

1. The Indian way of multiplying.

In ancient India, two methods of multiplication were used: grids and galleys. At first glance, they seem to be very difficult, but if you follow step by step in the proposed exercises, you can see that it is quite simple.

We multiply, for example, the numbers6827 and 345:

1. Draw a square grid and write one of the numbers above the columns, and the second in height. In the example provided, you can use one of these grids.

Grid 1 Grid 2

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

Mesh 1

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

Mesh 1

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

Mesh1

See how a number is compiled from the results of adding the digits along the diagonals (they are highlighted in yellow)2355315 which isproduct of numbers6827 and 345, that is, 6827 x 345 \u003d 2355315.

1. Egyptian multiplication method.

Ancient Egyptian multiplication is a sequential method of multiplying two numbers. To multiply numbers, they did not need to know the multiplication tables, and it was enough only to be able to decompose numbers into multiple bases, multiply these multiples and add. The Egyptian method involves expanding the smallest of two factors into multiples and then sequentially multiplying them by the second factor (see example). This method can be found today in very remote regions.

Decomposition. The Egyptians used a system to decompose the smallest factor into multiples that would add up to the original number.

To find the correct multiple number, you needed to know the following table of values:

1 x 2 \u003d 2 2 x 2 \u003d 4 4 x 2 \u003d 8 8 x 2 \u003d 16 16 x 2 \u003d 32

Example expansion of the number 25: The multiple factor for the number "25" is 16; 25 - 16 \u003d 9. The multiple factor for the number "9" is 8; 9 - 8 \u003d 1. The multiple factor for the number "1" is 1; 1 - 1 \u003d 0. Thus, "25" is the sum of three terms: 16, 8 and 1.

Example: multiply "13" by "238 ". It is known that 13 \u003d 8 + 4 + 1. Each of these terms must be multiplied by 238. We get: ✔ 1 x 238 \u003d 238 ✔ 4 x 238 \u003d 952 ✔ 8 x 238 \u003d 190413 × 238 \u003d (8 + 4 + 1) × 238 \u003d 8 x 238 + 4 × 238 + 1 × 238 \u003d 1904 + 952 + 238 \u003d 3094.

1. Chinese way of multiplication.

Now let's imagine a multiplication method that is widely discussed on the Internet, which is called Chinese. When multiplying numbers, the points of intersection of straight lines are considered, which correspond to the number of digits of each digit of both factors.

Example: multiply 21 by 13 ... In the first factor there are 2 tens and 1 unit, which means we build 2 parallel straight lines and 1 straight line at a distance.

The second factor has 1 tens and 3 units. We build parallel 1 and at some distance 3 straight lines intersecting the straight lines of the first factor.

The lines intersect at points, the number of which is the answer, that is21 x 13 \u003d 273

It's funny and interesting, but drawing 9 straight lines when multiplying by 9 is somehow long and uninteresting, and then counting the intersection points ... In general, you can't do without a multiplication table!

1. The Japanese way of multiplying.

The Japanese way of multiplying is a graphical way using circles and lines. No less funny and interesting than Chinese. Even something like him.

Example: multiply 12 by 34. Since the second factor is a two-digit number, and the first digit of the first factor1 , we build two single circles on the top line and two binary circles on the bottom line, since the second digit of the first factor is2 .

12 x 34

Since the first digit of the second factor3, and the second 4 , divide the circles of the first column into three parts, the second column into four.

12 x 34

The number of parts into which the circles are divided is the answer, that is 12 x 34 \u003d 408.

1. Conclusion.

While working on this topic, we learned that there are many different, funny and interesting ways to multiply. Some are still used in various countries. But not all methods are convenient to use, especially when multiplying multi-digit numbers. In general, you still need to know the multiplication table!

This work can be used for lessons in math circles, additional activities with children outside of school hours, as additional material in the lesson on the topic "Multiplication of natural numbers". The material is presented in an accessible and interesting way that will attract the attention and interest of students to the subject of mathematics.

1. Literature.

1. AND I. Depman, N. Ya. Vilenkin “Behind the Pages of a Mathematics Textbook”.
2. L.F. Magnitsky "Arithmetic".
3. Journal "Mathematics" №15 2011
4. Internet resources.

Research paper in elementary school mathematics

Brief abstract of the research paper
Every student knows how to multiply multidigit numbers in a column. In this paper, the author draws attention to the existence of alternative methods of multiplication available to primary schoolchildren, which can turn "boring" calculations into a fun game.
The paper examines six unconventional ways of multiplying multidigit 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 under study, to deepen knowledge in the field of mathematics.
Table of contents
Introduction 3
Chapter 1. Alternative methods of multiplication 4
1.1. A bit of history 4
1.2. Russian peasant multiplication method 4
1.3. Multiplication by 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 way of multiplying 6
1.7. Okoneshnikov table 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. Okoneshnikov table 8
2.7. Questionnaire 8
Conclusion 9
Appendix 10

“The subject of mathematics is so serious that it’s helpful to watch out for 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 actions on numbers, that is, to count. We multiply, divide, add and subtract in the usual ways that are taught in school. The question arose: are there any other alternative ways of computing? I wanted to study them in more detail. In search of an answer to these questions, this study was conducted.
Purpose of the research: identification of unconventional multiplication methods to study the possibility of their application.
In accordance with the set goal, we formulated the following tasks:
- Find as many unusual multiplication methods as possible.
- Learn to apply them.
- Choose for yourself the most interesting or easier ones than those offered at the school, and use them when counting.
- Check in practice multiplying multi-digit numbers.
- Conduct a questionnaire survey of 4th grade students
Object of study: various non-standard algorithms for multiplying multi-digit numbers
Research subject: the mathematical action "multiplication"
Hypothesis: If there are standard ways to multiply multidigit numbers, there may be alternative ways.
Relevance: spreading 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 multidigit numbers in several alternative ways. This may interest classmates to expand their mathematical horizons and will serve as the beginning of new experiments.

Chapter 1. Alternative methods of multiplication

1.1. A bit of history
The methods of computing that we use now have not always been so simple and convenient. In the old days, they used more cumbersome and slow methods. And if a modern schoolboy could go five hundred years ago, he would amaze everyone with the speed and accuracy of his calculations. Rumors about him would have spread throughout the surrounding schools and monasteries, eclipsing the glory of the most skillful enumerators of that era, and people would come from all sides to study with the new great master.
The actions of multiplication and division were especially difficult in the old days.
In the book by V. Bellustin "How people gradually got to real arithmetic" 27 methods of multiplication are set forth, and the author notes: "it is quite possible that there are more methods hidden in the caches of book depositories, scattered in numerous, mainly manuscript collections." And all these methods of multiplication competed with each other and were learned with great difficulty.
Let's consider the most interesting and simple ways to multiply.
1.2. Russian peasant way of multiplication
In Russia 2-3 centuries ago, among the peasants of some provinces, a method was widespread that did not require knowledge of the entire multiplication table. It was only necessary to know how to multiply and divide by 2. This method was called peasant.
To multiply two numbers, they were written side by side, and then the left number was divided by 2, and the right number was multiplied by 2. Write the results in a column until there is 1 on the left. The remainder is discarded. Cross out those lines in which there are even numbers on the left. Add the remaining numbers in the right column.
1.3. Multiplication by the "Little Castle" method
The Italian mathematician Luca Pacioli in his treatise The Sum of Knowledge in Arithmetic, Relations 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 most significant digits are determined from the very beginning, and this is important if you need to quickly estimate the value.
The digits of the upper number, starting with 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. The results are then added up.
1.4. Multiplication of numbers by the method of "jealousy" or "lattice multiplication"
The second way Luca Pacioli is called "jealousy" or "lattice multiplication".
First, a rectangle is drawn, divided into squares. Then the square cells are divided diagonally and “… a picture looks like a lattice shutter-jalousie,” Pacioli writes. "Such shutters were hung on the windows of Venetian houses, making it difficult for street passers-by to see 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 into the corresponding cells, placing tens above the diagonal, and units below it. The numbers of the work are obtained by adding the numbers in the oblique stripes. The results of additions are recorded under the table, as well as to the right of it.
1.5. Chinese way of multiplication
Now let's imagine a multiplication method that is widely discussed on the Internet, which is called Chinese. When multiplying numbers, the points of intersection of straight lines are considered, which correspond to the number of digits of each digit of both factors.
1.6. Japanese way of multiplication
The Japanese way of multiplying is a graphical way using circles and lines. No less funny and interesting than Chinese. Even something like him.
1.7. Okoneshnikov table
Vasily Okoneshnikov, PhD in Philosophy, who is also the inventor of a new oral counting system, believes that students will be able to learn orally to add and multiply millions, billions and even sextillions with quadrillions. According to the scientist himself, the most advantageous in this respect is the ninefold system - all the data are simply placed in nine cells, located 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 located according to the principle of a mini calculator: in the lower left corner "1", in the upper right corner "9". Each part is a multiplication table for numbers from 1 to 9 (according to the same "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-digit factor. We add the resulting numbers separately: the first digit remains unchanged, and all the rest are added in pairs. The resulting number will be the result of multiplication.
If the addition of two digits results in a number exceeding nine, then its first digit is added to the previous digit of the result, and the second is written in its "proper" place.
The new technique has been tested in several Russian schools and universities. The Ministry of Education of the Russian Federation allowed to publish a new multiplication table in notebooks in a box along with the usual Pythagorean table - so far just for acquaintance.
1.8. Column multiplication.
Not many people know that Adam Riese should be considered the author of our usual way of multiplying a multi-digit number by a multi-digit number (Appendix 7). This algorithm is considered the most convenient.
Chapter 2. Practical part
While mastering the listed methods of multiplication, many examples were solved, an album was designed with samples of various calculation algorithms. (Application). Let's consider the calculation algorithm using examples.
2.1. Peasant way
Multiply 47 by 35 (Appendix 1),
-write the numbers on one line, draw a vertical line between them;
-the left number will be divided by 2, the right number will be multiplied by 2 (if a remainder occurs during division, then we discard the remainder);
-division ends when one appears on the left;
- cross out those lines in which there are even numbers on the left;
- the numbers remaining on the right are added - this is the result.
35 + 70 + 140 + 280 + 1120 = 1645.
Output. 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). Output. The method is very similar to our modern "column". Moreover, the numbers of the most significant digits are immediately determined. This is important if you need to quickly estimate the value.
2.3. Multiplication of numbers by the method of "jealousy" or "lattice multiplication"
Let's multiply, for example, numbers 6827 and 345 (Appendix 3):
1. Draw a square grid and write one of the factors above the columns, and the second in height.
2. Multiply the number of each row sequentially by the numbers of each column. Sequentially 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 digits along the diagonals, the number 2355315 is compiled, which is the product of the numbers 6827 and 345, that is, 6827 ∙ 345 \u003d 2355315.
Output. The "lattice multiplication" method is no worse than the generally accepted one. It is even simpler, since numbers are entered into the cells of the table directly from the multiplication table without the 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.
Draw the first number - 12. To do this, from top to bottom, from left to right, draw:
one green stick (1)
and two orange ones (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, with a simple pencil, separate the intersection points and start counting them. We move from right to left (clockwise): 2, 5, 8, 3.
Read the result from left to right - 3852
Output. An interesting way, but drawing 9 lines when multiplying by 9 is somehow long and uninteresting, and then count the intersection points. Without skill, it is difficult to understand the division of a number into digits. In general, you can't do without the 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 construct two single circles on the top line and two binary circles on the bottom line, since the second digit of the first factor is 2.
Since the first digit of the second multiplier 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 were divided is the answer, that is, 12 x 34 \u003d 408.
Output. The method is very similar to the 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 table
It is required to multiply 15647 x 5. Immediately remember the big "button" 5 (it is in the middle) and on it we mentally find the small buttons 1, 5, 6, 4, 7 (they are also located, like on a calculator). They correspond to the numbers 05, 25, 30, 20, 35. The resulting numbers add up: the first digit 0 (remains unchanged), mentally add 5 with 2, we get 7 - this is the second digit of the result, 5 we add with 3, we get the third digit - 8 , 0 + 2 \u003d 2, 0 + 3 \u003d 3 and the last digit of the product remains - 5. The result is 78,235.
Output. 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 of fourth-graders was carried out. 26 people took part (Appendix 8). On the basis of the questionnaire, it was revealed that all the respondents are able to multiply in the traditional way. But most of the 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 lesson “Multiplication with enthusiasm” was held, where the children got acquainted with alternative multiplication algorithms. After that, a survey was conducted in order to identify the methods they liked most. The undisputed leader was the most modern method of Vasily Okoneshnikov. (Appendix 9)
Conclusion
Having learned to count in all the presented ways, I believe that the most convenient multiplication method is the "Little Castle" method - after all, it is so similar to our current one!
Of all the unusual counting methods I found, the Japanese method seemed to be the most interesting. The simplest method seemed to me to be the “doubling and doubling” method used by the Russian peasants. I use it when multiplying numbers that are not too large. It is very convenient to use it when multiplying two-digit numbers.
Thus, I achieved the goal of my research - I studied and learned to apply unconventional methods of multiplying multidigit numbers. My hypothesis was confirmed - I mastered six alternative methods and found out that these are not all possible algorithms.
The unconventional methods of multiplication that I have studied are very interesting and have a right to exist. And in some cases they are even easier to use. I believe 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, maybe in the future we ourselves will be able to discover new ways of multiplication. Also, I do not want to stop there and continue the study of unconventional methods of multiplication.
List of information sources
1. References
1.1. Harutyunyan E., Levitas G. Amusing mathematics. - M .: AST - PRESS, 1999 .-- 368 p.
1.2. Bellustina V. How people gradually came to real arithmetic. - LKI, 2012.-208 p.
1.3. Depman I. Stories about mathematics. - Leningrad .: Education, 1954 .-- 140 p.
1.4. Likum A. Everything about everything. T. 2. - M .: Philological Society "Slovo", 1993. - 512 p.
1.5. Olekhnik S. N., Nesterenko Yu. V., Potapov M. K .. Old entertaining problems. - M .: Science. Main edition of physical and mathematical literature, 1985. - 160 p.
1.6. Perelman Ya.I. Entertaining arithmetic. - M .: Rusanova, 1994 - 205s.
1.7. Perelman Ya.I. Fast counting. Thirty Easy Verbal Counting Techniques. L .: Lenizdat, 1941 - 12 p.
1.8. A.P. Savin Mathematical miniatures. Entertaining math for kids. - M .: Children's literature, 1998 - 175 p.
1.9. Encyclopedia for children. Maths. - M .: Avanta +, 2003 .-- 688 p.
1.10. I know the world: Children's encyclopedia: Mathematics / comp. Savin A.P., Stanzo V.V., Kotova A.Yu. - M .: OOO "AST Publishing House", 2000. - 480 p.
2. Other sources of information
Internet resources:
2.1. A.A. Korneev The phenomenon of Russian multiplication. History. [Electronic resource]

Image copyright Getty Images Image caption I wouldn't get a headache ...

"Mathematics is so difficult ..." You have probably heard this phrase more than once, and perhaps even pronounced it aloud yourself.

For many, mathematical calculations are not easy, but here are three easy ways to help you perform at least one arithmetic operation - multiplication. No calculator.

It is likely that at school you got acquainted with the most traditional method of multiplication: first you learned the multiplication table by memory, and only then you began to multiply each of the digits in a column, which are used to write multi-digit numbers.

If you need to multiply many-digit numbers, then finding the answer will take a large sheet of paper.

But if this long set of lines with numbers going one under another makes your head spin, then there are other, more visual methods that can help you in this matter.

But some artistic skills come in handy here.

Let's draw!

At least three methods of multiplication involve drawing intersecting lines.

1. Mayan way, or Japanese method

There are several versions regarding the origin of this method.

Is it hard to multiply in your mind? Try the Maya and Japanese method

Some say that it was invented by the Mayan Indians who inhabited areas of Central America before the arrival of the conquistadors there in the 16th century. It is also known as the Japanese multiplication method because teachers in Japan use this visual method when teaching multiplication to younger students.

The bottom line is that parallel and perpendicular lines represent the digits of those numbers that need to be multiplied.

Let's multiply 23 by 41.

To do this, we need to draw two parallel lines representing 2, and, backing slightly, three more lines representing 3.

Then, perpendicular to these lines, we will draw four parallel lines representing 4 and, slightly indented, another line for 1.

Well, is it really difficult?

2. Indian way, or Italian multiplication by "lattice" - "gelosia"

The origin of this method of multiplication is also not clear, but it is well known throughout Asia.

"The" Gelosia "algorithm was transmitted from India to China, then to Arabia, and from there to Italy in the XIV-XV centuries, where it was called" Gelosia ", because it looked like Venetian latticed shutters," writes Mario Roberto Canales Villanueva in his book on different ways of multiplying.

Image copyright Getty Images Image caption Indian or Italian multiplication system is similar to Venetian blinds

Let's take the example again, multiplying 23 by 41.

Now we need to draw a table of four cells - one cell per digit. Let us sign the corresponding number at the top of each cell - 2,3,4,1.

Then you need to divide each cell in two diagonally to get triangles.

Now we first multiply the first digits of each number, that is, 2 by 4, and write 0 in the first triangle, and 8 in the second.

Then we multiply 3x4 and write 1 in the first triangle and 2 in the second.

Let's do the same with the other two numbers.

When all the cells of our table are filled, we add the numbers in the sequence shown in the video and write down the resulting result.

Media playback is unsupported on your device

Is it hard to multiply in your mind? Try the Indian method

The first digit will be 0, the second 9, the third 4, the fourth 3. Thus, the result is 943.

Do you think this method is easier or not?

Let's try another multiplication method using a picture.

3. "Array", or table method

As in the previous case, this will require drawing a table.

Let's take the same example: 23 x 41.

Here we need to divide our numbers into tens and ones, so we write 23 as 20 in one column, and 3 in the other.

Vertically, we write 40 at the top, and 1 at the bottom.

Then we will multiply the numbers horizontally and vertically.

Media playback is unsupported on your device

Is it hard to multiply in your mind? Draw a table.

But instead of multiplying 20 by 40, we discard the zeros and simply multiply 2 x 4 to get 8.

Do the same by multiplying 3 by 40. We hold 0 in parentheses and multiply 3 by 4 to get 12.

Let's do the same with the bottom row.

Now let's add zeros: in the upper left cell we got 8, but we dropped two zeros - now we will add them and get 800.

In the upper right cell, when we multiplied 3 by 4 (0), we got 12; now we add zero to get 120.

Let's do the same for all other held zeros.

Finally, we add all four numbers obtained by multiplying in the table.

Result? 943. Well, how did it help?

Diversity is important

Image copyright Getty Images Image caption All methods are good, the main thing is that the answer converges

What can be said for sure is that all these different methods gave us the same result!

We still had to multiply a few things in the process, but each step was easier than when multiplying in the traditional way, and much more intuitive.

So why is it that few places in the world in mainstream schools teach these calculation methods?

One of the reasons may be the emphasis on teaching "mental calculations" - to develop mental abilities.

However, David Weese, a Canadian math teacher who works in public schools in New York, explains it differently.

"I recently read that the reason the traditional multiplication method is used is to save paper and ink. This method was not thought to be the easiest to use, but the most economical in terms of resources, as ink and paper were in short supply." , explains Wiz.

Image copyright Getty Images Image caption For some calculation methods, only the head is not enough, you also need felt-tip pens

Despite this, he believes that alternative methods of multiplication are very useful.

“I don’t think it’s useful to immediately teach schoolchildren to multiply, forcing them to learn the multiplication table, but without explaining to them where it came from. Because if they forget one number, how can they advance in solving the problem? The Mayan method or The Japanese method is necessary because with it you can understand the general structure of multiplication, which is a good start, "says Wiz.

There are a number of other methods of multiplication, for example, Russian or Egyptian, they do not require additional drawing skills.

According to the experts with whom we spoke, all these methods help to better understand the process of multiplication.

"It is clear that everything is for the good. Mathematics in today's world is open both inside and outside the classroom," sums up Andrea Vazquez, a math teacher from Argentina.