Learning different ways to multiply. Ways to quickly multiply numbers verbally The Italian way of multiplication

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

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

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

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

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

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:

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 “The 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.

Municipal budgetary educational institution

Secondary school with. Shlanly

Municipal district Aurgazinsky district of the Republic of Belarus

Research work

"UNUSUAL WAYS OF MULTIPLICATION"

Vasiliev Nikolay

Supervisor -

2013-2014 academic year G.

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

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

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

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

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

4) Pythagorean table ……………………………………………………

5) Okoneshnikov table…………………………………………….

6) Peasant method of multiplication……………………….………....

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

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

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

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

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

4. List of references……………………………………………………….

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 page on the Internet with an unusual method of multiplication that children in China use (as it is written there). I read, studied and liked this method. 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 unusual ways of multiplication.

Tasks:

Ø Find as many unusual calculation methods as possible.

Ø Learn to use them.

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

I was wondering if modern schoolchildren, my classmates and others, know other ways to perform arithmetic operations other than multiplication by a column and division by a “corner” and would like to learn new ways? I conducted an oral survey. 20 students in grades 5-7 were surveyed. This survey showed that modern schoolchildren do not know other ways to perform actions, since they rarely turn to material outside the school curriculum.

Survey results:

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

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

https://pandia.ru/text/80/266/images/image004_2.png" align="left" width="264 height=176" height="176">

3) would you like to know?

Unusual ways of multiplication.

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 intricate than the other, which a person of average abilities 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.

Multiply by 9

Multiplication for the number 9- 9·1, 9·2 ... 9·10 - is easier to erase 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).

calculations."

counting machine" the fingers may not necessarily protrude. Take, for example, 10 cells in a notebook. Cross out the 8th cell. 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.

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.

Pythagorean table

Let us recall the main rule of ancient Egyptian mathematics, which states 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 doubling of numbers and figures, but obtained by the modern method of folding “in a column”, known even in the elementary grades of school.

Okoneshnikov table

Students will be able to learn to verbally add and multiply millions, billions, and even sextillions and quadrillions. And Candidate of Philosophical Sciences Vasily Okoneshnikov, who is also the inventor of a new mental counting system, will help them with this. The scientist claims that a person is capable of remembering 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.

According to the scientist, before becoming a computing “computer”, it is necessary to memorize the table he created. The numbers in it are distributed in nine cells in an uneasy manner. According to Okoneshnikov, the human eye and his memory are so cleverly designed that information arranged 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: “1” in the lower left corner, “9” in the upper right corner. Each part is a table for multiplying numbers from 1 to 9 (again in the lower left corner by 1, next to the right by 2, etc., using the same “push-button” system). How to use them?
For example, you need to multiply 9 on 842 . We immediately remember the big “button” 9 (it’s at the top right and on it we mentally find the 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 is 7 ( remains unchanged), 2 is mentally added to 3, we get 5 - this is the second digit of the result, 6 is added to 1, we get the third digit -7, and the last digit of the desired number remains - 8. The result is 7578.
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.

Using Okoneshnikov’s matrix table, according to the author himself, you can study foreign languages and even the periodic table. 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.

Example : 15647 x 5

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

Multiplication using the “SMALL CASTLE” method

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.

Multiplying numbers using the “jealousy” method.

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

Grid 1

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

Grid1

From the results of adding the numbers along the diagonals (they are highlighted in yellow), a number is formed 2355315 , which is the product of numbers 6827 and 345, that is 6827 x 345 = 2355315.

Chinese way of multiplication

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

https://pandia.ru/text/80/266/images/image024_0.png" width="92" height="46"> Example : let's multiply 21 on 13 . The first factor contains 2 tens and 1 unit, which means we build 2 parallel lines and 1 straight line at a distance.

The lines intersect at points, the number of which is the answer, that is 21 x 13 = 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 method of multiplication is a graphical method using circles and lines. No less funny and interesting than Chinese. Even somewhat similar to him.

Example: let's multiply 12 on 34. Since the second factor is a two-digit number, and the first digit of the first factor 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 equal to 2 .

12 x 34

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

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 think that our method of multiplying by column 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. Story. http://numbernautics. ru/

3. “Ancient entertaining problems.” – M.: Science. Main editorial office of physical and mathematical literature, 1985. – 160 p.

4. Perelman account. Thirty simple mental counting techniques. L., 1941 - 12 p.

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

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

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

Back forward

Attention! Slide previews are for informational purposes only and may not represent all the features of the presentation. If you are interested in this work, please download the full version.

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

Target:

Learn ancient multiplication techniques.
Expand your knowledge of various multiplication techniques.
Learn to perform operations with natural numbers using ancient methods of multiplication.

The old way of multiplying by 9 on your fingers
Multiplication by Ferrol method.
Japanese way of multiplication.
Italian way of multiplication (“Grid”)
Russian method of multiplication.
Indian way of multiplication.

Progress of the lesson

The relevance of using fast counting techniques.

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

Ancient ways of multiplying numbers.

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

It's simple. To multiply any number from 1 to 9 by 9, look at your hands. Fold the finger that corresponds to the number being multiplied (for example, 9 x 3 - fold the third finger), count the fingers before the folded finger (in the case of 9 x 3, this is 2), then count after the folded finger (in our case, 7). The answer is 27.

2. Multiplication by the Ferrol method.

To multiply the units of the product of remultiplication, the units of the factors are multiplied; to obtain tens, the tens of one are multiplied by the units of the other and vice versa and the results are added; to obtain hundreds, the tens are multiplied. Using the Ferrol method, it is easy to multiply two-digit numbers from 10 to 20 verbally.

For example: 12x14=168

a) 2x4=8, write 8

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

c) 1x1=1, write 1.

3. Japanese way of multiplication

This technique is reminiscent of multiplication by a column, but it takes quite a long time.

Using the technique. Let's say we need to multiply 13 by 24. Let's draw the following figure:

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

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

(intersections in the figure are indicated by dots)

Number of crossings:

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

1) Intersections in the upper left edge (2) – the first number of the answer

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

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

It turns out: 2; 10; 12.

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

4. Italian way of multiplication (“Grid”)

In Italy, as well as in many Eastern countries, this method has gained great popularity.

Using the technique:

For example, let's multiply 6827 by 345.

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

2. Multiply the number of each row sequentially by the numbers of each column.

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

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

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

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

Answer: 2355315.

5. Russian method of multiplication.

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

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

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

However, what should you do if you have to divide an odd number in half? The folk method easily overcomes this difficulty. It is necessary, says the rule, in the case of an odd number, discard one and divide the remainder in half; but then 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 desired product. In practice, this is done in such a way that all lines with even left numbers are crossed out; Only those that contain an odd number to the left remain. Here's an example (asterisks indicate that this line should be crossed out):

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

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

17 + 34 + 272 = 323.

Answer: 323.

6. Indian way of multiplication.

This method of multiplication was used in Ancient India.

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

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

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

Let's repeat the whole process with other digits of the multiplier, following the same rules (C).

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

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

Conclusion

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

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

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

“Whoever studies mathematics from childhood develops attention, trains the brain, his will, and cultivates perseverance and perseverance in achieving goals.”(A. Markushevich)

Literature.

Encyclopedia for children. "T.23". Universal Encyclopedic Dictionary \ ed. board: M. Aksenova, E. Zhuravleva, D. Lyury and others - M.: World of Encyclopedias Avanta +, Astrel, 2008. - 688 p.
Ozhegov S.I. Dictionary of the Russian language: approx. 57,000 words / Ed. member - corr. ANSIR N.YU. Shvedova. – 20th ed. – M.: Education, 2000. – 1012 p.
I want to know everything! Large illustrated encyclopedia of intelligence / Transl. from English A. Zykova, K. Malkova, O. Ozerova. – M.: Publishing house ECMO, 2006. – 440 p.
Sheinina O.S., Solovyova G.M. Mathematics. School club classes 5-6 grades / O.S. Sheinina, G.M. Solovyova - M.: Publishing house NTsENAS, 2007. - 208 p.
Kordemsky B. A., Akhadov A. A. The amazing world of numbers: A book of students, - M. Education, 1986.
Minskikh E. M. “From game to knowledge”, M., “Enlightenment” 1982.
Svechnikov A. A. Numbers, figures, problems M., Education, 1977.
http://matsievsky. newmail. ru/sys-schi/file15.htm
http://sch69.narod. ru/mod/1/6506/hystory. html

Municipal educational institution

Staromaximkinskaya basic secondary school

Regional scientific and practical conference on mathematics

"Step into Science"

Research work

"Non-standard counting algorithms or quick counting without a calculator"

Supervisor: ,

mathematic teacher

With. Art. Maksimkino, 2010

Introduction…………………………………………………………………………………..…………….3

Chapter 1. Account history

1.2. Miracle counters………………………………………………………………………………...9

Chapter 2. Ancient methods of multiplication

2.1. Russian peasant method of multiplication…..…………….……………….……..The “lattice” method……………….…….. …………………………… …….………..13

2.3. Indian way of multiplication……………………………………………………..15

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

2.5. Multiplication on fingers……………………………………………………………..17

Chapter 3. Mental arithmetic - mental gymnastics

3.1. Multiplication and division by 4………………..……………………….………………….19

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

3.3. Multiplying by 25………………………………………………………………………………19

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

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

3.6. Multiplication by 11……………………………………………………………..…………….….20

3.7. Multiplying a three-digit number by 101…………………………………………21

3.7. Squaring a number ending in 5………………………21

3.8. Squaring a number close to 50……………….………………………22

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

Conclusion……………………………………………………………………………………….…24

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

Introduction

Is it possible to imagine a world without numbers? Without numbers you can’t make a purchase, you can’t find out the time, you can’t dial a phone number. And what about spaceships, lasers and all other technical achievements?! They would simply be impossible if it were not for the science of numbers.

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

Now, at the stage of rapid development of computer science and computer technology, modern schoolchildren do not want to bother themselves with mental arithmetic. Therefore we considered It is important to show not only that the process of performing an action itself can be interesting, but also that, having thoroughly mastered the techniques of quick counting, one can compete with a computer.

Object research are counting algorithms.

Subject research is the process of calculation.

Target: study non-standard calculation methods and experimentally identify the reason for the refusal to use these methods when teaching mathematics to modern schoolchildren.

Tasks:

Reveal the history of the origin of the account and the phenomenon of “Miracle counters”;

Describe ancient methods of multiplication and experimentally identify difficulties in their use;

Consider some oral multiplication techniques and use specific examples to show the advantages of their use.

Hypothesis: In the old days they said: “Multiplication is my torment.” This means that multiplication used to be complicated and difficult. Is our modern way of multiplying simple?

While working on the report I used the following methods :

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

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

Ø analysis data obtained during the study.

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

Behind the simple act of multiplication lies the secrets of the history of mathematics. Accidentally hearing the words “multiplication by lattice”, “chess method” intrigued me. I wanted to know these and other methods of multiplication and compare them with our multiplication action today.

In order to find out whether modern schoolchildren know other ways of performing arithmetic operations, in addition to multiplication by a column and division by a corner, and would like to learn new ways, an oral survey was conducted. 20 students in grades 5-7 were surveyed. This survey showed that modern schoolchildren do not know other ways to perform actions, since they rarely turn to material outside the school curriculum.

Survey results:

(The diagrams show the percentage of students’ affirmative answers).

1) Do modern people need to be able to perform arithmetic operations with natural numbers?

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

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

3) would you like to know?

Chapter 1. Account history

1.1. How did the numbers come about?

People learned to count objects back in the ancient Stone Age - Paleolithic, tens of thousands of years ago. How did this happen? At first, people only compared different quantities of identical objects by eye. They could determine which of two heaps had more fruit, which herd had more deer, etc. If one tribe exchanged caught fish for stone knives made by people of another tribe, there was no need to count how many fish and how many knives they brought. It was enough to place a knife next to each fish for the exchange between the tribes to take place.

To successfully engage in agriculture, arithmetic knowledge was needed. Without counting days, it was difficult to determine when to sow fields, when to start watering, when to expect offspring from animals. It was necessary to know how many sheep were in the herd, how many bags of grain were put in the barns.
And more than eight thousand years ago, ancient shepherds began to make mugs out of clay - one for each sheep. To find out if at least one sheep had gone missing during the day, the shepherd put aside a mug each time another animal entered the pen. And only after making sure that as many sheep had returned as there were circles, he calmly went to bed. But in his herd there were not only sheep - he grazed cows, goats, and donkeys. Therefore, I had to make other figures from clay. And farmers, using clay figurines, kept records of the harvest, noting how many bags of grain were placed in the barn, how many jugs of oil were squeezed from olives, how many pieces of linen were woven. If the sheep gave birth, the shepherd added new ones to the circles, and if some of the sheep were used for meat, several circles had to be removed. So, not yet knowing how to count, the ancient people practiced arithmetic.

Then numerals appeared in the human language, and people were able to name the number of objects, animals, days. Usually there were few such numerals. For example, the Murray River people of Australia had two prime numbers: enea (1) and petchewal (2). They expressed other numbers with compound numerals: 3 = “petcheval-enea”, 4 “petcheval-petcheval”, etc. Another Australian tribe, the Kamiloroi, had simple numerals mal (1), Bulan (2), Guliba (3). And here other numbers were obtained by adding less: 4 = “bulan - bulan”, 5 = “bulan - guliba”, 6 = “guliba - guliba”, etc.

For many peoples, the name of the number depended on the items being counted. If the inhabitants of the Fiji Islands counted boats, then the number 10 was called “bolo”; if they counted coconuts, the number 10 was called "karo". The Nivkhs living on Sakhalin and the banks of the Amur did exactly the same. Even in the last century, they called the same number with different words if they counted people, fish, boats, nets, stars, sticks.

We still use various indefinite numbers with the meaning “many”: “crowd”, “herd”, “flock”, “heap”, “bunch” and others.

With the development of production and trade exchange, people began to better understand what three boats and three axes, ten arrows and ten nuts have in common. Tribes often traded "item for item"; for example, they exchanged 5 edible roots for 5 fish. It became clear that 5 is the same for both roots and fish; This means that you can call it in one word.

Other peoples used similar methods of counting. This is how numberings based on counting in fives, tens, and twenties arose.

So far we have talked about mental counting. How were the numbers written down? At first, even before the advent of writing, they used notches on sticks, notches on bones, and knots on ropes. The wolf bone found in Dolní Vestonice (Czechoslovakia) had 55 notches made more than 25,000 years ago.

When writing appeared, numbers appeared to record numbers. At first, numbers resembled notches on sticks: in Egypt and Babylon, in Etruria and Phenice, in India and China, small numbers were written with sticks or lines. For example, the number 5 was written with five sticks. The Aztec and Mayan Indians used dots instead of sticks. Then special signs appeared for some numbers, such as 5 and 10.

At that time, almost all numberings were not positional, but similar to Roman numbering. Only one Babylonian sexagesimal numbering was positional. But for a long time there was no zero in it, as well as a comma separating the whole part from the fractional part. Therefore, the same number could mean 1, 60, or 3600. The meaning of the number had to be guessed according to the meaning of the problem.

Several centuries before the new era, a new way of writing numbers was invented, in which the letters of the ordinary alphabet served as numbers. The first 9 letters denoted the numbers tens 10, 20,..., 90, and another 9 letters denoted hundreds. This alphabetical numbering was used until the 17th century. To distinguish “real” letters from numbers, a dash was placed above the letters-numbers (in Rus' this dash was called a “titlo”).

In all these numberings it was very difficult to perform arithmetic operations. Therefore, the invention in the 6th century. By Indians, decimal positional numbering is rightfully considered one of the greatest achievements of mankind. Indian numbering and Indian numerals became known in Europe from the Arabs, and are usually called Arabic.

When writing fractions for a long time, the whole part was written in the new, decimal numbering, and the fractional part in sexagesimal. But at the beginning of the 15th century. Samarkand mathematician and astronomer al-Kashi began to use decimal fractions in calculations.

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

1.2 "Miracle - counters"

He understands everything at a glance and immediately formulates a conclusion to which an ordinary person, perhaps, will come through long and painful thought. He devolves books at an incredible speed, and in first place on his short list of bestsellers is a textbook on entertaining mathematics. At the moment of solving the most difficult and unusual problems, the fire of inspiration burns in his eyes. Requests to go to the store or wash the dishes go unheeded or are met with great dissatisfaction. The best reward is a trip to the lecture hall, and the most valuable gift is a book. He is as practical as possible and in his actions is mainly subject to reason and logic. He treats people around him coldly and would prefer a chess game with a computer to roller skating. As a child, he is precociously aware of his own shortcomings and is distinguished by increased emotional stability and adaptability to external circumstances.

This portrait is not based on a CIA analyst.
This is what, according to psychologists, a human calculator looks like, an individual with unique mathematical abilities that allow him to make the most complex calculations in his head in the blink of an eye.

Beyond the threshold of consciousness is a miracle - accountants, capable of performing unimaginably complex arithmetic operations without a calculator, have unique memory characteristics that distinguish them from other people. As a rule, in addition to huge lines of formulas and calculations, these people (scientists call them mnemonics - from the Greek word mnemonika, meaning “the art of memorization”) keep in their heads lists of addresses not only of friends, but also of casual acquaintances, as well as numerous organizations where they I had to be there once.

In the laboratory of the Research Institute of Psychotechnologies, where they decided to study the phenomenon, they conducted such an experiment. They invited a unique person - an employee of the Central State Archive of St. Petersburg. He was offered various words and numbers to remember. He had to repeat them. In just a couple of minutes he could fix up to seventy elements in his memory. Dozens of words and numbers were literally “downloaded” into Alexander’s memory. When the number of elements exceeded two hundred, we decided to test its capabilities. To the surprise of the experiment participants, the megamemory did not fail at all. Moving his lips for a second, he began to reproduce the entire series of elements with amazing accuracy, as if reading.

For example, another scientist-researcher conducted an experiment with Mademoiselle Osaka. The subject was asked to square 97 to obtain the tenth power of that number. She did it instantly.

Aron Chikashvili lives in the Van region of western Georgia. He quickly and accurately performs complex calculations in his head. Somehow, friends decided to test the capabilities of the “miracle counter”. The task was difficult: how many words and letters will the announcer say when commenting on the second half of the football match “Spartak” (Moscow) - “Dynamo” (Tbilisi). At the same time the tape recorder was turned on. The answer came as soon as the announcer said the last word: 17427 letters, 1835 words. It took….5 hours to check. The answer turned out to be correct.

It is said that Gauss's father usually paid his workers at the end of the week, adding overtime to each day's earnings. One day, after Gauss the father had finished his calculations, a three-year-old child who was following his father’s operations exclaimed: “Dad, the calculation is not correct!” This should be the amount." The calculations were repeated and we were surprised to see that the kid had indicated the correct amount.

Interestingly, many “miracle counters” have no idea how they count. “We count, that’s all! But as we think, God knows.” Some of the “counters” were completely uneducated people. The Englishman Buxton, a “virtuoso calculator,” never learned to read; American “negro accountant” Thomas Faller died illiterate at the age of 80.

Competitions were held at the Institute of Cybernetics of the Ukrainian Academy of Sciences. The competition was attended by the young “counter-phenomenon” Igor Shelushkov and the Mir computer. The machine performed many complex mathematical operations in a few seconds. The winner of this competition was Igor Shelushkov.

Most of these people have excellent memory and talent. But some of them have no ability in mathematics. They know the secret! And this secret is that they have mastered the techniques of quick counting well and memorized several special formulas. But a Belgian employee who, in 30 seconds, given a multi-digit number given to him, obtained by multiplying a certain number by itself 47 times, calls this number (extracts the root of the 47th

degrees from a multi-digit number), achieved such amazing success in counting as a result of many years of training.

So, many “counting phenomena” use special quick counting techniques and special formulas. This means that we can also use some of these techniques.

ChapterII. Ancient methods of multiplication.

2.1. Russian peasant method of multiplication.

In Russia, 2-3 centuries ago, a method was widespread 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 peasant(there is an opinion that it originates from Egyptian).

Example: multiply 47 by 35,

Let's write down the numbers on one line and draw a vertical line between them;

We will divide the left number by 2, multiply the right number by 2 (if a remainder arises during division, then we discard the remainder);

The division ends when a unit appears on the left;

We cross out those lines in which there are even numbers on the left;

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

2.2. Lattice method.

1). The outstanding Arab mathematician and astronomer Abu Mussa al-Khorezmi lived and worked in Baghdad. “Al - Khorezmi” literally means “from Khorezmi”, i.e. born in the city of Khorezm (now part of Uzbekistan). The scientist worked in the House of Wisdom, where there was a library and an observatory; almost all the major Arab scientists worked here.

There is very little information about the life and activities of Muhammad al-Khorezmi. Only two of his works have survived - on algebra and arithmetic. The last of these books gives four rules of arithmetic operations, almost the same as those used in our time.

2). In his "The Book of Indian Accounting" the scientist described a method invented in Ancient India, and later called "lattice method"(aka "jealousy"). This method is even simpler than the one used today.

Let's say we need to multiply 25 and 63.

Let's draw a table in which there are two cells in length and two in width, write down one number for the length and another for the width. In the cells we write the result of multiplying these numbers, at their intersection we separate the tens and ones with a diagonal. We add the resulting numbers diagonally, and the resulting result can be read along the arrow (down and to the right).

We have considered a simple example, however, this method can be used to multiply any multi-digit numbers.

Let's look at another example: multiply 987 and 12:

Draw a 3 by 2 rectangle (according to the number of decimal places for each factor);

Then we divide the square cells diagonally;

At the top of the table we write the number 987;

On the left of the table is the number 12 (see picture);

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

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

We write the result on the right and bottom of the table (see figure);

987 ∙ 12=11844

This algorithm for multiplying two natural numbers was common in the Middle Ages in the East and Italy.

We noted the inconvenience of this method in the laboriousness of preparing a rectangular table, although the calculation process itself is interesting and filling out the table resembles a game.

2.3 Indian way of multiplication

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

The Americans liked it so much that they even called it “The American Way.” However, it was used by the inhabitants of India back in the 6th century. n. e., and it would be more correct to call it the “Indian way.” Multiply any two two-digit numbers, say 23 by 12. I immediately write what happens.

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

First step: x23 I say: “2 x 3 = 6”

Second step: x23 I say: “2 x 2 + 1 x 3 = 7”

Third step: x23 I say: “1 x 2 = 2.”

12 I write 2 to the left of the number 7

276 we get 276.

We got acquainted with this method using a very simple example without going through a bit. However, our research has shown that it can also be used when multiplying numbers with transition through digit, as well as when multiplying multi-digit numbers. Here are some examples:

x528 x24 x15 x18 x317

123 30 13 19 12

In Rus', this method was known as the method of multiplication with a cross.

This “cross” is the inconvenience of multiplication; it is easy to get confused, and it is also difficult to keep in mind all the intermediate products, the results of which must then be added up.

2.4. Egyptian way of multiplication

The number notations that were used in ancient times were more or less suitable for recording the result of a count. But it was very difficult to perform arithmetic operations with their help, especially with regard to the operation of multiplication (try multiplying: ξφß*τδ). The Egyptians found a way out of this situation, so the method was called Egyptian. They replaced multiplication by any number with doubling, that is, adding a number to itself.

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

Since 5 = 4 + 1, then to get the answer it remained to add the numbers in the right column against the numbers 4 and 1, i.e. 136 + 34 = 170.

2.5. Multiplication on fingers

The ancient Egyptians were very religious and believed that the soul of the deceased in the afterlife was subjected to a finger counting test. This already speaks volumes about the importance that the ancients attached to this method of multiplying natural numbers (it was called finger counting).

They multiplied single-digit numbers from 6 to 9 on their fingers. To do this, they stretched out as many fingers on one hand as the first factor exceeded the number 5, and on the second they did the same for the second factor. The remaining fingers were bent. After this, they took as many tens as the length of the fingers on both hands, and added to this number the product of the bent fingers on the first and second hand.

Example: 8 ∙ 9 = 72

Later, finger counting was improved - they learned to show numbers up to 10,000 with their fingers.

Finger movement

Here’s another way to help your memory: use your fingers to remember the multiplication table by 9. Putting both hands side by side on the table, number the fingers of both hands in order as follows: the first finger on the left will be designated 1, the second one behind it will be designated 2, then 3 , 4... to the tenth finger, which means 10. If you need to multiply any of the first nine numbers by 9, then to do this, without moving your hands from the table, you need to lift up the finger whose number means the number by which nine is multiplied; then the number of fingers lying to the left of the raised finger determines the number of tens, and the number of fingers lying to the right of the raised finger indicates the number of units of the resulting product.

Example. Suppose we need to find the product 4x9.

With both hands on the table, raise the fourth finger, counting from left to right. Then there are three fingers (tens) before the raised finger, and 6 fingers (units) after the raised finger. The result of the product 4 by 9 is therefore equal to 36.

Another example:

Let's say we need to multiply 3 * 9.

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

To the right of the bent finger, 7 fingers will be straightened, they mean 7 units. Add 2 tens and 7 units and you get 27.

The fingers themselves showed this number.

// // /////

So, the ancient methods of multiplication we examined show that the algorithm used in school for multiplying natural numbers is not the only one and it was not always known.

However, it is quite fast and most convenient.

Chapter 3. Mental arithmetic - mental gymnastics

3.1. Multiplying and dividing by 4.

To multiply a number by 4, it is doubled.

For example,

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

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

To divide a number by 4, it is divided by 2 twice.

For example,

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

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

3.2. Multiplying and dividing by 5.

To multiply a number by 5, you need to multiply it by 10/2, that is, multiply by 10 and divide by 2.

For example,

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

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

To divide a number by 5, you need to multiply it by 0.2, that is, in double the original number, separate the last digit with a comma.

For example,

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

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

3.3. Multiply by 25.

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

For example,

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

3.4. Multiply by 1.5.

To multiply a number by 1.5, you need to add half of it to the original number.

For example,

26 * 1,5 = 26 + 13 = 39

228 * 1,5 = 228 + 114 = 342

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

3.5. Multiply by 9.

To multiply a number by 9, add 0 to it and subtract the original number. For example,

241 * 9 = 2410 – 241 = 2169

847 * 9 = 8470 – 847 = 7623

3.6. Multiply by 11.

1 way. To multiply a number by 11, add 0 to it and add the original number. For example:

47 * 11 = 470 + 47 = 517

243 * 11 = 2430 + 243 = 2673

Method 2. If you want to multiply a number by 11, then do this: write down the number that needs to be multiplied by 11, and between the digits of the original number insert the sum of these digits. If the sum turns out to be a two-digit number, then add 1 to the first digit of the original number. For example:

45 * 11 = * 11 = 967

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

3.7. Multiplying a three-digit number by 101.

For example 125 * 101 = 12625

(increase the first factor by the number of its hundreds and add the last two digits of the first factor to it on the right)

125 + 1 = 126 12625

Children easily learn this technique when writing calculations in a column.

x x125
101
+ 125
125 _
12625

x x348
101
+348
348 _
35148

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

3.8. Squaring a number ending in 5.

To square a number ending in 5 (for example, 65), multiply its tens number (6) by the number of tens increased by 1 (6+1 = 7), and add 25 to the resulting number

(6 * 7 = 42 Answer: 4225)

For example:

3.8. Squaring a number close to 50.

If you want to square a number that is close to 50 but greater than 50, then do this:

1) subtract 25 from this number;

2) add to the result in two digits the square of the excess of the given number over 50.

Explanation: 58 – 25 = 33, 82 = 64, 582 = 3364.

Explanation: 67 – 25 = 42, 67 – 50 = 17, 172 =289,

672 = 4200 + 289 = 4489.

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

1) subtract 25 from this number;

2) add to the result in two digits the square of the disadvantage of this number up to 50.

Explanation: 48 – 25 = 23, 50 – 48 =2, 22 = 4, 482 = 2304.

Explanation: 37 – 25 = 12,= 13, 132 =169,

372 = 1200 + 169 = 1369.

3.9. Games

Guessing the resulting number.

1. Think of a number. Add 11 to it; multiply the resulting amount by 2; subtract 20 from this product; multiply the resulting difference by 5 and subtract from the new product a number that is 10 times larger than the number you have in mind.

I guess: you got 10. Right?

2. Think of a number. Triple it. Subtract 1 from the result. Multiply the result by 5. Add 20 to the result. Divide the result by 15. Subtract the intended value from the result.

You got 1.

3. Think of a number. Multiply it by 6. Subtract 3. Multiply it by 2. Add 26. Subtract twice the intended value. Divide by 10. Subtract what you intended.

You got 2.

4. Think of a number. Triple it. Subtract 2. Multiply by 5. Add 5. Divide by 5. Add 1. Divide by intended. You got 3.

5. Think of a number, double it. Add 3. Multiply by 4. Subtract 12. Divide by what you intended.

You got 8.

Guessing the intended numbers.

Invite your comrades to think of any numbers. Let everyone add 5 to their intended number.

Let the resulting amount be multiplied by 3.

Let him subtract 7 from the product.

Let him subtract another 8 from the result obtained.

Let everyone give you the sheet with the final result. Looking at the piece of paper, you immediately tell everyone what number they have in mind.

(To guess the intended number, divide the result written on a piece of paper or told to you orally by 3)

Conclusion

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

By describing ancient methods of calculation and modern methods of quick calculation, we tried to show that both in the past and in the future, one cannot do without mathematics, a science created by the human mind.

The study of ancient methods of multiplication showed that this arithmetic operation was difficult and complex due to the variety of methods and their cumbersome implementation.

The modern method of multiplication is simple and accessible to everyone.

Upon reviewing the scientific literature, we discovered faster and more reliable methods of multiplication. Therefore, studying the action of multiplication is a promising topic.

It is possible that many people will not be able to quickly and immediately perform these or other calculations the first time. Let it not be possible to use the technique shown in the work at first. No problem. Constant computational training is needed. From lesson to lesson, from year to year. It will help you acquire useful mental arithmetic skills.

List of used literature

1. Wangqiang: Textbook for 5th grade. - Samara: Publishing house

"Fedorov", 1999.

2., Ahadov’s world of numbers: A book of students, - M. Education, 1986.

3. “From game to knowledge”, M., “Enlightenment” 1982.

4. Svechnikov, figures, problems M., Education, 1977.

5. http://matsievsky. *****/sys-schi/file15.htm

6. http://*****/mod/1/6506/hystory. html