Detailed solution part 1 (page) 3 around the world workbook for grade 4 students, authors Vinogradova N.F., Kalinova G.S. 2016

• Gdz around the world for grade 4 can be found
• Gdz workbook on the world around you for grade 4 can be found

1. Task. Consider the drawings. What important object of wildlife has not been drawn by a person? Draw this object.

Answer. This object is a person

2. Task. Complete the diagram.

3. Task. Write what substances the human body exchanges with the environment.

Answer. Nutrients - proteins, fats, carbohydrates, mineral salts, vitamins, water - enter the human body with food. In the process of breathing, oxygen enters the body, and oxygen is also partially absorbed by the skin.

From the body stand out: undigested food residues, urine, which is formed in the kidneys; in the process of respiration - carbon dioxide and water; skin secretes sweat, fat; the lacrimal gland secretes tear fluid that wets the eye; salivary glands - saliva.

4. Task. Cross out the names of those organs that do not belong to the nervous system.

Answer: heart (cross out), trachea (cross out), muscles (cross out).

5. Task. Complete the chart.

6. Task. Write the numbers that are indicated in the figure: brain, spinal cord, nerves.

Answer. Brain - 1, spinal cord - 2, nerves - 3.

7. Task. Explain why nerves are compared to electrical wires. Prepare an oral presentation.

Answer. In the human body, information is transmitted along the nerves along the nerves. A nerve impulse is nothing more than an electrical discharge. The peculiarity of the transmission is that this discharge is not transmitted from nerve to nerve directly, but through chemicals located on the border between the nerves.

Exercise. Express your opinion. From the brain and spinal cord, signals are transmitted to the organs at a very high speed. What significance does this have for a person?

Answer. Signals are transmitted at high speed in order to enable the body to respond in a timely manner to any stimuli. For example, a person touches a hot object and immediately withdraws his hand. The eye saw a flying mote and immediately closed. You were told something and you immediately answered. Thus, we protect ourselves from any danger, navigate the environment, lead a certain lifestyle.

8. Task. Label the parts of the skeleton indicated by numbers in the figure.

2. Spine

3. The ribs that make up the chest

4. Upper free limb (shoulder, forearm, hand)

5. Lower free limb (thigh, lower leg, foot)

9. Task. Answer the questions. Discuss the answers.

How do you understand the expression: "He has good posture"?

Asya spends all her free time in front of the TV or at the computer, and Alyosha loves to play football. Explain which of the children will be developed physically.

Good posture - this means the correct location of the parts of the skeleton relative to each other and in space. There is no curvature of the spine, defects of individual bones of the skeleton. This is achieved through physical education and sports, constant concern for one's physical form, adherence to work culture, and the ability to choose a working posture.

Alyosha, of course, is better developed physically. This is due to the fact that regular physical education and sports are necessary for the development of the skeleton and muscles (musculoskeletal system). When a person goes in for sports, blood vessels are regularly dilated in his muscles and bones, through which building substances (baks, fats, carbohydrates, mineral salts) enter, as well as oxygen, which ensures metabolism. Consequently, bones and muscles will grow. During physical education, the nervous system gives signals that satisfy the needs of the muscles in development. That is, the whole organism is tuned for development.

Asya is not physically engaged, so her development will lag behind Alyosha.

10. Task. Mark the correct answers to the question: “What contributes to the development of the human skeleton and muscles?”

Physical exercise and sports games (correct).

Proper nutrition (correctly).

Question. How was task 10 completed? Mark only one statement.

11. Task. Explain how you understand the words of the ancient Greek scientist Aristotle: "Nothing exhausts and destroys a person like prolonged physical inactivity."

Answer. In order for the human body to be in good physical shape, to maintain working capacity for a long time, to cope with various diseases, it is necessary to constantly engage in physical culture and sports. Classes allow the muscles to be in the right tone, the nervous system to be ready for a quick response to external manifestations, and to perform a large amount of physical work. In trained muscles, muscle fibers, blood vessels are elastic, the heart muscle is strong, the vital volume of the lungs is significant.

If you do not engage in physical culture, the muscles become flabby, more cells die than are formed, blood vessels are brittle and fragile. The vital volume of the lungs is constantly decreasing. Even a slight load causes shortness of breath, increased heart rate, fatigue.

12. Task. Underline the name of the products that you need to include in the menu in order to get the required amount of protein daily.

Answer. Fish, meat, eggs, cheese, milk.

13. Task. Fill in the table, arranging the names of the listed products in accordance with what vitamin they contain in large quantities.

14. Task. Write in the numbers that are indicated in the figure: stomach, esophagus, large intestine.

1. Esophagus

2. Stomach

3. Large intestine

Question. What other digestive organs are shown in the diagram? Write their names.

Answer. The oral cavity (it contains teeth, tongue, salivary glands), pharynx, pancreas, small intestine, liver.

Question. How was task 14 completed? Mark only one statement.

Quickly, correctly, independently.

15. Task. Prepare an answer to the question: “Prepare an answer to the question: “Why is it not recommended to read, watch TV, talk while eating?”

Answer. While eating, it is not recommended to read, watch TV, because when performing these actions, information enters the brain, which becomes the main one, and this leads to the fact that the secretion of saliva, gastric juice, digestive juices of the pancreas and liver is inhibited. The walls of the stomach and intestines work more slowly.

If you talk while eating, then food can get into the larynx or even the trachea, which is very dangerous.

Exercise. Let's work on the project.

Project topics

Definitely not. The thing is that the human body itself does not synthesize vitamins, but receives them from food. The amount of a particular vitamin can significantly affect a person's health and mood. You can buy vitamins in a pharmacy, but many scientists believe that this creates a load on the liver. Ideally, you need to eat the right balanced food. For example, citrus fruits are rich in vitamin C, fish have a lot of vitamin D, carrots have a lot of vitamin A, and so on. Lack of these substances in the body leads to diseases such as scurvy and rickets.

Scurvy is a disease caused by an acute deficiency of vitamin C (ascorbic acid). A lack of vitamin C leads to a violation of collagen synthesis, the connective tissue loses its strength. Symptoms - lethargy, fatigue, weakening of muscle tone, rheumatoid pain in the sacrum and extremities (especially the lower ones), loosening and loss of teeth; fragility of blood vessels leads to bleeding gums, hemorrhages in the form of dark red spots on the skin. Treatment and prevention - the normal supply of the body with vitamin C.

There is also evidence that sailors often suffered from scurvy due to a lack of table salt.

Saturation of food with vitamins is one of the conditions for a healthy diet that allows you to maintain physical and mental activity. Vitamins are substances that have certain similar properties:

- occupy an important place in the metabolism;

- are produced in the human body in small quantities, which necessitates their targeted intake;

- manifest their role in microscopic quantities.

The importance of vitamins for optimal human life is evidenced by the fact that when they are deficient in the body, diseases called avitaminosis and hypovitaminosis develop.

Causes of vitamin deficiency in humans:

1. The presence of diseases of the digestive system, as a result of which vitamins in food are poorly absorbed, partially destroyed, and also synthesized by the intestines in a low amount. For example, helminthic diseases are a serious obstacle to the absorption of vitamins. Some medicines inhibit the activity of vitamins.

2. Vitamin deficiency of the diet, due to:

Wrong set of products. Lack of fruits and vegetables leads to vitamin C deficiency. If you follow only a vegetarian diet, there is a lack of vitamin B12. If you prioritize refined foods (wheat flour, refined rice, sugar), you are more likely to be deficient in B vitamins.

Seasonal changes in the content of vitamins in foods. In spring and winter, the level of vitamin C in fruits decreases, and the assortment of this group of products also decreases. In the same period, eggs and milk are poor in vitamins A and D.

Improper cooking and storage of dishes, leading to a decrease in vitamins B, C, A in food. For example, with prolonged heat treatment of berries in the process of making jam, the amount of vitamin C is significantly reduced.

Diet imbalance. Vitamins in food may be present in sufficient quantities, but their absorption will be difficult due to the wrong amount (both excess and deficiency) of other vitamins, as well as due to a long-term shortage of complete proteins.

Special measures to prevent vitamin deficiency in food. In order to increase the value of some food products, they are specially fortified. This is how many products for baby food are enriched with vitamins: cereals, mashed potatoes, nutritional mixtures, drinks. For example, vitamin D2 is added to milk for children's consumption in such a way that half a liter of the drink contains the daily dose. The need for fortification of products also arises if they are intended for use in special conditions (on expeditions, during wintering). Special enrichment of food with vitamin C is carried out in sanatoriums, maternity hospitals, hospitals, dietary canteens, as well as canteen educational institutions.

16. Task. Underline the name of the organs of the digestive system.

Answer. Stomach, esophagus, teeth, small intestine.

17. Task. Mark the correct statements.

Caries is a disease of the teeth. (right)

Caries occurs in people who take poor care of their teeth. (right)

18. Task. Mark the correct statement.

In the process of digestion, proteins, fats and carbohydrates break down (split) into simpler substances. (right)

19. Task. Finish the offer.

Answer. In addition to proteins, fats and carbohydrates, our body needs water, vitamins, and minerals.

20. Question. In 1860, a dental drill appeared. What century was it? Could teeth be treated with a drill in the 16th century?

Answer. 1860 is the 19th century, so in the 16th century they could not treat their teeth with a drill.

21. Task. Mark the correct statements. Prepare explanations for your answers.

The liver cleanses the blood of harmful substances. (Blood is filtered in the liver, almost all blood is cleansed of harmful substances here). (right)

Bad teeth are a source of infection. (with food, pathogens of infectious diseases enter the esophagus and further the stomach, intestines). (right)

22. Task. Finish the offer.

Answer. In the nasal cavity, the air is warmed and purified. When breathing, oxygen is taken in and carbon dioxide is released.

23. Task. Note the rules for respiratory protection.

You need to breathe through your nose. (right)

No smoking. (right)

It is necessary to do wet cleaning of the room. (right)

You can not stay in an unventilated room for a long time. (right)

24. Task. Write the names of the organs of the respiratory system. Label them on the picture.

Answer: larynx, lungs, nasal cavity, trachea, bronchi.

On the image:

1. Nasal cavity

2. Larynx

Question. How was task 24 completed? Mark only one statement.

Quickly, correctly, independently. (+)

25. Task. Mark the correct answers to the questions.

How does tobacco ladies affect the respiratory system?

Reduces protective properties.

Why is it important to cover your nose with a tissue when sneezing and coughing?

so as not to infect others.

What gas is absorbed during respiration?

Oxygen.

Where is the air warmed and cleaned of dust and bacteria?

In the nasal cavity.

26. Task. Prepare a memo "How to protect the respiratory system."

1. It is necessary to breathe through the nose.

2. When coughing and sneezing, cover your nose with a handkerchief.

3. Systematically engage in physical culture and sports.

4. Ventilate the premises.

5. Do not smoke yourself and do not be in a room with smokers.

Exercise. Let's work on the project.

Project themes.

The consumption of oxygen and the release of carbon dioxide as a by-product is called the process of respiration. The main respiratory organs of fish are gills.

Fish have two sets of gills - one on each side of the body behind the head. These delicate organs are protected by hard plates called opercula.

Each set of gills includes four bony arches. Each of these arches supports two rows of feather-shaped gill fibers called primary lamellae (petals).

Each primary lamina, in turn, is lined with tiny lamellae (secondary lobes) through which narrow blood capillaries pass.

It is through the thin shell of the secondary lobes that gas exchange occurs between the blood and the external environment. The blood in the secondary lobes flows in the opposite direction to that of water flowing over the surfaces of the lamellae.

As a result, a large diffusion gradient of oxygen and carbon dioxide arises between these two liquids. This "counter-flow" system greatly increases the efficiency of gas exchange.

The respiratory system of amphibians is represented by lungs and skin, through which they are also able to breathe. The lungs are paired hollow sacs with a cellular inner surface, which is dotted with capillaries. This is where gas exchange takes place. The mechanism of respiration in frogs is forced and cannot be called perfect. The frog draws air into the oropharyngeal cavity, which is achieved by lowering the floor of the mouth and opening the nostrils. Then the bottom of the mouth rises, and the nostrils are again closed with valves, and air is forced into the lungs.

Let's take a whale as an example.

The skull of whales is adapted so that breathing takes place when the nostrils are exposed from the water without bending the neck (the nostrils are shifted to the top of the head).

The maxillary, intermaxillary, and mandibular bones are elongated due to the development of the sieve apparatus (whalebone) or numerous unimodal teeth. The nasal bones are reduced, the parietals are shifted to the sides so that the superior occipital bone is in contact with the frontal.

The blowhole - one or two external nasal openings - is located at the top of the head and opens only at the moment of a short respiratory act of exhalation - inhalation, produced immediately after emerging. In cool weather, when exhaling, condensed steam flies up, forming a so-called fountain, by which whalers distinguish between the type of whale.

Sometimes atomized sprays of water also take off with this steam. The rest of the time, while the respiratory pause lasts and the animal dives, the nostrils are tightly closed with valves that do not let water into the respiratory tract. Due to the special structure of the larynx, the airway is separated from the food. This allows you to breathe safely if water or food is in your mouth. The nasal canal of most species is connected to special air sacs and together with them plays the role of a sound-signaling organ.

The lungs of cetaceans are very resilient and elastic, adapted to rapid contraction and expansion, which provides a very short respiratory act and allows you to renew the air in one breath by 80-90% (in humans, only 15%). In the lungs, the musculature of the alveoli and cartilaginous rings are strongly developed, even in small bronchi, and in dolphins - in bronchioles.

Cetaceans can stay under water for a long time (sperm whales and bottlenose up to 1.5 hours) with the same air supply: a large lung capacity and a rich content of muscle hemoglobin allow them to carry away an increased amount of oxygen from the surface, which is consumed very economically: during diving, activity the heart (pulse) slows down by more than half and the blood flow is redistributed so that oxygen is supplied primarily to the brain and heart muscle. During prolonged immersion, these organs also receive oxygen with arterial blood from the reserves of the "wonderful network" - the thinnest branching of blood vessels.

Tissues less sensitive to oxygen starvation (especially the muscles of the body) are transferred to starvation rations. Muscle hemoglobin, which gives the muscles a dark color, supplies the muscles with oxygen during the respiratory pause.

Air enters the open tracheal system through spiracles, the number of which varies from one or two pairs to eight to ten pairs. The number and location of spiracles reflect the adaptation of insects to habitat conditions. Each spiracle leads to an atrial cavity, the walls of which form a closing apparatus and an air filtration system. The tracheae branch and entangle all the internal organs. The terminal branches of the trachea end in a stellate tracheal cell, from which the smallest branches extend, having a diameter of 1-2 microns (tracheoles). Their tips lie on the cell membranes or penetrate into the cells. Many well-flying insects have air sacs, which are extensions of the longitudinal tracheal trunks. Their cavity is not permanent and may collapse as air escapes. Air sacs take part in the ventilation of the wing muscles and perform an aerostatic function, reducing the specific gravity of flying insects.

27. Task. Label the picture with the names of the organs of the circulatory system. Using a picture, describe how blood moves through the body. Explain why the heart is compared to a pump?

1. Arteries

Blood moves throughout the body within the circulatory system. The human circulatory system is closed. It is made up of the heart and blood vessels. Blood vessels are divided into arteries, veins and capillaries. Arteries move blood away from the heart. Veins carry blood to the heart. Inside the organs, muscles, skin, blood moves through the capillaries. There are two circles of blood circulation - small and large.

The heart is compared to a pump, because the speed with which the blood will move through the body, pressure depends on its work. The heart has muscular walls and when it contracts, blood is released into the blood vessels. The heart beats about 100,000 times a day. Throughout life, the heart works and pumps tons of blood. That's why it's called a "pump".

28. Task. Finish the offer.

Answer. The circulatory system is made up of the heart and blood vessels - arteries, veins, capillaries.

Practical work

29. Task. Underline the names of the organs of the circulatory system.

Answer: heart, blood vessels.

30. Task. In 1908, the Russian scientist I.I. Mechnikov believed that white blood cells protect the human body from pathogenic microbes. What century was it.

Answer. It was in the XX (20) century.

31. Task. Draw a line between the name of the organ and its function.

32. Task. Mark the correct statements.

What is the main function of the circulatory system?

Transport of substances and gases. (+)

What should be done to stop bleeding from a cut?

Apply a bandage or clean handkerchief to the wound. (+)

33. Task. Write down the function of these organs.

Heart - performs the work of the "pump" of the circulatory system, pumps blood throughout the body.

Stomach - produces gastric juice, digests food.

The brain - processes information coming from the senses, "manages" the work of internal organs.

34. Task. Make a plan for a story on the topic "The human circulatory system."

Answer. Plan:

1. What is the importance of the circulatory system?

2. What organs make up the human circulatory system?

3. In what direction does the blood move through the blood vessels?

4. How does blood differ in composition?

5. What circles of blood circulation are there in the circulatory system?

6. How blood moves through the circulation.

7. What is the role of the heart in circulation?

8. What are the rules for hygiene of the circulatory organs?

35. Task. Underline the name of the excretory organs.

Answer: kidneys, ureters, bladder.

36. Task. Mark the correct statements.

What is the role of the kidneys in the body?

Removes waste products from the body. (+)

In which organ is urine produced?

In the kidneys. (+)

37. Task.

one). The microscope was invented in Holland in 1590. What do you think, could Peter I work with a microscope?

2) The famous Russian surgeon N.I. Pirogov was the first to use a plaster cast for fractures, as well as iodine and alcohol to treat wounds. This was in 1855. In what century did N.I. Pirogov?

Answer. N.I. Pirogov lived in the century.

38. Task. Mark the correct statement.

The skin does not allow pathogenic bacteria to enter the body. (+)

39. Task. Write down in the tables the methods of hardening of the body and the functions of the skin known to you.

Exercise. Make a drawing "The structure of the skin." Look at the diagram on p. 31 textbooks.

MOU "Kurovskaya secondary school No. 6"

ABSTRACT ON MATHEMATICS ON THE TOPIC:

« UNUSUAL MULTIPLICATION WAYS».

Completed by a student of 6 "b" class

Krestnikov Vasily.

Supervisor:

Smirnova Tatyana Vladimirovna

Introduction…………………………………………………………………………2

Main part. Unusual ways of multiplication…………………………3

2.1. A bit of history………………………………………………………………..3

2.2. Multiplication on fingers……………………………………………………………4

2.3. Multiplication by 9………………………………………………………………………5

2.4. Indian way of multiplication……………………………………………….6

2.5. Multiplication by the “Little Castle” method……………………………………………………7

2.6. Multiplication by the “Jealousy” method………………………………………………8

2.7. Peasant way of multiplication……………………………………………..9

2.8 New way…………………………………………………………………..10

Conclusion………………………………………………………………………… 11

References…………………………………………………………….1 2

I. Introduction.

It is impossible for a person to do without calculations in everyday life. Therefore, in mathematics lessons, we are first of all taught to perform operations on numbers, that is, to count. We multiply, divide, add and subtract in the usual ways for everyone that are studied at school.

Once I accidentally came across a book by S. N. Olekhnika, Yu. V. Nesterenko and M. K. Potapov "Old entertaining problems." Leafing through this book, my attention was drawn to a page called "Multiplication on the fingers." It turned out that you can multiply not only as they offer us in mathematics textbooks. I was wondering if there are any other ways to calculate. After all, the ability to quickly make calculations is frankly surprising.

The constant use of modern computing technology leads to the fact that students find it difficult to make any calculations without having tables or a calculating machine at their disposal. Knowledge of simplified calculation techniques makes it possible not only to quickly perform simple calculations in the mind, but also to control, evaluate, find and correct errors as a result of mechanized calculations. In addition, the development of computational skills develops memory, increases the level of mathematical culture of thinking, helps to fully assimilate the subjects of the physical and mathematical cycle.

Objective:

Show unusualmultiplication methods.

Tasks:

Find as many as possibleunusual ways of computing.

Learn to apply them.

Choose for yourself the most interesting or easier than those thatofferedat school, and use them when counting.

II. Main part. Unusual ways of multiplication.

2.1. A bit of history.

The methods of calculation that we use now were not always so simple and convenient. In the old days, more cumbersome and slower methods were used. And if a schoolboy of the 21st century could travel back five centuries, he would impress our ancestors with the speed and accuracy of his calculations. The rumor about him would have spread around the surrounding schools and monasteries, eclipsing the glory of the most skillful counters of that era, and people would come from all over to study with the new great master.

The operations of multiplication and division were especially difficult in the old days. At that time, there was no single technique worked out by practice for each action. On the contrary, almost a dozen different methods of multiplication and division were in use at the same time - methods one more intricate than the other, which a person of average ability could not remember. Each calculus teacher kept to his favorite method, each "master of division" (there were such specialists) praised his own way of performing this action.

In the book by V. Bellyustin “How people gradually reached real arithmetic”, 27 methods of multiplication are outlined, and the author notes: “it is quite possible that there are more methods hidden in the recesses of book depositories, scattered in numerous, mainly handwritten collections.”

And all these methods of multiplication - "chess or organ", "bending", "cross", "lattice", "back to front", "diamond" and others competed with each other and were assimilated with great difficulty.

Let's look at the most interesting and simple ways of multiplication.

2.2. 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.

2.3. 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).

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 figure below shows in detail the whole principle of “calculation”.

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.

2.4. 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

2.5 . multiplication way"LITTLE CASTLE".

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.

2.6. Number multiplicationjealousy method.

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.

2.7. Torustic 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

2.8 . A new way to multiply.

interesting a new way of multiplication that 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 number (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.

III. Conclusion.

Of all the unusual counting methods I found, the method of “lattice multiplication or jealousy” seemed to be the most interesting. I showed it to my classmates and they also liked it very much.

The simplest method seemed to me to be the “doubling and splitting” method used by Russian peasants. I use it when multiplying not too large numbers (it is very convenient to use it when multiplying two-digit numbers).

I was interested in a new way of multiplication, because it allows you to "turn" huge numbers in your mind.

I think that our method of multiplying by a column is not perfect either, and we can come up with even faster and more reliable methods.

Literature.

Depman I. "Stories about Mathematics". - Leningrad.: Education, 1954. - 140 p.

Korneev A.A. The phenomenon of Russian multiplication. Story. http://numbernautics.ru/

Olekhnik S. N., Nesterenko Yu. V., Potapov M. K. "Old entertaining problems." – M.: Science. Main edition of physical and mathematical literature, 1985. - 160 p.

Perelman Ya.I. Quick account. Thirty simple methods of mental counting. L., 1941 - 12 p.

Perelman Ya.I. Entertaining arithmetic. M.Rusanova, 1994–205p.

Encyclopedia “I know the world. Mathematics". – M.: Astrel Ermak, 2004.

Encyclopedia for children. "Mathematics". - M.: Avanta +, 2003. - 688 p.

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:

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, Relations and Proportionality (1494), lists 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.

Mincheva Anna, 6th grade student of MAOU secondary school No. 37, Ulan-Ude

The constant use of modern computing technology leads to the fact that students find it difficult to make any calculations without having tables or a calculating machine at their disposal. Relevance of the topic research consists in the fact that knowledge of simplified methods of calculations makes it possible not only to quickly perform simple calculations in the mind, but also to control, evaluate, find and correct errors as a result of mechanized calculations. In addition, the development of computational skills develops memory, increases the level of mathematical culture of thinking, helps to fully assimilate the subjects of the physical and mathematical cycle.

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MAOU "Secondary school No. 37"

Scientific-practical conference "Ordinary miracle"

Section: Arithmetic

"Different ways of multiplication: from antiquity to the present"

Performed:

Mincheva, Anna

6th grade student

Supervisor:

Koneva Galina Mikhailovna,

Mathematic teacher,

"Excellence in Education of the Russian Federation",

Winner of the Competition of the best teachers of Russia (2009)

Ulan-Ude

2017

Review.

I believe that the student has done a great job, and this report will be of interest to students who are fond of mathematics, future economists.

Teacher of the highest category: Koneva G.M.

Plan.

1. Introduction

2. The main part. Ways to multiply natural numbers

2.1. Reception of cross multiplication when operating with two-digit numbers

2.2. Multiplication by the method "Jealousy, or lattice multiplication"

2.3. Multiplication using the "Little Castle" method

2.4. Peasant way of multiplication

2.5. Indian way of multiplication

2.6.Geometric method of multiplication

2.7. The original way to multiply by 9 on the fingers

2.8 Okoneshnikov's method

3.Conclusion

“The subject of mathematics is so serious
what is useful not to miss opportunities to do
its a bit entertaining." B. Pascal

1. Introduction.

It is impossible for a person to do without calculations in everyday life. Therefore, in mathematics lessons, we are taught to perform operations on numbers, that is, to count. We multiply, divide, add and subtract in the usual ways for everyone that are studied at school.

At one of the lessons, a math teacher showed how you can multiply, for example, the number 23 by 11. To do this, you need to mentally push the numbers 2 and 3, and put the number 5 in this place, that is, the sum of the numbers 2 and 3. It turned out the number 253. I felt I wonder if there are any other ways to calculate. After all, the ability to quickly make calculations is frankly surprising.

The constant use of modern computing technology leads to the fact that students find it difficult to make any calculations without having tables or a calculating machine at their disposal.Relevance of the topicresearch consists in the fact that knowledge of simplified methods of calculations makes it possible not only to quickly perform simple calculations in the mind, but also to control, evaluate, find and correct errors as a result of mechanized calculations. In addition, the development of computational skills develops memory, increases the level of mathematical culture of thinking, helps to fully assimilate the subjects of the physical and mathematical cycle.

Objective:

Explore and learn unusual ways of multiplication.

Research objectives:

1. Find as many unusual ways of computing as possible.

2. Learn to apply them.

3. Choose for yourself the most interesting or easier ones than those offered at school, and use them when counting.

4. Teach your classmates various methods of multiplication, organize a competition - a mathematical battle in extracurricular activities.

Research methods:

Search method using scientific and educational literature, the Internet;

Research method in determining the methods of multiplication;

A practical method in solving examples.

II. From the history of computing practice

The methods of calculation that we use now were not always so simple and convenient. In the old days, more cumbersome and slower methods were used. And if a schoolboy of the 21st century could travel back five centuries, he would impress our ancestors with the speed and accuracy of his calculations.

The operations of multiplication and division were especially difficult in the old days. At that time, there was no single technique worked out by practice for each action. On the contrary, almost a dozen different methods of multiplication and division were in use at the same time - methods one more intricate than the other, which a person of average ability could not remember. Each teacher of counting kept to his favorite technique, each "master of division" praised his own way of doing this action.

In V. Bellyustin’s book “How People Gradually Came to True Arithmetic”, 27 methods of multiplication are outlined, and the author notes: “It is quite possible that there are more methods hidden in the recesses of book depositories, scattered in numerous, mainly handwritten collections.”

And all these multiplication techniques - “chess or organ”, “bending”, “cross”, “lattice”, “back to front”, “diamond” and others competed with each other and were assimilated with great difficulty.

I began to study and research some of these methods and chose the most interesting ones.

III. Various ways to multiply.

3.1. The method of cross multiplication when operating with two-digit numbers

The ancient Greeks and Hindus in the old days called the method of cross multiplication "the method of lightning" or "multiplication by a cross."

Example: 52 x 23 = 1173 5 1

We sequentially perform the following actions:

1. 1 x 3 = 3 is the last digit of the result.

2. 5 x 3 = 15; 1x 2 = 2; 15 + 2 = 17.

7 - the penultimate digit in the answer, we remember the unit.

3. 5 x 2 \u003d 10, 10 + 1 \u003d 11 - these are the first digits in the answer.

Answer: 1173.

3.2. The ancient way of Luca Pacioli: "Jealousy, or lattice multiplication"

Over the millennia of the development of mathematics, many methods of multiplication have been invented. Except for the multiplication table, they are all bulky, complicated and hard to remember. It was believed that to master the art of rapid multiplication, you need a special natural talent. Ordinary people who do not have a special mathematical gift, this art is not available.

Let's multiply the number 987 by the number 1998.

We draw a rectangle, divide it into squares, divide the squares diagonally. The result is a picture similar to the lattice shutters of Venetian houses. This is where the name of the method comes from.

At the top of the table we write the number 987, and from the bottom left to the top - 1998 (Fig. 1).

In each square we write the product of the numbers located in the same row and one column with this square. Tens are located in the lower triangle, and ones in the upper one. The numbers are added along each diagonal. Results are written to the right and left of the table .

Rice. 1 "Jealousy, or lattice multiplication."

Answer: 1972026.

3.3. Another way of Luca Pacioli: "Little Castle"

One number is written under another as in column multiplication (Fig. 2). Then the digits of the upper number are multiplied in turn by the lower number, starting with the most significant digit and adding the required number of zeros each time.

The resulting numbers are added together.

Rice. 2 "Little Castle"

Answer: 1972026.

Conclusion:

Let's compare the results obtained by multiplying the numbers 987 and 1998 by these two methods. The answers are 1972026.

Obviously, these ancient multiplication methods are really very complex and require knowledge of the multiplication table.

3.4. Russian peasant way of multiplication

In Russia, a method was common among peasants that did not require knowledge of the entire multiplication table. All you need is the ability to multiply and divide numbers by 2.

Let's write one number on the left and another on the right on one line (Fig. 3). We will divide the left number by 2, and multiply the right number by 2 and write the results in a column.

If a remainder occurs during division, then it is discarded. Multiplication and division by 2 continue until 1 remains on the left.

Then we cross out those lines from the column in which there are even numbers on the left. Now let's add the remaining numbers in the right column.

Rice. 3 "Russian peasant way"

Answer: 1972026.

Conclusion: This method of multiplication is much simpler than the previously considered methods of multiplication by Luca Pacioli. But it is also very bulky.

3.5. 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:

537 6

(5 ∙ 6 =30) 30

537 6

(300 + 3 ∙ 6 = 318) 318

537 6

(3180 +7 ∙ 6 = 3222) 3222. Answer: 3222

3.6. Geometric multiplication method

This method uses a geometric figure - a circle.

First, let's look at this method with an example. Let's multiply, for example, the number 13 by 24.

1) We draw circles. Since the first factor is a two-digit number, then two lines; the second multiplier is also a two-digit number, then two columns. Since the number of tens in the first factor is 1, then in the first line we draw one circle, that is, we do not change anything. Since the number of units of the first factor is 3, we draw three circles in the second line. (Fig. 4).

Rice. 4

2) The second factor is the number 24, then the circles, which are divided into two parts in the first column, and the circles, which are divided into four parts in the second column

(Fig. 5).

Rice. 5

3) We draw straight lines and count the points (Fig. 6).

Rice. 6 Fig. 7

The answer is written as follows (Fig. 7), we look from the bottom up the number of points is 12, 2 is the last digit of the result, one in the mind, the number of points in the second area is 10 and +1, that is 11, we write 1 and one in the mind, the number of points in third area 2 and +1, total 3. Answer: 312.

I solved many examples this way. Then she generalized particular examples andmade a rule:

1. Draw circles. The number of digits in the first multiplier indicates the number of rows, and the number of digits in the second factor indicates the number of columns.

If the number contains 0, the circle denoting zero is drawn with a dotted line. This is an imaginary line, there are no points on it.

2. The first digit of the first multiplier means the number of concentric circles in the first line, the second digit of the first multiplier means the number of circles in the second line

3. The numbers of the second multiplier indicate how many parts the circles need to be divided into: the first number is for the first column, the second number is for the second, etc.

4. Let's get the circles divided into parts. We put a point in each part.

6. We write down the answer according to the principle considered in the example.

3.6. The original way to multiply by 9 on your fingers

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).

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".

3.7. Okoneshnikov's modern method

interesting a new way of multiplication that 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.

III. Conclusion.

Of all the unusual counting methods I found, the method of “lattice multiplication or jealousy” seemed to be the most interesting. I showed it to my classmates and they also liked it very much.

The simplest method seemed to me to be the “doubling and splitting” method used by Russian peasants. I use it when multiplying not too large numbers (it is very convenient to use it when multiplying two-digit numbers).

I was interested in a new way of multiplication, because it allows you to "turn" huge numbers in your mind.

I think that our method of multiplying by a column is not perfect either, and we can come up with even faster and more reliable methods.

Literature.

Literature.

Depman I. "Stories about Mathematics". - Leningrad.: Education, 1954. - 140 p.

Korneev A.A. The phenomenon of Russian multiplication. Story. http://numbernautics.ru/

Olekhnik S. N., Nesterenko Yu. V., Potapov M. K. "Old entertaining problems." – M.: Science. Main edition of physical and mathematical literature, 1985. - 160 p.

Perelman Ya.I. Quick account. Thirty simple methods of mental counting. L., 1941 - 12 p.

Perelman Ya.I. Entertaining arithmetic. M.Rusanova, 1994-205p.

Encyclopedia “I know the world. Mathematics". – M.: Astrel Ermak, 2004.

Encyclopedia for children. "Mathematics". - M.: Avanta +, 2003. - 688 p.