Pervomaisky branch

Municipal educational institution Podbelskaya secondary school

Pokhvistnevsky district

Samara region

Plan - summary of extracurricular activities

in 2nd grade

"The Club of Cheerful Mathematicians"

Teacher: Tikhomirova T.P.

With. Pervomaisk

2008/2009 academic year

Club of cheerful mathematicians.

Leading: Friends, the MCU is having fun.

We have come to visit you again.

We were really looking forward to this meeting

And they tried their best.

(BAM team comes out)

Welcome to the BAM team.

Our motto: “Let's think actively.”

Team captain : Hello friends! Today at school

Big and interesting day

We have prepared a fun

Our school evening MCU.

MCU - competition

In wit and knowledge.

So that this evening MCU

Everyone liked you,

You need to have solid knowledge,

Be cheerful and resourceful.

And this MCU now

Dedicated to science

What mathematics do we have?

It's called with love.

She will help raise

Such precision of thought,

To know everything in our life,

Measure and count.

(Team PUPS comes out)

Welcome to the PUPS team.

Our motto: “Let the mind conquer the force.”

Team captain:we are funny guys

And we don't like to be bored.

It's our pleasure to join you

We'll play in the MCU.

We answer together

And here there is no doubt.

Today there will be friendship

Mistress of victories.

And let the struggle rage more intensely,

Stronger competition.

Success is not decided by fate,

But only our knowledge.

And, competing with you,

We remain friends.

So let the fight rage on

And our friendship grows stronger with her.

Team warm-up.

(Each team receives 3 tasks)

(For the BAM team)

1. Find the essential.

Sum (minus, plus, equality, addend, divisor)

Geometry (figure, point, properties, theorem, equation).

1. Checking definitions.

Having defined a particular concept, you must be sure that it is correct. Correctness can be checked by swapping the condition and conclusion in the definition. If the sentence remains true when changing places, then we have given the definition correctly.

Check the definitions are correct:

A square is a quadrilateral.

Addition is a mathematical operation.

a) 2.4, 7, 9, 6;

b) 13, 18, 25, 33, 48, 57.

(For the PUPS team)

1. Find the essential.

Triangle (plane, vertex, center, side, perpendicular)

Difference (subtraction, plus, minus, sum, addend)

1. Check definitions:

A circle is a geometric figure.

An even number is a natural number.

1. Name a group of numbers in one word:

a) 2, 4, 8, 12, 44, 56;

b) 1, 13, 77, 83, 95.

Competition "six-cell logion"

(For the BAM team)

a) 6 1 7

14 4 ?

b) 9 2 11

26 8 ?

c) 35 7 5

48 8 ?

d) 92 46 2

72 ? 8

(For the PUPS team)

a) 16 7 9

36 11 ?

b) 44 18 26

33 14

c) 32 8 4

56 ? ?

d) 22 4 88

12 ? 96

Let's work on a computer.

A computer is depicted on the board. The computer performs all four arithmetic operations. The number 36 appeared on the display. What number was included in the machine?

X 3 -19 +10: 9 +86: 3 +

← 2: 41+

While the team finds the right number, the fans guess the charades.

The first letter is in the word "marmot"

But it is not in the word “lesson”.

Among the smart guys you will find anyone.

Mom can use two letters without embarrassment,

But in general you will get the result from addition. (Sum)

The preposition is at the beginning of mine,

At the end is a country house.

And we decided everything

Both at the blackboard and at the table. (Task)

At the beginning of the word there is an oral count,

Then the consonant sound comes.

Coarse animal hair then,

But in general we will find the result. (Difference)

Compositor

Make as many words as possible from the letters in the given word. Which team will make up the most words faster?

For the BAM team - addition

For the PUPS team– subtraction

Problem solving

(For the BAM team)

The centipede mother bought boots for her three daughters. How many pairs of boots did mom have to buy?

To find his bride, the prince forced his soldiers to go around 12 settlements. Each of them had 40 girls. How many girls in total tried on the shoe?

How to write the number 100 in five units? (111 – 11 =100)

For the PUPS team

The hare had 4 sons and a sweetheart - a daughter. One day he brought home a bag of 60 apples. How many apples did each hare get if the hare divided them equally between them?

The brave little tailor killed 7 flies with one blow. How many flies did he kill if he made 11 strikes?

The guys and their dogs went for a walk. One grandfather tells them: “Look, guys, don’t lose your heads and don’t break your legs.” One boy said: “We only have 36 legs and 13 heads, so we won’t get lost.” How many dogs and how many boys? (5 dogs and 8 boys)

Fairytale tasks.

An unknown number doubled, looked at yourself in the mirror and saw 811 there. What was the number before the increase?

In the elevator, the button for the first floor is located at a height of 1m20cm from the floor. The button for each next floor is 10 cm higher than the previous one. To what floor can a little boy, whose height is 90 cm, get in the elevator if, by jumping, he can reach a height that is 45 cm higher than his height?

Little Red Riding Hood helped her mother bake pies for her grandmother. Mom kneaded the dough from 2 cups of flour and said that it should make 30 pies. Little Red Riding Hood asked to bake 60 pies. How much flour will this require?

Captain Flirt decided to reward his pirates. He had 720 coins. He decided to keep half for himself, and divided the remaining coins equally between 9 pirates. How many coins did each pirate receive?

Challenges for ingenuity.

The boy Sasha has as many sisters as brothers, and his sister has half as many sisters as brothers. How many brothers and all sisters are there? (4 brothers and 3 sisters)

There were 36 jackdaws sitting on three trees. When 6 jackdaws flew from the first tree to the second, and 4 jackdaws flew from the second to the third, then there were equal numbers of jackdaws on all three trees. How many jackdaws were originally on each tree? (18, 10, 8)

Igor was asked how old he was. He thought and said: “I’m three times younger than dad, but twice as old as my brother Vitalka.” And Vitalka came running and said that he was 35 years younger than dad. How old are Igor, Vitalik and dad?

Igor is 14 years old, Vitalik is 7 years old, dad is 42 years old)

The grandson asked his grandfather: “How old are you?” GRANDFATHER ANSWERED: “If I live another half of what I lived, and another year, then it will be 100 years.” How old is grandpa? (66 years old)

Teacher: Tikhomirova T.P.

Mathematics plays a special role in the system of educational subjects. It equips students with the necessary knowledge, skills and abilities that are used in the study of other school disciplines, especially in the study of geometry, algebra, physics and computer science. When studying this subject, students require a lot of volitional and mental effort, developed imagination, and concentration; mathematics develops the student’s personality. In addition, studying mathematics significantly contributes to the development of logical thinking and broadens the horizons of schoolchildren.

Mathematics is one of the most important sciences on earth and it is with it that a person encounters every day in his life. That is why the teacher needs to develop children’s interest in this science and subject. In my opinion, it is possible to develop cognitive interest in mathematics through the use of various types of mental calculation, and the involvement of students in preparing and conducting this stage of the lesson and the lesson as a whole.

Oral arithmetic in mathematics lessons can be represented by various forms of work with the class and students (mathematical, arithmetic and graphic dictations, mathematical lotto, puzzles, crosswords, tests, conversations, surveys, warm-ups, “circular” examples and much more). It includes algebraic and geometric material, solving simple problems and ingenuity problems, the properties of actions on numbers and quantities and other issues are considered, with the help of mental calculation you can create a problem situation, etc.

Oral arithmetic is not a random stage of the lesson, it is in a methodological connection with the main topic and is problematic in nature.

To achieve accuracy and fluency in oral calculations, each mathematics lesson allocates 5-10 minutes for exercises in oral calculations.
Oral arithmetic activates the mental activity of students. When they are performed, memory, speech, attention, the ability to perceive what is said by ear, and speed of reaction are activated and developed.

This stage is an integral part of the structure of a mathematics lesson. It helps the teacher, firstly, to switch the student from one activity to another, secondly, to prepare students to study a new topic, thirdly, tasks for repeating and summarizing the material covered can be included in the oral calculation, fourthly, it increases students' intelligence.

Goals At this stage of the lesson, the following can be determined:

1) achieving the set goals of the lesson;
2) development of computing skills;
3) development of mathematical culture and speech;
4) the ability to generalize and systematize, transfer acquired knowledge to new tasks.

Since oral exercises or oral counting is a stage of the lesson, it has its own tasks:

1. Reproduction and adjustment of certain knowledge, skills and abilities of students necessary for their independent activity in the lesson or conscious perception of the teacher’s explanation.
2. Teacher’s control over the state of students’ knowledge.
3. Psychological preparation of students to perceive new material.
4. Increasing cognitive interest.

When conducting oral counting, each teacher adheres to the following: requirements:

• Exercises for mental counting are not chosen randomly, but purposefully.
• The tasks should be varied, the proposed tasks should not be easy, but they should not be “cumbersome”.
• Texts of exercises, drawings and notes, if required, must be prepared in advance.
• All students should be involved in mental counting.
• When conducting an oral count, evaluation criteria (reward) must be thought out.

An oral count can be constructed in the following form:

• Tasks for the development and improvement of attention. Such as: find a pattern and solve an example, continue the series.
• Tasks for the development of perception and spatial imagination. For example, draw an ornament, a pattern; count how many lines.
• Tasks to develop observation skills (find a pattern, what is superfluous?)
• Oral exercises using didactic games.

Mental calculation skills are developed as students perform a variety of exercises. Let's look at their main types:

1) Finding the values ​​of mathematical expressions.

A mathematical expression is proposed in one form or another, and you need to find its value. These exercises have many variations. You can offer numerical mathematical expressions and alphabetic ones (an expression with a variable), while the letters are given numerical values ​​and the numerical value of the resulting expression is found.

2) Comparison of mathematical expressions.

These exercises have a number of variations. Two expressions can be given, but it is necessary to establish whether their values ​​are equal, and if not equal, then which of them is greater or less.
Exercises may be offered in which the relation sign and one of the expressions are already given, and another expression must be composed or supplemented: 8 · (10 + 2) = 8 · 10 + ...
Expressions in such exercises can include various numerical materials: single-digit, two-digit, three-digit numbers and quantities. Expressions can have different actions.

The main role of such exercises is to facilitate the acquisition of theoretical knowledge about arithmetic operations, their properties, equalities, inequalities, etc. They also help develop computational skills.

3) Solving equations.

These are, first of all, the simplest equations (x + 2 = 10) and more complex ones (15 x – 9 = 51)

The equation can be presented in different forms:

• From what number must 18 be subtracted to get 40?
• solution to the equation x 8 = 72;
• find the unknown number: 77 + x = 77 + 25
• Nikolai thought of a number, multiplied it by 5 and got 125. What number did Nikolai think of?

The purpose of such exercises is to develop the ability to solve equations and help students understand the connections between the components and results of arithmetic operations.

4) Problem solving.

For oral work, both simple and compound tasks are offered.

These exercises are included with the aim of developing problem-solving skills; they help master theoretical knowledge and develop computational skills.
A variety of exercises arouses children's interest and activates their mental activity.

Forms of perception of oral counting

1) Fluent auditory (read by a teacher, student, audio recording) – when perceiving a task by ear, a large load is placed on memory, so students quickly get tired. However, such exercises are very useful: they develop auditory memory.

2) Visual (tables, posters, cards, notes on the board, computer) – writing down the task makes calculations easier (no need to memorize numbers). Sometimes it is difficult and even impossible to complete a task without recording. For example, you need to perform an action with quantities expressed in units of two names, fill out a table, or perform actions when comparing expressions.

3) Combined.

• feedback (showing answers using cards, mutual testing, guessing keywords, checking using the Microsoft Power Point computer program).
• assignments based on options (ensure independence).
• exercises in the form of a game (“Dialogue”, “Mathematical duel”, “Magic squares”, “Maze of factors”, “Quiz”, “Magic number”, “Individual lotto”, “Best counter”, “Coded exercises”, “Chip” ”, “Who is faster”, “Flower, sunshine”, “Number mill”, “Number fireworks”, “Mathematical phenomenon”, “Silence”, “Mathematical relay race”). The ways and forms of using the listed games in mathematics lessons are discussed in the work of V. P. Kovalenko “Didactic games in mathematics lessons.”

Organization of mental arithmetic classes

When preparing for a lesson, the teacher must clearly determine (based on the goals of the lesson) the scope and content of oral tasks. If the purpose of the lesson is to present a new topic, then at the beginning of the lesson you can carry out oral calculations on the material covered, and you can also organize the work so that there is a smooth transition to the new topic. After presenting a new topic, it is appropriate to offer students oral tasks to develop skills on this topic. If the goal of the lesson is repetition, then both the teacher and students should prepare for oral calculations in the classroom. Students, with the advice of the teacher, can perform mental calculations themselves at each lesson.
Oral counting can be combined with checking homework, consolidating the material studied, offered during a survey, and also specially allotted 5-7 minutes in class for mental counting. Material for this can be selected from textbooks, special collections, mathematical encyclopedias or books, or you can invite students to come up with tasks themselves.
Oral exercises must correspond to the topic and purpose of the lesson and help master the material being studied in this lesson or previously covered. Depending on this, the teacher determines the place of oral calculation in the lesson. If oral exercises are intended to review material, develop computational skills, and prepare for learning new material, then it is better to conduct them at the beginning of the lesson before learning new material. If oral exercises are aimed at consolidating what has been learned in this lesson, then it is necessary to carry out oral calculations after studying new material.
When selecting exercises for a lesson, it should be taken into account that preparatory exercises and the first exercises for consolidation, as a rule, should be simpler and more straightforward. Here there is no need to strive for special diversity in formulations and methods of work. Exercises for practicing knowledge and skills and, especially for applying them in different conditions, on the contrary, should be more monotonous. The wording of tasks, if possible, should be designed so that they are easily perceived by ear. To do this, they must be clear and concise, formulated easily and definitely, and not allow for different interpretations.
In addition to the fact that mental calculation in mathematics lessons contributes to the development and formation of strong computational skills and abilities, it also plays an important role in instilling and increasing children’s cognitive interest in mathematics lessons, as one of the most important motives for educational and cognitive activity, the development of logical thinking, and development of the child’s personal qualities. In my opinion, by arousing interest and instilling a love for mathematics through various types of oral exercises, the teacher will help students actively work with educational material, awaken in them the desire to improve methods of calculations and problem solving, replacing less rational ones with more advanced ones. And this is the most important condition for conscious assimilation of the material.
If a student likes a subject, then he will always be interested and enthusiastically learn more and more knowledge, and increasing interest in mathematics lessons can be achieved in the following way:

1) Enrichment of the content with material on the history of science, which is often found on the pages of the textbook.
2) Solving problems of increased difficulty and non-standard problems. The selection of tasks is carried out from workbooks and didactic materials.
3) Emphasizing strength and grace, the rationality of methods of calculation, evidence, transformation and research.
4) The variety of lessons, their non-standard construction, the inclusion in lessons of elements that give each lesson a unique character, the solution of problem situations, the use of technical teaching aids (interactive whiteboard, computer, etc.), visual aids, a variety of oral calculations.
5) Activation of students’ cognitive activity in the classroom using forms of independent and creative work.
6) Using various forms of feedback: systematically conducting surveys, short-term oral and written tests, various tests, mathematical dictations, tests, along with tests provided for in the plan.
7) Variety of homework. For example, invite students to write a fairy tale about a geometric figure, a poem about a fraction, a degree.
8) Establishing internal and interdisciplinary connections by showing and explaining the application of mathematics in life and in production.

For example, when studying triangles, you can tell that triangles are used in the game of billiards and bowling; during the construction of iron structures (Shukhov Tower on Shabolovka); railway bridges; high-voltage power lines; introduce legends about the Bermuda Triangle, Pascal's triangle, Penrose's triangle and much more.

Students like to take part in preparing for the lesson, so in addition to homework, if desired, you can give the task to independently prepare an oral calculation for the lesson in accordance with the topic, and carry it out yourself at the next lesson (play the role of a teacher). You can also give students the task of preparing an essay, report, coming up with a puzzle, rebus, game (see. Annex 1 ).

The children prepare and conduct oral work in class very responsibly and diligently. When completing this task, they put a lot of effort, since they need to come up with tasks that will be interesting for the class, and that the tasks correspond to the topic of the lesson.

Saturating lessons with varied, entertaining and useful computational tasks with a high density of current theoretical material on the topics being studied is possible only through improving the system of oral exercises in lessons. This will allow, first of all, to teach students to learn, to delve into the meaning of what is being studied at every step of learning so much as to be able to independently solve emerging problems.
This gives them self-confidence and encourages them to improve their results; children begin to work actively in the lesson and they begin to like this subject.
It is also important to note the following, that primary and secondary school students quickly count, calculate in their heads, orally, but for some reason in high school, mental calculation is done using a calculator or with great difficulty without a calculator. It seems to me that we need to strive to ensure that this does not happen. And this, of course, can be achieved using oral counting as an important and necessary element of the lesson.
Oral arithmetic as a mandatory stage of the lesson should be carried out in mathematics lessons both in primary school and in middle and high school.

Bibliography:

1. Berimets V.I.“The use of various types of oral exercises as a means of increasing cognitive interest in a mathematics lesson.”
2. V. P. Kovalenko“Didactic games in mathematics lessons.”
3. Zaitseva O.P. The role of mental arithmetic in the formation of computational skills and in the development of a child’s personality // Primary school, 2001 No. 1
4. N.K. Vinokurova: “Let’s think together,” M. “Growth.”

Department of Education of the Okhinsky Urban District

Municipal budgetary educational institution

Secondary school No. 1, Okha

Techniques

mental counting

The work was completed by:

Students of grade 5 "A"

Turboevskaya Eva

Bezinsky Stanislav

Project Manager:

mathematic teacher

Kravchuk Maria Arkadyevna

2017

CONTENT

Chapter 1. ACCOUNT HISTORY……………………………………………………………….....

Chapter 2. MULTIPLICATION TABLE ON YOUR FINGERS …………………………

2.1 Multiplication table by 9

2.2 Multiplying numbers from 6 to 9

Chapter 3. DIFFERENT METHODS OF MULTIPLICATION……………………….....

3.1 Multiplying a number by 9

3.2 Multiplying two-digit numbers by 11

3.3 Multiplying two-digit numbers by 111, 1111, etc.

3.4 Multiplying a two-digit number by 101, 1001, etc.

3.5 Multiplication by 5; 25; 125

3.7 Multiplying by 37

3.8 Multiplying a number by 1.5

Chapter 4.SQUAREING A TWO-DIGIT NUMBER …………...

4.1 Squaring a two-digit number ending in 5

4.2 Squaring a two-digit number starting with 5

CONCLUSION ……………………………………………………………….....

BIBLIOGRAPHY ………………………………………………………

ANNEX 1 ………………………………………………………………..

APPENDIX 2………………………………………………………………..

INTRODUCTION

At all times, mathematics has been and remains one of the main subjects in school, because mathematical knowledge is necessary for all people. Not every student, while studying at school, knows what profession he will choose in the future, but everyone understands that mathematics is necessary for solving many life problems: calculations in a store, paying for utilities, calculating the family budget, etc. In addition, all schoolchildren must take exams in the 9th grade and in the 11th grade, and for this, studying from the 1st grade, it is necessary to master mathematics well and, above all, to learn to count.

The relevance of our project is that nowadays, calculators are increasingly coming to the aid of students, and an increasing number of students cannot count orally.

But the study of mathematics develops logical thinking, memory, flexibility of mind, accustoms a person to accuracy, to the ability to see the main thing, and provides the necessary information for understanding complex problems that arise in various fields of activity of modern man.

Objective of the project: study mental calculation techniques, show the need for their use to simplify calculations.

In accordance with the goal, we determinedtasks:

To investigate whether schoolchildren use mental counting techniques.

Learn mental counting techniques that can be used to simplify calculations.

Create a memo for students in grades 5-6 to use quick mental counting techniques.

Object of study: oral counting techniques.

Subject of study : calculation process.

Hypothesis: If you show that the use of fast mental calculation techniques makes calculations easier, then you can ensure that students’ computing culture improves and it will be easier for them to solve practical problems.

The following were used to carry out the work:techniques and methods : survey (questioning), analysis (statistical data processing), work with sources of information, practical work.

To begin with, we conducted a survey in the 5th and 6th grades of our school. We asked the guys simple questions.Why do you need to be able to count?When studying which school subjects will you need to count correctly?Do you know mental counting techniques?Would you like to learn fast mental counting techniques to count quickly?Annex 1

105 people took part in the survey. After analyzing the results, we concluded that the majority of studentsbelievethat the ability to count is useful in life and to be literate, especially when studying mathematics (100%), physics (68%), chemistry (50%), computer science (63%). A small number of students know mental counting techniques and almost all of them would like to learn quick mental counting (63%).Appendix 2

Having studied a number of articles, we discovered very interesting historical facts about unusual methods of mental counting, as well as many patterns and unexpected results.Therefore, in our work we will show how you can count quickly and correctly and that the process of performing these actions can be not only useful, but also an interesting activity.

Chapter 1. ACCOUNT HISTORY

People learned to count objects back in the ancient Stone Age - Paleolithic, tens of thousands of years ago. How did this happen? At first, people only compared different quantities of identical objects by eye. They could determine which of two piles had more fruit, which herd had more deer, etc. If one tribe exchanged caught fish for stone knives made by people of another tribe, there was no need to count how many fish and how many knives they brought. It was enough to place a knife next to each fish for the exchange between the tribes to take place.

To successfully engage in agriculture, arithmetic knowledge was needed. Without counting days, it was difficult to determine when to sow fields, when to start watering, when to expect offspring from animals. It was necessary to know how many sheep were in the herd, how many bags of grain were placed in the barns.
And more than eight thousand years ago, ancient shepherds began to make mugs out of clay - one for each sheep. To find out if at least one sheep had gone missing during the day, the shepherd put aside a mug each time another animal entered the pen. And only after making sure that as many sheep had returned as there were circles, he calmly went to bed. But in his herd there were not only sheep - he grazed cows, goats, and donkeys. Therefore, we had to make other figures from clay. And farmers, using clay figurines, kept records of the harvest, noting how many bags of grain were placed in the barn, how many jugs of oil were squeezed from olives, how many pieces of linen were woven. If the sheep gave birth, the shepherd added new ones to the circles, and if some of the sheep were used for meat, several circles had to be removed. So, not yet knowing how to count, the ancient people practiced arithmetic.

Then numerals appeared in the human language, and people were able to name the number of objects, animals, days. Usually there were few such numerals. For example, the Murray River people of Australia had two prime numbers: enea (1) and petchewal (2). They expressed other numbers with compound numerals: 3 = “petcheval-enea”, 4 “petcheval-petcheval”, etc. Another Australian tribe, the Kamiloroi, had simple numerals mal (1), Bulan (2), Guliba (3). And here other numbers were obtained by adding smaller ones: 4 = “Bulan-Bulan”, 5 = “Bulan-Guliba”, 6 = “Guliba-Guliba”, etc.

For many peoples, the name of the number depended on the items being counted. If the inhabitants of the Fiji Islands counted boats, then the number 10 was called “bolo”; if they counted coconuts, the number 10 was called "karo". The Nivkhs living on Sakhalin on the banks of the Amur did exactly the same thing. Also inXIXcentury, they called the same number with different words if they counted people, fish, boats, nets, stars, sticks.

We still use various indefinite numbers with the meaning “many”: “crowd”, “herd”, “flock”, “heap”, “bunch” and others.

With the development of production and trade exchange, people began to better understand what three boats and three axes, ten arrows and ten nuts have in common. Tribes often traded "item for item"; for example, they exchanged 5 edible roots for 5 fish. It became clear that 5 is the same for both roots and fish; This means that you can call it in one word.

Other peoples used similar methods of counting. This is how numberings based on counting in fives, tens, and twenties arose.

So far I have talked about mental counting. How were the numbers written down? At first, even before the advent of writing, they used notches on sticks, notches on bones, and knots on ropes. The wolf bone found in Dolní Vestonice (Czechoslovakia) had 55 incisions made more than 25,000 years ago.

When writing appeared, numbers appeared to record numbers. At first, numbers resembled notches on sticks: in Egypt and Babylon, in Etruria and Phenice, in India and China, small numbers were written with sticks or lines. For example, the number 5 was written with five sticks. The Aztec and Mayan Indians used dots instead of sticks. Then special signs appeared for some numbers, such as 5 and 10.

At that time, almost all numberings were not positional, but similar to Roman numbering. Only one Babylonian sexagesimal numbering was positional. But for a long time there was no zero in it, as well as a comma separating the whole part from the fractional part. Therefore, the same number could mean 1, 60, or 3600. The meaning of the number had to be guessed according to the meaning of the problem.

Several centuries before the new era, a new way of writing numbers was invented, in which the letters of the ordinary alphabet served as numbers. The first 9 letters denoted the numbers tens 10, 20,..., 90, and another 9 letters denoted hundreds. This alphabetical numbering was used until the 17th century. To distinguish “real” letters from numbers, a dash was placed above the letters-numbers (in Rus' this dash was called a “titlo”).

In all these numberings it was very difficult to perform arithmetic operations. Therefore, the invention inVIcentury by Indians, decimal positional numbering is rightfully considered one of the greatest achievements of mankind. Indian numbering and Indian numerals became known in Europe from the Arabs, and are usually called Arabic.

When writing fractions for a long time, the whole part was written in the new decimal numbering, and the fractional part in sexagesimal. But at the beginningXVV. Samarkand mathematician and astronomer al-Kashi began to use decimal fractions in calculations.

The numbers we work with are positive and negative numbers. But it turns out that these are not all the numbers that are used in mathematics and other sciences. And you can learn about them without waiting for high school, but much earlier if you study the history of the emergence of numbers in mathematics.

Chapter 2. MULTIPLICATION TABLE ON YOUR FINGERS

2.1 Multiplication table by 9.

Finger movement - this is one way to help your memory: use your fingers to remember the multiplication table by 9. Putting both hands side by side on the table, we number the fingers of both hands in order as follows: the first finger on the left will be designated 1, the second one behind it will be designated 2, then 3, 4... to the tenth finger, which means 10. If you need to multiply any of the first nine numbers by 9, then to do this, without moving your hands from the table, you need to bend the finger whose number means the number by which nine is multiplied. The number of fingers lying to the left of the bent finger determines the number of tens, and the number of fingers lying to the right indicates the number of units of the resulting product.

3 9= 27

Try multiplying yourself using this method:6 · 9, 9 · 7.

2.2 Multiplying numbers from 6 to 9.

The ancient Egyptians were very religious and believed that the soul of the deceased in the afterlife was subjected to a finger counting test. This already speaks volumes about the importance that the ancients attached to this method of multiplying natural numbers (it was calledfinger counting ).

They multiplied single-digit numbers from 6 to 9 on their fingers. To do this, they stretched out as many fingers on one hand as the first factor exceeded the number 5, and on the second they did the same for the second factor. The remaining fingers were bent. After this, they took as many tens as the length of the fingers on both hands, and added to this number the product of the bent fingers on the first and second hand.

Example: 8 ∙ 9 = 72

Thus,7 7 = 49.

Chapter 3. DIFFERENT WAYS OF MULTIPLICATION

3.1 Multiplying a number by 9.

To multiply a number by 9, you need to add 0 to it and subtract the original number.

For example: 72 · 9 = 720 – 72 = 648.

3.2 Multiplying two-digit numbers by 11.

To multiply a number by 11, you need to mentally expand the digits of this number and put the sum of these digits between them.

45 ∙ 11 = 495

53 ∙ 11 = 583

“Fold the edges, put them in the middle” - these words will help you easily remember this method of multiplying by 11.

To multiply by 11 a number whose sum of digits is 10 or more than 10, you need to mentally move apart the digits of this number, put the sum of these digits between them, and then add 1 to the first digit, leaving the second and third digits unchanged.

87 ∙ 11 = 957

94 ∙ 11 = 1024

This method is only suitable for multiplying two-digit numbers.

3.3 Multiplying two-digit numbers by 111, 1111, etc., knowing the rules for multiplying a two-digit number by the number 11.

If the sum of the digits of the first factor is less than 10, you need to mentally expand the digits of this number by 2, 3, etc. step, add these numbers and write down their sum between the spread out numbers the appropriate number of times. Please note that the number of steps is always less than the number of units by 1.

Example:

24 111=2 (2+4) (2+4) 4 = 2664 (number of steps - 2)

24 1111=2 (2+4) (2+4) (2+4) 4 = 26664 (number of steps - 3)

42 · 111 111 = 4 (4+2) (4+2) (4+2) (4+2) (4+2) 2 = 4666662. (number of steps – 5)

If there are 6 units, then there will be 1 fewer steps, that is, 5.

If there are 7 units, then there will be 6 steps, etc.

It is a little more difficult to perform mental multiplication if the sum of the digits of the first factor is 10 or more than 10.

Examples:

86 · 111 = 8 (8+6) (8+6) 6 = 8 (14) (14) 6 = (8+1) (4+1) 46 = 9546.

In this case, you need to add 1 to the first digit 8, we get 9, then 4+1 = 5; and leave the last numbers 4 and 6 unchanged. We get the answer 9546.

3.4 Multiplying a two-digit number by 101, 1001, etc.

Perhaps the simplest rule: assign your number to yourself. Multiplication is complete. Example:

32 · 101 = 3232;

47 · 101 = 4747;

324 · 1001 = 324 324;

675 · 1001 = 675 675;

6478 · 10001 = 64786478;

846932 · 1000001 = 846932846932.

3.5 Multiplication by 5; 25; 125.

First multiply by 10, 100, 1000 and divide the result by 2, 4, 8

32 5 = 32 10: 2 = 320: 2 = 160

84 25 = 84 100: 4 = 8400: 4 = 2100

24 125 = 24 1000: 8 = 24000: 8 = 3000

Another way: 32 5 = 32: 2 10 = 160

3.6 Multiplication by 22, 33, …, 99

To multiply a two-digit number by 22.33,..., 99, this factor must be represented as the product of a single-digit number (from 2 to 9) by 11, that is, 33 = 3 x 11; 44 = 4 x 11, etc. Then multiply the product of the first numbers by 11.

Examples:

18 · 44 = 18 · 4 · 11 = 72 · 11 = 792;

42 · 22 = 42 · 2 · 11 = 84 · 11 = 924;

13 · 55 = 13 · 5 · 11 = 65 · 11 = 715;

24 · 99 = 24 · 9 · 11 = 216 · 11 = 2376.

3.7 Multiplying by 37

Before learning how to verbally multiply by 37, you need to know well the sign of divisibility and the multiplication table by 3. To verbally multiply a number by 37, you need to divide this number by 3 and multiply by 111.

Examples:

24 · 37 = (24: 3) · 37 · 3 = 8 · 111 = 888;

· 37 = (18: 3) · 111 = 6 · 111 = 666.

3.8 Multiplying a number by 1.5.

To multiply a number by 1.5, you need to add half of it to the original number.

For example:

34 · 1.5 = 34 + 17 = 51;

146 · 1.5 = 146 + 73 = 219.

Chapter 4.SQUAREING A TWO-DIGIT NUMBER

4.1 Squaring a two-digit number ending in 5.

To square a two-digit number ending in 5, you need to multiply the tens digit by the digit greater than one, and add the number 25 to the right of the resulting product.

25 25 = 625

2 · (2 ​​+ 1) = 2 · 3 = 6, write 6; 5 5 = 25, write 25.

35 35 = 1225

3 · (3 + 1) = 3 · 4 = 12, write 12; 5 5 = 25, write 25.

4.2 Squaring a two-digit number starting with 5.

To square a two-digit number starting with five, you need to add the second digit of the number to 25 and add the square of the second digit to the right, and if the square of the second digit is a single-digit number, then you need to add the digit 0 in front of it.

For example:
52 2 = 2704, because 25 +2 = 27 and 2 2 = 04;
58 2 = 3364, because 25 + 8 = 33 and 8 2 = 64.

CONCLUSION

As we see, quick mental counting is no longer a sealed secret, but a scientifically developed system. Since there is a system, it means it can be studied, it can be followed, it can be mastered.

All the oral multiplication methods we have considered indicate the long-term interest of scientists and ordinary people in playing with numbers.

Using some of these methods in the classroom or at home, you can develop the speed of calculations, instill interest in mathematics, and achieve success in studying all school subjects. In addition, mastering these skills develops the student’s logic and memory.

Knowledge of quick counting techniques allows you to simplify calculations, save time, and develop logical thinking and mental flexibility.

There are practically no quick counting techniques in school textbooks, so the result of this work - a reminder for quick mental counting - will be very useful for students in grades 5-6.

We chose the topic “Tricks of mental calculation”because we love mathematics and would like to learn how to count quickly and correctly, without resorting to using a calculator.

LIST OF REFERENCES USED

Vantsyan A.G. Mathematics: Textbook for 5th grade. - Samara: Publishing house "Fedorov", 1999.

Kordemsky B.A., Akhadov A.A. The wonderful world of numbers: A book of students, - M. Education, 1986.

Oral counting, Kamaev P. M. 2007

“Mal arithmetic – mental gymnastics” G.A. Filippov

"Verbal counting". E.L.Strunnikov

Bill Handley “Count in your head like a computer”, Minsk, Potpourri, 2009.

Annex 1

QUESTIONNAIRE

1 . Why do you need to be able to count?

a) useful in life, for example, counting money;

b) to do well at school; c) to decide quickly;

d) to be literate; e) it is not necessary to be able to count.

2. List which school subjects you will need to count correctly when studying?

a) mathematics; b) physics; c) chemistry; d) technology; e) music; f) physical culture;

g) life safety; h) computer science; i) geography; j) Russian language; k) literature.

3. Do you know quick counting techniques?

a) yes, a lot; b) yes, several; c) no, I don’t know.

4. Would you like to learn fast counting tricks to count quickly?

a) yes; b) no.

Appendix 2

STATISTICAL DATA PROCESSING

1) Why do you need to be able to count?

Useful in life

To do well in school

To decide quickly

To be literate

You don't have to be able to count

Number of students

65

32

36

60

0

%

62%

30%

34%

57%

0%

2) When studying which school subjects will you need to count correctly?

Mathematics

Physics

Chemistry

Technology

Music

Physical Culture

life safety fundamentals

Computer science

Geography

Russian language

Literature

Number of students

105

71

55

37

5

26

7

66

39

18

12

%

100%

68%

52%

35%

5%

25%

7%

63%

No,

Don't know

Number of students

18

21

66

%

17%

20%

63%

4) Would you like to learn quick counting techniques to solve quickly?

Yes

No

Number of students

91

9

%

91%

9%

mastering mental arithmetic

This list of a few little-known math tricks will show you how to quickly do math in your head in cases more complicated than 5 times 10, and also allow your friends to use you as a calculator.

1. Multiply by 11
We all know how to quickly multiply a number by 10, you just need to add a zero at the end, but did you know that there is a trick to easily multiply a two-digit number by 11?
Let's say we need to multiply 63 by 11. Take the two-digit number that needs to be multiplied by 11 and imagine the space between its two digits:
6_3
Now add the first and second digit of this number and place it in this place:
6_(6+3)_3
And our multiplication result is ready:
63*11=693
If the result of adding the first and second digits is a two-digit number, insert only the second digit, and add one to the first digit of the original number:
79*11=
7_(7+9)_9
(7+1)_6_9
79*11=869

2. Quickly squaring a number ending in 5
If you need to square a two-digit number ending in 5, you can do it very simply in your head. Multiply the first digit of the number by itself plus one and add 25 at the end, and that's it:
45*45=4*(4+1)_25=2025

3. Multiply by 5
For most people, multiplying by 5 is easy for small numbers, but how can you quickly count large numbers multiplied by 5 in your head?
You need to take this number and divide by 2. If the result is an integer then add 0 to it at the end, if not, discard the remainder and add 5 at the end:
1248*5=(1248/2)_(0 or 5)=624_(0 or 5)=6240 (the result of division by 2 is an integer)
4469*5=(4469/2)_(0 or 5)=(2234.5)_(0 or 5)=22345 (the result of division by 2 with a remainder)

4. Multiply by 4
This is a very simple and, at first glance, obvious trick for multiplying any number by 4, but despite this, people do not realize it at the right time. To simply multiply any number by 4, you need to multiply it by 2, and then multiply it by 2 again:
67*4=67*2*2=134*2=268

5. Calculate 15%
If you need to mentally calculate 15% of a number, there is an easy way to do it. Take 10% of the number (dividing the number by 10) and add half of the resulting 10% to that number.
15% of 884 rubles=(10% of 884 rubles)+((10% of 884 rubles)/2)=88.4 rubles + 44.2 rubles = 132.6 rubles

6. Multiplying large numbers
If you need to multiply large numbers in your head and one of them is even, then you can use the method of simplifying factors by halving the even number and doubling the second:
32*125 is
16*250 is
8*500 is
4*1000=4000

7. Division by 5
Dividing a large number by 5 is very easy in your head. All you need to do is multiply the number by 2 and move the decimal place back one place:
175/5
Multiply by 2: 175*2=350
Shift by one sign: 35.0 or 35
1244/5
Multiply by 2: 1244*2=2488
Shift by one sign: 248.8

8. Subtraction from 1000
To subtract a large number from a thousand, follow a simple technique: subtract all digits of the number from 9 except the last one, and subtract the last digit of the number from 10:
1000-489=(9-4)_(9-8)_(10-9)=511

Of course, to learn how to quickly count in your head, you need to practice using these techniques many times in order to bring them to automaticity; a one-time reading will leave only zeros in your head.

beginning of mental counting

One-time action

One (about quantity, when counting)

. "... in a year and the stick shoots"

. "... on... there is no need"

. "... on... there is no need" (pron)

. "... let's get to work - I wanted a drink"

. "..., two, they took it!" (loader's cry)

. "...-two, grief is not a problem!" (movie)

. "Here are those..."

. "One" into the microphone

. "Eh..., and also...!"

. "Stay where you are, ...-two"

And forever

Two and done

Two three

. "do...!"

. "many, many more..."

. "first... to first class"

. "eh..., still..."

M. krata, reception, finally; unit, one. One, two, three, etc. Not once, not once, no matter how many times it was ordered. I see him for the first time, for the first time or for the first time. You can’t do it all at once or all at once. At once, at once or immediately, not to leave, in one mud, blow. You won’t guess right away, suddenly, soon. He was found at once, suddenly, instantly. Give him one! hit, give a punch. Grandma will give it to you once or twice! about an unpleasant accident. Count times, times, times. Take it at once! suddenly, together, amicably, swoop, at once, in full swing, whoosh; strike from here. It’s better to sing at once (all together), but to speak separately. Once this way, once that way, it’s different. Ten times (ten) example, cut once (one again). For the first time, this time I forgive, but next time (nother times) don’t get caught. Once in a while, always, every time. If only you could visit them once again, sometimes. Time after time, in a row, time after time, every time. the king dines at once, song of the south. zap. together. once and a lot. For some it doesn’t take long, but for us it’s just that. It doesn't happen once at a time. One (first) time doesn't count. Once doesn't count. Not at once, but not too far ahead. Once I lost my mind, I was known as a fool forever; Once you steal, you become a thief forever. Born twice, never baptized, sang and sang and died. Born twice, never baptized, ordained as a sexton (rooster). Yes, not all at once (not all at once)! said the drunken Cossack, who climbed onto his horse, asking for help from the saints, and threw himself over the saddle to the ground. Once upon a time, once upon a time, somehow, someday. Once, on Epiphany evening, the girls were wondering, Zhukovsky. Once, once, once, once, once, once, once. Once, southern, pastenok, stennik, erroneous. insole, one layer of honeycomb. Each layer of honeycomb is called at once; disposable honey, cell phone. One-time, once, times related. One-time money, payment, according to the condition, actor or writer, for each time of the game, performance

adv. more than once, more than once, repeatedly, repeatedly, many times, often

Designation of a single action (when counting, indicating quantity)

Single action; one (about quantity, when counting)

Slap (colloquial)

An isolated case

First word into the microphone

Just like..., two, three

Ras, grew, once, is a combined preposition, meaning: a) the end of an action, like all prepositions in general: to make you laugh, to wake you up; b) division, singularity, difference: break, distribute, discern, disperse; into destruction, remaking again: to develop, to grow; to warm; d strong, highest degree of action or state: to decorate, to offend; subtle, beautiful, reasonable; run away, go wild. The spelling of this preposition, like others in z, is shaky. Once it changes into roses and grew when the stress is transferred to the preposition: but our surrounding population generally loves roses more: rozinya, to develop; unbend, etc. the cursing Little Russian says roses, cursing Belarus: once; Southern Great Russians, including Moscow, while northern and eastern ones are mostly roses, although literacy smooths out these pronunciations more. It will be enough to explain some of the words of this beginning with examples; but there cannot be completeness here: in the meaning of the highest degree, since it can be attached to all verbs and to most names; eg It's a beaver hat, beaver! “Even if it’s a beaver, or even if it’s a beaver, I won’t buy it!” Razgrisha, razvanyushka, razdarushka, vm. Grisha, Vanya, Daria, humorously and affectionately, sometimes reproachfully

Seven...measure

The case of phenomena in a series of single-row actions, manifestations of something

Oral start of counting

The film "..., two woe is not a problem!"

Film "Do...!"

Yuzovsky's film "..., two - no trouble!"

. "... and forever"

. “here are those...”

Yuzovsky's film "..., two - grief is not a problem!"

. "first... to first class"

. "... on... there is no need"

. "many, many more..."

. “do...!”

. “stay where you are, ...-two”

. “eh..., and also...!”

. “eh..., still...”

The film "..., two sorrows are not a problem!"

Film "Do...!"

. "... let's get to work - I wanted a drink"

. "one" into the microphone

. “... on... there is no need” (pogov)

. "... in a year and the stick shoots"

. “..., two, they took it!” (loader's cry)

. "Eh..., and also...!"

. "... and forever" (express.)

. “... and forever” (express.)