Tatiana Petrova
Newspaper for children and caring parents on the formation of elementary mathematical representations"Why"
T. F. Petrova
Dear readers: children and adults ( parents and teachers, in front of you newspaper« Why» .
IN newspaper there will be pages for children where they will find interesting tasks and fun coloring books, puzzles, puzzles, pages for moms and dads, which will contain tips on formation of elementary mathematical concepts, development of thinking, memory, and many more interesting and useful things.
Some tips:
Do not complete all tasks with your child at once.
Completing tasks should bring joy to the child.
Get your child interested, but don't force him.
Easier tasks offer do it yourself, but do the difficult ones together; the child really needs your help and support.
Do not tell your child that he completed the task incorrectly, refrain from making offensive comments, focus on success, and rejoice in it with your child.
Good luck to you and your child.
The meaning of preschool age
"Children's mathematics teaches in simple
mental games to develop your mind,
create, create, produce."
Formation of elementary mathematical concepts there is only a means mental development child, his cognitive abilities. The desire to know the world inherent in man, the same desire exists in every child. However, cognition is not only a function of human intelligence. Cognition is a function of his personality; it is not possible without such qualities as activity and independence, self-confidence and self-confidence. For children younger age You need a feeling of security and safety. Therefore, the kind of atmosphere the teacher creates in the group determines how much interest in the world around each child will manifest and develop, the desire to learn and learn new things.
IN at different ages cognitive activity children different from each other. For example, thinking children from 2 to 3 years is predominantly visual and effective in nature. Basic shape cognitive activity is substantively- a manipulative game. What it is? This is a child’s independent play, during which he, by manipulating objects, gets acquainted with their internal structure, correlating them by size and form. It is very important to create positive conditions for this game in the group, since it is in this game that intelligence develops children of the third year of life.
For this it is necessary: * create a positive atmosphere in the group; * provide variety subject-development environment; * provide free access to a subject-development environment; * encourage independence and curiosity children.
Thinking children From 3 to 4 years old, children are different; they are already fluent enough to express their thoughts in words and not in gestures. They have a good command of nouns and verbs, and now the main task is to master adjectives. To do this, it is necessary to teach the child to identify individual signs items, such as color, size, form. For the child to learn this, the teacher needs to pay attention children for signs of objects and use them in your speech. However, there is no difference between cognition and play. The child learns as he lives. His world is the world "Here" And "Now". His attention is absorbed by real things and people surrounding him in this moment. While playing, a child at this age gains a wealth of experience interacting with the world, and he often needs the teacher to explain the experience to him.
Thinking children from 4 to 5 years is the age « Whychek» . At this age children want to know everything "For what?", « Why?» etc. They are capable of mentally imagine that, which has never been seen. They love to listen to adults' stories and ask a lot of questions. Thinking takes a huge leap forward. Now children begin to be interested in processes as ordered systems of events. The main way of learning for a child of this age is through adult stories. Therefore, the teacher needs to tell the children as much as possible, answer their questions, and ask the children themselves, that is, encourage them to think and reflect. When looking for answers, you need to think out loud with your children. As an adult thinks, so will children think.
It is important to get acquainted children with math problems concepts occurred in ordinary real life, on regular ones, not on special ones subjects so that children can see that mathematical concepts describe real world, and do not exist on their own. Thus - elementary mathematical representations V kindergarten should not destroy the naturalness of life children. The teacher’s task is to reveal to the child the beauty and richness of the world around him, and any knowledge is only a means of solving this task. When planning his work, the teacher should try to include mathematics not forced into different types of activities. This will allow you to safely avoid frontal math classes that are so tiring children. Then small children will learn without knowing what it is mathematics.
Pedagogical commandments that can guide your work.
- J. J. Rousseau wrote: “...what they are not in a hurry to achieve, they usually achieve for sure and very quickly.” Each child has his own time and hour of comprehension.
Maximum attention must be paid to children who are lagging behind. New material you need to start learning with them earlier than with the whole group children(get ahead, not catch up with the group).
It is necessary to constantly encourage all the efforts of the child and his very desire to learn new things, to learn new things.
In preschool age, negative assessments of the child and the results of his activities should be avoided.
You can compare the results of a child’s work only with his own achievements, but not with the achievements of others. children.
It is very important to answer all questions children and do things with them that they like.
Forced training is useless.
Only by having good personal contact with a child can you teach him something.
They hear better those who speak more quietly.
WHERE IS WHOSE LUNCH?
Give each hat a pair of mittens.
Draw the missing one in each square item.
Organization substantively– development environment for formation of elementary
mathematical concepts in children preschool age.
Mathematics- a serious and complex science, especially for children preschool age. On the success of teaching preschoolers mathematical the beginnings are influenced not only by the content proposed material, but also form of its presentation, which can arouse the interest of the child and his cognitive activity. Need to organize pedagogical process so that the child plays, develops and learns at the same time.
Carrying out activities in this direction, I came to the conclusion that it is more interesting for a preschooler to learn everything on his own, in a practical way, transferring his life into a fairy tale, overcoming obstacles artificially created by adults, and simultaneously mastering not only clear mathematical skills, but also learning about the world around us.
An indispensable condition for development mathematical abilities in preschoolers is enriched subject-development environment.
To achieve development goals children through entertaining material, was in the group math corner decorated« Entertaining mathematics» . The organization of the corner was carried out with the active participation children, which created a positive attitude towards them material, interest, desire to play. In artistic registration corner, geometric ornaments and plot images from geometric figures, heroes of children's literature were used. Game selection material was determined age capabilities and level of development children group. The corner houses a variety of entertaining material for that so that each of children I was able to choose the game for myself. This
Board and printed games ( "Pick a pattern", "Collect the number","Fun Cube", etc);
Games for logical development thinking: ("Games with Cuisenaire's sticks", "Games with Dienesh blocks" etc.);
Puzzles ( "Labyrinth", "Games with Counting Sticks", "Puzzles" etc.);
Logic problems ( “What numbers have changed?”, "Find a similar figure", "Only one property" etc.);
Games for composing a whole from parts, for recreating figures - silhouettes from special sets of figures ( "Matryoshka", "Geometric mosaic" etc.)
Games to develop spatial orientation ( "Find something similar").
They are all interesting and entertaining. Particularly popular with children enjoy plane geometric games character: "Tangram", "Cubes for everyone" etc. Children can come up with new, more complex silhouettes not only from one, but also from 2 - 3 sets for the game.
As children master games, more complex games are introduced with new entertaining material.
The main task of the teacher is: stimulating the manifestation of independence in games, maintaining and further development of children's interest in entertaining games
In achieving independent activity, I was guided by the following rules:
1. Explanation of the rules of the game, familiarization with general methods of action.
2. Playing together with a child, with a subgroup children. Children learn game actions, their methods, and approaches to solving problems.
3. Creation elementary problematic - search situation in joint play activities with the child.
4. Organization of various forms activities in corner: competitions, competitions (for the best logic problem, labyrinth, figure - silhouette, evenings of leisure, mathematical entertainment
Organization of a corner in a group entertaining mathematical material gave positive results: children learned to reason, justify the progress of searching for solutions to problems; find several solutions to problem problems mathematical situations . There was a desire to occupy one's own free time not only entertaining, but also games that require mental stress and intellectual effort.
Math on a walk
Mathematical The development of preschool children is a complex process, it is not only the ability to count and solve arithmetic problems, but also the development of the ability to see relationships, dependencies, and operate in the world around them. objects, signs, symbols.
Our task is to develop these abilities, to give the child the opportunity to explore the world at every stage of his growing up.
Richest source for expansion children's mathematical horizons are walks.
If you don't give your child a chance to look around math facts, then he will not notice them and will not show interest in them on his own. The attention of a preschooler is selective, and if it is not directed to something special, it "something" he may not notice. Therefore, it is important to set a simple question: "What do you see?" Be sure to give your child time to look around again, do not rush him.
While walking down the street, in the park, in the forest, pay attention to the quantity, size, form, spatial arrangement of objects (count how many cars have passed; compare the height of a tree and a house, the size of a dove and a sparrow; how many floors are in the house to the right or left of you; which birch leaf shapes).
Suggest child look around and find steam rooms items: the bird has 2 wings, 2 legs; in a dog (cats) 2 eyes, 2 ears. Ask what people want two: two arms, two ears, two eyes, two shoulders, two elbows, two feet, two heels. The child can not only name them, but also show them.
Playing in the sandbox suggest for your baby to make Easter cakes out of wet sand using molds of different sizes. Compare them by size. Find the same ones. Ask how many Easter cakes there are? Which Easter cakes are there more or less?
You can collect fallen leaves together into small bouquets. Then try to guess which bouquet has more leaves and justify your answer. Don't tell me how to do it. Let the child find a way on his own solutions: arrange the leaves one under the other or put the leaves of one of the bouquets on leaves from the other.
Suggest draw a triangle on the ground or asphalt, and then think and say that it could be like this forms(kerchief, balalaika, road sign).
While walking in the park, draw your child’s attention to thin and thick tree trunks. Suggest, clasping them with his hands, define which ones are thicker? You can look together for thick and thin branches, high and low items.
In winter, children love to make snowmen, take a little time, make your child happy, and then ask how big the balls were they made? Which ball is below? Which one is at the top? Which is the largest ball? Which ball is smaller?
Draw wide and narrow paths in the snow with sticks. Suggest child to jump over them. Ask which paths are easier to jump over. Why?
While watching children slide down the slide, ask how long children came down, who was first, third, fifth, etc. Who climbed higher than everyone else, who climbed lower? Who was the first to climb the hill, who was the second?
So, in an immediate environment, sacrificing a small amount of time, you can introduce your child to many mathematical concepts, contribute to their better assimilation, maintaining and developing interest in mathematics.
Help the butterfly
Who does she look like?
Number 2 was walking along the path and heard someone crying under a bush.
- I-I-I, I got lost.
Deuce looked under the bush and saw a large gray chick there.
- Who is your mom? – number 2 asked the chick.
– My mother is a beautiful and big bird. “She looks like you,” the chick squeaked.
Don’t cry, we will find her,” said number 2.
She put the chick on her tail, and they went to look for their mother.
Soon Deuce saw a beautiful flat bird with a long tail above the meadow.
– Isn’t this your chick, beautiful bird? – asked Deuce.
“I’m not a bird, but a kite.” I don't even have wings.
“Pee-pee, this is not my mother, my mother looks like you,” said the chick.
Meaning elementary math concepts for children preschool age... 3
Games for little ones why....6
Organization substantively– development environment for formation of elementary mathematical concepts in children preschool age... eleven
Mathematics for a walk…15
Mom, read a fairy tale... 18
classes "Math cafe"
evenings,
dedicated to closed and math week
mathematical lottery.
Questions for the game
What is the result of addition called?
How many minutes in one hour?
What is the name of the angle measuring device?
What does half an apple look like?
What is the smallest three-digit number?
Three horses ran 30 km. How far did each horse run?
What is the modulus of the number -6?
What is the name of a fraction in which the numerator is equal to the denominator?
What is the sum of adjacent angles?
Name the number that “separates” positive and negative numbers.
72:8.
One hundredth of a number.
Third month of summer holidays.
Another name for the independent variable.
Smallest even natural number.
How many kids were there in a goat with many children?
A triangle with two equal sides?
What shaft is depicted in Aivazovsky’s painting?
Zero's rival.
Part of a line bounded by two points?
The reciprocal of 2.
Subtraction result.
What is the name of the segment that extends from the vertex of a triangle and bisects the opposite side?
The opposite number is 5.
A rectangle with all sides equal.
One hundredth of a meter.
Divide 50 by half.
What is the name of the device for measuring segments?
What is the result of multiplication called?
How many seconds are in one minute?
What is the largest three-digit number?
Name the modulus of the number -4.
What is the name of a fraction in which the numerator is greater than the denominator?
What is the straight angle?
Name an integer greater than -1 but less than 1.
60:5.
Last month of the school year.
The reciprocal of 5.
The name of the graph of a function of direct proportionality.
Day of the week preceding Friday.
One tenth of a decimeter.
How many sides does a square have?
The opposite number is -7.
Unit of measurement of angles.
Which lines intersect at right angles?
The first month of winter.
How to find an unknown multiplier?
What are the equal sides of an isosceles triangle called?
The number by which given number divided without remainder.
A figure formed by two rays with a common origin.
How many negative factors must a product have for it to be a negative number?
1/60th of a degree?
The player's friend.
What is the value of the dependent variable called?
Angle equal to 180.
The number that makes an equation true.
What is the result of division called?
How many months are there in a year?
What is the name of the device for measuring the length of segments?
Name the largest single digit number.
A number that cannot be divided by.
Name the modulus of the number -2.
First month of the year.
A triangle whose two sides are equal.
The opposite number is -4.
First month of autumn.
What is the largest integer that can divide any integer without leaving a remainder?
Highest grade in school.
Smallest even number.
Equality with a variable.
What is the graph of the function y=kx+b?
Volume of a kilogram of water?
The sum of the lengths of all sides of a polygon?
A part of a line bounded by two points.
How to find an unknown dividend?
Property of vertical angles.
How many negative factors must a product have for it to be a positive number?
One hundredth of a kilometer.
Not a school day of the week.
1/60th of a minute.
Lowest grade in school.
The number of heights in a triangle.
The largest five-digit number.
An angle equal to 90 degrees.
What is the result of subtraction called?
How many hours are there in a day?
What is the name of the tool for drawing a circle?
The largest two-digit number.
Module of number 15.
What is the name of a fraction in which the numerator is less than the denominator?
What is a right angle?
A number that is neither positive nor negative?
One seventh of a week.
The first month of the new school year.
The name of the graph of a linear function.
Smallest positive integer.
A triangle with all sides equal.
The reciprocal of 3.
What is the name of the ray that comes out from the vertex and divides it in half?
One tenth of a decimeter.
What comes after Tuesday?
The opposite number is 9.
What is heavier than 1 kg of cotton wool or 1 kg of iron?
First month of summer?
In what case is the product equal to zero?
How to find an unknown subtrahend?
A line connecting two neighboring peaks triangle.
1/180 part of a developed angle.
The coming week at our school is dedicated to the oldest and youngest, forever young science -mathematics.
Mathematics has always accompanied a person in life. It helps the development of other sciences, it develops in a person such important qualities personalities like:
Logical thinking;
Determination, strong will;
Sustained attention, concentration;
Good memory;
Ability to think logically: compare, contrast, classify;
Ability for creativity and scientific imagination;
Sense of foresight;
Ability to estimate and evaluate results;
Performance;
Clarity and realism in your judgments and conclusions;
Resourcefulness and ingenuity;
Sense of humor.
And qualities such as intuition, inspiration, insight lead to great discoveries in science.« IN any opening There is 99% labor And sweating And only 1% talent And abilities », - saidL. Magnitsky. « Inspiration – This like this guest , which Not loves visit lazy », - he remarked.
Systematic studies of mathematics enrich a person and ennoble him. Anyone who has at least once experienced the joyful feeling of solving a difficult problem, has known the joy of a small, but still discovery, since every problem in mathematics is a problem to which humanity has sometimes gone for hundreds and millennia, will strive to learn more. and use the acquired knowledge in life.
In many modern professions mathematical knowledge is needed: an agronomist and an engineer, a worker and a milkmaid, an astronaut and a diplomat, a salesman and a cashier. Even for a housewife - for housekeeping, for repairing an apartment, for visiting a store, post office, telegraph, etc.
Great Charles Gauss said in the 18th century:« Mathematics – queen everyone sciences , A arithmetic – queen mathematicians ».
Leonty Magnitsky published the first Russian textbook in 1703“Arithmetic is the science of numbers”. On the cover of the textbook he depicted the Temple of Sciences. On the throne is Queen Mathematics, the columns of the temple are applied sciences: astronomy, algebra, physics, geology, geometry, trigonometry, geography, and arithmetic is the initial stages of the entire temple: addition, subtraction, division, multiplication.
From grades 1 to 6 at school you study arithmetic - those steps on which the throne of the Queen of Mathematics stands, i.e. you entered along these steps into the temple of sciences. In the 7th grade, you begin to study algebra, geometry, physics, and your success in new sciences, in each of which mathematics is invisibly present, will depend on how strong your levels are.
Mathematics - this is a tool with the help of which a person learns and conquers the world around him. To make a discovery in mathematics, you must love it as each of the great mathematicians loved it, as dozens and hundreds of other people loved and love it. Do even a small part of what each of them did, and the world will forever remain grateful to you. Love math!
Mathematics is the language spoken by all the exact sciences, especially physics and astronomy. All physical laws written in mathematical formulas. All laws of motion of planets, stars and galaxies are subject to mathematical laws.
The role of mathematics in biologyis that all research is based on logical conclusions. From simple observation to abstract thinking. Mathematical methods analysis and synthesis, establishing connections between phenomena help to discover the laws of the development of living nature. This serves new science – mathematical biology.
Chemist – a technologist of our days uses the apparatus of higher mathematics in his practical work. The following branches of science have appeared:physical chemistry, chemical thermodynamics and others.
Geography– an interesting subject, but unthinkable without mathematics. Until the second century AD, geography was a descriptive science, then the ancient Greek scientist Ptolemy first used degrees of a circle and, using a degree network, drew a map that was used for several centuries. The call signs "sos ! People are in distress at sea. Their voice has been heard, but how can we find them? The victims provide their coordinates. What it is? And these are the azimuths. Again mathematics came to the rescue, because azimuth is nothing more than a sector of a circle. The graphs and diagrams that geography is so rich in are comparative values. You can't measure distance on a map without resorting to mathematics.
Many of you have heard about machine translation, about poems composed by machines, about mathematicians deciphering the languages of disappeared peoples. This is a new science -
mathematical linguistics. There are many facts about the combination of artistic and mathematical talents of some authors. A. Griboyedov, the author of “Woe from Wit,” studied at the university in three faculties, including physics and mathematics. The famous Soviet mathematician A. Ya. Khinchin did not become a professional poet, although in his youth he published four books of his poems. And the outstanding Russian woman - mathematician S. V. Kovalevskaya wrote and published the books “Childhood Memories”, “Nihilist” and others.
In Syracuse, in Greecethere is an Archimedes site. He was not only a great scientist, but also a great patriot. He used his inventions to protect his hometown from the Romans. Archimedes burned their ships with the help of huge magnifying glasses that he himself constructed. History remembers many scientists not only for their mathematical discoveries, but also for their civic position, their spiritual generosity and beauty.
In his youth, Carl Gauss was equally interested in ancient languages and mathematics. And if it weren’t for the regular 17-gon, which he built with a compass and ruler at the age of 19, perhaps Gauss would have been known not as a mathematician, but as a linguist. After becoming acquainted with the works of N. I. Lobachevsky, Gauss, in the 62nd year of his life, began studying the Russian language. And after 2 years I was already freely reading Russian scientific and fiction literature.Now transfers from foreign languages special machines do this.
The great Leonardo da Vinci developed in the 16th centurymathematical theory of painting. In his paintings he used the laws of the “golden ratio”, the laws of perspective, the laws of parallel and rectangular projection. His great paintings “The Last Supper”, a portrait of Mona Lisa (the so-called “La Gioconda”) and others adorn the best museums in the world. Mathematics is one of the most important subjects when teaching an artist.
Back in 1660, the great fencing master Spaniard Luis Pachena de Narvaez developed fencing theory based on mathematical principles, in the book "Great Steps". Today mathematics is persistently knocking on the door of sports. This includes analysis of assessments in sports, analysis of the abilities of future athletes, calculation of permissible loads, etc.
Music also has his own theory. The first theory originated with the ancient Greeks. It is based on mathematics. All sounds are arranged strictly sequentially according to the steps of the natural series in the duodecimal system. Our music theory is based on fractional numbers 1, which indicate the duration of any note. These fractions can be converted into binary, which is the basis of the language of computing.
Do you know the talented Descartes -
Creator of coordinate systems.
You know Lobachevsky, he, brother,
Copernicus of geometry, creator, sculptor.
Chebyshev is still a great titan,
And Sofya Kovalevskaya is a wonderful “mermaid”!
They were given a mighty talent,
They were given genius ingenuity.
Remember what Gauss told everyone:
“The science of mathematics is the queen of all sciences,”
It was not for nothing that he bequeathed -
Create in the fire of labor and torment.
Her role in discovery of laws,
In creating machines, airships,
Perhaps it would be difficult for us without the Newtons,
What history has given us to this day.
May you not become Pythagoras,
How I would like it to be!
But you will be a worker, maybe also a scientist,
And you will serve your Motherland honestly!
Song to the tune of “What do they teach at school?”
HYMN TO MATHEMATICS.
Solve equations, calculate radicals -
Interesting algebra problem!
extract integrals,
Fractions divide and multiply
If you try, good luck will come to you!
Geometry is needed, but it’s so complicated!
Either the figure or the body - you can’t tell.
Axioms are needed there,
Theorems are so important
Teach them - and you will achieve results!
All sciences are good
For the development of the soul.
You all know them yourself, of course.
Mathematics is necessary for the development of the mind,
It was, it will be, it is forever.
Final words from the teacher.
Mathematics is a tool with the help of which a person learns and conquers the world around him. To make a discovery in mathematics, you must love it as each of the great mathematicians loved it, as dozens and hundreds of other people loved and love it. Do even a small part of what each of them did, and the world will forever remain grateful to you. Love math!
Musical pause. Song to the tune of “My Bunny”.
You are my plus, I am your minus,
You are the cosine, I am your sine,
You are an axiom, I am a theorem,
The consequence is you, and I am the lemma.
Ma-te-ma-ti-ka my...
Chorus:
I don't sleep well at night,
I love mathematics so much
I have loved mathematics for so, so long.
I don’t even sleep during the day now,
I don't sleep even in the evening,
I keep learning, learning, learning, learning, learning.
You are knowledge, I am a cheat sheet,
If you are a zero, then I am a stick.
You are the ordinate, then I am the abscissa,
You are a corner, I am a bisector.
Ma - te - ma - ti - ka is mine...
Particular you, I am the divider,
You are the denominator, I am the numerator.
You are my circle, I am your sector,
You are my module, I am your vector.
Ma - te - ma - ti - ka is mine...
The sum is mine, and I am the difference,
You are long, and I am multiplicity,
You are the hypotenuse, I am your leg,
You and I have enough terms.
Ma - te - ma - ti - ka is mine...
Preview:
Preview:
Mathematics in Ancient Greece
The concept of ancient Greek mathematics covers the achievements of Greek-speaking mathematicians who lived between the 6th century BC. e. and V century AD e.
Until the 6th century BC. e. Greek mathematics was not famous for anything outstanding. As usual, counting and measurement were mastered. We know about the achievements of early Greek mathematicians mainly from the comments of later authors, mainly Euclid, Plato and Aristotle.
In the 6th century BC. e. The “Greek miracle” begins: two scientific schools appear at once: Ionians (Thales of Miletus) and Pythagoreans (Pythagoras).
Thales, a wealthy merchant, apparently learned Babylonian mathematics and astronomy well during his trading trips. The Ionians gave the first proofs of geometric theorems . However the main role in the creation of ancient mathematics belongs Pythagoreans.
Pythagoras, the founder of the school, like Thales, traveled a lot and also studied with Egyptian and Babylonian sages. It was he who put forward the thesis “Numbers rule the world", and worked on its justification.
The Pythagoreans made a lot of progress in the theory of divisibility, but were overly carried away by games with “triangular”, “square”, “perfect”, etc. numbers, to which, apparently, they attached mystical significance. Apparently, the rules for constructing “Pythagorean triplets” were already discovered then; comprehensive formulas for them are given by Diophantus. The theory of greatest common divisors and least common multiples is also apparently of Pythagorean origin. They probably built it general theory fractions (understood as ratios (proportions), since the unit was considered indivisible), learned to perform comparisons with fractions (reduction to a common denominator) and all 4 arithmetic operations.
Athens School of Pythagoras
From the history of mathematics
Mathematics in the East
Al-Khwarizmi or Muhammad ibn Musa Khwarizmi (c. 783 - c. 850) - great Persian mathematician, astronomer and geographer, founder of classical algebra.
Book about algebra and almukabal
Al-Khorezmi is best known for his “Book of Complementation and Opposition” (“Al-kitab al-mukhtasar fi hisab al-jabr wa-l-mukabala”), from the title of which the word “ algebra".
In the theoretical part of his treatise, al-Khwarizmi gives a classification equations 1st and 2nd degrees and distinguishes six types:
- squares are equal to roots (example 5 x 2 = 10 x);
- squares equal a number (example 5 x 2 = 80);
- the roots are equal to the number (example 4 x = 20);
- squares and roots are equal to a number (example x 2 + 10 x = 39);
- squares and numbers are equal to roots (example x 2 + 21 = 10 x );
- roots and numbers are equal to the square (example 3 x + 4 = x 2 ).
This classification is explained by the requirement that both sides of the equation contain positive members. Having characterized each type of equations and showing with examples the rules for solving them, al-Khwarizmi gives geometric proof of these rules for the last three species, when the solution is not reduced to simple extraction of the root.
To bring quadratic equation general view Al-Khwarizmi introduces two actions to one of the six canonical types. The first of them, al-jabr, consists of transferring negative member from one part to the other to obtain positive terms in both parts. The second action - al-mukabala - consists of bringing similar terms in both sides of the equation. In addition, al-Khwarizmi introduces the multiplication rule polynomials . He shows the application of all these actions and the rules introduced above using the example of 40 problems.
Persian Gulf
Euclidean geometry
Euclid
ancient Greek mathematician
(365-300 BC)
Almost nothing is known about Euclid, where he was from, where and with whom he studied.
The Pope of Alexandria (3rd century) claimed that he was very friendly to all those who made at least some contribution to mathematics. Correct, in highest degree decent and completely devoid of vanity. Once King Ptolemy I asked Euclid if there was a shorter way to study geometry than studying the Elements. To this Euclid boldly replied that “in geometry there is no royal road.” Euclid, like other great Greek geometers, studied astronomy, optics and music theory.
We know much more about the mathematical creativity of Euclid. First of all, Euclid is for us the author of the Elements, from which mathematicians all over the world studied. This amazing book has survived more than two millennia, but has still not lost its significance not only in the history of science, but also in mathematics itself. The system of Euclidean geometry created there is now studied in all schools of the world and underlies almost all practical activities of people. Classical mechanics is based on Euclid’s geometry; its apotheosis was the appearance in 1687 of “Newton’s mathematical principles of natural philosophy, where the laws of earthly and celestial mechanics and physics are established in the absolute Euclidean space.
"N The beginnings of Euclid consist of 15 books. The 1st formulates the initial provisions of geometry, and also contains the fundamental theorems of planimetry, including the theorem on the sum of the angles of a triangle and the Pythagorean theorem. The 2nd book sets out the foundations of geometric algebra. The 3rd book is devoted to properties of the circle, its tangents and chords. In the 4th book, regular polygons are considered...
Geometry of the Middle Ages
The geometry of the Greeks, called today Euclidean, or elementary, was concerned with the study of the simplest forms: straight lines, planes, segments, regular polygons and polyhedrons, conic sections, as well as spheres, cylinders, prisms, pyramids and cones. Their areas and volumes were calculated. The transformations were mainly limited to similarities.
Muse of Geometry, Louvre.
The Middle Ages gave a little to geometry, and the next great event in its history was the discovery by Descartes in the 17th century coordinate method(“Discourse on Method”, 1637). Sets of numbers are associated with points, this allows one to study the relationships between shapes using algebraic methods. This is how analytical geometry appeared, studying figures and transformations that are specified in coordinates algebraic equations. Around the same time, Pascal and Desargues began researching the properties flat figures, which do not change when projecting from one plane to another. This section is called projective geometry. The coordinate method underlies the differential geometry that appeared somewhat later, where figures and transformations are still specified in coordinates, but by arbitrary, fairly smooth functions.
In geometry we can roughly distinguish the following sections:
- Elementary geometry - the geometry of points, lines and planes, as well as figures on a plane and bodies in space. Includes planimetry and stereometry.
- Analytical geometry - geometry of the coordinate method. Studies lines, vectors, figures and transformations that are given by algebraic equations in affine or Cartesian coordinates, using algebraic methods.
- Differential geometry and topology studies lines and surfaces defined by differentiable functions, as well as their mappings.
- Topology is the science of the concept of continuity in its most general form.
The study of Euclid's axiom system in the second half of the 19th century showed its incompleteness. In 1899, D. Hilbert proposed the first sufficiently strict axiomatics of Euclidean geometry.
Lobachevsky geometry
Nikolai Ivanovich Lobachevsky (November 20, 1792 – February 12, 1856), great Russian mathematician
The reason for the invention of Lobachevsky’s geometry was Euclid’s V postulate: “Through a point not lying on a given line there passes only one straight line that lies with the given line in the same plane and does not intersect it" The relative complexity of its formulation gave rise to a feeling of its secondary nature and gave rise to attempts to derive it from the rest of Euclid’s postulates.
Attempts to prove Euclid's fifth postulate were carried out by such scientists as the ancient Greek mathematician Ptolemy (2nd century), Proclus (5th century), Omar Khayyam (11th - 12th centuries), and the French mathematician A. Legendre (1800).
Attempts were made to use proof by contradiction: the Italian mathematician G. Saccheri (1733), the German mathematician I. Lambert (1766). Finally, an understanding began to emerge that it was possible to construct a theory based on the opposite postulate:German mathematicians F. Schweickart (1818) and F. Taurinus (1825) (however, they did not realize that such a theory would be logically just as harmonious).
Lobachevsky in his work “On the Principles of Geometry” (1829), his first published work on non-Euclidean geometry, clearly stated that the V postulate cannot be proven on the basis of other premises of Euclidean geometry, and that the assumption of a postulate opposite to the postulate of Euclid allows one to construct a geometry so as meaningful as Euclidean, and free from contradictions.
In 1868, E. Beltrami published an article on Lobachevsky’s interpretations of geometry. Beltrami determined the metric of the Lobachevsky plane and proved that it has constant negative curvature everywhere. Such a surface was already known at that time - this is the Minding pseudosphere. Beltrami concluded that locally the Lobachevsky plane is isometric to a section of the pseudosphere.
The consistency of Lobachevsky's geometry was finally proven in 1871, after the appearance of Klein's model.
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DIVIDENT VALUE
PRIVATE
PRIVATE
MULTIPLIER MULTIPLIER VALUE
WORKS
WORK
SUBTRACT VALUE
DIFFERENCES
DIFFERENCE
TERM TERM VALUE
AMOUNTS
SUM
1 km = 1000m
1m = 10 dm
1 dm = 10cm
1cm = 10mm
1m = 100cm =1000mm
1 century = 100 years
1 year = 12 months
1 year = 365(366) days
1 day = 24 hours
1 hour = 60 minutes
1 minute = 60 seconds
1 t = 1000kg
1kg = 1000g
1c = 100kg
1t = 10c
R straight.
= a+b+a+b
R straight.
= (a+b) 2
R straight.
= a 2 + b 2
P square = a+a+a+a
P squared = a 4
a – length S = a b
b – width a = S b
S – area b = S a
(m, cm, etc.)
S – area b = S a
Increase
in time
b – width a = S b
Decrease
(m, cm, etc.)
Decrease
How many times
More less
1. ()
Preview:
by... units
Sophistry is a deliberately false conclusion that has the appearance of being correct. Whatever the sophistry, it necessarily contains one or more disguised errors. Especially often in mathematical sophisms “forbidden” actions are performed or the conditions of applicability of theorems, formulas and rules are not taken into account. Sometimes reasoning is carried out using an erroneous drawing or is based on “obviousness” leading to erroneous conclusions. There are sophisms containing other errors.
How are sophisms useful for students of mathematics? What can they give? Analysis of sophisms, first of all, develops logical thinking, that is, instills the skills of correct thinking. To discover an error in sophism means to realize it, and awareness of the error prevents it from being repeated in other mathematical reasoning. Analysis of sophisms helps the conscious assimilation of the mathematical material being studied, develops observation, thoughtfulness and a critical attitude towards what is being studied.
TRY YOUR STRENGTH
1) 4 rubles = 40,000 kopecks. Let’s take the correct equality: 2p = 200 k. Let’s square it piece by piece. We will receive: 4 rubles = 40,000 k. What is the mistake?
2) 5=6. Let's try to prove that 5=6. For this purpose, let's take a numerical identity:
35+10-45=42+12-54. Let's take the common factors of the left and right sides out of brackets. We get: 5(7+2-9)=6(7+2-9). Let's divide both sides of this equality by a common factor (enclosed in parentheses). We get 5=6. Where is the mistake?
3) . 2*2=5. Find the error in the following reasoning. We have the correct numerical equality: 4:4=5:5. Let us take its common factor out of brackets in each part. We get: 4(1:1)=5(1:1). The numbers in brackets are equal, so 4=5, or 2*2=5.
4) All numbers are equal to each other.Let m=n. Let's take the identity: m 2 -2mn+n 2 =n 2 -2mn+m 2 . We have: (m-n) 2 = (n-m) 2 . Hence m-n=n-m? or 2m=2n, which means m=n. Where is the mistake?
WE ARE LEARNING
REALIZE!
- A plane from Moscow flies to Kyiv and returns back to Moscow. In what weather will this plane make the entire journey faster: in calm weather; with the wind blowing with the same force in the direction Moscow-Kyiv?
- From a conversation on September 1: “How much longer do you have to study?” - “As much as you have already studied. And you?" - “One and a half times more.” Who went to what grade?
- In the notation KTS+KST=TSK, each letter has its own number. Find what the number TSC is equal to!
PROVE!
- The square of an odd number is an odd number.
- The square of an even number is a multiple of 4.
- Difference of squares of two consecutive odd numbers divisible by 8.
- The sum of the product of two consecutive natural numbers and the larger of them is equal to the square of that larger number.
- If you take some two-digit number With different numbers, rearrange the numbers in it and subtract the resulting number from the taken number, then the difference will be divided by 9.Will this be true for three-digit numbers(end numbers are rearranged)?
WONDERFUL CURVES
Archimedes' spiral. Imagine that along the radius of a uniformly rotating disk with constant speed a fly crawls. The path described by the fly is a curve called the Archimedes spiral. Draw some kind of Archimedes spiral.
Sine wave. Make a tube out of thick paper by folding it several times. Cut this tube at an angle. Look at the cut line if you unfold one of the parts of this tube. Redraw this line onto a piece of paper. You'll end up with one of those wonderful curves called a sine wave. You encounter it especially often when studying electrical and radio engineering.
Cardioid. Take two equal circles cut out of plywood (you can take two identical coins). Secure one of these circles. Attach the second one to the first one, mark point A on its edge, which is farthest from the center of the first circle. Then roll the movable circle along the stationary one without sliding and observe what line point A describes. Draw this line. It is one of Pascal's snails and is called a cardioid. In technology, this curve is often used to design cam mechanisms.
Geometric puzzles
- Fold three equal squares: 1) from 11 matches; 2) from 10 matches.
- The figure shown in the figure needs to be divided into 6 parts by drawing only 2 straight lines. How to do it?
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Rules of conduct for students
in the office
The mathematics room is equipped modern equipment for conducting training sessions: PC, projector, screen, printing device.
This equipment does not tolerate dust and requires careful handling.
The first requirement in the office is TB compliance. | |
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Prohibiting rules of behavior
in the office
Two other requirements in the cabinet -discipline and cleanliness. | |
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Rules of conduct for students
at the lesson
- When the teacher enters the classroom, students stand up. They sit down after greeting and permission from the teacher. Students also greet any adult who enters the classroom during class. When the teacher leaves the class, the students also stand up.
- During the lesson, the teacher sets rules for behavior in the lesson.
- During the lesson, you must not make noise, be distracted yourself or distract your comrades from their studies with conversations, games and other matters not related to the lesson.
- If a student wants to say something, ask the teacher a question, or answer a question, he raises his hand and, after permission, speaks. The teacher may set other rules.
- The end of lesson bell is given to the teacher. He determines the end time of the lesson and announces to the students its end.
- If a student missed lessons at school, he must present to the class teacher medical certificate or note from parents. Missing or being late for lessons without good reason is not permitted.
Rules of conduct for students
at a break
- At the end of the lesson, students are required to:
- tidy up your workplace;
- leave the class;
- obey the requirements of the teacher and students on duty.
- During recess, students are in the hallway. There are two attendants in the classroom who:
- ventilate the classroom
- erased from the board,
- prepare chalk and rag,
- make sure that no one is in the class during breaks,
- help the teacher prepare material for the lesson,
- Allow students to enter the classroom two minutes before the bell and with the permission of the teacher.
- During a break it is prohibited:
- run in places unsuitable for play, push each other;
- use obscene expressions and gestures, make noise, disturb others from resting or preparing for a lesson.
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Preview:
Will make it through the road
going,
And mathematics -
thinking!
Did you know that the first calculating device was the abacus?
The first “computing devices” that people used in ancient times were fingers and pebbles. In Ancient Egypt and Ancient Greece, long before our era, they used an abacus - a board with stripes along which pebbles moved. This was the first device specifically designed for computing. Over time, the abacus was improved - in the Roman abacus, pebbles or balls moved along grooves. The abacus lasted until the 18th century, when it was replaced by written calculations. Russian abacus - abacus appeared in the 16th century. They are still used today. The big advantage of Russian abacus is that it is based on the decimal number system, and not on the five-digit number system, like all other abaci.
Algorithm for working on a task
- I read the whole problem.
- I read the condition and highlight the data.
- I read the question and highlight what I'm looking for.
- I determine the structure of the task (simple or complex).
- I find the missing datum (if compound).
- I am bringing the decision to the end.
- I'm re-reading the question.
- I answer it.
Comic problems
- Firefighters are trained to put on their pants in three seconds. How many pants can a well-trained firefighter put on in 1 minute?
- There is one hole in a bagel, and twice as many in a pretzel. How many fewer holes are there in 7 bagels than in 12 pretzels?
- If baby Kuzya is weighed together with his grandmother, the result will be 59 kg. If you weigh grandma without Kuzya, you get 54 kg. How much does Kuzya weigh without his grandmother?
- A boxer, a karateka, and a weightlifter chased a cyclist at a speed of 12 km/h. Will they catch up with a cyclist if he, having covered 45 km at a speed of 15 km/h, lies down to rest for an hour?.
- Katya's height is 1 m 75 cm. Stretched out to her full height, she sleeps under a blanket whose length is 155 cm. How many centimeters does Katya stick out from under the blanket?.
- How many holes will there be in an oilcloth if you pierce it 12 times with a 4-tooth fork during lunch?.
- At a math lesson in the 7th group, there were students who had 56 ears, the teacher had 54 fewer ears. How many ears can you count during a math lesson?
- The area of one elephant's ear is 10,000 sq.cm. Find out in apt. m., area 2 elephant ears..
- Let's say that you decide to jump into water from a height of 8 meters. And, having flown 5 meters, he changed his mind. How many more meters will you have to fly involuntarily?
- Baby Kuzya screams like crazy 5 hours a day. Sleeps like the dead 16 hours a day. The rest of the time, baby Kuzya enjoys life in all ways available to him. How many hours a day does baby Kuzya enjoy life?
- Koschey the Immortal was born in 1123, and received a passport only in 1936. How many years did he live without a passport?
- Hungry Vasya eats it in 9 minutes. 3 bars, a well-fed Vasya spends 3 baht. 15 minutes. How much min. Is hungry Vasya faster with one candy bar?
- Baby Kuzi has 4 more teeth, but his grandmother only has 3. How many teeth do the grandmother and grandson have?
- Who will be heavier after dinner: the first is the cannibal, who weighed 48 kg before dinner and ate the 2nd cannibal for dinner, or the second, who weighed 52 kg and ate the first.
Rules of conduct in the mathematics classroom
- Enter the office only with the permission of the teacher. Students must enter the office in a change of shoes and without outerwear
- Students should enter the classroom calmly, without jostling, and maintaining order. Loud conversations and disputes over the workplace are prohibited
- You cannot touch any device in the office without permission, open cabinets, or touch projection equipment.
- It is prohibited to bring things not intended for study into the office. It is prohibited to use a cell phone
- Chewing gum, no matter how tasty it may seem, is strictly prohibited for use in the classroom, both in class and during recess.
- The main and most important requirement in the office is discipline. Dust raised in the classroom is harmful to both equipment and students
- You cannot bring bread, nuts, sweets, or seeds into the office. Lunch in the dining room must be eaten at the dining room table
Thanks for following the rules!
Preview:
In the world of mathematics
PERIMETER consists of two Greek words peri (around) and metreō (measure). Compare it with the words periscope (ckopeo – look), periphery (phero – carry), pericardia (kardia – heart), period (hogjs – way, road)
CHORD (Greek chordē) translated from Greek - string. The origin of this term in geometry is associated with the manufacture of a bow, in which a tightly stretched string - a bowstring - tightens its ends.
The words SECTOR and SEGMENT , it turns out, are related, because they come from the same Latin word (like the word axe), which is translated into Russian as cut. So, the sector and the segment dissect the circle, but each in its own way.
MEDIAN , mediator, physician - cognate. They come from the word medium - intermediary, average. A mediator is an object that allows a musician to extract sound from his musical instrument; physician - a doctor with the help of whom the patient is healed.
Word RHOMBUS comes from the Greek rhombos meaning tambourine. It turns out that in ancient times, tambourines - musical instruments - were not round, as they are now, but had the shape of a quadrangle with equal sides.
In the word BISEXTER the root is sectr - (familiar truth), and the prefix "bis" - which means repeat, twice. So, by the very structure of the word “bisector” it is easy to determine its meaning, and also understand why a double consonant should be written in this word With .
The word CATET is the same root as the words catacombs, cataract. The root kata is of Greek origin, meaning down, to fall. The word cataract (clouding of the eye lens) was previously used in the form of cataracts and had 2 meanings: a waterfall in the mountains, as well as movable barriers in the fortress gates. Catacombs – kata under; down + kumbē bowl.
The word HYPOTENUSE translated from Greek as to be opposite, i.e. the side of a triangle opposite its right angle.
Rebuses
Answers:
- Task
- Axiom
- Apothem
Answers:
- Vector
- Cone
- Pyramid
Preview:
Golden Ratio
Geometry has two treasures:
one of them is the Pythagorean theorem,
another is the division of a segment in average and extreme ratio.
I. Kepler
There are things that cannot be explained. So you come to an empty bench and sit down on it. Where will you sit - in the middle? Or maybe from the very edge? No, most likely, neither one nor the other. You will sit so that the ratio of one part of the bench to the other, relative to your body, will be approximately 1.62. A simple thing, absolutely instinctive... Sitting on a bench, you produced the “golden ratio”. The golden ratio was known back in ancient Egypt and Babylon, in India and China. The great Pythagoras created a secret school where the mystical essence of the “golden ratio” was studied. Euclid used it when creating his geometry, and Phidias - his immortal sculptures. Plato said that the Universe is arranged according to the “golden ratio”. And Aristotle found a correspondence between the “golden ratio” and the ethical law. The highest harmony of the “golden ratio” will be preached by Leonardo da Vinci and Michelangelo, because beauty and the “golden ratio” are one and the same thing. And Christian mystics will draw pentagrams of the “golden ratio” on the walls of their monasteries, fleeing from the Devil. At the same time, scientists - from Pacioli to Einstein - will search, but will never find its exact meaning. An endless series after the decimal point - 1.6180339887... Everything living and everything beautiful - everything obeys the divine law, whose name is the “golden ratio”.
Angel de Coitiers
Golden ratio in mathematics
In mathematics, proportion call the equality of two relations: a : b = c : d .
Line segment AB can be divided into two parts in the following ways:
- into two equal parts - AB: AC = AB: BC;
- into two unequal parts in any respect (such parts do not form proportions);
- thus, when AB: AC = AC: BC.
The latter is the golden division or division of a segment in extreme and average ratio.
The golden ratio is such a proportional division of a segment into unequal parts, in which the entire segment is related to the larger part as the larger part itself is related to the smaller one; or in other words, the smaller segment is to the larger as the larger is to the whole
a: b = b: c or c: b = b: a.
Practical acquaintance with the golden ratio begins with dividing a straight line segment in the golden proportion using a compass and ruler.
From point B a perpendicular equal to half is restored AB . Received point WITH connected by a line to a point A . A segment is plotted on the resulting line Sun ending with a dot D. Segment AD transferred to direct AB . The resulting point E divides segment AB in the golden ratio ratio.
Segments of the golden ratio are expressed as an infinite irrational fraction AE = 0.618..., if AB take as one BE = 0.382... For practical purposes, approximate values of 0.62 and 0.38 are used. If the segment AB taken as 100 parts, then the larger part of the segment is 62, and the smaller part is 38.
The properties of the golden ratio are described by the equation:
x 2 – x – 1 = 0.
Solution to this equation:
Golden Triangle
To find segments of the golden proportion of the ascending and descending series, you can use pentagram.
To build a pentagram, you need to build a regular pentagon. The method of its construction was developed by the German painter and graphic artist Albrecht Durer (1471...1528). Let O – center of the circle, A – a point on a circle and E – the middle of the segment OA . Perpendicular to radius OA , restored at the point ABOUT , intersects the circle at the point D . Using a compass, plot a segment on the diameter CE = ED . Length of a side inscribed in a circle regular pentagon equal to DC . Lay out segments on the circle DC and we get five points to draw a regular pentagon. We connect the corners of the pentagon through one another with diagonals and get a pentagram. All diagonals of the pentagon divide each other into segments connected by the golden ratio.
We draw straight AB. From point A lay a segment on it three times ABOUT arbitrary value, through the resulting point R draw a perpendicular to the line AB , on the perpendicular to the right and left of the point R set aside the segments ABOUT . Received points d and d 1 connect with straight lines to a point A . Segment dd 1 put on line Ad 1, getting point C . She split the line Ad 1 in proportion to the golden ratio. Lines Ad 1 and dd 1 used to construct a “golden” rectangle. |
Golden ratio in architecture
One of the most beautiful works of ancient Greek architecture is the Parthenon (5th century BC).
Visible in the pictures whole line patterns associated with the golden ratio. The proportions of the building can be expressed through various powers of the number Ф=0.618...
All architectural structures, temples and even homes from Ancient Egypt and Ancient Greece to this day were created and are being created in the harmony of numbers - according to the rules of the “Golden Section”.
Golden ratio in sculpture
The golden proportion was used by many ancient sculptors. Known golden ratio statues of Apollo Belvedere: the height of the person depicted is divided by the umbilical line in the golden ratio.
Back in the Renaissance, artists discovered that any picture has certain points that involuntarily attract our attention, the so-called visual centers. In this case, it does not matter what format the picture has - horizontal or vertical. There are only four such points; they divide the image size horizontally and vertically in the golden ratio, i.e. they are located at a distance of approximately 3/8 and 5/8 from the corresponding edges of the plane.
Golden ratio in fonts and household items
Golden ratio in biology
Rostock
Among the roadside herbs grows an unremarkable plant - chicory. Let's take a closer look at it. A shoot has formed from the main stem. The first leaf was located right there.
The shoot makes a strong ejection into space, stops, releases a leaf, but this time it is shorter than the first one, again makes an ejection into space, but with less force, releases a leaf of an even smaller size and is ejected again. If the first emission is taken as 100 units, then the second is equal to 62 units, the third is 38, the fourth is 24, etc. The length of the petals is also subject to the golden proportion. In growing and conquering space, the plant maintained certain proportions. The impulses of its growth gradually decreased in proportion to the golden ratio.
Golden ratio in body parts
By comparing the lengths of the phalanges of the fingers and the hand as a whole, as well as the distances between individual parts of the face, one can also find the “golden” ratios:
Sculptors claim that the waist divides the perfect human body in relation to the golden ratio. Measurements of several thousand human bodies have revealed that for adult men this ratio averages approximately 13/8 = 1.625
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5-6 grades
Warm-up
1. An orange is not lighter than a pear, and an apple is not lighter than an orange. Can a pear be heavier than an apple? Isn't it lighter than an apple?
2. A sister has four times as many brothers as sisters. And the brother has one more brothers than sisters. How many brothers and how many sisters are there in the family?
3. Two diggers dig a 2 m ditch in 2 hours. How many diggers will dig a 5 m ditch in 5 hours?
Comparison problems
Weighing problems
- Available pan scales without weights and three coins, one of them is counterfeit- easier others. Detect a counterfeit coin with one weighing.
- Solve the previous problem if there are 4 coins; 5; 6; 8; 9 and two weigh-ins.
Transfusion tasks
- There are 18 liters of gasoline in a barrel. There is a scoop with a volume4 l and two buckets of 7 l, inwhich you need to pour 6 liters of gasoline. Howto carry out a spill?
Number problems
Problems on “Graphs”
- The figure shows a diagram of bridges in the city of Königsberg. Is it possible to take a walk so that you cross each bridge exactly once?
Getting ready for the Olympics
We enter a university based on the results of olympiads
5-6 grades
Small Olympiad (autumn round)
1. Puss in Boots caught four pike and half the catch. How many pikes did Puss in Boots catch?
2. The hares sawed several logs. They made 10 cuts and got 16 logs. How many logs did they cut?
3.
What do you think - even or odd - will be the sum:
a) two even numbers;
b) two odd numbers;
c) even and odd numbers;
d) odd and even numbers?
4. The guys brought a full basket of mushrooms from the forest. A total of 289 mushrooms were collected, with the same amount in each basket. How many guys were there?
5. The boy had 10 coins worth 1 ruble. and 5 rub. He counted 57 rubles. Was the boy wrong?
6. From a barrel containing at least 10 l gasoline, pour exactly 6 l, using a can with the capacity of a nine-liter bucket.
7. 7 chocolates are more expensive than 8 packs of cookies. What is more expensive - 8 chocolates or 9 packs of cookies?
9. There are less than 100 apples in the basket. They can be divided between two, three or five children, but cannot be divided equally between four children. How many apples are in the basket?
10. Rumor reached King Gorokh that, finally, someone had killed the Serpent Gorynych. The Tsar guessed that this was the work of either Ilya Muromets, or Dobrynya Nikitich, or Alyosha Popovich. He invited them to the court and began to question them. Each hero made a speech three times. And they said this:
Ilya Muromets: 1) I did not kill Zmey Gorynych. 2) I went to overseas countries. 3) And Alyosha Popovich killed the Serpent Gorynych.
Nikitich:4) Snake Gorynych was killed by Alyosha Popovich. 5) But even if I had killed, I would not have confessed. 6) There is still a lot of evil spirit left.
Alesha Popovich: 7) It was not I who killed Zmey Gorynych. 8) I have been looking for a long time for some feat to accomplish. 9) Indeed, Ilya Muromets left for overseas countries.
Then King Gorokh learned that twice each hero spoke the truth, and once he was disingenuous. So who killed Zmey Gorynych?
7-8 grades
Invariant
Invariant - a term used in mathematics, physics, and also in programming, denotes something unchangeable.
All tasks, united by the conventional name “invariant”, have the following form: certain objects are given on which certain operations are allowed to be performed. As a rule, the problem asks, is it possible to obtain another from one object using these operations? If possible, then you need to give an example of how to do this. If it is not possible, you need to prove that it is impossible.
A variety of quantities can act as an invariant: parity, sum, product, remainder, etc.
Problem 1
The change machine exchanges one coin for five others. Is it possible to use it to exchange one coin for 27 coins?
Solution. After each such exchange, the number of coins increases by 4, while the remainder of the number of coins when divided by 4 remains unchanged. At first we had 1 coin, which means the remainder will always be 1. The number 27 when divided by 4 has a remainder of 3, so you cannot exchange one coin for 27 coins.
Problem 2
The bully Vasya tore the wall newspaper, and he tore each piece he came across into four parts. Could it have been 2009 pieces? What if each piece was torn into 4 or 10 pieces?
Solution. No. The number of pieces changes each time by 3 or 9, that is, the remainder when divided by 3 is invariant. Initially there was one newspaper, which means that the number of pieces must have a remainder of 1 modulo 3, and 2009 is divided by 3 with a remainder of 2.
Problem 3
The numbers 1, 2, 3,..., 100 are written in a row. You can swap any two numbers between which there is exactly one. Is it possible to get the series 100, 99, 98,..., 2, 1?
Solution. Note that during permitted operations, either only even numbers or only odd numbers are swapped. In this case, even numbers will always be in even places. This means that it is impossible to get a row in which 100 is in the first place.
Problem 4
80 tons of peaches, which contained 99% water, were transported from Astrakhan to Moscow. On the way, they dried out and began to contain 98% water. How many tons of peaches were brought to Moscow?
Solution. In this problem, the invariant is the weight of the “dry residue”, i.e. the difference between the weight of peaches and the weight of the water they contain. In Astrakhan, peaches contained 1%, i.e. 8 tons of “dry residue”, in Moscow these 8 tons already accounted for 2% of the peaches brought. Then the weight of peaches is 8:2-100 = 40t. Weight has halved!
Problem 5
You can add the sum of its digits to a number. Is it possible to get the number 20092009 from three in a few steps?
Solution . With each step, the number increases by the sum of the digits. Note that the number and the sum of its digits have the same remainder when divided by 3. Three is divisible by 3 without a remainder, which means that the numbers that can be obtained from it by such an operation will also be divisible by 3. And the number 20092009 is not a multiple of 3.
Answer: no.
Problem 6
An 8x8 table is given, in which numbers from 1 to 64 are written. 8 cells are shaded so that in each horizontal and in each vertical there is exactly one shaded cell. Prove that the sum of the numbers written in these 8 cells does not depend on the set of shaded cells.
Solution. Let us number the columns in the table from left to right with numbers from 1 to 8. Then we will represent the numbers in the first row as the sum of 0 and the column number; numbers written in the second line as 8+column no.; in the third line: 16+ No., etc. Since exactly one cell is shaded in each row and each column, then, regardless of the choice, the sum of the eight numbers in the set is equal to: (0 + 8 + 16 + ... + 56) + (1 + 2 + ... + 8) = 260.
Problem 7
Solve the equation in whole numbers x 2 +y 2 +z 2 =8k - 1.
Solution. Let's look at the remainder full squares when divided by 8. The square of an even number can give remainders 0 and 4, and the square of an odd number always gives remainder 1, since(2k + 1) 2 = 4k 2 + 4k + 1 = 4k(k + 1) + 1. The sum of the remainders of three complete squares can be either even, or 1, or 3. But 8k - 1 is divisible by 8 with a remainder of 7. This means that this equation has no solutions.
Problem 8
Given a convex quadrilateral with diagonals 10 cm and 7 cm. Prove thatthat when cutting such a quadrangle, it is impossible to pave a 6x6 cm square with the resulting pieces.
Solution. The area of such a quadrilateral is 5∙7 sinα (α - angle between diagonals). Therefore, the area of a figure equivalent to a given quadrilateral cannot exceed 35. The area of a 6x6 square is 36.
7-8 grades
Problems to solve independently
2.1. There are 50 glasses in the dining room, 25 of them are upside down. Will the person on duty, turning over 4 glasses at a time, be able to get all the glasses standing correctly, that is, on the bottom?
2.2. The numbers 1,2,..., 2009 are written on the board. You are allowed to erase any two numbers and write the difference of these numbers instead. Is it possible to ensure that all the numbers on the board are zeros?
2.4. Ivan Tsarevich has two magic swords, one of which can cut off 21 heads of the Serpent Gorynych, and the second - 4 heads, but then the Serpent Gorynych grows 2008 heads. Note that if the Serpent Gorynych has, for example, only three heads left, then it is impossible to chop them with either one or the other sword. Can Ivan Tsarevich cut off all the heads of the Serpent Gorynych, if at the very beginning he had 100 heads?
2.5. On a chessboard, you are allowed to recolor all the cells in one row or one column in one move. Can there be exactly one white cell left after several moves?
2.7. There are two letters in the alphabet of the language of the UYU tribe: U and Y, and this language has an interesting property: if you remove the adjacent letters UY and UYUU from a word, the meaning of the word will not change. In the same way, the meaning of the word does not change when the letter combinations УУ, ыыУУыы and Уыыу are added anywhere in the word. a) Is it possible to say that the words UYY and UYYY have the same meaning? In this problem, the expressions “have the same meaning” and “obtain from each other by transformation” are equivalent, b) Do the words UYY and UYY have the same meaning?
2.8. There are only two letters in the alphabet - A and Z. Combinations of letters AYA and YAYA, YA and AAYA, YAYA and AAA in any word can be replaced with each other. Is it possible to get the word YAA from the word AYA?
2.10. Numbers from 1 to 20 are written on the board. Any pair of numbers can be(x, y) replace with a number x + y + 5xy. Could it end up being 20082009?
2.17. There is a pile of 1001 stones on the table. The first move is to throw a stone out of the pile and then divide it into two. Each subsequent move consists of throwing away a stone from any pile containing more than one stone, and then one of the piles is again divided into two. Is it possible to leave only piles of three stones on the table after a few moves?
2.18. Prove that numbers of the form 2009n + 3 and 2009n + 4 cannot be represented as the sum of two cubes of natural numbers.
2.20. The entire set of dominoes was laid out according to the rules of the game. It is known that five comes first. What is the last number?
2.23. There are 100 pros and 100 cons written on the board. You can replace any 2 minuses with a plus, a plus and a minus with a minus, two pluses with a plus. Prove that the sign that remains at the end does not depend on the order of operations.
2.26. Prove that the equation 15x 2 - 7y 2 = 9 has no solutions in integers.
2.27. Prove that the equation x 2 - 7y = 10 has no integer solutions.
In China, Korea and Japan, the number 4 is considered unlucky, as it is consonant with the word “death”. In these countries, floors with numbers ending in four are almost always absent.
- How do Arabs write and read numbers?
The Arabs use their own signs to write numbers, although the Arabs of Europe and North Africa use the “Arabic” numbers that are familiar to us. However, no matter what the signs of the numbers are, the Arabs write them, like letters, from right to left, but starting from the lower digits. It turns out that if we encounter familiar numbers in an Arabic text and read the number in the usual way from left to right, we will not be mistaken.
- How many legs do centipedes have?
A centipede does not necessarily have 40 legs. Centipede is a common name different types arthropods, scientifically united into the superclass centipedes. Different species of centipedes have from 30 to 400 or more legs, and this number can vary even among individuals of the same species. In English, two names have been established for these animals - centipede (“centipede” translated from Latin) and millipede (“millipede”). Moreover, the difference between them is significant - millipedes are not dangerous to humans, but centipedes bite very painfully.
- Where did the Olympic Games take place, on the emblem of which the year of the event was indicated by five numbers?
On the emblems Olympic Games the year is usually indicated by two (for example, Barcelona 92) or four digits (for example, Beijing 2008). But once the year was indicated by five digits. This happened in 1960, when the Olympics were held in Rome - the number 1960 was written as MCMLX.
- On satellite images Which Ukrainian city can you see the number 666?
In the 522 microdistrict of Kharkov, according to the plan, a block of residential buildings was to be built so that from the air they would form the letters of the USSR. However, after construction three letters C and the vertical line of the letter P were amended to the plan. As a result, these houses can now be seen as the number 666.
- How strange are the numbers 70, 80 and 90 called? French?
In the majority European languages The names of numerals from 20 to 90 are formed according to a standard scheme - consonant with the base numbers from 2 to 9. However, in French, the names of some numbers have a strange logic. Thus, the number 70 is pronounced 'soixante-dix', which translates as 'sixty and ten', 80 is pronounced 'quatre-vingts' ('four times twenty'), and 90 is pronounced 'quatre-vingt-dix' ('four times twenty and ten'). "). The situation is similar in Georgian and Danish. In the latter, the number 70 is literally translated as "halfway from three times twenty to four times twenty."
- Why does the name of the number 40 stand out from the similar names “twenty”, “thirty”, “fifty”, etc.?
In Russian, the names of numerals up to 100, divisible by 10, are formed by adding the name of the number and “ten”: twenty, thirty, fifty, etc. An exception to this series is the number “forty”. This is explained by the fact that in ancient times the conventional unit of trade in fur pelts was a bundle of 40 pieces. The fabric in which these skins were wrapped was called “sorok” (the word “shirt” comes from the same root). Thus, the name “forty” replaced the more ancient “four deste”.
The numbers on the calculator increase from bottom to top, and on the phone keyboard - from top to bottom. This is because calculators evolved from mechanical counting machines, where numbers have historically been arranged from bottom to top. Telephones were equipped with a dial for a long time, and when it became possible to produce push-button devices with tone dialing, they decided to arrange the numbers on the buttons by analogy with the dial - in ascending order from top to bottom with a zero at the end.
From the history of mathematics
Subject week mathematics.
the date of the
Solve number puzzles where the same letters correspond to the same numbers, and different - different.
***
Carl Gauss stood out for his sharp mind even at school. One day the teacher told him:- Karl, I wanted to ask you two questions. If you answer the first question correctly, then you don’t have to answer the second. So, how many needles are on the school Christmas tree?“65,786 needles, Mr. Teacher,” Gauss answered immediately.- Okay, but how did you know this? - asked the teacher.“And this is the second question,” the student quickly answered.
Read the statement of the outstanding
mathematics Galileo!
find the correct answer for example
Mathematical puzzle
Questions for Chinaword.
1. 2.1. Geometric figure. 1. Measure of area.2. Regular polygon. 2. Place occupied by a digit in a number notation 3. Number. 3. Number defining the length of the line4. Ancient measure length. 4. 100 square meters.5. A relation connecting two numbers. 5. A segment connecting a point on a circle with its6. Part of a straight line limited by two centers dots. 6. Number.7. School team. 7. Rhombus with equal angles.8. Mathematical operation. 8. One hundred tens.9. A segment whose length is 1. 9. Part of mathematics, the science of numbers.
Pythagoras (c. 570 - c. 500 BC)
The judges of one of the first Olympics in history did not want to allow a young man with a strong physique to participate in sports competitions, since he was not tall enough. But he not only became a participant in the Olympics, but also defeated all his opponents. This is the legend... This young man was Pythagoras, the famous mathematician.
His whole life is a legend, or rather, a layering of many legends. He was born on the island of Samos, off the coast of Asia Minor. Only five kilometers of water separated this island from big land. Pythagoras left his homeland when he was very young. He walked along the roads of Egypt, lived for 12 years in Babylon, where he listened to the speeches of the priests who revealed to him the secrets of astronomy and astrology, then for several years in Italy. Already in adulthood, Pythagoras moved to Sicily and there, in Crotone, created an amazing school,
which will be called Pythagorean. Here are the “commandments” of the Pythagoreans:
Do only what will not upset you later and will not force you to repent.Never do what you don’t know, but learn everything you need to know.
Don't neglect the health of your body.
Learn to live simply and without luxury.
Before you go to bed, analyze your actions for the day.
Pythagoras did not write down his teachings. It is known only in the retellings of Aristotle and Plato.
How many triangles do you see
on the image?
