Indian way of multiplying multi-digit numbers. Ways to quickly multiply numbers verbally

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.

Purpose of work: Explore and show unusual ways of multiplication Tasks: Find unusual ways of multiplication. Learn to use them. Choose for yourself the most interesting or easier ones than those offered at school, and use them when counting. Teach classmates to use a new way of multiplication

Methods: search method using scientific and educational literature, as well as searching for the necessary information on the Internet; a practical method of performing calculations using non-standard counting algorithms; analysis of the data obtained during the study. The relevance of this topic lies in the fact 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

In math lessons we learned an unusual way of multiplying by columns. We liked it and decided to learn other ways to multiply natural numbers. We asked our classmates if they knew other ways to count? Everyone talked only about those methods that are studied at school. It turned out that all our friends knew nothing about other methods. In the history of mathematics, about 30 methods of multiplication are known, differing in the notation scheme or the course of the calculation itself. The columnar multiplication method that we study at school is one of the methods. But is this the most effective way? Let's get a look! Introduction

This is one of the most commonly used methods, which Russian merchants have successfully used for many centuries. The principle of this method: multiplying single-digit numbers from 6 to 9 on the fingers. 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 (23=6), you will get, respectively, the numbers of tens and units of the desired product 56. This way you can calculate the product of any single-digit numbers greater than 5.

Multiplication for the number 9 is very easy to reproduce “on your 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. Let's say we want to multiply 9 by 6. We bend the finger with a number equal to the number by which we will multiply nine. In our example, we need to bend the finger with number 6. The number of fingers to the left of the bent finger shows us the number of tens in the answer, the number of fingers to the right shows the number of ones. On the left we have 5 fingers not bent, on the right - 4 fingers. Thus, 9·6=54.

“Small Castle” multiplication method The advantage of the “Small Castle” multiplication method is that the most significant digits are determined from the very beginning, and this is 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.

“Jealousy” or “lattice multiplication” First, a rectangle is drawn, divided into squares, and the dimensions of the sides of the rectangle correspond to the number of decimal places of the multiplicand and the multiplier. Then the square cells are divided diagonally, and “... the result is a picture similar to lattice shutters, - writes Pacioli. “Such shutters were hung on the windows of Venetian houses...”

Lattice multiplication = +1 +2

Peasant method This is the method of Great Russian peasants. Its essence lies in the fact that the multiplication of any numbers comes down to a series of successive divisions of one number in half, while simultaneously doubling another number……….32 74……………….8 296……….4 592……… ………1 3732=1184

Peasant way (odd numbers) 47 x =1645

Step 1. first number 15: Draw the first number – with one line. Draw the second number with five lines. Step 2. second number 23: Draw the first number with two lines. Draw the second number with three lines. Step 3. Count the number of points in groups. Step 4. Result – 345. Multiply two two-digit numbers: 15*23

Indian way of multiplying (cross) 24 and X 3 2 1)4x2=8 - the last digit of the result; 2)2x2=4; 4x3=12; 4+12=16 ; 6 is the penultimate digit of the result, remember the unit; 3) 2x3 = 6 and also a number held in mind, we have 7 - this is the first digit of the result. We get all the numbers of the product: 7,6,8. Answer: 768.

Indian way of multiplication = = = = 3822 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. We start the multiplication from the highest digit, and write down the incomplete products just above the multiplicand, bit by bit. In this case, the most significant digit of the complete product is immediately visible and, in addition, missing any digits is eliminated. The multiplication sign was not yet known, so a small distance was left between the factors

Reference number Multiply 18*19 20 (reference number) * 2 1 (18-1)*20 = Answer: 342 Short notation: 18*19 = 20*17+2 = 342

New way to multiply X = , 5+2, 5+3, 0+2, 0+3, 5

Conclusion: Having learned to count using all the presented methods, we came to the conclusion that the simplest methods are those that we study at school, or maybe we are just used to them. Of all the unusual methods of counting considered, the method of graphic multiplication seemed more interesting. We showed it to our classmates and they really liked it too. The simplest method seemed to be “doubling and splitting”, which was used by Russian peasants. After working with literature and materials on the Internet, we realized that we had considered a very small number of multiplication methods, which means that a lot of interesting things await us ahead

Conclusion 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 complicated due to the variety of methods and their cumbersome implementation. The modern method of multiplication is simple and accessible to everyone. But we think that our method of multiplying by column is not perfect and we can come up with even faster and more reliable methods. It is possible that many people will not be able to quickly, right away, perform these or other calculations the first time. It doesn’t matter. Constant computational training is needed. It will help you acquire useful mental arithmetic skills!

Materials used: html Encyclopedia for children. "Mathematics". – M.: Avanta +, – 688 p. Encyclopedia “I explore the world. Mathematics". – M.: Astrel Ermak, Perelman Ya.I. Quick count. Thirty simple mental counting techniques. L., p.

second way of multiplication:

IN Rus', peasants did not use multiplication tables, but they perfectly calculated the product of multi-digit numbers.

In Rus', starting from ancient times and almost until the eighteenthcenturies, Russian people in their calculations did without multiplication anddivision. They used only two arithmetic operations - addition andsubtraction. Moreover, the so-called “doubling” and “bifurcation”. Butthe needs of trade and other activities required the productionmultiplication of fairly large numbers, both two-digit and three-digit.For this purpose, there was a special way of multiplying such numbers.

The essence of the old Russian method of multiplication is thatmultiplication of any two numbers was reduced to a series of successive divisionsone number in half (sequential bifurcation) with simultaneousdoubling another number.

For example, if in the product 24 ∙ 5 the multiplicand 24 is reduced by twotimes (double), and the multiplicand is doubled (double), i.e. takethe product is 12 ∙ 10, then the product remains equal to the number 120. Thisour distant ancestors noticed the quality of the work and learnedapply it when multiplying numbers in your special old Russianway of multiplication.

Let's multiply this way 32 ∙ 17..
32 ∙ 17
16 ∙ 34
8 ∙ 68
4 ∙ 136
2 ∙ 272
1 ∙544 Answer: 32 ∙ 17 = 544.

In the analyzed example, division by two - “bifurcation” occurswithout a trace. But what if the multiplier is not divisible by two without a remainder? ANDthis seemed within the capabilities of the ancient calculators. In this case, we did this:
21 ∙ 17
10 ∙ 34
5 ∙ 68
2 ∙ 136
1 ∙ 272
357 Answer: 357.

From the example it is clear that if the multiplicand is not divisible by two, then from itfirst we subtracted one, then we doubled the resulting result” and so on5 to go. Then all lines with even multiplicands were crossed out (2nd, 4th,6th, etc.), and all the right parts of the remaining lines were added and receivedthe desired work.

How did the ancient calculators reason, justifying their method?calculations? That's how: 21 ∙ 17 = 20 ∙ 17 + 17.
The number 17 is remembered, and the product 20 ∙ 17 = 10 ∙ 34 (we bifurcate -double) and write it down. Product 10 ∙ 34 = 5 ∙ 68 (we bifurcate –double), and cross out the extra product 10∙34. Since 5 * 34= 4 ∙ 68 + 68, then the number 68 is remembered, i.e. the third line is not crossed out, but4 ∙ 68 = 2 ∙ 136 = 1 ∙ 272 (we double - we double), with the fourththe line containing, as it were, an extra product 2 ∙ 136, is crossed out, andthe number 272 is remembered. So it turns out that to multiply 21 by 17,you need to add the numbers 17, 68 and 272 - these are exactly equal parts of the linesnamely with odd multiplicands.
The Russian method of multiplication is both elegant and extravagant at the same time

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 to 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 working 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, more than once or twice, to my shame, I noticed some slowdowns, indicating that my multiplication table was rusty in some places and I 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.

P.S.: Glory and praise to the native column!
In terms of construction, the method is unpretentious and compact, very fast, It trains your memory and prevents you from forgetting the multiplication tables.

And therefore, I strongly recommend that you and yourself, if possible, forget about calculators on phones and computers; and periodically treat yourself to multiplication by column. 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!

LEARNING THE MULTIPLICATION TABLE!!!

The world of mathematics is very big, but I have always been interested in methods of multiplication. While working on this topic, I learned a lot of interesting things and learned to select the material I needed from what I read. I learned how to solve certain entertaining problems, puzzles and examples of multiplication in different ways, as well as what arithmetic tricks and intensive calculation techniques are based on.

ABOUT MULTIPLICATION

What stays in most people's minds from what they once studied in school? Of course, it’s different for different people, but everyone probably has a multiplication table. In addition to the efforts made to “drill down” it, let’s remember the hundreds (if not thousands) of problems we solved with its help. Three hundred years ago in England, a person who knew the multiplication tables was already considered a learned person.

Many methods of multiplication have been invented. The Italian mathematician of the late 15th - early 16th centuries, Luca Pacioli, in his treatise on arithmetic, gives 8 different methods of multiplication. In the first, which is called the “small castle,” the digits of the upper number, starting with the highest, 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. The advantage of this method over the usual one is that the numbers of the most significant digits are determined from the very beginning, and this can be important for rough calculations.

The second method has the no less romantic name “jealousy” (or lattice multiplication). A lattice is drawn into which the results of intermediate calculations are then entered, more precisely, numbers from the multiplication table. The grid is a rectangle divided into square cells, which in turn are divided in half by diagonals. The first factor was written on the left (top to bottom), and the second one at the top. At the intersection of the corresponding row and column, the product of the numbers in them was written. Then the resulting numbers were added along the drawn diagonals, and the result was written at the end of such a column. The result was read along the bottom and right sides of the rectangle. “Such a lattice,” writes Luca Pacioli, “reminiscent of the lattice shutters that were hung on Venetian windows, preventing passers-by from seeing the ladies and nuns sitting at the windows.”

All the multiplication methods described in Luca Pacioli's book used a multiplication table. However, Russian peasants knew how to multiply without a table. Their method of multiplication used only multiplication and division by 2. To multiply two numbers, they were written side by side, and then the left number was divided by 2, and the right was multiplied by 2. If the division resulted in a remainder, it was discarded. Then those lines in the left column containing even numbers were crossed out. The remaining numbers in the right column were added together. The result was the product of the original numbers. Check on several pairs of numbers that this is indeed the case. The proof of the validity of this method is shown using the binary number system.

An ancient Russian method of multiplication.

From ancient times and almost until the eighteenth century, Russian people did their calculations without multiplication and division: they used only two arithmetic operations - addition and subtraction, and also the so-called “doubling” and “bifurcation”. The essence of the ancient Russian method of multiplication is that the multiplication of any two numbers is reduced to a series of successive divisions of one number in half (sequential, bifurcation) while simultaneously doubling the other number. If in a product, for example 24 X 5, the multiplicand is reduced by 2 times (“double”), and the multiplier is increased by 2 times

(“double”), then the product will not change: 24 x 5 = 12 X 10 = 120. Example:

Dividing the multiplicand in half continues until the quotient turns out to be 1, while doubling the multiplier. The last doubled number gives the desired result. So 32 X 17 = 1 X 544 = 544.

In those ancient times, doubling and bifurcation were even taken as special arithmetic operations. Just how special they are. actions? After all, for example, doubling a number is not a special action, but just adding a given number to itself.

Note that numbers are divisible by 2 all the time without a remainder. But what if the multiplicand is divisible by 2 with a remainder? Example:

If the multiplicand is not divisible by 2, then one is first subtracted from it, and then divided by 2. The lines with even multiplicands are crossed out, and the right parts of the lines with odd multiplicands are added.

21 X 17 = (20 + 1) X 17 = 20 X 17+17.

Let us remember the number 17 (the first line is not crossed out!), and replace the product 20 X 17 with the equal product 10 X 34. But the product 10 X 34, in turn, can be replaced with the equal product 5 X 68; so the second line is crossed out:

5 X 68 = (4 + 1) X 68 = 4 X 68 + 68.

Let us remember the number 68 (the third line is not crossed out!), and replace the product 4 X 68 with the equal product 2 X 136. But the product 2 X 136 can be replaced with the equal product 1 X 272; therefore the fourth line is crossed out. This means that in order to calculate the product 21 X 17, you need to add the numbers 17, 68, 272 - the right sides of the lines with odd multiplicands. Products with even multiplicands can always be replaced by doubling the multiplicand and doubling the factor by equal products; therefore, such lines are excluded from the calculation of the final product.

I tried to multiply myself the old-fashioned way. I took the numbers 39 and 247, and this is what I got:

The columns will turn out to be even longer than mine if we take the multiplicand more than 39. Then I decided, the same example in a modern way:

It turns out that our school method of multiplying numbers is much simpler and more economical than the old Russian method!

Only we must know, first of all, the multiplication table, but our ancestors did not know it. In addition, we must know well the rule of multiplication itself, but they only knew how to double and double numbers. As you can see, you can multiply much better and faster than the most famous calculator in ancient Rus'. By the way, several thousand years ago the Egyptians performed multiplication almost exactly the same way as the Russian people did in the old days.

It's great that people from different countries multiplied in the same way.

Not so long ago, just about a hundred years ago, learning the multiplication tables was very difficult for students. To convince students of the need to know tables by heart, authors of mathematical books have long resorted to. to poetry.

Here are a few lines from a book unfamiliar to us: “But for multiplication you need to have the following table, just have it firmly in your memory, so that each number, having multiplied with it, without any delay in speech, say or write, also 2 times 2 is 4 , or 2 times 3 is 6, and 3 times 3 is 9 and so on.”

If someone does not repeat the table and is proud in all science, he is not free from torment,

Koliko cannot know without teaching by number that multiplying Tuna will depress him

True, in this passage and verses not everything is clear: it is somehow not written quite in Russian, because all this was written more than 250 years ago, in 1703, by Leonty Filippovich Magnitsky, a wonderful Russian teacher, and since then the Russian language has changed noticeably .

L. F. Magnitsky wrote and published the first printed arithmetic textbook in Russia; before him there were only handwritten mathematical books. The great Russian scientist M.V. Lomonosov, as well as many other prominent Russian scientists of the eighteenth century, studied from L. F. Magnitsky’s “Arithmetic.”

How did they multiply in those days, in the time of Lomonosov? Let's see an example.

As we understand, the action of multiplication was then written down almost the same way as in our time. Only the multiplicand was called “quantity”, and the product was called “product” and, in addition, the multiplication sign was not written.

How did they explain multiplication then?

It is known that M.V. Lomonosov knew by heart the entire “Arithmetic” of Magnitsky. In accordance with this textbook, little Misha Lomonosov would explain the multiplication of 48 by 8 as follows: “8 times 8 is 64, I write 4 under the line, against 8, and have 6 decimals in my mind. And then 8 times 4 is 32, and I keep 3 in my mind, and to 2 I will add 6 decimals, and it will be 8. And I will write this 8 next to 4, in a row to my left hand, and while 3 is in my mind, I will write in a row near 8, to the left hand. And from the multiplication of 48 with 8 the product will be 384.”

Yes, and we explain it almost the same way, only we speak in modern, not ancient, and, in addition, we name the categories. For example, 3 should be written in third place because it will be hundreds, and not just “in a row next to 8, to the left hand.”

The story “Masha is a magician.”

“I can guess not only the birthday, as Pavlik did last time, but also the year of birth,” Masha began.

Multiply the number of the month in which you were born by 100, then add your birthday. , multiply the result by 2. , add 2 to the resulting number; multiply the result by 5, add 1 to the resulting number, add zero to the result. , add another 1 to the resulting number and, finally, add the number of your years.

Done, I got 20721. - I say.

* Correct,” I confirmed.

And I got 81321,” says Vitya, a third grade student.

“You, Masha, must have been mistaken,” Petya doubted. - How does it happen: Vitya is from the third grade, and was also born in 1949, like Sasha.

No, Masha guessed correctly,” Vitya confirms. Only I was sick for a long time for one year and therefore went to second grade twice.

* And I got 111521,” reports Pavlik.

How is it possible, asks Vasya, Pavlik is also 10 years old, like Sasha, and he was born in 1948. Why not in 1949?

But because it’s September now, and Pavlik was born in November, and he’s still only 10 years old, although he was born in 1948,” Masha explained.

She guessed the birth dates of three or four other students and then explained how she did it. It turns out that she subtracts 111 from the last number, and then the remainder is added to three sides from right to left, two digits each. The middle two digits indicate the birthday, the first two or one indicate the month, and the last two digits indicate the number of years. Knowing how old a person is, it is not difficult to determine the year of birth. For example, I got the number 20721. If you subtract 111 from it, you get 20610. This means that I am now 10 years old, and I was born on February 6th. Since it is now September 1959, it means I was born in 1949.

Why do you need to subtract 111 and not some other number? - we asked. -And why are the birthday, month number and number of years distributed exactly this way?

But look,” Masha explained. - For example, Pavlik, fulfilling my requirements, solved the following examples:

1)11 X 100 = 1100; 2) 1100 + J4 = 1114; 3) 1114 X 2 =

2228; 4) 2228 + 2 = 2230; 57 2230 X 5 = 11150; 6) 11150 1 = 11151; 7) 11151 X 10 = 111510

8)111510 1 1-111511; 9)111511 + 10=111521.

As you can see, he multiplied the month number (11) by 100, then by 2, then by another 5 and, finally, by another 10 (he added a sack), and in total by 100 X 2 X 5 X 10, that is, by 10,000. This means , 11 became tens of thousands, that is, they constitute the third side, if you count two digits from right to left. This is how they find out the number of the month in which you were born. He multiplied his birthday (14) by 2, then by 5 and, finally, by another 10, and in total by 2 X 5 X 10, that is, by 100. This means that the birthday must be looked for among hundreds, in the second face, but here there are hundreds of strangers. Look: he added the number 2, which he multiplied by 5 and 10. This means that he got an extra 2x5x10=100 - 1 hundred. I subtract this 1 hundred from the 15 hundreds in the number 111521, resulting in 14 hundreds. This is how I find out my birthday. The number of years (10) was not multiplied by anything. This means that this number must be looked for among the units, in the first face, but there are extraneous units here. Look: he added the number 1, which he multiplied by 10, and then added another 1. This means that he only got an extra 1 x TO + 1 = 11 units. I subtract these 11 units from the 21 units in the number 111521, it turns out 10. This is how I find out the number of years. And in total, as you can see, from the number 111521 I subtracted 100 + 11 = 111. When I subtracted 111 from the number 111521, then it turned out to be PNU. Means,

Pavlik was born on November 14th and is 10 years old. Now the year is 1959, but I subtracted 10 not from 1959, but from 1958, since Pavlik turned 10 last year, in November.

Of course, you won’t remember this explanation right away, but I tried to understand it with my own example:

1) 2 X 100 = 200; 2) 200 + 6 = 206; 3) 206 X 2 = 412;

4) 412 + 2 = 414; 5) 414 X 5 = 2070; 6) 2070 + 1 = 2071; 7) 2071 X 10 = 20710; 8) 20710 + 1 = 20711; 9) 20711 + + 10 = 20721; 20721 - 111 = 2 "Obto; 1959 - 10 = 1949;

Puzzle.

First task: At noon, a passenger steamer leaves Stalingrad for Kuibyshev. An hour later, a goods and passenger ship leaves Kuibyshev for Stalingrad, moving slower than the first ship. When the ships meet, which one will be further from Stalingrad?

This is not an ordinary arithmetic problem, but a joke! The steamships will be at the same distance from Stalingrad, as well as from Kuibyshev.

And here is the second task: Last Sunday, our squad and the fifth grade squad planted trees along Bolshaya Pionerskaya Street. The teams had to plant an equal number of trees on each side of the street. As you remember, our team came to work early, and before the fifth-graders arrived, we managed to plant 8 trees, but, as it turned out, not on our side of the street: we got excited and started work in the wrong place. Then we worked on our side of the street. The fifth graders finished their work early. However, they did not remain in debt to us: they came over to our side and first planted 8 trees (“they paid off the debt”), and then 5 more trees, and we completed the work.

The question is, how many more trees have fifth-graders planted than we have?

: Of course, the fifth-graders planted only 5 trees more than us: when they planted 8 trees on our side, they thereby repaid the debt; and when they planted 5 more trees, it was as if they had given us 5 trees on loan. So it turns out that they planted only 5 more trees than us.

No, the reasoning is wrong. It is true that the fifth graders did us a favor by planting 5 trees for us. But then, in order to get the correct answer, we need to reason like this: we underfulfilled our task by 5 trees, while the fifth-graders exceeded theirs by 5 trees. So it turns out that the difference between the number of trees planted by fifth-graders and the number of trees planted by us is not 5, but 10 trees!

And here is the last puzzle task, Playing ball, 16 students were placed on the sides of a square area so that there were 4 people on each side. Then 2 students left. The rest moved so that there were again 4 people on each side of the square. Finally, 2 more students left, but the rest settled down so that there were still 4 people on each side of the square. How could this happen? Decide.

Two tricks for quick multiplication

One day a teacher offered his students this example: 84 X 84. One boy quickly answered: 7056. “What did you count?” - the teacher asked the student. “I took 50 X 144 and rolled 144,” he replied. Well, let’s explain how the student thought.

84 x 84 = 7 X 12 X 7 X 12 = 7 X 7 X 12 X 12 = 49 X 144 = (50 - 1) X 144 = 50 X 144 - 144, and 144 fifty is 72 hundred, so 84 X 84 = 7200 - 144 =

Now let’s calculate in the same way how much 56 X 56 is.

56 X 56 = 7 X 8 X 7 X 8 = 49 X 64 = 50 X 64 - 64, that is, 64 fifty, or 32 hundreds (3200), without 64, i.e., to multiply a number by 49, you need this number multiply by 50 (fifty), and subtract this number from the resulting product.

Here are examples for another method of calculation, 92 X 96, 94 X 98.

Answers: 8832 and 9212. Example, 93 X 95. Answer: 8835. Our calculations gave the same number.

You can count so quickly only when the numbers are close to 100. We find the complements up to 100 to these numbers: for 93 there will be 7, and for 95 there will be 5, from the first given number we subtract the complement of the second: 93 - 5 = 88 - this will be in the product hundreds, multiply the additions: 7 X 5 = 3 5 - this is how much will be in the product of units. This means 93 X 95 = 8835. And why exactly this should be done is not difficult to explain.

For example, 93 is 100 without the 7, and 95 is 100 without the 5. 95 X 93 = (100 - 5) x 93 = 93 X 100 - 93 x 5.

To subtract 5 times 93, you can subtract 5 times 100, but add 5 times 7. Then it turns out:

95 x 93 = 93 x 100 - 5 x 100 + 5 x 7 = 93 cells. - 5 hundred. + 5 X 7 = (93 - 5) cells. + 5 x 7 = 8800 + 35= = 8835.

97 X 94 = (97 - 6) X 100 + 3 X 6 = 9100 + 18 = 9118, 91 X 95 = (91 - 5) x 100 + 9 x 5 = 8600 + 45 = 8645.

Multiplication c. domino

With the help of dominoes it is easy to depict some cases of multiplying multi-digit numbers by a single-digit number. For example:

402 X 3 and 2663 X 4

The winner will be the one who, within a certain time, will be able to use the largest number of dominoes, making up examples of multiplying three- and four-digit numbers by a single-digit number.

Examples for multiplying four-digit numbers by one-digit numbers.

2234 X 6; 2425 X 6; 2336 X 1; 526 X 6.

As you can see, only 20 dominoes were used. Examples have been compiled for multiplying not only four-digit numbers by a single-digit number, but also three-, five-, and six-digit numbers by a single-digit number. 25 dice were used and the following examples were compiled:

However, all 28 dice can still be used.

Stories about how well old Hottabych knew arithmetic.

The story “I get a “5” in arithmetic.”

As soon as I went to Misha the next day, he immediately asked: “What was new or interesting in the circle class?” I showed Misha and his friends how smart the Russian people were in the old days. Then I asked them to mentally calculate how much 97 X 95, 42 X 42 and 98 X 93 would be. They, of course, could not do this without a pencil and paper and were very surprised when I almost instantly gave the correct answers to these examples. Finally, we all solved the problem given for home together. It turns out that it is very important how the dots are located on a sheet of paper. Depending on this, you can draw one, four, or six straight lines through four points, but no more.

Then I invited the children to create examples of multiplication using dominoes, just as they did on the mug. We managed to use 20, 24 and even 27 dice, but out of all 28 we were never able to create examples, although we sat at this task for a long time.

Misha remembered that today the movie “Old Man Hottabych” was being shown at the cinema. We quickly finished doing arithmetic and ran to the cinema.

What a picture! Even though it’s a fairy tale, it’s still interesting: it tells about us boys, about school life, and also about the eccentric sage - Genie Hottabych. And Hottabych made a big mistake when he gave Volka some geography tips! Apparently, in long-gone times, even the Indian sages - the genies - knew geography very, very poorly. I wonder how old Hottabych would have given advice if Volka had passed the arithmetic exam? Hottabych probably didn’t even know arithmetic properly.

Indian way of multiplication.

Let's say we need to multiply 468 by 7. We write the multiplicand on the left and the multiplier on the right:

The Indians did not have a multiplication sign.

Now I multiply 4 by 7, we get 28. We write this number above the digit 4.

Now we multiply 8 by 7, we get 56. We add 5 to 28, we get 33; Let's erase 28, write down 33, write 6 above the number 8:

It turned out to be quite interesting.

Now we multiply 6 by 7, we get 42, we add 4 to 36, we get 40; We’ll erase 36 and write down 40; Let’s write 2 above the number 6. So, multiply 486 by 7, you get 3402:

The solution was correct, but not very quickly and conveniently! This is exactly how the most famous calculators of that time multiplied.

As you can see, old Hottabych knew arithmetic quite well. However, he recorded his actions differently than we do.

Long ago, more than one thousand three hundred years ago, Indians were the best calculators. However, they did not yet have paper, and all calculations were carried out on a small black board, writing on it with a reed pen and using very liquid white paint, which left marks that were easily erased.

When we write with chalk on a blackboard, it is a little reminiscent of the Indian way of writing: white marks appear on a black background, which are easy to erase and correct.

The Indians also made calculations on a white tablet sprinkled with red powder, on which they wrote signs with a small stick, so that white characters appeared on a red field. Approximately the same picture is obtained when we write with chalk on a red or brown board - linoleum.

The multiplication sign did not yet exist at that time, and only a certain gap was left between the multiplicand and the multiplier. The Indian way would be to multiply starting with units. However, the Indians themselves performed multiplication starting from the highest 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.

An example of multiplication in the Indian way.

Arabic method of multiplication.

Well, how, in the date itself, can you perform multiplication in the Indian way, if you write it down on paper?

This method of multiplication for writing on paper was adapted by the Arabs. The famous ancient Uzbek scientist Muhammad ibn Musa Alkhwariz-mi (Muhammad son of Musa from Khorezm, a city located on the territory of the modern Uzbek SSR) more than a thousand years ago performed multiplication on parchment like this:

Apparently, he did not erase unnecessary numbers (it is already inconvenient to do this on paper), but crossed them out; He wrote down the new numbers above the crossed out ones, of course, bit by bit.

An example of multiplication in the same way, making notes in a notebook.

This means 7264 X 8 = 58112. But how to multiply by a two-digit number, by a multi-digit number?

The method of multiplication remains the same, but the recording becomes much more complicated. For example, you need to multiply 746 by 64. First, multiply by 3 tens, it turns out

So 746 X 34 = 25364.

As you can see, crossing out unnecessary digits and replacing them with new digits when multiplying even by a two-digit number leads to too cumbersome recording. What happens if you multiply by a three- or four-digit number?!

Yes, the Arabic method of multiplication is not very convenient.

This method of multiplication persisted in Europe until the eighteenth century, for a full thousand years. It was called the cross method, or chiasmus, since the Greek letter X (chi) was placed between the numbers being multiplied, which was gradually replaced by an oblique cross. Now we clearly see that our modern method of multiplication is the simplest and most convenient, probably the best of all possible methods of multiplication.

Yes, our school method of multiplying multi-digit numbers itself is very good. However, multiplication can be written in another way. Perhaps the best way would be to do it, for example, like this:

This method is really good: multiplication begins from the highest digit of the multiplier, the lowest digit of incomplete products is written under the corresponding digit of the multiplier, which eliminates the possibility of error in the case when a zero occurs in any digit of the multiplier. This is approximately how Czechoslovakian schoolchildren write the multiplication of multi-digit numbers. That's interesting. And we thought that arithmetic operations can only be written in the way that is customary in our country.

A few more puzzles.

Here's your first, simple task: A tourist can walk 5 km in an hour. How many kilometers will he walk in 100 hours?

Answer: 500 kilometers.

And this is another big question! We need to know more precisely how the tourist walked for these 100 hours: without rest or with breaks. In other words, you need to know: 100 hours is the time a tourist travels or simply the time he spends on the road. A person is probably not able to be on the move for 100 hours in a row: that’s more than four days; and the speed of movement would decrease all the time. It’s another matter if the tourist walked with breaks for lunch, sleep, etc. Then in 100 hours of movement he can cover the entire 500 km; only he should be on the road not for four days, but for about twelve days (if he covers an average of 40 km per day). If he was on the road for 100 hours, then he could only cover approximately 160-180 km.

Various answers. This means that something needs to be added to the problem statement, otherwise it is impossible to give an answer.

Let us now solve the following problem: 10 chickens eat 1 kg of grain in 10 days. How many kilograms of grain will 100 chickens eat in 100 days?

Solution: 10 chickens eat 1 kg of grain in 10 days, which means that 1 chicken eats 10 times less in the same 10 days, that is, 1000 g: 10 = 100 g.

In one day, the chicken eats another 10 times less, that is, 100 g: 10 = 10 g. Now we know that 1 chicken eats 10 g of grain in 1 day. This means that 100 chickens a day eat 100 times more, that is

10 g X 100 = 1000 g = 1 kg. In 100 days they will eat another 100 times more, that is, 1 kg X 100 = 100 kg = 1 kg. This means that 100 chickens eat a whole centner of grain in 100 days.

There is a faster solution: there are 10 times more chickens and they need to be fed 10 times longer, which means that the total grain needed is 100 times more, that is, 100 kg. However, there is one omission in all these arguments. Let's think and find an error in reasoning.

: -Let us pay attention to the last reasoning: “100 chickens eat 1 kg of grain in one day, and in 100 days they will eat 100 times more. »

After all, in 100 days (that’s more than three months!) the chickens will grow up noticeably and will no longer eat 10 grams of grain per day, but 40-50 grams, since an ordinary chicken eats about 100 grams of grain per day. This means that in 100 days, 100 chickens will eat not 1 quintal of grain, but much more: two or three quintals.

And here is the last puzzle task for you about tying a knot: “There is a piece of rope stretched out in a straight line on the table. You need to take one end of it with one hand, the other end with the other hand and, without letting go of the ends of the rope from your hands, tie a knot. “It’s a well-known fact that some problems are easy to analyze, going from the data to the problem question, while others, on the contrary, are going from the problem question to the data.

Well, so we tried to analyze this problem, going from the question to the data. Let there already be a knot on the rope, and its ends are in your hands and are not released. Let's try to return from the solved problem to its data, to the original position: the rope lies stretched out on the table, and its ends are not released from the hands.

It turns out that if you straighten the rope without letting go of its ends from your hands, then the left hand, going under the outstretched rope and above the right hand, holds the right end of the rope; and the right hand, going above the rope and under the left hand, holds the left end of the rope

I think after this analysis of the problem, it became clear to everyone how to tie a knot on a rope; you need to do everything in reverse order.

Two more quick multiplication techniques.

I'll show you how to quickly multiply numbers such as 24 and 26, 63 and 67, 84 and 86, etc. p., that is, when there are equal numbers of tens in the factors, and ones together make exactly 10. Give examples.

* 34 and 36, 53 and 57, 72 and 78,

* You get 1224, 3021, 5616.

For example, you need to multiply 53 by 57. I multiply 5 by 6 (1 more than 5), it turns out 30 - so many hundreds in the product; I multiply 3 by 7, it turns out 21 - that’s how many units there are in the product. So 53 X 57 = 3021.

* How to explain this?

(50 + 3) X 57 = 50 X 57 + 3 X 57 = 50 X (50 + 7) +3 X (50 + 7) = 50 X 50 + 7 X 50 + 3 x 50 + 3 X 7 = 2500 + + 50 X (7 + 3) + 3 X 7 = 2500 + 50 X 10 + 3 X 7 = =: 25 hundred. + 5 hundred. +3 X 7 = 30 cells. + 3 X 7 = 5 X 6 cells. + 21.

Let's see how you can quickly multiply two-digit numbers within 20. For example, to multiply 14 by 17, you need to add the units 4 and 7, you get 11 - that is how many tens there will be in the product (that is, 10 units). Then you need to multiply 4 by 7, you get 28 - that’s how many units there will be in the product. In addition, exactly 100 must be added to the resulting numbers 110 and 28. This means that 14 X 17 = 100 + 110 + 28 = 238. In fact:

14 X 17 = 14 X (10 + 7) = 14 X 10 + 14 X 7 = (10 + + 4) X 10 + (10 + 4) X 7 = 10 X 10 + 4 X 10 + 10 X 7 + 4 X 7 = 100 +(4 + 7) X 10 + 4 X 7 = 100+ 110 + + 28.

After that, we solved the following examples: 13 x 16 = 100 + (3 + 6) X 10 + 3 x 6 = 100 + 90 + + 18 = 208; 14 X 18 = 100 + 120 + 32 = 252.

Multiplication on abacus

Here are a few techniques that, using them, anyone who knows how to quickly add on an abacus will be able to quickly perform examples of multiplication encountered in practice.

Multiplication by 2 and 3 is replaced by double and triple addition.

When multiplying by 4, first multiply by 2 and add this result to itself.

Multiplying a number by 5 is done on an abacus like this: move the entire number one wire higher, that is, multiply it by 10, and then divide this 10-fold number in half (like dividing by 2 using an abacus.

Instead of multiplying by 6, multiply by 5 and add what is being multiplied.

Instead of multiplying by 7, multiply by 10 and subtract the multiplied three times.

Multiplying by 8 is replaced by multiplying by 10 minus two multiplied.

They multiply by 9 in the same way: they replace it by multiplying by 10 minus one being multiplied.

When multiplying by 10, transfer, as we have already said, all the numbers one wire higher.

The reader will probably figure out for himself how to proceed when multiplying by numbers greater than 10, and what kind of substitutions will be most convenient here. The factor 11 must, of course, be replaced by 10 + 1. The factor 12 must be replaced by 10 + 2 or practically 2 + 10, that is, first they set aside the doubled number and then add the tenfold one. The multiplier of 13 is replaced by 10 + 3, etc.

Let's look at a few special cases for the first hundred multipliers:

It is easy to see, by the way, that with the help of abacus it is very convenient to multiply by numbers such as 22, 33, 44, 55, etc.; Therefore, when dividing factors, we must strive to use similar numbers with the same digits.

Similar techniques are also used when multiplying by numbers greater than 100. If such artificial techniques are tedious, then, of course, we can always multiply using abacus according to the general rule, multiplying each digit of the multiplier and writing down the partial products - this still gives some reduction in time .

"Russian" method of multiplication

You cannot multiply multi-digit numbers, even double-digit ones, unless you memorize all the results of multiplying single-digit numbers, that is, what is called the multiplication table. In the ancient “Arithmetic” of Magnitsky, which we have already mentioned, the need for a solid knowledge of the multiplication tables is glorified in the following verses (alien to modern ears):

Unless someone repeats tables and is proud, he cannot know by number what to multiply

And according to all sciences, I am not free from torment, Koliko does not teach Tuna and depresses me

And it won’t be beneficial if he forgets.

The author of these verses obviously did not know or overlooked that there is a way to multiply numbers without knowing the multiplication table. This method, similar to our school methods, was used in the everyday life of 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. Here's an example:

Dividing in half continues until) the pitch in the quotient turns out to be 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 repeating this operation many times, the desired product is obtained.

However, what to do if at the same time... Is it possible 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, throw one and divide the remainder in half; but then to the one number in the right column you will need to add all those numbers in this column that are opposite the odd numbers in the left column - the sum will be what you are looking for? l work. 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):

By adding up the numbers that have not been crossed out, we get a completely correct result: 17 + 34 + 272 = 32 What is this technique based on?

The correctness of the technique will become clear if we take into account that

19X 17 = (18+ 1)X 17= 18X17+17, 9X34 = (8 + 1)X34=; 8X34 + 34, etc.

It is clear that the numbers 17, 34, etc., lost when dividing an odd number in half, must be added to the result of the last multiplication to obtain the product.

Examples of accelerated multiplication

We mentioned earlier that there are also convenient ways to perform those individual multiplication operations into which each of the above techniques breaks down. Some of them are very simple and conveniently applicable; they make calculations so easy that it doesn’t hurt to remember them at all in order to use them in ordinary calculations.

This is, for example, the technique of cross multiplication, which is very convenient when working with two-digit numbers. The method is not new; it goes back to the Greeks and Hindus and in ancient times was called the “lightning method”, or “multiplication by a cross”. Now it is forgotten, and it doesn’t hurt to remind about it1.

Suppose you want to multiply 24X32. Mentally arrange the numbers according to the following scheme, one below the other:

Now we perform the following steps sequentially:

1)4X2 = 8 is the last digit of the result.

2)2X2 = 4; 4X3=12; 4+12=16; 6 - penultimate digit of the result; 1 remember.

3)2X3 = 6, and also the unit retained in mind, we have

7 is the first digit of the result.

We get all the digits of the product: 7, 6, 8 -- 768.

After a short exercise, this technique is learned very easily.

Another method, which consists in the use of so-called “additions”, is conveniently used in cases where the numbers being multiplied are close to 100.

Let's say you want to multiply 92X96. The “addition” for 92 to 100 will be 8, for 96 - 4. The action is carried out according to the following scheme: multipliers: 92 and 96 “additions”: 8 and 4.

The first two digits of the result are obtained by simply subtracting the “complement” of the multiplicand from the multiplier or vice versa; i.e., 4 is subtracted from 92 or 8 is subtracted from 96.

In both cases we have 88; the product of “additions” is added to this number: 8X4 = 32. We get the result 8832.

That the result obtained must be correct is clearly seen from the following transformations:

92x9b = 88X96 = 88(100-4) = 88 X 100-88X4

1 4X96= 4 (88 + 8)= 4X 8 + 88X4 92x96 8832+0

Another example. You need to multiply 78 by 77: factors: 78 and 77 “additions”: 22 and 23.

78 - 23 = 55, 22 X 23 = 506, 5500 + 506 = 6006.

Third example. Multiply 99 X 9.

multipliers: 99 and 98 “extras”: 1 and 2.

99-2 = 97, 1X2= 2.

In this case, we must remember that 97 here means the number of hundreds. So we add it up.

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 common among peasants in some provinces that did not require knowledge of the entire multiplication table. You just had to be able to multiply and divide by 2. This method was called 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 when it came to 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!”

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.

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 play 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