8 ways to multiply. Project on the topic: "Unusual ways of multiplication"

problem: understand the types of multiplication

Target: familiarization with various methods of multiplying natural numbers not used in lessons, and their application in calculating numerical expressions.
Tasks:
1. Find and analyze different methods of multiplication.
2. Learn to demonstrate some methods of multiplication.
3. Talk about new ways of multiplication and teach students how to use them.
4. Develop independent work skills: searching for information, selecting and processing the material found.
5. Experiment “which method is faster”
Hypothesis:Do I need to know the multiplication table?
Relevance: Recently, students trust gadgets more than themselves. And this is why they count only on calculators. We wanted to show that there are different ways of multiplication, so that it would be easier for students to count and interesting to learn.
INTRODUCTION
You won't be able to multiply multiple-digit numbers—even double-digit ones—if you don't memorize all the results for single-digit multiplication, that is, what's called the multiplication table.
At different times, different peoples had different ways of multiplying natural numbers.
Why do all peoples now use one method of multiplication “column”?
Why did people abandon old methods of multiplication in favor of modern ones?
Do forgotten methods of multiplication have a right to exist in our time?
To answer these questions I did the following work:
1. Using the Internet, I found information about some methods of multiplication that were used before.;
2. Studied the literature suggested by the teacher;
3. I solved a couple of examples using all the studied methods in order to find out their shortcomings;
4) Identified the most effective ones among them;
5. Conducted an experiment;
6. Drew conclusions.
1. Find and analyze different methods of multiplication.
Multiplication on fingers.

The Old Russian method of multiplying on fingers is one of the most commonly used methods, which was successfully used by Russian merchants for many centuries. They learned to multiply single-digit numbers from 6 to 9 on their fingers. In this case, it was enough to have basic finger counting skills in “units”, “pairs”, “threes”, “fours”, “fives” and “tens”. 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 remaining fingers were bent. Then the number (total) of extended fingers was taken and multiplied by 10, then the numbers were multiplied, showing how many fingers were bent, and the results were added up.

For example, let's multiply 7 by 8. In the example considered, 2 and 3 fingers will be bent. If you add up the number of bent fingers (2+3=5) and multiply the number of not bent ones (2 3=6), you will get the numbers of tens and ones of the desired product 56, respectively. This way you can calculate the product of any single-digit numbers greater than 5.

Methods of multiplying numbers in different countries

Multiply by 9.

Multiplication for the number 9 - 9 1, 9 2 ... 9 10 - is easier to forget from memory and more difficult to recalculate manually using the addition method, however, specifically for the number 9, multiplication is easily reproduced “on the fingers”. Spread your fingers on both hands and turn your hands with your palms facing away from you. Mentally assign numbers from 1 to 10 to your fingers, 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).

Who invented multiplication on fingers

Let's say we want to multiply 9 by 6. We bend the finger with a number equal to the number by which we will multiply nine. In our example, we 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 shows the number of ones. On the left we have 5 fingers not bent, on the right - 4 fingers. Thus, 9·6=54. The figure below shows in detail the entire principle of “calculation”.

Multiplying in an unusual way

Another example: you need to calculate 9·8=?. Along the way, let’s say that the fingers cannot necessarily act as a “calculating machine”. Take, for example, 10 cells in a notebook. Cross out the 8th box. There are 7 cells left on the left, 2 cells on the right. So 9·8=72. Everything is very simple.

7 cells 2 cells.

Indian way of multiplication.

The most valuable contribution to the treasury of mathematical knowledge was made in India. The Hindus proposed the method 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 represents units, tens, hundreds or thousands, depending on where the digit occupies. The occupied space, in the absence of any digits, is determined by the zeros assigned to the numbers.

The Indians were great at counting. They came up with a very simple way to multiply. They performed multiplication starting from the most significant digit, and wrote down incomplete products just above the multiplicand, bit by bit. In this case, the most significant digit of the complete product was immediately visible and, in addition, the omission of any digit was eliminated. The multiplication sign was not yet known, so they left a small distance between the factors. For example, let's multiply them using the method 537 by 6:

(5 ∙ 6 =30) 30

(300 + 3 ∙ 6 = 318) 318

(3180 +7 ∙ 6 = 3222) 3222

6
Multiplication using the “SMALL CASTLE” method.

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

Over the millennia of development of mathematics, many ways of multiplying numbers have been invented. The Italian mathematician Luca Pacioli, in his treatise “Summa of Arithmetic, Ratios and Proportionality” (1494), gives eight different methods of multiplication. The first of them is called “Little Castle”, and the second is no less romanticly called “Jealousy or lattice multiplication”.

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

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

Methods of multiplying numbers in different countries

Multiplying numbers using the “jealousy” method.

“Methods of multiplication The second method has the romantic name 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 of the multiplicand and the multiplier. Then the square cells are divided diagonally, and “... the result is a picture similar to lattice shutters,” writes Pacioli. “Such shutters were hung on the windows of Venetian houses, preventing street passers-by from seeing the ladies and nuns sitting at the windows.”

Let's multiply 347 by 29 in this way. Let's draw a table, write the number 347 above it, and the number 29 on the right.

In each line we will write the product of the numbers above this cell and to the right of it, while we will write the tens digit of the product above the slash, and the units digit below it. Now we add the numbers in each oblique strip, performing this operation, from right to left. If the amount is less than 10, then we write it under the bottom number of the strip. If it turns out to be greater than 10, then we write only the units digit of the sum, and add the tens digit to the next sum. As a result, we obtain the desired product 10063.

Peasant method of multiplication.

The most “native” and easiest way of multiplication, in my opinion, is the method used by Russian peasants. This technique does not require knowledge of the multiplication table beyond the number 2 at all. Its essence is that the multiplication of any two numbers is reduced to a series of successive divisions of one number in half while simultaneously doubling the other number. Dividing in half continues until the quotient reaches 1, while simultaneously doubling the other number. The last doubled number gives the desired result.

If the number is odd, remove one and divide the remainder in half; but to the last number of the right column you will need to add all those numbers of this column that stand opposite the odd numbers of the left column: the sum will be the required product

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

37 ∙ 32 = 1184 ∙ 1 = 1184

In the case when one of the numbers is odd or both numbers are odd, proceed as follows:

384 ∙ 1 = 384

24 ∙ 17 = 24∙(16+1)=24 ∙ 16 + 24 = 384 + 24 = 408
A new way to multiply.

An interesting new method of multiplication has recently been reported. The inventor of the new mental counting system, Candidate of Philosophy Vasily Okoneshnikov, claims that a person is able to remember a huge amount of information, the main thing is how to arrange this information. According to the scientist himself, the most advantageous in this regard is the nine-fold system - all data is simply placed in nine cells, located like buttons on a calculator.

It is very easy to calculate using such a table. For example, let's multiply the number 15647 by 5. In the part of the table corresponding to five, select the numbers corresponding to the digits of the number in order: one, five, six, four and seven. We get: 05 25 30 20 35

We leave the left digit (zero in our example) unchanged, and add the following numbers in pairs: five with a two, five with a three, zero with a two, zero with a three. The last digit is also unchanged.

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

If, when adding two digits, a number greater than nine is obtained, then its first digit is added to the previous digit of the result, and the second is written in its “own” place.

Conclusion.

While working on this topic, I learned that there are about 30 different, fun and interesting ways to multiply. Some are still used in various countries. I have chosen some interesting ways for myself. But not all methods are convenient to use, especially when multiplying multi-digit numbers.

Multiplication methods

Research work on mathematics in primary school

Brief summary of the research work
Every schoolchild knows how to multiply multi-digit numbers in a column. In this work, the author draws attention to the existence of alternative methods of multiplication available to primary schoolchildren, which can turn “tedious” calculations into a fun game.
The work examines six unconventional methods of multiplying multi-digit numbers, used in various historical eras: Russian peasant, lattice, small castle, Chinese, Japanese, according to the table of V. Okoneshnikov.
The project is intended to develop cognitive interest in the subject being studied and to deepen knowledge in the field of mathematics.
Table of contents
Introduction 3
Chapter 1. Alternative methods of multiplication 4
1.1. A little history 4
1.2. Russian peasant method of multiplication 4
1.3. Multiplication using the “Small Castle” method 5
1.4. Multiplying numbers using the “jealousy” or “lattice multiplication” method 5
1.5. Chinese way of multiplying 5
1.6. Japanese way of multiplying 6
1.7. Okoneshnikov table 6
1.8.Multiplication by column. 7
Chapter 2. Practical part 7
2.1. Peasant way 7
2.2. Little castle 7
2.3. Multiplying numbers using the “jealousy” or “lattice multiplication” method 7
2.4. Chinese way 8
2.5. Japanese method 8
2.6. Okoneshnikov table 8
2.7. Questioning 8
Conclusion 9
Appendix 10

“The subject of mathematics is such a serious subject that it is good to take every opportunity 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 with numbers, that is, to count. We multiply, divide, add and subtract in the usual ways that are studied at school. The question arose: are there any other alternative methods of calculation? I wanted to study them in more detail. In search of an answer to these questions, this study was conducted.
Purpose of the study: to identify unconventional methods of multiplication 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 use 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 algorithms for multiplying multi-digit numbers
Subject of study: 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. During the work, many examples were solved and an album was created, which included examples with various algorithms for multiplying multi-digit numbers in several alternative ways. This may interest classmates to expand their mathematical horizons and serve as the beginning of new experiments.

Chapter 1. Alternative Methods of Multiplication

1.1. A little history
The methods of calculation that we use now were not always so simple and convenient. In the old days, more cumbersome and slower techniques 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. Rumors about him would have spread throughout the surrounding schools and monasteries, eclipsing the glory of the most skilled calculators 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 V. Bellustin’s book “How people gradually reached real arithmetic,” 27 methods of multiplication are outlined, and the author notes: “it is very possible that there are other methods hidden in the recesses of book depositories, scattered in numerous, mainly handwritten collections.” And all these multiplication techniques competed with each other and were learned with great difficulty.
Let's look at the most interesting and simple ways of multiplication.
1.2. Russian peasant method of multiplication
In Russia, 2-3 centuries ago, a method was common among peasants in some provinces that did not require knowledge of the entire multiplication table. You just had 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 number was multiplied by 2. The results were written in a column until 1 remained on the left. The remainder was discarded. Cross out those lines that have even numbers on the left. We add up the remaining numbers in the right column.
1.3. Multiplication using the “Small Castle” method
The Italian mathematician Luca Pacioli, in his treatise “Summa of 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 leading digits are determined from the very beginning, and this can be important if you need to quickly estimate a value.
The digits of the upper number, starting from the most significant digit, are multiplied in turn by the lower number and written in a column with the required number of zeros added. The results are then 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 “... the result is a picture similar to lattice shutters,” writes Pacioli. “Such shutters were hung on the windows of Venetian houses, preventing street passers-by from seeing the ladies and nuns sitting at the windows.”
By 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 ones below it. The digits of the product are obtained by adding the digits in oblique stripes. The results of additions are written below the table, as well as to the right of it.
1.5. Chinese way of multiplication
Now let's introduce the multiplication method, which is vigorously discussed on the Internet, which is called Chinese. When multiplying numbers, the intersection points of the lines are calculated, which correspond to the number of digits of each digit of both factors.
1.6. Japanese way of multiplication
The Japanese method of multiplication is a graphical method using circles and lines. No less funny and interesting than Chinese. Even somewhat similar to him.
1.7. Okoneshnikov table
Candidate of Philosophy Vasily Okoneshnikov, part-time inventor of a new mental counting system, believes that schoolchildren will be able to learn to verbally add and multiply millions, billions and even sextillions and quadrillions. According to the scientist himself, the most advantageous in this regard is the nine-fold system - all data is 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: “1” in the lower left corner, “9” in the upper right corner. Each part is a multiplication table for numbers from 1 to 9 (using 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-digit multiplier. 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, when adding two digits, a number greater than nine is obtained, then its first digit is added to the previous digit of the result, and the second is written in its “own” place.
The new technique was tested in several Russian schools and universities. The Ministry of Education of the Russian Federation has allowed the publication of a new multiplication table in checkered notebooks along with the usual Pythagorean table - for now, 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 Riese (Appendix 7). This algorithm is considered the most convenient.
Chapter 2. Practical part
Mastering the listed methods of multiplication, many examples were solved, and an album was prepared with samples of various calculation algorithms. (Application). Let's look at the calculation algorithm using examples.
2.1. Peasant way
Multiply 47 by 35 (Appendix 1),
-write down 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 arises during division, then the remainder will be discarded);
- division ends when a unit appears on the left;
-cross out those lines in which there are even numbers on the left;
-we add up the remaining numbers on the right - this is the result.
35 + 70 + 140 + 280 + 1120 = 1645.
Conclusion. The method is convenient in that it is enough to know the table only for 2. However, when working with large numbers it is very cumbersome. Convenient for working with two-digit numbers.
2.2. Little castle
(Appendix 2). Conclusion. The method is very similar to our modern “column”. Moreover, the numbers of the highest digits are immediately determined. This can be important if you need to quickly estimate a 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. Draw a square grid and write one of the factors above the columns, and the second - along the 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 the numbers following the diagonal stripes. If the sum of one diagonal contains tens, then add them to the next diagonal.
From the results of adding the numbers along the diagonals, the number 2355315 is formed, 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 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 present in the standard method.
2.4. Chinese way
Suppose you need to multiply 12 by 321 (Appendix 4). On a sheet of paper we draw lines one by one, 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).
Draw the second number – 321, from bottom to top, from left to right:
three blue sticks (3);
two red (2);
one lilac (1).
Now, using a simple pencil, we separate the intersection points and begin to count them. We move from right to left (clockwise): 2, 5, 8, 3.
Let's read the result from left to right - 3852
Conclusion. An interesting way, but drawing 9 straight lines when multiplying by 9 is somehow long and uninteresting, and then counting the intersection points. Without skill, it is difficult to understand the division of numbers into digits. In general, you can’t do without a multiplication table!
2.5. Japanese way
Let's 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 in the top line and two binary circles in the bottom line, 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, and the circles of the second column into four parts.
The number of parts into which the circles were 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 table
You need to multiply 15647 x 5. We immediately remember the big “button” 5 (it’s in the middle) and mentally find the small buttons 1, 5, 6, 4, 7 on it (they are also located like 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 it by heart or always have a table at hand.
2.7. Student survey
A survey of fourth-graders was conducted. 26 people took part (Appendix 8). Based on the survey, it was revealed that all respondents knew how to multiply in the traditional way. But most guys don’t know about non-traditional methods of multiplication. And there are people who want to get to know them.
After the initial survey, an extracurricular lesson “Multiplication with Passion” was held, where the children became acquainted with alternative multiplication algorithms. After that, a survey was conducted to identify the methods that we liked most. The undisputed leader was the most modern method of Vasily Okoneshnikov. (Appendix 9)
Conclusion
Having learned to count using all the methods presented, I believe that the most convenient method of multiplication 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 more interesting. The simplest method seemed to me to be “doubling and splitting”, which was used by Russian peasants. I use it when multiplying not too large numbers. It is very convenient to use when multiplying two-digit numbers.
Thus, I achieved the goal of my research - I studied and learned to use unconventional methods 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 non-traditional multiplication methods 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 already known methods of multiplication. But who knows, perhaps in the future we ourselves will be able to discover new ways of multiplication. Also, I don’t want to stop there and continue to study unconventional methods of multiplication.
List of information sources
1. References
1.1. Harutyunyan E., Levitas G. Entertaining mathematics. - M.: AST - PRESS, 1999. - 368 p.
1.2. Bellustina V. How people gradually reached 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. Olehnik S.N., Nesterenko Yu.V., Potapov M.K.. Old entertaining problems. – M.: Science. Main editorial office of physical and mathematical literature, 1985. – 160 p.
1.6. Perelman Ya.I. Interesting arithmetic. - M.: Rusanova, 1994 – 205 p.
1.7. Perelman Ya.I. Quick count. Thirty simple mental counting techniques. L.: Lenizdat, 1941 - 12 p.
1.8. Savin A.P. Mathematical miniatures. Entertaining mathematics for children. - M.: Children's literature, 1998 - 175 p.
1.9. Encyclopedia for children. Mathematics. – M.: Avanta +, 2003. – 688 p.
1.10. I explore the world: Children's encyclopedia: Mathematics / comp. Savin A.P., Stanzo V.V., Kotova A.Yu. - M.: AST Publishing House LLC, 2000. - 480 p.
2. Other sources of information
Internet resources:
2.1. Korneev A.A. The phenomenon of Russian multiplication. Story. [Electronic resource]

published 20.04.2012
Dedicated to Elena Petrovna Karinskaya ,
to my school math teacher and class teacher
Almaty, ROFMSH, 1984–1987
“Science only reaches perfection when it manages to use mathematics”. Karl Heinrich Marx
these words were inscribed above the blackboard in our math classroom ;-)
Computer science lessons(lecture materials and workshops)

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

Let's try to do this action
enjoyable and exciting ;-)

METHODS OF MULTIPLICATION WITHOUT MULTIPLICATION TABLES (gymnastics for the mind)

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

This method was common among Russian peasants and was inherited by them 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 simultaneously doubling the other number, There is no need for a multiplication table in this case :-)

Dividing in half continues until the quotient turns out to be 1, while at the same time doubling the other number. The last doubled number gives the desired result(picture 1). 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 should you do if you have to halve an odd number? In this case, we remove one from the odd number and divide the remainder in half, while to the last number of the right column we will need to add all those numbers in this column that stand opposite the odd numbers in the left column - the sum will be the required product (Figures: 2, 3).
In other words, we cross out all lines with even left numbers; leave and then add up numbers not crossed out right column.

For Figure 2: 192 + 48 + 12 = 252
The correctness of the reception will become clear if we take into account that:
5× 48 = (4 + 1) × 48 = 4 × 48 + 48
21× 12 = (20 + 1) × 12 = 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 obtain the product.
The Russian method of multiplication is both elegant and extravagant at the same time ;-)

§ Logical problem about Zmeya Gorynych and famous Russian heroes on green page “Which of the heroes defeated the Serpent Gorynych?”
solving logical problems using logical algebra
For those who love to learn! For those who are happy gymnastics for the mind ;-)
§ Solving logical problems using a tabular method

Let's continue the conversation :-)

Chinese??? Drawing method of multiplication

My son introduced me to this method of multiplication, putting at my disposal several pieces of paper from a notebook with ready-made solutions in the form of intricate drawings. The process of deciphering the algorithm began to boil a drawing way of multiplication :-) For clarity, I decided to resort to the help of colored pencils, and... the ice was broken gentlemen of the jury :-)
I bring to your attention three examples in color pictures (in the upper right corner check post).

Example #1: 12 × 321 = 3852
Let's draw first number from top to bottom, from left to right: one green stick ( 1 ); two orange sticks ( 2 ). 12 drew :-)
Let's draw second number from bottom to top, from left to right: three little blue sticks ( 3 ); two red ones ( 2 ); one lilac one ( 1 ). 321 drew :-)

Now, using a simple pencil, we will walk through the drawing, divide the intersection points of the stick numbers into parts and begin counting the dots. 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 nuances in this example;-) When counting the points in the first part, it turned out 16 . We send one and add it to the dots of the second part ( 20 + 1 )…

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

At first, it seemed to me somewhat pretentious, but at the same time intriguing and surprisingly harmonious. In the fifth example, I caught myself thinking that multiplication is taking off :-) and it works in autopilot mode: draw, count dots, We don’t remember the multiplication table, it’s like we don’t know it at all :-)))

To be honest, when checking drawing method of multiplication and turning to column multiplication, and more than once or twice, to my shame, I noted some slowdowns, indicating that my multiplication table was rusty in some places: - (and you shouldn’t forget it. When working with more “serious” numbers drawing method of multiplication became too bulky, and multiplication by column it was a 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, Trains your memory - prevents you from forgetting the multiplication table :-) And therefore, I strongly recommend that you and yourself, if possible, forget about calculators on phones and computers ;-) and periodically indulge yourself in multiplication. Otherwise the plot from the film “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..., knock on wood... :-))) ...and most importantly Don't forget about mental gymnastics!

For the curious: Multiplication indicated by [×] or [·]
The [×] sign was introduced by an English mathematician William Oughtred 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 a × b or a · b write ab.

To the webmaster's piggy bank: Some mathematical symbols in HTML

°	° or °	degree
±	± or ±	plus or minus
¼	¼ or ¼	fraction - one quarter
½	½ or ½	fraction - one half
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 ≈	approximate equal sign
≠	≠ or ≠	not equal sign
≡	≡ or ≡	identically
>	> or >	more
<	< или	less
≥	≥ or ≥	more or equal
≤	≤ or ≤	less or equal
∑	∑ or ∑	summation sign
√	√ or √	square root (radical)
∞	∞ or ∞	infinity
Ø	Ø or Ø	diameter
∠	∠ or ∠	corner
⊥	⊥ or ⊥	perpendicular

Municipal educational institution "Kurovskaya secondary school No. 6"

ABSTRACT ON MATHEMATICS ON THE TOPIC:

« UNUSUAL WAYS OF MULTIPLICATION».

Completed by a student of grade 6 “b”

Krestnikov Vasily.

Supervisor:

Smirnova Tatyana Vladimirovna.

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

Main part. Unusual ways of multiplication…………………………3

2.1. A little history……………………………………………………………..3

2.2. Multiplication on fingers………………………………………………………4

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

2.4. Indian way of multiplication…………………………………………….6

2.5. Multiplication using the “Small Castle” method…………………………………7

2.6. Multiplication using the “Jealousy” method………………………………………………………8

2.7. Peasant method of multiplication……………………………………………..9

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

Conclusion…………………………………………………………………………………11

References…………………………………………………………….1 2

I. 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 that are studied at school.

One day I accidentally came across a book by S. N. Olekhnik, Yu. V. Nesterenko and M. K. Potapov, “Old Entertaining Problems.” Leafing through this book, my attention was drawn to a page called “Multiplication on the fingers.” It turned out that you can multiply not only as suggested to us in mathematics textbooks. I was wondering if there were any other methods of calculation. After all, the ability to quickly perform calculations is frankly surprising.

The constant use of modern computer 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 calculations. In addition, mastering computational skills develops memory, increases the level of mathematical culture of thinking, and helps to fully master the subjects of the physical and mathematical cycle.

Goal of the work:

Show unusualmethods of multiplication.

Tasks:

Find as many as possibleunusual methods of calculations.

Learn to use them.

Choose for yourself the most interesting or easier ones than those thatare offeredat school, and use them when counting.

II. Main part. Unusual ways of multiplication.

2.1. A little history.

The methods of calculation that we use now were not always so simple and convenient. In the old days, more cumbersome and slower techniques were used. And if a schoolchild 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 throughout the surrounding schools and monasteries, eclipsing the glory of the most skilled calculators 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. Then there was no one method developed by practice for each action. On the contrary, there were almost a dozen different methods of multiplication and division in use at the same time - techniques one more complicated than the other, which a person of average ability was not able to remember. Each teacher of counting stuck to his favorite technique, each “master of division” (there were such specialists) praised his own way of performing this action.

In V. Bellustin’s book “How people gradually reached real arithmetic,” 27 methods of multiplication are outlined, and the author notes: “it is very possible that there are other methods hidden in the recesses of book depositories, scattered in numerous, mainly handwritten collections.”

And all these methods of multiplication - “chess or organ”, “folding”, “cross”, “lattice”, “back to front”, “diamond” and others competed with each other and were learned with great difficulty.

Let's look at the most interesting and simple ways of multiplication.

2.2. Multiplication on fingers.

2.3. Multiply by 9.

Multiplication for the number 9– 9·1, 9·2 ... 9·10 – is easier to forget from memory and more difficult to recalculate manually using the addition method, however, specifically for the number 9, multiplication is easily reproduced “on the fingers.” Spread your fingers on both hands and turn your hands with your palms facing away from you. Mentally assign numbers from 1 to 10 to your fingers, 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).

Let's say we want to multiply 9 by 6. We bend the finger with a number equal to the number by which we will multiply nine. In our example, we 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 shows the number of ones. On the left we have 5 fingers not bent, on the right we have 4 fingers. Thus, 9·6=54. The figure below shows in detail the entire principle of “calculation”.

7 cells 2 cells.

2.4. Indian way of multiplication.

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

(5 ∙ 6 =30) 30

(300 + 3 ∙ 6 = 318) 318

(3180 +7 ∙ 6 = 3222) 3222

2.5 . Multiplication way"LITTLE CASTLE".

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

2.6. Multiplying numbersusing the "jealousy" method.

The second method has the romantic name “jealousy”, or “lattice multiplication”.

Let's multiply 347 by 29 in this way. Let's draw a table, write the number 347 above it, and the number 29 on the right.

2.7. TOpeasant method of multiplication.

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

37 ∙ 32 = 1184 ∙ 1 = 1184

In the case when one of the numbers is odd or both numbers are odd, proceed as follows:

384 ∙ 1 = 384

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

2.8 . A new way to multiply.

Interesting a new method of multiplication that has recently been reported. The inventor of the new mental counting system, Candidate of Philosophy Vasily Okoneshnikov, claims that a person is able to remember a huge amount of information, the main thing is how to arrange this information. According to the scientist himself, the most advantageous in this regard is the nine-fold system - all data is simply placed in nine cells, located like buttons on a calculator.

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

If, when adding two digits, a number greater than nine is obtained, then its first digit is added to the previous digit of the result, and the second is written in its “own” place.

III. Conclusion.

Of all the unusual counting methods I found, the “lattice multiplication or jealousy” method seemed more interesting. I showed it to my classmates and they really liked it too.

The simplest method seemed to me to be “doubling and splitting”, which was 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).

I was interested in the new method of multiplication, because it allows me to “toss around” huge numbers in my mind.

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

Literature.

Depman I. “Stories about mathematics.” – Leningrad: Education, 1954. – 140 p.

Korneev A.A. The phenomenon of Russian multiplication. Story. http://numbernautics.ru/

Olehnik S. N., Nesterenko Yu. V., Potapov M. K. “Old entertaining problems.” – M.: Science. Main editorial office of physical and mathematical literature, 1985. – 160 p.

Perelman Ya.I. Quick count. Thirty simple mental counting techniques. L., 1941 - 12 p.

Perelman Ya.I. Entertaining arithmetic. M. Rusanova, 1994–205 p.

Encyclopedia “I explore the world. Mathematics". – M.: Astrel Ermak, 2004.

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