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“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.

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

1. 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.
2. 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.
3. 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.
4. 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.
5. Kordemsky B. A., Akhadov A. A. The amazing world of numbers: A book of students, - M. Education, 1986.
6. Minskikh E. M. “From game to knowledge”, M., “Enlightenment” 1982.
7. Svechnikov A. A. Numbers, figures, problems M., Education, 1977.
8. http://matsievsky. newmail. ru/sys-schi/file15.htm
9. http://sch69.narod. ru/mod/1/6506/hystory. html

Some quick ways oral multiplication We’ve already figured it out, now let’s take a closer look at how to quickly multiply numbers in your head using various auxiliary methods. You may already know, and some of them are quite exotic, such as the ancient Chinese way of multiplying numbers.

Layout by ranks

It is the simplest technique for quickly multiplying two-digit numbers. Both factors need to be divided into tens and ones, and then all these new numbers must be multiplied by each other.

This method requires the ability to hold up to four numbers in memory at the same time, and to do calculations with these numbers.

For example, you need to multiply numbers 38 And 56 . We do it this way:

38 * 56 = (30 + 8) * (50 + 6) = 30 * 50 + 8 * 50 + 30 * 6 + 8 * 6 = 1500 + 400 + 180 + 48 = 2128 It will be even easier to do oral multiplication of two-digit numbers in three operations. First you need to multiply the tens, then add two products of ones by tens, and then add the product of ones by ones. It looks like this: 38 * 56 = (30 + 8) * (50 + 6) = 30 * 50 + (8 * 50 + 30 * 6) + 8 * 6 = 1500 + 580 + 48 = 2128 In order to successfully use this method, you need to know the multiplication table well, be able to quickly add two-digit and three-digit numbers, and switch between mathematical operations without forgetting intermediate results. The last skill is achieved through help and visualization.

This method is not the fastest and most effective, so it is worth exploring other methods of oral multiplication.

Fitting the numbers

You can try to bring the arithmetic calculation to a more convenient form. For example, the product of numbers 35 And 49 can be imagined this way: 35 * 49 = (35 * 100) / 2 — 35 = 1715
This method may be more effective than the previous one, but it is not universal and is not suitable for all cases. It is not always possible to find a suitable algorithm to simplify the problem.

On this topic, I remembered an anecdote about how a mathematician sailed along the river past a farm and told his interlocutors that he was able to quickly count the number of sheep in the pen, 1358 sheep. When asked how he did it, he said it was simple - you need to count the number of legs and divide by 4.

Visualization of columnar multiplication

This is one of the most universal ways of oral multiplication of numbers, developing spatial imagination and memory. First, you should learn to multiply two-digit numbers by single-digit numbers in a column in your head. After this, you can easily multiply two-digit numbers in three steps. First, a two-digit number must be multiplied by the tens of another number, then multiplied by the units of another number, and then sum the resulting numbers.

It looks like this: 38 * 56 = (38 * 5) * 10 + 38 * 6 = 1900 + 228 = 2128

Visualization with number arrangement

A very interesting way to multiply two-digit numbers is as follows. You need to sequentially multiply the digits in numbers to get hundreds, ones and tens.

Let's say you need to multiply 35 on 49 .

First you multiply 3 on 4 , you get 12 , then 5 And 9 , you get 45 . Recording 12 And 5 , with a space between them, and 4 remember.

You receive: 12 __ 5 (remember 4 ).

Now you multiply 3 on 9 , And 5 on 4 , and sum up: 3 * 9 + 5 * 4 = 27 + 20 = 47 .

Now we need to 47 add 4 which we remember. We get 51 .

We write 1 in the middle and 5 add to 12 , we get 17 .

In total, the number we were looking for is 1715 , it is the answer:

35 * 49 = 1715
Try multiplying in your head in the same way: 18 * 34, 45 * 91, 31 * 52 .

Chinese or Japanese multiplication

In Asian countries, it is customary to multiply numbers not in a column, but by drawing lines. For Eastern cultures, the desire for contemplation and visualization is important, which is probably why they came up with such a beautiful method that allows you to multiply any numbers. This method is complicated only at first glance. In fact, greater clarity allows you to use this method much more effectively than multiplying by column.

In addition, knowledge of this ancient oriental method increases your erudition. Agree, not everyone can boast that they know the ancient multiplication system that the Chinese used 3000 years ago.

Video about how the Chinese multiply numbers

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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.

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

“Math is so difficult...” You’ve probably heard this phrase more than once, and maybe even said it out loud yourself.

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

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

If you need to multiply multi-digit numbers, you will need a large sheet of paper to find the answer.

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

But this is where some artistic skills come in handy.

Let's draw!

At least three methods of multiplication involve drawing intersecting lines.

1. Mayan way, or Japanese method

There are several versions regarding the origin of this method.

Having trouble multiplying in your head? Try the Mayan and Japanese Method

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

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

Let's multiply 23 by 41.

To do this, we need to draw two parallel lines representing 2, and, stepping back a little, three more lines representing 3.

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

Well, is it really difficult?

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

The origin of this method of multiplication is also unclear, but it is well known throughout Asia.

“The Gelosia algorithm was transmitted from India to China, then to Arabia, and from there to Italy in the 14th and 15th centuries, where it was called Gelosia because it was similar in appearance to Venetian lattice shutters,” writes Mario Roberto Canales Villanueva in his book on various methods of multiplication.

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

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

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

Then you need to divide each cell in half diagonally to make triangles.

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

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

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

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

Media playback is unsupported on your device

Having trouble multiplying in your head? Try the Indian method

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

Do you think this method is easier or not?

Let's try another multiplication method using drawing.

3. "Array", or table method

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

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

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

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

Then we will multiply the numbers horizontally and vertically.

Media playback is unsupported on your device

Having trouble multiplying in your head? Draw a table.

But instead of multiplying 20 by 40, we'll drop the zeros and just multiply 2 x 4 to get 8.

We will do the same thing by multiplying 3 by 40. We keep 0 in parentheses and multiply 3 by 4 and get 12.

Let's do the same with the bottom row.

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

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

Let's do the same with all other retained zeros.

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

Result? 943. Well, did it help?

Variety is important

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

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

We did have to multiply a few things along the way, but each step was easier than traditional multiplication and much more visual.

So why are few places in the world teaching these methods of calculation in regular schools?

One reason may be the emphasis on teaching “mental arithmetic” to develop mental abilities.

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

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

Illustration copyright Getty Images Image caption For some calculation methods, just a head is not enough; you also need felt-tip pens

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

"I don't think it's helpful to teach schoolchildren multiplication right away, by making them learn the multiplication table without telling them where it comes from. Because if they forget one number, how can they make any progress in solving the problem? Mayan method or The Japanese method is necessary because with it you can understand the general structure of multiplication, and that is a good start,” says Weese.

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

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

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

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 the 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