Nations that use hieroglyphs have a different type of thinking. Does it affect their lives? Hard to tell. Such people are visual by nature, they perceive figuratively the world. And this system of perception does not bypass even the exact sciences. It will be interesting for everyone to know how the Japanese multiply. Firstly, you don’t have to frantically search for a calculator, and secondly, this is a very exciting activity.

Let's draw

It's amazing, but Japanese children can multiply even without knowing about the multiplication table. How do the Japanese multiply? They do it very simply, so simply that they use only basic drawing and counting skills. It’s easier to show with an example how this happens.

Let's say you need to multiply 123 by 321. First you need to draw one, two and three parallel lines that will be placed diagonally from the upper left corner to the lower right. On the created groups of parallels, draw three, two and one line, respectively. They will also be placed diagonally from the bottom left to the top right.

As a result, we get a so-called rhombus (as in the figure above). If anyone hasn't figured it out yet, the number of lines in a group depends on the numbers that need to be multiplied.

We count

So how do the Japanese multiply numbers? The next stage is counting the intersection points. First, we separate with a semicircle the intersection of three lines with one and count the number of points. We write the resulting number under the diamond. Then, in exactly the same way, we separate the areas where two lines intersect with three and one. We also count the points of contact and write them down, then we count the points that remain in the center. You should get a result similar to the figure below.

It is worth paying attention to the fact that if the central number is two-digit, then the first digit must be added to the number that was obtained when counting the points of contact in the area to the left of the center. Thus multiplying 123 by 321, we get 39,483.

This method can be used to multiply both two-digit and three-digit numbers. One problem is that if you have to count numbers like 999, 888, 777, etc., you will need to draw a lot of lines.

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?”
solution logical problems using the algebra of logic
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, placing at my disposal several pieces of paper from a notebook with ready-made solutions in the form of intricate designs. 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 (on the right top 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

What is mental arithmetic and why every person needs it.

Mental arithmetic is a program for the comprehensive development of children's intelligence and thinking, based on the formation of the skill of rapid mental calculation

Children learn in class quick counting using a special counting board (abacus, soroban). Teachers explain how to correctly move knuckles on knitting needles so that kids can almost instantly get an answer to complex example. Gradually, the attachment to the abacus weakens and the children imagine the actions they performed with the abacus in their minds.

The program is designed for 2-2.5 years. First, the children master addition and subtraction, then multiplication and division. A skill is acquired and developed through repeated repetition of the same actions. The method is suitable for almost all children, the teaching principle is from simple to complex.

Classes take place once or twice a week and last one to two hours.

The ancient abacus abacus, which children use to count, has been known for more than 2.5 thousand years.

In Japan, abacus counting is included in the official school curriculum.

For more than 50 years, mental arithmetic has been part of the public education system in Japan. It is interesting that after finishing school people continue to improve their mental arithmetic skills. In the Land of the Rising Sun, mental arithmetic is considered something like a sport. There are even competitions held on it. In Russia, international tournaments in Mental Arithmetic are now also held annually.

Mental arithmetic develops mechanical and photographic memory

When children count, they use both sides of their brain at once. Mental arithmetic develops photographic and mechanical memory, imagination, observation, improves concentration.

Increasing general level intelligence. This means that it is easier for children to absorb large amounts of information in a short time. Successes are immediately visible foreign languages. Now you don’t have to spend the whole day memorizing poetry and prose.

Slower students have faster reaction times. They begin not only to count at lightning speed, but to think faster and make decisions not related to arithmetic.

There are also unexpected results. One day a boy came to the center and played tennis. The mother said that her son has problems with coordination of movements. Unexpectedly, they were solved precisely through intensive mental arithmetic courses.

Mental arithmetic is more difficult for adults; the optimal age for starting classes is 5-14 years

You can develop your brain using mental arithmetic at any age, but the best results can be achieved before the age of 12–14. The children's brain is very plastic and mobile. IN at a young age it is where neural connections are most actively formed, which is why our program is easier for children under 14 years of age.

The older a person is, the more difficult it is for him to abstract from his experience and knowledge and simply trust the abacus. I mastered this technique at the age of 45 and constantly doubted whether I was doing it right or whether there was a mistake. This greatly interferes with learning.

But the more difficult it is for a person to master this account, the more useful it is. It’s as if a person overcomes himself, and every time he does it better and better. The classes are not in vain; the brain of an adult is also actively developing.

Just don’t expect the same results from an adult as from a child. We can learn the technique, but we won’t be able to count as quickly as a second grader does. As experience shows, the optimal age at which it is better to start classes is 6 and 7 years.

The best results are achieved by those who regularly exercise at home.

A prerequisite for classes is daily training on the abacus. Just 10-15 minutes. Children need to practice the formula that the teacher gave them in class and bring their actions to automaticity. Only in this case will the child learn to count quickly. The organizational role of parents, who need to monitor regular training, is important here.

Children do not get tired in class due to constant change of activities

The main activity in mental arithmetic is counting on the abacus. Children count different ways: by ear, in workbooks, at the school board on a demonstration abacus, using the electronic simulator “Jolly Soroban”, on a mental map (this graphic image abacus, with the help of which children imagine how to move the bones on an abacus).

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.

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 different situations. Possession of computing skills and abilities has great importance to master the material being studied, it allows you to cultivate valuable work qualities: a responsible attitude to your work, the ability to detect and correct mistakes made in your work, careful execution of a task, a creative attitude to work. However, in Lately 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 to the left remain odd number. 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. Amazing world numbers: 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