Tretyakova Anastasia, Tyomkina Alina
Purpose and objectives of the project:
Target: familiarization with various ways of multiplying natural numbers that are not used in the lessons, and their application in calculating numerical expressions.
Tasks:
Hypothesis: “Knowledge is revealed only by those.
Who knows with different numbers !!!
Municipal budgetary educational institution
secondary school No. 35 of the city district of Samara
Project on:
"Methods of multiplication
natural numbers"
The work was completed: students of the 5th "A" class
Tretyakova Anastasia,
Tyomkina Alina.
Scientific adviser:
mathematic teacher
Ruzanova I.M.
Samara, 2014
Purpose and objectives of the project:
Target: familiarization with various ways of multiplying natural numbers that are not used in the lessons, and their application in calculating numerical expressions.
Tasks:
Hypothesis: “Knowledge is revealed only by those.
Who knows with different numbers !!!
Pythagoras.
….. You will not be able to perform multiplications of multi-digit numbers - even two-digit ones - if you do not remember by heart all the results of multiplication of single-digit numbers, that is, what is called the multiplication table. In the old "Arithmetic" by Magnitsky, the need for a solid knowledge of the multiplication table is sung in such - it must be confessed, alien to modern hearing - verses:
Who else does not insist
tables and proud
Can't know
the number to multiply
And in all science, not free from torment,
Koliko does not take into account the Xia depresses
And it will not be in favor if I forget.
Magnitsky himself, the author of these verses, obviously did not know or lost sight of the fact that there are ways to multiply numbers even without knowing the multiplication table. These methods are not similar to our school methods, some were used in the everyday life of the Great Russian peasants and inherited by them from ancient times, some are used in our time.
At school, they study the multiplication table, and then teach children to multiply numbers in a column. Of course, this is not the only way to multiply. In fact, there are several dozen ways to multiply multi-digit numbers. In this work, we will give several methods of multiplication, perhaps they will seem simpler and you will use them.
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 another number. Example: 32 x 13
Multiplier =32 | Multiplier = 13 |
Table 1.
Bisection (see the left half of Table 1) is continued until the quotient is 1, while doubling another number in parallel (the right side of Table 1). 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 therefore clear that as a result of repeated repetition of this operation, the desired product is obtained:(32 x 13) = (1 x 416)
Those who are especially attentive will notice "What about odd numbers that are not a multiple of 2?".
So let's say we need to multiply two numbers: 987 and 1998. We write one on the left, and the second on the right on one line. We will divide the left number by 2, and multiply the right number by 2 and write the results in a column. If there is a remainder during division, it is discarded.
We continue the operation until 1 remains on the left. Then we cross out those lines in which there are even numbers on the left and add the remaining numbers in the right column. This is the desired work. Given a graphic illustration of this description. (See Table 2.)
Table 2.
1 2 3
4 5 6
7 8 9
This is the well-known Pythagorean Square, reflecting the world number system, consisting of nine digits: from 1 to 9. In modern terms, it is a nine-bit numerical matrix in which the numbers that are the basis for further calculations of any complexity are arranged in ascending order. The square of Pythagoras is also called the Ennead, and the triple of numbers is called the triad. You can consider triplets of numbers located horizontally (123, 456, 789) and vertically (147, 258, 369). Moreover, written in this way, triples of numbers begin to designate special numbers that obey the laws of mathematical proportion and harmony.
Recall the main rule of ancient Egyptian mathematics, which says that multiplication is done by doubling and adding the results obtained; that is, each doubling is the addition of a number to itself. Therefore, it is interesting to look at the result of such a doubling of numbers and numbers, but obtained by the modern method of folding "in a column", known even in the elementary grades of the school. This will resemble the Egyptian number system, in fact, with the difference that all numbers or numbers are written in one column (without indicating this or that action in the next column - like the Egyptians).
Let's start with the numbers that make up the Pythagorean Square: from 1 to 9.
1 2 3 4 5 6 7 8 9
2 4 6 8 10 12 14 16 18
3 6 9 12 15 18 21 24 27
4 8 12 16 20 24 28 32 36
5 10 15 20 25 30 35 40 45
6 12 18 24 30 36 42 48 54
7 14 21 28 35 42 49 56 63
8 16 24 32 40 48 56 64 72
9 18 27 36 45 54 63 72 81
10 20 30 40 50 60 70 80 90
Digit 1: the usual consecutive series of numbers.
Number 9: left column - a clear ascending row ("stream").
the right column is a clear descending row of consecutive digits. Let us agree to call an ascending series, the values of the numbers in which increase from top to bottom; in descending - on the contrary: the values of numbers decrease from top to bottom.
Digit 2: even numbers 2,4,6,8 ("in the period") are repeated in the right column.
Number 8: the same repetition - only in reverse order - 8,6,4,2.
Numbers 4 and 6: even numbers "in the period" 4,8,2,6 and 6,2,8,4.
Number 5: obeys the rule of adding the number 5 - alternating 5 and 0.
Number 3: the right column is a descending row no longer of numbers, but of numbers forming triples of vertical rows in the Pythagorean square - 369, 258, 147. Moreover, the countdown goes “from the right corner of the square” or from right to left. The ascending-descending series rule adopted above also applies here. But the ascending row is a movement from the triple of numbers 147 to the triple of 369; descending - from 369 to 147.
Digit 7: ascending row of numbers 147,258,369 from the "left corner" or from left to right. However, it all depends on how the nine-digit numerical matrix itself is depicted - where to put the number 1.
Students will be able to learn how to verbally add and multiply millions, billions and even sextillions with quadrillions. And the candidate of philosophical sciences Vasily Okoneshnikov, who is also the inventor of a new system of mental counting, will help them in this. The scientist claims that a person is able to memorize a huge amount of information, the main thing is how to arrange this information.
According to the scientist himself, the nine-decimal system is the most advantageous in this regard - all data is simply placed in nine cells arranged like buttons on a calculator.
According to the scientist, before becoming a computing "computer", it is necessary to memorize the table he created. The numbers in it are distributed in nine cells is not easy. According to Okoneshnikov, the human eye and his memory are so cunningly arranged that the information arranged according to his method is remembered, firstly, faster, and secondly, tightly.
The table is divided into 9 parts. They are arranged according to the principle of a mini calculator: on the left in the lower corner "1", on the right in the upper corner "9". Each part is a multiplication table of numbers from 1 to 9 (again, in the lower left corner by 1, next to the right by 2, etc., using the same "push-button" system). How to use them?
For example , it is required to multiply 9 by 842 . We immediately recall the big “button” 9 (it is at the top right and on it we mentally find small buttons 8,4,2 (they are also located like on a calculator). The numbers 72, 36, 18 correspond to them. We add the resulting numbers especially: the first digit is 7 ( remains unchanged), 2 is mentally added to 3, we get 5 - this is the second digit of the result, 6 is added to 1, we get the third digit -7, and the last digit of the desired number remains - 8. The result is 7578.
If the addition of two digits results in a number greater than nine, then its first digit is added to the previous digit of the result, and the second is written in its “own” place.
With the help of Okoneshnikov's matrix table, according to the author himself, it is possible to study foreign languages, and even the periodic table. The new methodology was tested in several Russian schools and universities. The Ministry of Education of the Russian Federation allowed the publication of a new multiplication table in squared notebooks along with the usual Pythagorean table - so far just for acquaintance.
Example: 15647 x 5
In ancient India, two methods of multiplication were used: grids and galleys. At first glance, they seem very complicated, but if you follow the steps in the suggested exercises, you can see that it is quite simple.
We multiply, for example, numbers 6827 and 345:
1. We draw a square grid and write one of the numbers above the columns, and the second in height. In the proposed example, one of these grids can be used.
Mesh 1 Mesh 2
2. Having chosen the grid, we multiply the number of each row sequentially by the numbers of each column. In this case, we sequentially multiply 3 by 6, by 8, by 2 and by 7. Look at this diagram how the product is written in the corresponding cell.
Grid 1
3. See what the grid looks like with all the filled cells.
Grid 1
4. Finally, add up the numbers, following the diagonal stripes. If the sum of one diagonal contains tens, then we add them to the next diagonal.
Mesh1
See how the results of adding numbers along the diagonals (they are highlighted in yellow) form a number 2355315 , which isproduct of numbers 6827 and 345, i.e. 6827 x 345 = 2355315.
Ancient Egyptian multiplication is a sequential method of multiplying two numbers. To multiply numbers, they did not need to know multiplication tables, but it was enough just to be able to decompose numbers into multiple bases, multiply these multiples and add. The Egyptian method involves decomposing the smallest of two factors into multiples and then multiplying them sequentially by the second factor (see example). This method can still be found today in very remote regions.
Decomposition. The Egyptians used a system of expanding the smallest factor into multiples, the sum of which would be the original number.
To correctly select a multiple, you had to know the following table of values:
1 x 2 = 2 2 x 2 = 4 4 x 2 = 8 8 x 2 = 16 16 x 2 = 32
Example expansions of the number 25: The multiple for the number "25" is 16; 25 - 16 = 9. The multiple for the number "9" is 8; 9 - 8 = 1. The multiple for the number "1" is 1; 1 - 1 = 0. Thus, "25" is the sum of three terms: 16, 8 and 1.
Example: multiply "13" by "238 » . It is known that 13 = 8 + 4 + 1. Each of these terms must be multiplied by 238. We get: ✔ 1 x 238 = 238 ✔ 4 x 238 = 952 ✔ 8 x 238 = 190413 x 238 = (8 + 4 + 1) x 238 = 8 x 238 + 4 x 238 + 1 x 238 = 1904 + 952 + 238 = 3094.
Now let's imagine the method of multiplication, vigorously discussed on the Internet, which is called Chinese. When multiplying numbers, the points of intersection of lines are considered, which correspond to the number of digits of each digit of both factors.
Example: multiply 21 by 13 . In the first multiplier there are 2 tens and 1 unit, so we build 2 parallel lines and 1 line at a distance.
The second multiplier has 1 ten and 3 units. We build parallel lines 1 and at a distance 3 lines intersecting the lines of the first factor.
The lines intersect at points, the number of which is the answer, that is 21 x 13 = 273
It’s funny and interesting, but it’s somehow long and uninteresting to draw 9 straight lines when multiplied by 9, and then count the intersection points ... In general, you can’t do without a multiplication table!
The Japanese multiplication method is a graphical method using circles and lines. No less funny and interesting than Chinese. Even something like him.
Example: multiply 12 by 34. Since the second factor is a two-digit number, and the first digit of the first factor 1 , we build two single circles in the top row and two binary circles in the bottom row, since the second digit of the first factor is 2 .
12 x 34
Since the first digit of the second factor 3 and the second 4 , we divide the circles of the first column into three parts, the second column into four.
12 x 34
The number of parts into which the circles are divided is the answer, that is 12 x 34 = 408.
While working on this topic, we learned that there are many different, fun and interesting ways to multiply. Some are still in use today in various countries. But not all methods are convenient to use, especially when multiplying multi-digit numbers. In general, you still need to know the multiplication table!
This work can be used for classes in mathematical circles, additional classes with children after school hours, as additional material in the lesson on the topic "Multiplication of natural numbers". The material is presented in an accessible and interesting way, which will attract the attention and interest of students to the subject of mathematics.
published 20.04.2012
Dedicated to Elena Petrovna Karinskaya
,
to my school math teacher and class teacher
Alma-Ata, ROFMSH, 1984–1987
“Science only achieves perfection when it succeeds in using mathematics”. Karl Heinrich Marx
these words were inscribed over the blackboard in our math classroom ;-)
Informatics lessons(lecture materials and workshops)
What is multiplication?
This is the addition action.
But not very pleasant
Because many times...
Tim Sobakin
Let's try to do this
nice and fun ;-)
I offer green pages readers two multiplication methods that do not use the multiplication table ;-) I hope that computer science teachers will like this material, which they can use when conducting extracurricular activities.
This method was common in everyday life of Russian peasants and 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 doubling the other number, multiplication table in this case unnecessarily :-)
Bisection is continued until the quotient is 1, while another number is doubled in parallel. 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 therefore clear that as a result of repeated repetition of this operation, the desired product is obtained.
However, what to do if you have to divide an odd number in half? In this case, we discard one from the odd number and divide the remainder in half, while to the last number of the right column it will be necessary to add all those numbers of this column that are against the odd numbers of the left column - the sum will be the desired product (Figures: 2, 3).
In other words, we cross out all lines with even left numbers; leave and then add not crossed out numbers 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 get the product.
The Russian way of multiplication is both elegant and extravagant at the same time ;-)
§ Logic puzzle about Serpent Gorynych and famous Russian heroes on the green page “Which of the heroes defeated the Serpent Gorynych?”
solution of logical problems by means of logic algebra
For those who love to learn! For those who are happy gymnastics for the mind ;-)
§ Solving logical problems in a tabular way
We continue the conversation :-)
My son introduced me to this method of multiplication by giving me a few sheets from a notebook with ready-made solutions in the form of intricate drawings. The process of decoding the algorithm began to boil drawing method of multiplication :-) For clarity, I decided to resort to the help of colored pencils, and ... the ice has broken gentlemen of the jury :-)
I bring to your attention three examples in color pictures (in the upper right corner test post).
Example #1: 12
× 321
= 3852
We draw first number top to bottom, left to right: one green stick ( 1
); two orange sticks ( 2
). 12
drew :-)
We draw second number from bottom to top, left to right: three blue sticks ( 3
); two red ones 2
); one lilac ( 1
). 321
drew :-)
Now we will walk along the drawing with a simple pencil, divide the points of intersection of stick numbers into parts and proceed to counting the points. Moving from right to left (clockwise): 2 , 5 , 8 , 3 . number-result 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 points 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 the multiplication is flying :-) and it works in autopilot mode: draw, count dots, we don’t remember about the multiplication table, it seems like we don’t know it at all :-)))
To be honest, by checking drawing method of multiplication and turning to multiplication by a column, and more than once or twice, to my shame, I noticed some slowdowns, indicating that my multiplication table had rusted in some places: - (and you shouldn’t forget it. When working with more “serious” numbers drawing method of multiplication became too cumbersome, and multiplication by a column went to 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, memory trains - the multiplication table does not allow to be forgotten :-) And therefore, I strongly recommend that you and yourself, if possible, forget about calculators in phones and computers ;-) and periodically indulge yourself with multiplication by a column. Otherwise, the hour is not even and the plot from the movie "Rise of the Machines" will unfold not on the cinema screen, but in our kitchen or lawn next to the house ...
Three times over the left shoulder ... knock on wood ... :-))) ... and most importantly Don't forget about gymnastics for the mind!
For the curious: Multiplication denoted by the sign [ × ] or [ · ]
The sign [ × ] was introduced by an English mathematician William Outred in 1631.
The sign [ ] was introduced by a German scientist Gottfried Wilhelm Leibniz in 1698.
In the literal designation, these signs are omitted and instead of a × b or a · b write ab.
To the webmaster's box: Some mathematical symbols in HTML
| ° | ° or ° | degree |
| ± | ± or ± | plus or minus |
| ¼ | ¼ or ¼ | fraction - one quarter |
| ½ | ½ or ½ | fraction - one second |
| 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 stroke - seconds and inches |
| ≈ | ≈ or ≈ | approximate equal sign |
| ≠ | ≠ or ≠ | sign not equal |
| ≡ | ≡ 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 |
problem: understand the types of multiplication
Target: familiarization with various ways of multiplying natural numbers that are not used in the lessons, and their application in calculating numerical expressions.
Tasks:
1. Find and analyze various ways of multiplication.
2. Learn to demonstrate some of the methods of multiplication.
3. Talk about new methods of multiplication and teach students how to use them.
4. Develop independent work skills: information search, selection and design of the material found.
5. Experiment "which way is faster"
Hypothesis Q: Do I need to know the multiplication table?
Relevance: Recently, students trust gadgets more than themselves. And that's why they count only on calculators. We wanted to show that there are different ways of multiplication, so that it would be easier for students to count, and it would be interesting to learn.
INTRODUCTION
You can't do multi-digit multiplications - even two-digit multiplications - unless you memorize all the results of single-digit multiplications, that is, what is called the multiplication table.
At different times, different peoples owned different ways of multiplying natural numbers.
Why now all peoples use one method of multiplying by a “column”?
Why did people abandon the old methods of multiplication in favor of the modern one?
Do the forgotten methods of multiplication have the right to exist in our time?
To answer these questions, I did the following:
1. Using the Internet, I found information about some of the multiplication methods that were used before .;
2. Studied the literature offered by the teacher;
3. I solved a couple of examples in all the ways studied in order to find out their shortcomings;
4) Identified among them the most effective;
5. Conducted an experiment;
6. Draw conclusions.
1. Find and analyze various ways of multiplication.
Finger multiplication.
The ancient Russian method of multiplying on fingers is one of the most common methods that Russian merchants have successfully used for many centuries. They learned to multiply single-digit numbers from 6 to 9 on their fingers. At the same time, it was enough to master the initial skills of finger counting in “ones”, “pairs”, “triples”, “fours”, “fives” and “tens”. The fingers here served as an auxiliary computing device.
To do this, on one hand they extended as many fingers as the first factor exceeds the number 5, and on the second they did the same for the second factor. The rest of the fingers were bent. Then the number (total) of outstretched fingers was taken and multiplied by 10, then the numbers were multiplied showing how many fingers were bent on the hands, and the results were added up.
For example, let's multiply 7 by 8. In the considered example, 2 and 3 fingers will be bent. If we add the number of bent fingers (2+3=5) and multiply the number of not bent fingers (2 3=6), then we will get the numbers of tens and units of the desired product, respectively 56 . So you can calculate the product of any single-digit numbers greater than 5.
Ways to multiply numbers in different countries
Multiply by 9.
Multiplication for the number 9 - 9 1, 9 2 ... 9 10 - is easier to fade from memory and more difficult to manually recalculate by addition, but it is for the number 9 that multiplication is easily reproduced "on the fingers". Spread your fingers on both hands and turn your palms away from you. Mentally assign numbers from 1 to 10 to the fingers, starting with the little finger of the left hand and ending with the little finger of the right hand (this is shown in the figure). 
Who invented finger multiplication
Let's say we want to multiply 9 by 6. We bend a finger with a number equal to the number by which we will multiply the nine. In our example, you 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 - the number of units. On the left, we have 5 fingers not bent, on the right - 4 fingers. Thus, 9 6=54. The figure below shows in detail the whole principle of "calculation". 
Multiplication in an unusual way
Another example: you need to calculate 9 8=?. Along the way, we will say that fingers may not necessarily act as a "calculating machine". Take, for example, 10 cells in a notebook. We cross out the 8th cell. There are 7 cells on the left, 2 cells on the right. So 9 8=72. Everything is very simple.
7 cells 2 cells.
Indian way of multiplication.
The most valuable contribution to the treasury of mathematical knowledge was made in India. The Hindus proposed the way we use to write numbers using ten signs: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.
The basis of this method is the idea that the same digit stands for units, tens, hundreds or thousands, depending on where this figure occupies. The place occupied, in the absence of any digits, is determined by zeros assigned to the numbers.
The Indians thought well. They came up with a very simple way to multiply. They performed multiplication, starting with the highest order, and wrote down incomplete products just above the multiplicand, bit by bit. At the same time, the senior digit of the complete product was immediately visible and, in addition, the omission of any digit was excluded. The multiplication sign was not yet known, so they left a small distance between the factors. For example, let's multiply them in the way 537 by 6:
(5 ∙ 6 =30) 30
(300 + 3 ∙ 6 = 318) 318
(3180 +7 ∙ 6 = 3222) 3222
6
Multiplication using the "LITTLE CASTLE" method.
Multiplication of numbers is now studied in the first grade of the school. But in the Middle Ages, very few mastered the art of multiplication. A rare aristocrat could boast of knowing the multiplication table, even if he graduated from a European university.
Over the millennia of the development of mathematics, many ways to multiply numbers have been invented. The Italian mathematician Luca Pacioli in his treatise "The sum of knowledge in arithmetic, ratios and proportionality" (1494) gives eight different methods of multiplication. The first of them is called "Little Castle", and the second is no less romantic called "Jealousy or Lattice Multiplication".
The advantage of the “Little Castle” multiplication method is that the digits of the highest digits are determined from the very beginning, and this can be important if you need to quickly estimate the value.
The digits of the upper number, starting from the most significant digit, are alternately multiplied by the lower number and written in a column with the addition of the required number of zeros. Then the results are added up. 
Ways to multiply numbers in different countries
Multiplying numbers using the "jealousy" method.
"Methods of multiplication The second method is romantically called 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 for the multiplier and multiplier. Then the square cells are divided diagonally, and “... it turns out a picture that looks like lattice shutters, blinds,” writes Pacioli. “Such shutters were hung on the windows of Venetian houses, preventing passers-by from seeing the ladies and nuns sitting at the windows.”
Let's multiply 347 by 29 in this way. Let's draw a table, write the number 347 above it, and the number 29 on the right.
In each line we write the product of the numbers above this cell and to the right of it, while the number of tens of the product is written above the slash, and the number of units is below it. Now add up the numbers in each slash by doing this operation, from right to left. If the amount is less than 10, then we write it under the bottom number of the band. If it turns out to be more than 10, then we write only the number of units of the sum, and add the number of tens to the next amount. As a result, we get the desired product 10063.
Peasant way of multiplication.
The most, in my opinion, "native" and easy way of multiplication is the method used by Russian peasants. This technique generally does not require knowledge of the multiplication table beyond the number 2. Its essence is that the multiplication of any two numbers is reduced to a series of successive divisions of one number in half while doubling another number. Bisection continues until the quotient is 1, while doubling another number in parallel. The last doubled number gives the desired result.
In the case of an odd number, one must discard the unit and divide the remainder in half; but on the other hand, to the last number of the right column it will be necessary to add all those numbers of this column that are against the odd numbers of the left column: the sum will be the desired product
The product of all pairs of corresponding numbers is the same, so
37 ∙ 32 = 1184 ∙ 1 = 1184
In the case when one of the numbers is odd or both numbers are odd, proceed as follows:
384 ∙ 1 = 384
24 ∙ 17 = 24∙(16+1)=24 ∙ 16 + 24 = 384 + 24 = 408
A new way to multiply.
An interesting new way of multiplying has recently been reported. Vasily Okoneshnikov, the inventor of the new mental counting system, claims that a person is able to memorize a huge amount of information, the main thing is how to arrange this information. According to the scientist himself, the nine-decimal system is the most advantageous in this regard - all data is simply placed in nine cells arranged like buttons on a calculator.
It is very easy to count according to such a table. For example, let's multiply the number 15647 by 5. In the part of the table corresponding to the five, we select the numbers corresponding to the digits of the number in order: one, five, six, four and seven. We get: 05 25 30 20 35
The left digit (in our example, zero) is left unchanged, and the following numbers are added in pairs: five with two, five with three, zero with two, zero with three. The last digit is also unchanged.
As a result, we get: 078235. The number 78235 is the result of multiplication.
If, when adding two digits, a number exceeding nine is obtained, then its first digit is added to the previous digit of the result, and the second is written in “its” place.
Conclusion.
While working on this topic, I learned that there are about 30 different, fun and interesting ways to multiply. Some are still in use today in various countries. I chose some interesting ways for myself. But not all methods are convenient to use, especially when multiplying multi-digit numbers.
Multiplication methods
The purpose of the work: To explore and show unusual ways of multiplication. Tasks: To find unusual ways of multiplication. Learn to apply 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 for 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 methods in the formation of computational skills enhances students' interest in mathematics and contributes to the development of mathematical abilities
In math class, we learned an unusual way of multiplication by a column. We liked it and decided to learn other ways to multiply natural numbers. We asked our classmates if they knew other ways of counting? Everyone spoke only about those methods that are studied at school. It turned out that all our friends do not know anything about other methods. In the history of mathematics, about 30 methods of multiplication are known, differing in the recording scheme or in the very course of the calculation. The multiplication method "in a column", which we study at school, is one of the ways. But is it the most efficient way? Let's get a look! Introduction
This is one of the most common methods that Russian merchants have successfully used for many centuries. The principle of this method: multiplication on the fingers of single-digit numbers from 6 to 9. 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 rest of the fingers were bent. Then the number (total) of outstretched fingers was taken and multiplied by 10, then the numbers were multiplied showing how many fingers were bent on the hands, and the results were added up. For example, let's multiply 7 by 8. In the considered example, 2 and 3 fingers will be bent. If we add the number of bent fingers (2 + 3 = 5) and multiply the number of not bent fingers (23 = 6), then we get the numbers of tens and units of the desired product 56, respectively. So you can calculate the product of any single-digit numbers greater than 5.
The multiplication for the number 9 is very easy to reproduce "on the fingers". Spread the fingers on both hands and turn the palms away from you. Mentally assign the numbers from 1 to 10 to the fingers in sequence, starting with the little finger of the left hand and ending with the little finger of the right hand. Let's say we want to multiply 9 by 6. We bend a finger with a number equal to the number by which we will multiply the nine. In our example, you 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 - the number of ones. On the left, we have 5 fingers not bent, on the right - 4 fingers. Thus, 9 6=54.
The "Little Castle" multiplication method The advantage of the "Little Castle" multiplication method is that the high-order digits are determined from the very beginning, which is important if you need to quickly estimate the value. The digits of the upper number, starting from the most significant digit, are alternately multiplied by the lower number and written in a column with the addition of the required number of zeros. Then the results are 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 for the multiplier and multiplier. Then the square cells are divided diagonally, and “... a picture is obtained that looks like lattice shutters-blinds, - writes Pacioli. - Such shutters were hung on the windows of Venetian houses ... "
Lattice multiplication = +1 +2
Peasant method This is the method of the Great Russian peasants. Its essence lies in the fact that the multiplication of any numbers is reduced to a series of successive divisions of one number in half, while 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 - in one line. We draw the second figure - five lines. Step 2. second number 23: Draw the first number - two lines. We draw the second figure - three lines. Step 3. Count the number of points in groups. Step 4. The result is 345. Let's multiply two two-digit numbers: 15 * 23
Indian multiplication method (cross) 24 and X 3 2 1)4x2=8 - the last digit of the result; 2)2x2=4; 4x3=12; 4+12=16 ; 6 - the penultimate digit of the result, remember the unit; 3) 2x3=6 and even the number kept in mind, we have 7 - this is the first digit of the result. We get all the digits of the product: 7,6,8. Answer: 768.
Indian method of multiplication = = = = 3822 The basis of this method is the idea that the same digit stands for units, tens, hundreds or thousands, depending on where this figure occupies. The place occupied, in the absence of any digits, is determined by zeros assigned to the numbers. we start the multiplication from the highest order, 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, the omission of any digit is excluded. The multiplication sign was not yet known, so a small distance was left between the factors
Base Number Multiply 18*19 20 (base number) * 2 1 (18-1)*20 = Answer: 342 Short Note: 18*19 = 20*17+2 = 342
New multiplication method X = , 5+2, 5+3, 0+2, 0+3, 5
Conclusion: Having learned to count in all the presented ways, we came to the conclusion that the simplest methods are those that we study at school, or maybe we just got used to them Of all the considered unusual counting methods, the method of graphic multiplication seemed more interesting. We showed it to our classmates and they also liked it very much. The “doubling and doubling” method used by Russian peasants seemed to be the simplest.
Conclusion Describing the ancient methods of calculations and modern methods of fast counting, 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. Studying the 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. But, we think that our method of multiplication in a column is not perfect and you can come up with even faster and more reliable methods. It is possible that the first time many will not be able to quickly, on the go, perform these or other calculations. It does not matter. Constant computational training is needed. It will help you develop useful mental counting skills!
Materials used: html Encyclopedia for children. "Mathematics". – M.: Avanta +, – 688 p. Encyclopedia “I know the world. Mathematics". - M .: Astrel Ermak, Perelman Ya.I. Quick account. Thirty simple methods of mental counting. L., s.
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“Counting and calculations are the basis of order in the head.”
Pestalozzi
Target:
Lesson progress
The relevance of using quick counting techniques.
In modern life, each person often has to perform a huge amount of calculations and calculations. Therefore, the purpose of my work is to show easy, fast and accurate counting methods that will not only help you during any calculations, but will cause considerable surprise among friends and comrades, because the free performance of counting operations can largely indicate the extraordinaryness of your intellect. A fundamental element of a computing culture is conscious and strong computing skills. The problem of forming a computational culture is relevant for the entire school course in mathematics, starting from the primary grades, and requires not just mastering computational skills, but using them in various situations. The possession of computational skills and abilities is of great importance for the assimilation of the material being studied, it allows one to cultivate valuable labor qualities: a responsible attitude to one's work, the ability to detect and correct mistakes made in the work, accurate execution of the task, and a creative attitude to work. However, recently the level of computational skills, expression transformations has a pronounced downward trend, students make a lot of mistakes when calculating, they increasingly use a calculator, do not think rationally, which negatively affects the quality of education and the level of mathematical knowledge of students in general. One of the components of computing culture is verbal counting which is of great importance. The ability to quickly and correctly make simple calculations “in the mind” is necessary for every person.
Ancient ways of multiplying numbers.
1. The old way of multiplying by 9 on your fingers
It's simple. To multiply any number between 1 and 9 by 9, look at the hands. Bend the finger that corresponds to the number being multiplied (for example 9 x 3 - bend the third finger), count the fingers up to the crooked finger (in the case of 9 x 3 it is 2), then count after the crooked finger (in our case 7). The answer is 27.
2. Multiplication by the Ferrol method.
To multiply the units of the multiplication product, multiply the units of factors, to get tens, multiply the tens of one by the units of the other and vice versa and add the results, to get hundreds, multiply the tens. Using the Ferrol method, it is easy to verbally multiply two-digit numbers from 10 to 20.
For example: 12x14=168
a) 2x4=8, write 8
b) 1x4+2x1=6, write 6
c) 1x1=1, write 1.
3. Japanese multiplication method
This technique resembles multiplication by a column, but it takes quite a long time.
Use of reception. Let's say we need to multiply 13 by 24. Let's draw the following picture:
This drawing consists of 10 lines (the number can be any)
(intersections in the figure are indicated by dots)
Number of crossings:
1) Crossings in the upper left edge (2) - the first number of the answer
2) The sum of the intersections of the lower left and upper right edges (6 + 4) - the second number of the answer
3) Intersections in the lower right edge (12) - the third number of the answer.
It turns out: 2; 10; 12.
Because the last two numbers are two-digit and we cannot write them down, then we write down only units, and add tens to the previous one.
4. Italian way of multiplication (“Grid”)
In Italy, as well as in many countries of the East, this method has become very famous.
Reception usage:
For example, let's multiply 6827 by 345.
1. We draw a square grid and write one of the numbers above the columns, and the second in height.
2. Multiply the number of each row sequentially by the numbers of each column.
If multiplication produces a single-digit number, we write 0 at the top, and this number at the bottom.
(As in our example, when multiplying 2 by 3, we got 6. At the top, we wrote 0, and at the bottom 6)
3. Fill in the entire grid and add up the numbers following the diagonal stripes. We begin to fold from right to left. If the sum of one diagonal contains tens, then we add them to the units of the next diagonal.
Answer: 2355315.
5. Russian way of multiplication.
This multiplication technique was used by Russian peasants about 2-4 centuries ago, and was developed in ancient times. The essence of this method is: “By how much we divide the first factor, we multiply the second by so much.” Here is an example: We need to multiply 32 by 13. This is how our ancestors would have solved this example 3-4 centuries ago:
Bisection continues until the quotient is 1, while doubling another number in parallel. The last doubled number gives the desired result. It is not difficult to understand what this method is based on: the product does not change if one factor is halved, and the other is doubled. It is clear, therefore, that as a result of repeated repetition of this operation, the desired product is obtained
However, what to do if you have to divide an odd number in half? The popular way easily gets out of this difficulty. It is necessary, - says the rule, - in the case of an odd number, discard the unit and divide the remainder in half; but on the other hand, to the last number of the right column it will be necessary to add all those numbers of this column that stand against the odd numbers of the left column: the sum will be the desired product. In practice, this is done in such a way that all lines with even left numbers are crossed out; only those that contain an odd number to the left remain. Here is an example (asterisks indicate that this line should be crossed out):
Adding the uncrossed numbers, we get a completely correct result:
Answer: 323.
6. Indian way of multiplication.
This method of multiplication was used in ancient India.
To multiply, for example, 793 by 92, we write one number as a multiplier and under it another as a factor. To make it easier to navigate, you can use the grid (A) as a reference.
Now we multiply the left digit of the multiplier by each digit of the multiplicand, that is, 9x7, 9x9 and 9x3. We write the resulting products in the grid (B), bearing in mind the following rules:
Repeat the whole process with other multiplier numbers, following the same rules (C).
Then we add the numbers in the columns and get the answer: 72956.
As you can see, we get a large list of works. The Indians, who had great practice, wrote each figure not in the corresponding column, but on top, as far as possible. Then they added up the numbers in the columns and got the result.
Conclusion
We have entered the new millennium! Grandiose discoveries and achievements of mankind. We know a lot, we can do a lot. It seems something supernatural that with the help of numbers and formulas one can calculate the flight of a spaceship, the “economic situation” in the country, the weather for “tomorrow”, describe the sound of notes in a melody. We know the saying of the ancient Greek mathematician, 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 that reigns in the world, the soul of the cosmos.
Describing the ancient methods of calculations and modern methods of quick counting, I tried to show that both in the past and in the future, one cannot do without mathematics, a science created by the human mind.
“Whoever has been involved in mathematics since childhood develops attention, trains the brain, his will, cultivates perseverance and perseverance in achieving the goal.”(A.Markushevich)
Literature.
