Our children and we are used to multiplying numbers in the traditional way, writing the factor numbers in a column.
However, in Asia, children are taught a completely different technique for multiplication.
When multiplying “in a column”, the child has to keep large amounts of data in his head. Japanese multiplication is useful to show to all children, they especially love this method visual and kinesthetic. After all, they can see the multiplication!
In this video we show you how to multiply in Japanese:
This method allows the child to visualize multiplication and solve examples within and outside the multiplication table.
For example, we need to multiply 12 by 12.
Step 1 - Draw the lines of the first number horizontally. For each number, its own number of lines is drawn. Tens and ones are separated by spaces. For example, for the number 12, the unit is drawn with one line. Two - just below two parallel lines. For the number 36, 3 is drawn with three lines, 6 with six parallel lines below, etc.
Step 2 By analogy with step 1, draw the second number 12 using vertical lines:
Unit - one line
Two - slightly stepping back to the right in two lines
Step 3 Place dots at the intersections of lines
Step 4 Count the number of points in three groups, dividing them into “Fish”: tail, head and body
Upper left corner – 1 (hundreds)
Upper right and lower left corners (Diagonal) – 4 (tens)
Lower right corner – 4 (units)
Step 5 Write down the result: 144. If the units or tens have a two-digit number, then the first digit is added to the next digit.
With Japanese multiplication, children can easily calculate any multiplication table example. What cannot be done with the traditional approach.
Having forgotten, a child can easily multiply 6 by 4 or 7 by 8.
All he needs to do is draw the multiplication!
Use this technique and method. And if a child forgets the multiplication table out of excitement during a test, he will be able to do the math “in Japanese” on his draft!
Last Sunday, during classes at the School of Special Agents, the children really enjoyed this method. After drawing the multiplication, the children used two more techniques that allowed them to learn the multiplication table once and for all.
There are a huge number of unique techniques in the world that simplify life and help you fall in love with learning.
We have collected them all, combined them with knowledge of child psychology, added an individual approach and teach children the world's best teaching techniques.
Let's give your children a New Year's gift - teach them to learn with pleasure. Because learning can be a lot of fun.
You can rejoice in this:
- How great you did with everyone complicated words and you write correctly
- How easy math problems are
- How to use the Hamburger technique to retell texts in a fun way
- You can enjoy learning the multiplication tables, cases, Russian rules and mathematics
- Can defeat difficult bosses such as physics or history
- And all this, using special agent techniques, unusual and exciting. In the standard package, children will learn how to easily cope with the most difficult tasks on the world around us, mathematics, the Russian language and answer beautifully at the board.
- In the PRO package, children will learn how to effectively learn and understand English, physics, history, and computer science.
- Learn to write essays and presentations.
Our students do not cram the rules, they learn to think, be attentive, believe in themselves and not be afraid of difficulties, write competently and quickly and count correctly.
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
During classes, children learn to count quickly 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.
Rising 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 in different ways: by ear, in workbooks, at the blackboard on a demonstration abacus, using the “Jolly Soroban” electronic simulator, on a mental map (this graphic image abacus, with the help of which children imagine how to move the bones on an abacus).
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 from odd number we discard one and divide the remainder in half, while to the last number of the right column we will need to add all those numbers of this column that stand opposite the odd numbers of the left column - the sum will be the required product (pictures: 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, 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 (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 |
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.
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 numbers in a column. 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.
Some say that it was invented by the Mayan Indians who inhabited the areas Central America before the conquistadors arrived there in the 16th century. It is also known as the Japanese multiplication method because teachers in Japan use this visual method when teaching junior schoolchildren multiplication.
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 to XIV-XV centuries, where it was called “gelosia” because it looked similar to Venetian lattice shutters,” writes Mario Roberto Canales Villanueva in his book on different methods of multiplication.
Illustration copyright Getty Images Image caption Indian or Italian multiplication system is similar to Venetian blindsLet'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 methodThe 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 agreesWhat can be said for sure is that all these different ways 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 working 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 pensDespite 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, which is a good start", says Wiz.
There are a number of other methods of multiplication, for example, Russian or Egyptian, they do not require additional drawing skills.
According to the experts 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.