Non-standard tasks. Non-standard tasks as a means of forming students' interest in mathematics

NON-STANDARD TASKS IN MATHEMATICS LESSONS

Primary school teacher Shamalova S.V.

Each generation of people makes its own demands on the school. An ancient Roman proverb says: "Not for school, but for life, we learn." The meaning of this proverb is still relevant today. Modern society dictates to the education system an order for the education of a person who is ready for life in constantly changing conditions, for continuing education, capable of learning throughout his life.

Among the spiritual abilities of man there is one that for many centuries has been the subject of close attention of scientists and which, at the same time, is still the most difficult and mysterious subject of science. This is the ability to think. We constantly encounter it in work, in teaching, in everyday life.

Any activity of a worker, schoolchild and scientist is inseparable from mental work. In any real matter, it is necessary to break one's head, to throw one's mind, that is, in the language of science, one must carry out a mental action, intellectual work. It is known that the problem can be solved, and not solved, one will cope with it quickly, the other thinks for a long time. There are tasks that are feasible even for a child, and whole teams of scientists have been struggling over some for years. So, there is the ability to think. Some are better at it, others worse. What is this skill? In what ways does it arise? How to purchase it?

No one will argue with the fact that every teacher must develop the logical thinking of students. This is stated in the methodological literature, in the explanatory notes to the curriculum. However, we teachers do not always know how to do this. Often this leads to the fact that the development of logical thinking is largely spontaneous, so most students, even high school students, do not master the initial methods of logical thinking (analysis, comparison, synthesis, abstraction, etc.).

According to experts, the level of logical culture of schoolchildren today cannot be considered satisfactory. Experts believe that the reason for this lies in the lack of work on the purposeful logical development of students in the early stages of education. Most modern manuals for preschoolers and primary schoolchildren contain a set of various tasks that focus on such methods of mental activity as analysis, synthesis, analogy, generalization, classification, flexibility and variability of thinking. In other words, the development of logical thinking occurs largely spontaneously, so most students do not master the techniques of thinking even in the senior grades, and these techniques must be taught to younger students.

In my practice I use modern educational technologies, various forms of organization of the educational process, a system of developing tasks. These tasks should be of a developmental nature (teach certain thinking techniques), they should take into account the age characteristics of students.

In the process of solving educational problems, children develop such a skill as being distracted from irrelevant details. This action is given to younger students with no less difficulty than highlighting the essential. As a result of studying at school, when it is necessary to regularly complete tasks without fail, younger students learn to control their thinking, to think when necessary. First, logical exercises accessible to children are introduced, aimed at improving mental operations.

In the process of performing such logical exercises, students practically learn to compare various objects, including mathematical ones, to build correct judgments on the basis of accessible and simple proofs based on their life experience. Logic exercises are gradually becoming more difficult.

I also use non-standard developing logical tasks in my practice. There is a large number of such problems; especially a lot of such specialized literature has been published in recent years.

In the methodological literature, the following names have been assigned to developing tasks: tasks for ingenuity, tasks for ingenuity, tasks with a "zest". In all its diversity, it is possible to single out into a special class such tasks that are called tasks - traps, provocative tasks. In the conditions of such tasks, there are various kinds of references, indications, hints that push to choose the wrong path of solution or the wrong answer. I will give examples of such tasks.

Tasks that impose one, quite definite answer.

Which of the numbers 333, 555, 666, 999 is not divisible by 3?

Tasks that encourage you to make the wrong choice of an answer from the proposed correct and incorrect answers.

One donkey is carrying 10 kg of sugar, and the other is carrying 10 kg of popcorn. Who had the heaviest load?

Tasks, the conditions of which push you to perform some action with given numbers, when you do not need to perform this action at all.

A Mercedes car has traveled 100 km. How many miles did each wheel travel?

Petya once told his friends: "The day before yesterday I was 9 years old, and next year I will be 12 years old." What date was Petya born?

Solving logical problems using reasoning.

Vadim, Sergey and Mikhail study various foreign languages: Chinese, Japanese, Arabic. When asked what language each of them studied, one answered: “Vadim is studying Chinese, Sergey is not studying Chinese, and Mikhail is not studying Arabic.” Subsequently, it turned out that in this statement only one statement is true. What language does each of them study?

The shorties from the Flower City planted a watermelon. For its watering exactly 1 liter of water is required. They have only two empty cans with a capacity of 3 liters. And 5 l. How to use these cans. Dial exactly 1 liter from the river. water?

How many years did Ilya Muromets sit on the stove? It is known that if he sat 2 more times for so many, then his age would be the largest two-digit number.

Baron Munchausen counted the number of magical hairs in old Hottabych's beard. It turned out to be equal to the sum of the smallest three-digit number and the largest two-digit number. What is this number?

When learning to solve non-standard problems, I observe the following conditions:in first , tasks should be introduced into the learning process in a certain system with a gradual increase in complexity, since an overwhelming task will have little effect on the development of students;in o second , it is necessary to provide students with maximum independence in finding solutions to problems, to give them the opportunity to go to the end along the wrong path in order to make sure of the error, return to the beginning and look for another, right way of solving;third , you need to help students understand some of the ways, techniques and general approaches to solving non-standard arithmetic problems. Most often, the proposed logical exercises do not require calculations, but only force children to make correct judgments and give simple proofs. The exercises themselves are entertaining, so they contribute to the emergence of interest in children in the process of mental activity. And this is one of the cardinal tasks of the educational process at school.

Examples of tasks used in my practice.

Find a pattern and continue the garlands

Find a pattern and continue the series

a B C D E F, …

1, 2, 4, 8, 16,…

The work began with the development in children of the ability to notice patterns, similarities and differences with the gradual complication of tasks. For this purpose, I have chosenassignments to identify patterns, dependencies and formulate a generalizationwith a gradual increase in the level of difficulty of tasks.Work on the development of logical thinking should become the object of serious attention of the teacher and be systematically carried out in mathematics lessons. For this purpose, exercises on logic should be constantly included in the oral work in the lesson. For example:

Find the result using this equation:

3+5=8

3+6=

3+7=

3+8=

Compare the expressions, find common ground in the resulting inequalities, formulate a conclusion:

2+3*2x3

4+4*3x4

4+5*4x5

5+6*5x6

Continue with the numbers.

3. 5, 7, 9, 11…

1, 4, 7, 10…

Think of a similar example for each given example.

12+6=18

16-4=12

What is common in writing the numbers of each line?

12 24 20 22

30 37 13 83

Given numbers:

23 74 41 14

40 17 60 50

What number is missing in each line?

In elementary school, I often use counting sticks in my math classes. These are problems of a geometric nature, since in the course of solving, as a rule, there is a transfiguration, the transformation of one figure into another, and not just a change in their number. They cannot be solved in any previously learned way. In the course of solving each new problem, the child is included in an active search for a solution, while striving for the ultimate goal, the required modification of the figure.

Exercises with counting sticks can be combined into 3 groups: tasks for drawing up a given figure from a certain number of sticks; tasks for changing figures, for the solution of which it is necessary to remove or add the specified number of sticks; tasks, the solution of which is to shift the sticks in order to modify, transform a given figure.

Exercises with counting sticks.

Tasks for drawing figures from a certain number of sticks.

Make two different squares of 7 sticks.

Tasks for changing the figure, where you need to remove or add the specified number of sticks.

Given a figure of 6 squares. You need to remove 2 sticks so that 4 squares remain"

Tasks for shifting sticks for the purpose of transformation.

Move two sticks so that you get 3 triangles.

Regular exercise is one of the conditions for the successful development of students. First of all, from lesson to lesson, it is necessary to develop the child's ability to analyze and synthesize; short-term training in logical concepts does not give an effect.

Solving non-standard problems forms the ability of students to make assumptions, check their reliability, and justify them logically. Speaking for the purpose of evidence, contributes to the development of speech, the development of skills to draw conclusions, draw conclusions. In the process of using these exercises in the classroom and in extracurricular work in mathematics, a positive dynamics of the influence of these exercises on the level of development of students' logical thinking appeared.

Tests and questionnaires Grade 3.

It is known that the solution of text problems presents great difficulties for students. It is also known which stage of the solution is especially difficult. This is the very first stage - the analysis of the text of the problem. Students are poorly oriented in the text of the problem, in its conditions and requirements. The text of the task is a story about some life facts: “Masha ran 100 m, and towards her ...”,

“The students of the first grade bought 12 carnations, and the students of the second…”, “The master made 20 parts in a shift, and his student…”.

Everything is important in the text; and actors, and their actions, and numerical characteristics. When working with a mathematical model of the problem (numerical expression or equation), some of these details are omitted. But we are precisely teaching the ability to abstract from some properties and use others.

The ability to navigate in the text of a mathematical problem is an important result and an important condition for the overall development of the student. And you need to do this not only in mathematics lessons, but also in reading and fine arts lessons. Some tasks are good subjects for drawings. And any task is a good topic for retelling. And if there are theater lessons in the class, then some mathematical problems can be staged. Of course, all these techniques: retelling, drawing, staging - can also take place in the mathematics lessons themselves. So, work on the texts of mathematical problems is an important element in the overall development of the child, an element of developmental learning.

But are the tasks that are available in the current textbooks and the solution of which is included in the mandatory minimum sufficient for this? No, not enough. The mandatory minimum includes the ability to solve problems of certain types:

about the number of elements of a certain set;

about movement, its speed, path and time;

about price and value;

about work, its time, volume and productivity.

These four themes are standard. It is believed that the ability to solve problems on these topics can teach how to solve problems in general. Unfortunately, this is not so. Good students who can solve practically

any problem from the textbook on the listed topics, often fail to understand the condition of the problem on another topic.

The way out is not to be limited to any topic of text tasks, but to solve non-standard tasks, that is, tasks whose subject matter is not in itself an object of study. After all, we do not limit the plots of stories in reading lessons!

Non-standard problems need to be solved in the classroom every day. They can be found in mathematics textbooks for grades 5-6 and in the journals Primary School, Mathematics at School, and even Kvant.

The number of tasks is such that you can choose from them tasks for each lesson: one per lesson. Problems are solved at home. But very often you need to disassemble them in the classroom. Among the proposed tasks there are those that a strong student solves instantly. Nevertheless, it is necessary to demand sufficient reasoning from strong children, explaining that on easy tasks a person learns the methods of reasoning that will be needed when solving difficult problems. It is necessary to educate in children a love for the beauty of logical reasoning. As a last resort, strong students can be forced to make such reasoning by requiring them to construct an explanation that is understandable for others - for those who do not understand the quick solution.

Among the tasks there are absolutely the same type in mathematical terms. If the kids see this, great. The teacher can show it himself. However, it is unacceptable to say: we solve this problem like that one, and the answer will be the same. The fact is that, firstly, not all students are capable of such analogies. And secondly, in non-standard problems, the plot is no less important than the mathematical content. Therefore, it is better to emphasize the connections between tasks with a similar plot.

Not all problems need to be solved (there are more of them than there are math lessons in the school year). You may want to change the order of the tasks or add a task that is not here.

Lyabina T.I.

Mathematics teacher of the highest category

MOU "Moshok Secondary School"

Non-standard tasks as a means of developing logical thinking

What problem in mathematics can be called non-standard? A good definition is given in the book

Non-standard tasks are those for which there are no general rules and regulations in the course of mathematics that determine the exact program for their solution. They should not be confused with tasks of increased complexity. The conditions of problems of increased complexity are such that they allow students to quite easily select the mathematical apparatus that is needed to solve a problem in mathematics. The teacher controls the process of consolidating the knowledge provided by the training program by solving problems of this type. But a non-standard task implies the presence of an exploratory nature. However, if the solution of a problem in mathematics for one student is non-standard, since he is unfamiliar with the methods for solving problems of this type, then for another, the solution of the problem occurs in a standard way, since he has already solved such problems and more than one. The same task in mathematics in the 5th grade is non-standard, and in the 6th grade it is ordinary, and not even of increased complexity.

So, if the student does not know what theoretical material to rely on to solve the problem, he also does not know, then in this case the problem in mathematics can be called non-standard for a given period of time.

What are the methods of teaching solving problems in mathematics, which we currently consider non-standard? Unfortunately, no one has come up with a universal recipe, given the uniqueness of these tasks. Some teachers, as they say, train in template exercises. This happens as follows: the teacher shows the way to solve, and then the student repeats this when solving problems many times. At the same time, students' interest in mathematics is being killed, which is at least sad.

You can teach children to solve problems of a non-standard type if you arouse interest, in other words, offer tasks that are interesting and meaningful for a modern student. Or replace the wording of the question using problematic life situations. For example, instead of the task “solve the Diaphantian equation”, offer to solve the following problem. Can

student to pay for a purchase worth 19 rubles, if he has only three-ruble bills, and the seller has ten-ruble bills?

The method of selecting auxiliary tasks is also effective. This means of teaching problem solving indicates a certain level of achievement in problem solving. Usually in such cases, the thinking student tries on his own, without the help of a teacher, to find auxiliary problems or to simplify and modify the conditions of these problems.

The ability to solve non-standard problems is acquired by practice. No wonder they say that you can't learn math by watching your neighbor do it. Self-study and the help of a teacher is the key to fruitful learning.

1.Non-standard tasks and their characteristics.

Observations show that mathematics is mainly loved by those students who know how to solve problems. Therefore, by teaching children to master the ability to solve problems, we will have a significant impact on their interest in the subject, on the development of thinking and speech.

Non-standard tasks contribute to the development of logical thinking to an even greater extent. In addition, they are a powerful means of activating cognitive activity, that is, they arouse great interest and desire in children to work. Let us give an example of non-standard tasks.

I. Tasks for ingenuity.

1. The mass of a heron standing on one leg is 12 kg. How much will a heron weigh if it stands on 2 legs?

2. A pair of horses ran 40 km. How far did each horse run?

3. Seven brothers have one sister. How many children are in the family?

4. Six cats eat six mice in six minutes. How many cats does it take to eat 100 mice in 100 minutes?

5. There are 6 glasses, 3 with water, 3 empty. How to arrange them so that glasses of water and empty ones alternate? Only one glass is allowed to be moved.

6. Geologists found 7 stones. Weight of each stone: 1 kg, 2 kg, 3 kg, 4 kg, 5 kg, 6 kg and 7 kg. These stones were laid out in 4 backpacks so

that in each backpack the mass of stones turned out to be the same.

How did they do it?

7. There are as many combed girls in the class as uncombed boys. Who is more in the class, girls or unkempt students?

8. Ducks flew: one in front and two behind, one behind and two in front, one between two and three in a row. How many ducks flew in total?

9. Misha says: “The day before yesterday I was 10 years old, and next year I will be 13 years old.” Is it possible?

10. Andrey and Borya have 11 candies, Boris and Vova have 13 candies, and Andrey and Vova have 12. How many candies do the boys have in total?

11. A father with two sons rode bicycles: two-wheeled and three-wheeled. They had 7 wheels in total. How many bicycles were there, and which ones?

12. Chickens and piglets in the yard. They all have 5 heads and 14 legs. How many chickens and how many pigs?

13. Chickens and rabbits walk around the yard. They have 12 legs in total. How many chickens and how many rabbits?

14. Each Martian has 3 hands. Can 13 Martians join hands in such a way that there are no free hands left?

15. While playing, each of the three girls - Katya, Galya, Olya - hid one of the toys - a bear, a hare and an elephant. Katya did not hide the hare, Olya did not hide either the hare or the bear. Who hid the toy?

II. Entertaining tasks.

1. How to arrange 6 chairs against 4 walls so that each wall has 2 chairs.

2. Dad and his two sons went camping. On their way they met a river. There is a raft on the shore. He stands on the water one dad or two sons. How to cross to the other side of the father with his sons?

3. For one horse and two cows, 34 kg of hay are given daily, and for two horses and one cow - 35 kg of hay. How much hay is given daily to one horse and how much to one cow?

4. Four ducklings and five goslings weigh 4kg100g, and five ducklings and four goslings weigh 4kg. How much does one duck weigh?

5. The boy had 22 coins - five-ruble and ten-ruble, totaling 150 rubles. How many five-ruble and ten-ruble coins were there?

6. Three kittens live in apartment No. 1, 2, 3: white, black and red. It was not a black kitten that lived in apartments 1 and 2. The white kitten did not live in apartment number 1. In which apartment did each of the kittens live?

7. For five weeks, the pirate Yerema is able to drink a barrel of rum. And it would take the pirate Emelya two weeks to do it. In how many days will the pirates finish the rum, acting together?

8. A horse eats a cartload of hay in a month, a goat in two months, a sheep in three months. How long will it take a horse, a goat, a sheep to eat the same load of hay together?

9. Two people peeled 400 potatoes; one cleared 3 pieces per minute, the other -2. The second one worked 25 minutes more than the first one. How long did each work?

10. Among soccer balls, red is heavier than brown, and brown is heavier than green. Which ball is heavier: green or red?

11. Three pretzels, five gingerbread and six bagels cost 24 rubles together. Which is more expensive: a pretzel or a bagel?

12. How to find one counterfeit (lighter) coin out of 20 coins by three weighings on a pan balance without weights?

13. From the top corner of the room, two flies crawled down the wall. Having descended to the floor, they crawled back. The first fly crawled in both directions at the same speed, and the second, although it ascended twice as slowly as the first, but descended twice as fast as it. Which of the flies will crawl back first?

14. There are pheasants and rabbits in the cage. All animals have 35 heads and 94 legs. How many rabbits in a cage and how many pheasants?

15. They say that when asked how many students he had, the ancient Greek mathematician Pythagoras answered this way: “Half of my students study mathematics, a fourth study nature, a seventh spend time in silent reflection, the rest are 3 virgins” How many students were at Pythagoras?

III. Geometric problems.

1. Divide the rectangular cake into two slices so that they have a triangular shape. How many parts did it make?

2. Draw a figure without lifting the tip of the pencil from the paper and without drawing the same line twice.

3. Cut the square into 4 parts and fold them into 2 squares. How to do it?

4. Remove 4 sticks so that 5 squares remain.

5. Cut the triangle into two triangles, a quadrilateral and a pentagon, by drawing two straight lines.

6. Can a square be divided into 5 parts and assemble an octagon?

IV. Logic squares.

1. Fill in the square (4 x 4) with the numbers 1, 2, 3, 6 so that the sum of the numbers in all rows, columns and diagonals is the same. Numbers in rows, columns and diagonals should not be repeated.

2. Color the square with red, green, yellow and blue colors so that the colors in the rows, columns and diagonals do not repeat.

3. In the square, you need to place more numbers 2,2,2,3,3,3 so that for all lines you get a total of 6.

5. In the cells of the square, put the numbers 4,6,7,9,10,11,12 so that in the columns, in the rows and along the diagonals you get the sum 24.

v. Combinatorial tasks.

1. Dasha has 2 skirts: red and blue, and 2 blouses: striped and polka dot. How many different outfits does Dasha have?

2. How many two-digit numbers are there in which all digits are odd?

3. Parents purchased a ticket to Greece. Greece can be reached using one of three modes of transport: plane, boat or bus. Make up all possible options for using these modes of transport.

4. How many different words can be formed using the letters of the word "connection"?

5. From the numbers 1, 3, 5, make up various three-digit numbers so that there are no identical numbers in the number.

6. Three friends met: the sculptor Belov, the violinist Chernov and the artist Ryzhov. “It's great that one of us is blond, the other is brunette, and the third is red-haired. But not one of them has hair of the color indicated by his last name, ”said the brunette. "You're right," Belov said. What color is the artist's hair?

7. Three friends went out for a walk in white, green and blue dresses and shoes of the same colors. It is known that only Anya has the same color of dress and shoes. Neither shoes nor Vali's dress were white. Natasha was wearing green shoes. Determine the color of the dress and shoes on each of the friends.

8. A cashier, a controller and a manager work in a bank branch. Their surnames are Borisov, Ivanov and Sidorov. The cashier has no brothers or sisters and is the shortest of all. Sidorov is married to Borisov's sister and taller than the controller. Give the names of the controller and manager.

9. For a picnic, sweet tooth Masha took sweets, cookies and a cake in three identical boxes. The boxes were labeled "Candy", "Cookie", and "Cake". But Masha knew that her mother liked to joke and always put food in

boxes with inscriptions that do not match their contents. Masha was sure that the sweets were not in the box with "Cake" written on it. What box is the cake in?

10. Ivanov, Petrov, Markov, Karpov are sitting in a circle. Their names are Andrey, Sergey, Timofey, Alexey. It is known that Ivanov is neither Andrei nor Alexei. Sergei sits between Markov and Timofey. Petrov sits between Karpov and Andrey. What are the names of Ivanova, Petrov, Markov and Karpov?

VI. Transfusion tasks.

1. Is it possible, having only two vessels with a capacity of 3 and 5 liters, to draw 4 liters of water from a water tap?

2. How to divide equally between two families 12 liters of bread kvass, located in a twelve-liter vessel, using two empty vessels for this: an eight-liter and a three-liter?

3. How, having two vessels with a capacity of 9 liters and 5 liters, to draw exactly 3 liters of water from a reservoir?

4. A can with a capacity of 10 liters is filled with juice. There are still empty vessels of 7 and 2 liters. How to pour juice into two vessels of 5 liters each?

5. There are two vessels. The capacity of one of them is 9 liters, and the other 4 liters. How to use these vessels to collect 6 liters of some liquid from the tank? (The liquid can be drained back into the tank).

An analysis of the proposed text tasks shows that their solution does not fit into the framework of a particular system of typical tasks. Such problems are called non-standard (I. K. Andronov, A. S. Pchelko, etc.) or non-standard (Yu. M. Kolyagin, K. I. Neshkov, D. Poya, etc.)

Summarizing the various approaches of methodologists in understanding standard and non-standard tasks (D. Poya, Ya. M. Fridman, etc.), under non-standard task we understand such a task, the algorithm of which is not familiar to the student and is not further formed as a program requirement.

An analysis of textbooks and teaching aids in mathematics shows that each text task under certain conditions can be non-standard, and in others - ordinary, standard. A standard problem in one course of mathematics may be non-standard in another course.

For example. “There were 57 planes and 79 helicopters at the airfield, 60 cars took off. Can it be argued that there is: a) at least 1 aircraft in the air; b) at least 1 helicopter?

Such tasks were optional for all students, they were intended for the most capable of mathematics.

“If you want to learn how to solve problems, then solve them!” - advises D. Poya.

The main thing in this case is to form such a general approach to solving problems, when the problem is considered as an object for research, and its solution - as the design and invention of a solution method.

Naturally, such an approach requires not a thoughtless solution of a huge number of problems, but a leisurely, attentive and thorough solution of a much smaller number of problems, but with a subsequent analysis of the solution.

So, there are no general rules for solving non-standard problems (that's why these problems are called non-standard). However, outstanding mathematicians and teachers (S.A. Yanovskaya, L.M. Fridman,

E.N. Balayan) found a number of general guidelines and recommendations that can be followed in solving non-standard problems. These guidelines are usually called heuristic rules or, simply, heuristics. The word "heuristics" is of Greek origin and means "the art of finding the truth."

Unlike mathematical rules, heuristics are in the nature of optional recommendations, advice, following which may (or may not) lead to solving the problem.

The process of solving any non-standard task (according to

S.A. Yanovskaya) consists in the sequential application of two operations:

1. reduction by means of transformations of a non-standard task to another, similar to it, but already a standard task;

2. splitting a non-standard task into several standard subtasks.

There are no specific rules for reducing a non-standard task to a standard one. However, if you carefully, thoughtfully analyze, solve each problem, fixing in your memory all the methods by which solutions were found, by what methods the problems were solved, then skill is developed in such information.

Consider an example task:

Along the path, along the bushes, walked a dozen tails,

Well, my question is this - how many roosters were there?

And I would be glad to know - how many pigs were there?

If it is not possible to solve this problem, we will try to reduce it to a similar one.

Let's reformulate:

1. Let's invent and solve a similar, but simpler one.

2. We use its solution to solve this one.

The difficulty is that there are two types of animals in the problem. Let everyone be piglets, then there will be 40 legs.

Let's create a similar problem:

Along the path, along the bushes, walked a dozen tails.

It was together that roosters and piglets were going somewhere.

Well, my question is - how many roosters were there?

And I would be glad to know - how many pigs were there?

It is clear that if there are 4 times more legs than tails, then all animals are piglets.

In a similar problem, 40 legs were taken, and in the main one there were 30. How to reduce the number of legs? Replace the piglet with a cockerel.

Solution to the main problem: if all animals were piglets, then they would have 40 legs. When we replace a pig with a cockerel, the number of legs is reduced by two. In total, you need to make five substitutions to get 30 legs. So, 5 cockerels and 5 piglets walked.

How to come up with a "similar" problem?

2 way to solve the problem.

In this problem, you can apply the principle of equalization.

Let all the pigs stand on their hind legs.

10 * 2 \u003d 20 so many feet walk along the path

30 - 20 \u003d 10 so many front legs of piglets

10:2 = 5 pigs walked along the path

Well, cockerels 10 -5 \u003d 5.

Let us formulate several rules for solving non-standard problems.

1. "Easy" rule: don't skip the easiest task.

Usually a simple task is overlooked. And you have to start with her.

2. The “next” rule: if possible, the conditions should be changed one by one. The number of conditions is a finite number, so sooner or later everyone will have a turn.

3. The “unknown” rule: after changing one condition, designate the other associated with it by x, and then choose it so that the auxiliary problem is solved at a given value and not solved when x is increased by one.

3. "Interesting" rule: make the conditions of the problem more interesting.

4. “Temporary” rule: if some process is going on in the task and the final state is more definite than the initial one, it is worth starting the time in the opposite direction: consider the last step of the process, then the penultimate one, etc.

Consider the application of these rules.

Task number 1. Five boys found nine mushrooms. Prove that at least two of them have found equal numbers of mushrooms.

1st step There are a lot of boys. Let them be 2 less in the next problem.

“Three boys found x mushrooms. Prove that at least two of them found mushrooms equally.

To prove this, let us establish for which x the problem has a solution.

For x=0, x=1, x=2 the problem has a solution, for x=3 the problem has no solution.

Let's formulate a similar problem.

Three boys found 2 mushrooms. Prove that at least two of them have found equal numbers of mushrooms.

Let all three boys find a different number of mushrooms. Then the minimum number of mushrooms is 3, because 3=0+1+2. But according to the condition, the number of mushrooms is less than 3, so two out of three boys found the same number of mushrooms.

When solving the original problem, the reasoning is exactly the same. Let everyone, five boys, find a different number of mushrooms. The minimum number of mushrooms should then be 10. (10 =0+1+2+3+4). But according to the condition, the number of mushrooms is less than 10, so the two boys found the same number of mushrooms.

When solving, the "unknown" rule was used.

Task number 2. Swans flew over the lakes. Half of the swans and half a swan landed on each, the rest flew on. All sat down on the seven lakes. How many swans were there?

1st step There is a process, the initial state is not defined, the final state is zero, i.e. there were no flying swans.

We start time in the opposite direction, having come up with the following task:

Swans flew over the lakes. On each took off half a swan and as many more as now flew. All took off from seven lakes. How many swans were there?

2 step. We start from scratch:

(((((((0+1/2)2+1/2)2+1/2)2+1/2)2+1/2)2+1/2)2+1/2)2 =127.

Task number 3.

A loafer and a devil met at the bridge across the river. The loafer complained about his poverty. In response, the devil suggested:

I can help you. Every time you cross this bridge, your money will double. But every time you cross the bridge, you will have to give me 24 kopecks. The loafer crossed the bridge three times, and when he looked into the wallet, it was empty. How much money did the slacker have?

(((0+24):2+24):2+24):2= 21

When solving problems No. 2 and No. 3, the "temporary" rule was used.

Task number 4. A blacksmith shoves one hoof in 15 minutes. How long will it take 8 blacksmiths to shoe 10 horses. (A horse cannot stand on two legs.)

1st step There are too many horses and blacksmiths, let's reduce their number proportionally, making up the problem.

A farrier shods one hoof in five minutes. How long does it take four blacksmiths to shoe five horses?

It is clear that the minimum possible time is 25 minutes, but can it be reached? It is necessary to organize the work of blacksmiths without downtime. Let's act without breaking the symmetry. Arrange five horses in a circle. After four blacksmiths have shod one horse's hoof each, the blacksmiths move one horse in a circle. To get around the full circle, it will take five cycles of work for five minutes. During 4 cycles, each horse will be shod and one cycle will be rested. As a result, all horses will be shod in 25 minutes.

2 step. Returning to the original problem, note that 8=2*4 and 10=2*5. Then 8 blacksmiths need to be divided into two brigades

4 people each, and horses - two herds of 5 horses each.

In 25 minutes, the first team of blacksmiths will forge the first herd, and the second - the second.

When solving, the “next” rule was used.

Of course, there may be a problem to which none of the above rules can be applied. Then you need to invent a special method for solving this problem.

It must be remembered that solving non-standard problems is an art that can be mastered only as a result of constant introspection of actions to solve problems.

2. Educational functions of non-standard tasks.

The role of non-standard tasks in the formation of logical thinking.

At the present stage of education, there has been a tendency to use tasks as a necessary component of teaching students mathematics. This is explained, first of all, by the increasing requirements aimed at strengthening the developmental functions of training.

The concept of "non-standard task" is used by many methodologists. So, Yu. M. Kolyagin expands on this notion as follows: non-standard understood task, upon presentation of which, students do not know in advance either the method of solving it, or what educational material the solution is based on.

Based on the analysis of the theory and practice of using non-standard tasks in teaching mathematics, their general and specific role has been established.

Non-standard tasks:

They teach children to use not only ready-made algorithms, but also independently find new ways to solve problems, that is, they contribute to the ability to find original ways to solve problems;

Influence the development of ingenuity, ingenuity of students;

prevent the development of harmful cliches when solving problems, destroy incorrect associations in the knowledge and skills of students, involve not so much the assimilation of algorithmic techniques, but finding new connections in knowledge, to the transfer

knowledge in new conditions, to mastering various methods of mental activity;

They create favorable conditions for increasing the strength and depth of students' knowledge, ensure the conscious assimilation of mathematical concepts.

Non-standard tasks:

They should not have ready-made algorithms memorized by children;

Should be accessible in content to all students;

Must be interesting in content;

To solve non-standard tasks, students should have enough knowledge acquired by them in the program.

3.Methodology for the formation of the ability to solve non-standard tasks.

Task number 1.

A caravan of camels slowly moves through the desert, there are 40 in total. If you count all the humps of these camels, you get 57 humps. How many one-humped camels are in this caravan?

How many humps can camels have?

(there may be two or one)

Let's attach a flower to each camel on one hump.

How many flowers will you need? (40 camels - 40 flowers)

How many camels will be left without flowers?

(There will be 57-40=17 of them. These are the second humps of two-humped camels).

How many bactrian camels? (17)

How many one humped camels? (40-17=23)

What is the answer to the problem? (17 and 23 camels).

Task number 2.

There were cars and motorcycles with sidecars in the garage, all together 18. Cars and motorcycles had 65 wheels. How many motorcycles with sidecars were in the garage if the cars had 4 wheels and the motorcycle had 3 wheels?

Let's reformulate the problem. The robbers who came to the garage, where there were 18 cars and motorcycles with sidecars, removed three wheels from each car and each motorcycle and carried them away. How many wheels were left in the garage if there were 65? Do they belong to a car or motorcycle?

How many wheels did the robbers take? (3*18=54wheels)

How many wheels are left? (65-54=11)

How many cars were in the garage?

There were 18 cars and motorcycles with a sidecar in the garage. Cars and motorcycles have 65 wheels. How many motorcycles are in the garage if they put a spare tire in each sidecar?

How many wheels did cars and motorcycles have together? (4*18=72)

How many spare wheels did you put in each stroller? (72-65=7)

How many cars are in the garage? (18-7=1)

Task number 3.

For one horse and two cows, 34 kg of hay are given daily, and for two horses and one cow - 35 kg of hay. How much hay is given to one horse and how much to one cow?

Let's write a brief condition of the problem:

1 horse and 2 cows -34kg.

2 horses and 1 cow -35kg.

Is it possible to know how much hay is needed for 3 horses and 3 cows? (for 3 horses and 3 cows - 34+35=69 kg)

Is it possible to know how much hay is needed for one horse and one cow? (69: 3 - 23kg)

How much hay is needed for one horse? (35-23=12kg)

How much hay is needed for one cow? (23 -13 =11kg)

Answer: 12kg and 11kg

Task number 4.

- Geese flew: 2 in front, 1 behind, 1 in front, 2 behind.

How many geese flew?

How many geese flew, as stated in the condition? (2 ahead, 1 behind)

Draw it with dots.

Draw with dots.

Count what you got (2 in front, 1, 1, 2 behind)

Is that what the condition says? (No)

So you drew extra geese. You can tell from your drawing that 2 is in front and 4 is behind, or 4 is in front and 2 is behind. And this is not a condition. What needs to be done? (remove last 3 dots)

What will happen?

So how many geese flew? (3)

Task number 5.

Four ducklings and five goslings weigh 4kg 100g, five ducklings and four goslings weigh 4kg. How much does one duck weigh?

Let's reformulate the problem.

Four ducklings and five goslings weigh 4kg 100g, five ducklings and four goslings weigh 4kg.

How much do one duckling and one gosling weigh together?

How much do 9 ducklings and 9 goslings weigh together?

Apply the solution of the auxiliary problem to solve the main one, knowing how much 3 ducklings and 3 caterpillars weigh together?

Tasks with elements of combinatorics and ingenuity.

Task number 6.

Marina decided to have breakfast at the school cafeteria. Look at the menu and tell me how many ways can she choose a drink and a confection?

Let's assume that Marina chooses tea from drinks. What confectionery can she choose for tea? (tea - cheesecake, tea - cookies, tea - roll)

How many ways? (3)

And if compote? (also 3)

So how do you know how many ways Marina can use to choose her lunch? (3+3+3=9)

Yes you are right. But to make it easier for us to solve such a problem, we will use graphs. Let's denote drinks and confectionery with dots and connect the pairs of those dishes that Marina chooses.

tea milk compote

cheesecake cookies bun

Now let's count the number of lines. There are 9 of them. So, there are 9 ways to choose dishes.

Task number 7.

Three heroes - Ilya Muromets, Alyosha Popovich and Dobrynya Nikitich, defending their native land from the invasion, cut down all 13 heads of the Serpent Gorynych. Ilya Muromets cut down the most heads, and Alyosha Popovich cut the fewest. How many heads could each of them cut down?

Who can answer this question?

(teacher asks several people - everyone has different answers)

Why are there different answers? (because it is not said specifically how many heads were cut down by at least one of the heroes)

Let's try to find all possible solutions to this problem. The table will help us with this.

What condition must we meet when solving this problem? (All the heroes cut down a different number of heads, and Alyosha had the least, Ilya had the most)

How many possible solutions does this problem have? (eight)

Such problems are called problems with multiple solutions.

Compose your problem with multiple solutions.

Task number 8.

-In the battle with the three-headed and three-tailed Serpent Gorynych

Ivan Tsarevich with one blow of the sword can cut down either one head, or two heads, or one tail, or two tails. If you cut one head, a new one will grow; if you cut one tail, two new ones will grow; if you cut two tails, a head will grow; if you cut two heads, nothing will grow. Advise Ivan Tsarevich what to do so that he can cut off all the heads and tails of the Serpent.

What will happen if Ivan Tsarevich cuts off one head? (a new head will grow)

Does it make sense to cut off one head? (no, nothing will change)

So, cutting off one head is excluded - an extra waste of time and effort.

What happens if one tail is cut off? (two new tails will grow)

And if you cut off two tails? (head grows)

What about two heads? (nothing will grow)

So, we cannot cut off one head, because nothing will change, the head will grow again. It is necessary to achieve such a situation that there are an even number of heads, and no tails. But for this it is necessary that there be an even number of tails.

How can you achieve the desired result?

one). 1st hit: cut down 2 tails - there will be 4 heads and 1 tail;

2nd hit: cut down 1 tail - there will be 4 heads and 2 tails;

3rd hit: cut down 1 tail - there will be 4 heads and 3 tails;

4th hit: cut down 1 tail - there will be 4 heads and 4 tails;

5th hit: cut down 2 tails - there will be 5 heads and 2 tails;

6th hit: cut down 2 tails - there will be 6 heads and 0 tails;

7th hit: cut down 2 heads - there will be 4 heads;

2). 1st hit: cut down 2 heads - becomes 1 head and 3 tails;

2nd hit: cut down 1 tail - there will be 1 head and 4 tails;

3rd hit: cut down 1 tail - there will be 1 head and 5 tails;

4th hit: cut down 1 tail - there will be 1 head and 6 tails;

5th hit: cut down 2 tails - there will be 2 heads and 4 tails;

6th hit: cut down 2 tails - there will be 3 heads and 2 tails;

7th hit: cut down 2 tails - there will be 4 heads;

8th hit: cut down 2 heads - there will be 2 heads;

9th hit: cut down 2 heads - becomes 0 heads.

Task number 9.

There are four children in the family: Seryozha, Ira, Vitya and Galya. They are 5, 7, 9 and 11 years old. How old is each of them, if one of the boys goes to kindergarten, Ira is younger than Serezha, and the sum of the girls' years is divisible by 3?

Repeat the problem statement.

In order not to get confused in the process of reasoning, we draw a table.

What do we know about one of the boys? (goes to kindergarten)

How old is this boy? (5)

Could this boy's name be Seryozha? (no, Seryozha is older than Ira, so his name is Vitya)

Let's put the sign "+" in the line "Vitya", column "5". So, the youngest child's name is Vitya and he is 5 years old.

What do we know about Ira? (she is younger than Serezha, and if we add the age of another sister to her age, then this amount will be divided by 3)

Let's try to calculate all the sums of the numbers 7, 9 and 11.

16 and 20 are not divisible by 3, but 18 is divisible by 3.

So the girls are 7 and 11 years old.

How old is Seryozha? (nine)

And Ire? (7, because she is younger than Serezha)

And Gale? (11 years)

Entering data into a table:

What is the answer to the problem? (Vita is 5 years old, Ira is 7 years old, Serezha is 9 years old, and Galya is 11 years old)

Task number 10.

Katya, Sonya, Galya and Toma were born on March 2, May 17, June 2, March 20. Sonya and Galya were born in the same month, while Galya and Katya had the same birthday. Who, what date, and in what month was born?

Read the task.

What do we know? (that Sonya and Galya were born in the same month, and Galya and Katya were born on the same date)

So what month is Sonya and Galya's birthday? (in March)

And what can you say about Galya, knowing that she was born in March, and even her number matches the number of Katya? (Galya was born on March 2)

The concept of "non-standard task" is used by many methodologists. So, Yu. M. Kolyagin reveals this concept as follows: “Under non-standard understood task, upon presentation of which, students do not know in advance either the method of solving it, or what educational material the solution is based on.

The definition of a non-standard problem is also given in the book “How to learn to solve problems” by the authors L.M. Fridman, E.N. Turkish: " Non-standard tasks- these are those for which there are no general rules and regulations in the course of mathematics that determine the exact program for their solution.

Do not confuse non-standard tasks with tasks of increased complexity. The conditions of problems of increased complexity are such that they allow students to quite easily select the mathematical apparatus that is needed to solve a problem in mathematics. The teacher controls the process of consolidating the knowledge provided by the training program by solving problems of this type. But a non-standard task implies the presence of an exploratory nature. However, if the solution of a problem in mathematics for one student is non-standard, since he is unfamiliar with the methods for solving problems of this type, then for another, the solution of the problem occurs in a standard way, since he has already solved such problems and more than one. The same task in mathematics in the 5th grade is non-standard, and in the 6th grade it is ordinary, and not even of increased complexity.

An analysis of textbooks and teaching aids in mathematics shows that each text problem under certain conditions can be non-standard, and in others - ordinary, standard. A standard problem in one course of mathematics may be non-standard in another course.

Based on the analysis of the theory and practice of using non-standard tasks in teaching mathematics, one can establish their general and specific role. Non-standard tasks:

• · teach children not only to use ready-made algorithms, but also to independently find new ways to solve problems, i.e. contribute to the ability to find original ways to solve problems;
• influence the development of ingenuity, ingenuity of students;
• They prevent the development of harmful clichés when solving problems, destroy incorrect associations in the knowledge and skills of students, involve not so much the assimilation of algorithmic techniques, but the discovery of new connections in knowledge, the transfer of knowledge to new conditions, and the mastery of various methods of mental activity;
• create favorable conditions for increasing the strength and depth of knowledge of students, ensure the conscious assimilation of mathematical concepts.

Non-standard tasks:

• should not have ready-made algorithms memorized by children;
• should be accessible to all students in terms of content;
• must be interesting in content;
• To solve non-standard problems, students should have enough knowledge acquired by them in the program.

Solving non-standard tasks activates the activity of students. Students learn to compare, classify, generalize, analyze, and this contributes to a stronger and more conscious assimilation of knowledge.

As practice has shown, non-standard tasks are very useful not only for lessons, but also for extracurricular activities, for Olympiad tasks, since this opens up the opportunity to truly differentiate the results of each participant. Such tasks can be successfully used as individual tasks for those students who easily and quickly cope with the main part of independent work in the lesson, or for those who wish as additional tasks. As a result, students receive intellectual development and preparation for active practical work.

There is no generally accepted classification of non-standard tasks, but B.A. Kordemsky identifies the following types of such tasks:

• · Tasks related to the school mathematics course, but of increased difficulty - such as tasks of mathematical Olympiads. They are intended mainly for schoolchildren with a definite interest in mathematics; thematically, these tasks are usually associated with one or another specific section of the school curriculum. The exercises related to this deepen the educational material, supplement and generalize the individual provisions of the school course, expand the mathematical horizons, and develop skills in solving difficult problems.
• · Problems of the type of mathematical entertainment. They are not directly related to the school curriculum and, as a rule, do not require much mathematical preparation. This does not mean, however, that the second category of tasks includes only easy exercises. Here there are problems with a very difficult solution and such problems, the solution of which has not yet been obtained. “Non-standard tasks, presented in a fun way, bring an emotional moment to mental activities. Not connected with the need to apply memorized rules and techniques to solve them every time, they require the mobilization of all accumulated knowledge, teach them to search for original, non-standard ways of solving, enrich the art of solving with beautiful examples, make them admire the power of the mind.

These types of tasks include:

a variety of numerical puzzles ("... examples in which all or some of the numbers are replaced by asterisks or letters. The same letters replace the same numbers, different letters - different numbers" .) and puzzles for ingenuity;

logical tasks, the solution of which does not require calculations, but is based on the construction of a chain of exact reasoning;

tasks, the solution of which is based on a combination of mathematical development and practical ingenuity: weighing and transfusions under difficult conditions;

mathematical sophistry is a deliberate, false conclusion that has the appearance of being correct. (Sophism is a proof of a false statement, and the error in the proof is skillfully disguised. Sophism in Greek means a cunning invention, trick, puzzle);

joke tasks;

combinatorial problems, in which various combinations of given objects are considered that satisfy certain conditions (B.A. Kordemsky, 1958).

No less interesting is the classification of non-standard problems given by I.V. Egorchenko:

• tasks aimed at finding relationships between given objects, processes or phenomena;
• tasks that are unsolvable or unsolvable by means of a school course at a given level of knowledge of students;
• Tasks that require:

conducting and using analogies, determining the differences between given objects, processes or phenomena, establishing the opposite of given phenomena and processes or their antipodes;

implementation of a practical demonstration, abstraction from certain properties of an object, process, phenomenon or concretization of one or another side of this phenomenon;

establishment of causal relationships between given objects, processes or phenomena;

construction of causal chains in an analytical or synthetic way with subsequent analysis of the resulting options;

the correct implementation of a sequence of certain actions, avoiding errors-"traps";

implementation of the transition from a planar to a spatial version of a given process, object, phenomenon, or vice versa (I.V. Egorchenko, 2003).

So, there is no unified classification of non-standard tasks. There are several of them, but the author of the work used the classification proposed by I.V. Egorchenko.

The collection contains materials on the formation of students' skills to solve non-standard problems. The ability to solve non-standard problems, i.e. those whose solution algorithm is not known in advance, is an important component of school education. How to teach students to solve non-standard problems? One of the possible options for such training - a constant competition for solving problems was described on the pages of the application "Mathematics" (No. 28-29, 38-40 / 96). The set of tasks brought to your attention can also be used in extracurricular activities. The material was prepared at the request of teachers from the city of Kostroma.

The ability to solve problems is the most important (and most easily controlled) component of the mathematical development of students. This is not about typical tasks (exercises), but about tasks non-standard, the algorithm for solving which is not known in advance (the boundary between these types of tasks is conditional, and what is non-standard for a sixth grade student may be familiar to a seventh grade student!. The 150 tasks proposed below (direct continuation of non-standard tasks for fifth graders) are intended for annual competition in the 6th grade. These tasks can also be used in extracurricular activities.

Commentary on tasks

All tasks can be divided into three groups:

1.Tasks for ingenuity. To solve such problems, as a rule, deep knowledge is not required, only quick wit and a desire to overcome the difficulties encountered on the way to a solution are needed. Among other things, this is a chance to interest students who do not show much zeal for learning, and, in particular, for mathematics.

2.Tasks for fixing the material. From time to time, it is necessary to solve problems designed solely to consolidate the learned ideas. Note that it is desirable to check the degree of assimilation of new material some time after its study.

3.Tasks for propaedeutics of new ideas. Tasks of this type prepare students for the systematic study of program material, and the ideas and facts contained in them receive a natural and simple generalization in the future. So, for example, the calculation of various numerical sums will help students understand the derivation of the formula for the sum of an arithmetic progression, and the ideas and facts contained in some of the text tasks of this set prepare for the study of topics: Systems of linear equations, "Uniform motion", etc. How experience shows that the longer the material is studied, the easier it is to learn.

About problem solving

We note the most important points:

1. We provide "purely arithmetic" solutions to text problems where possible, even if students can easily solve them using equations. This is due to the fact that the reproduction of material in verbal form requires much greater logical effort and therefore most effectively develops the thinking of students. The ability to present material in verbal form is the most important indicator of the level of mathematical thinking.

2. The studied material is better absorbed if in the minds of students it is associated with other material, therefore, as a rule, we refer to already solved problems (such links are typed in italics).

3. It is useful to solve problems in different ways (a positive assessment is given for any way of solving). Therefore, for all word problems except arithmetic considered algebraic solution (equation). The teacher is recommended to conduct a comparative analysis of the proposed solutions.

Task Conditions

▼ 1.1. What single-digit number must be multiplied by so that the result is a new number written in one units?

1.2. If Anya goes to school on foot and goes back by bus, then she spends 1.5 hours on the road. If she goes both ways by bus, then the whole journey takes her 30 minutes. How much time will Anna spend on the road if she walks both to and from school?

1.3. Potato prices fell by 20%. How many percent more potatoes can you buy for the same amount?

1.4. A six-liter bucket contains 4 liters of kvass, and a seven-liter bucket contains 6 liters. How to divide all the available kvass in half using these buckets and an empty three-liter jar?

1.5. Is it possible to move the chess knight from the bottom left corner of the board to the top right corner, visiting each square exactly once? If it is possible, then indicate the route, if not, then explain why.

▼ 2.1. Is the statement true: If you add the square of the same number to a negative number, will you always get a positive number?

2.2. I walk from home to school in 30 minutes, and my brother takes 40 minutes. In how many minutes will I catch up with my brother if he left the house 5 minutes before me?

2.3. The student wrote on the board an example for multiplying two-digit numbers. He then erased all the numbers and replaced them with letters. The result is an equality: . Prove that the student was wrong.

2.4. The pitcher balances the decanter and the glass, two jugs weigh as much as three cups, and the glass and cup balance the decanter. How many glasses balance the decanter?

▼ 3.1. The passenger, having traveled half of the entire journey, went to bed and slept until half of the journey was left that he had traveled sleeping. How much of the journey did he travel sleeping?

3.2. What word is encrypted in the notation of a number if each letter is replaced by its number in the alphabet?

3.3. 173 numbers are given, each of which is equal to 1 or -1. Is it possible to divide them into two groups so that the sums of the numbers in the groups are equal?

3.4. The student read the book in 3 days. On the first day, he read 0.2 of the entire book and 16 more pages, on the second day, 0.3 of the remainder and 20 more pages, and on the third day, 0.75 of the new balance and the last 30 pages. How many pages are in the book?

3.5. A painted cube with an edge of 10 cm is cut into cubes with an edge of 1 cm. How many cubes will there be among them with one painted face? With two painted edges?

▼ 4.1. From the numbers 21, 19, 30, 25, 3, 12, 9, 15, 6, 27 choose three such numbers, the sum of which is 50.

4.2. The car is traveling at a speed of 60 km/h. By how much should you increase your speed in order to travel a kilometer one minute faster?

4.3. One cell has been added to the tic-tac-toe board (see picture). How should the first player play in order to secure a win for sure?

4.4. 7 people participated in the chess tournament. Each chess player played one game with each. How many games were played?

4.5. Is it possible to cut a chessboard into 3x1 rectangles?

▼ 5.1. I paid 5000 for the book. And it remains to pay as much as would be left to pay if they paid for it as much as was left to pay. How much does a book cost?

5.2. The nephew asked his uncle how old he was. The uncle replied: “If you add 7 to half of my years, then you will find out my age 13 years ago.” How old is uncle?

5.3. If you enter 0 between the digits of a two-digit number, then the resulting three-digit number is 9 times greater than the original one. Find this two digit number.

5.4. Find the sum of the numbers 1 + 2 + ... + 870 + 871.

5.5. There are 6 sticks, each 1 cm long, 3 sticks - 2 cm each, 6 sticks - 3 cm each, 5 sticks - 4 cm each. Is it possible to make a square from this set using all the sticks without breaking them and not putting one on top of the other?

▼ 6.1. The multiplier is increased by 10%, and the multiplier is decreased by 10%. How did this change the work?

6.2. Three runners BUT , B and AT competed in the 100m. When BUT ran to the end of the race B lagged behind him by 10 m, when B ran to the finish line AT lagged behind him by 10 m. How many meters lagged behind AT from BUT , when BUT finished?

6.3. The number of absent students in the class equals the number of those present. After one student left the class, the number of those absent became equal to the number of those present. How many students are in the class?

6.4 . Watermelon balances melon and beets. Melon balances cabbage and beets. Two watermelons weigh as much as three cabbages. How many times heavier is a melon than a beetroot?

6.5. Can a 4x8 rectangle be cut into 9 squares?

▼ 7.1. The price of the product was reduced by 10%, and then again by 10%. Will a product become cheaper if its price is immediately reduced by 20%?

7.2. A rower, while sailing down the river, lost his hat under the bridge. After 15 minutes, he noticed the loss, returned and caught the hat 1 km from the bridge. What is the speed of the river?

7.3. It is known that one of the coins is false and it is lighter than the others. How many weighings on a balance pan without weights can determine which coin is counterfeit?

7.4. Is it possible, according to the rules of the game, to lay out all 28 dominoes in a chain so that at one end there is a "six" and at the other - "five"?

7.5. There are 19 telephones. Is it possible to connect them in pairs so that each is connected to exactly thirteen of them?

▼ 8.1. 47 boxers participate in competitions according to the Olympic system (the loser is eliminated). How many fights do you have to fight to determine the winner?

8.2. Apple and cherry trees grow in the garden. If we take all the cherries and all the apple trees, then both those and other trees will remain equally, and in total there are 360 ​​trees in the garden. How many apple and cherry trees were there in the garden?

8.3. Kolya, Borya, Vova and Yura took the first four places in the competition, and no two boys shared any places among themselves. When asked who wilted which place, Kolya replied: “Neither the first, nor the fourth.” Borya said: “Second”, and Vova noticed that he was not the last. What place did each of the boys take if they all told the truth?

8.4. Is the number divisible by 9?

8.5. Cut a rectangle with a length of 9 cm and a width of 4 cm into two equal parts so that they can be folded into a square.

▼ 9.1. Collected 100 kg of mushrooms. It turned out that their humidity is 99%. When the mushrooms are dried, the humidity

dropped to 98%. What was the mass of mushrooms after drying?

9.2. Is it possible to make a table of 3 rows and 4 columns from the numbers 1, 2, 3, ..., 11, 12 such that the sum of the numbers in each of the columns is the same?

9.3. What digit does the sum 135x + 31y + 56x+y end with if x and y – integers?

9.4. Five boys Andrei, Borya, Volodya, Gena and Dima have different ages: one is 1 year old, the other is 2 years old, the rest are 3, 4 and 5 years old. Volodya is the smallest, Dima is as old as Andrei and Gena together. How old is Bora? Whose age can be determined?

9.5. Two fields have been sawn off from the chessboard: the lower left and the upper right. Is it possible to cover such a chessboard with "bones" of 2x1 dominoes?

▼ 10.1. Is it possible from the numbers 1,2,3,…. 11.12 make a table of 3 rows and 4 columns such that the sum of the numbers in each of the three rows is the same?

10.2. The plant manager usually arrives by train in the city at 8 o'clock. Just at this time, a car arrives and takes him to the plant. One day the director arrived at the station at 7 o'clock and went to the factory on foot. Having met the car, he got into it and arrived at the plant 20 minutes earlier than usual. What time did the clock show when the director met the machine?

10.3 . Two bags contain 140 kg of flour. If 1/8 of the flour in the first bag is transferred from the first bag to the second, then the flour will be equally in both bags. How much flour was originally in each bag?

10.4. In one month, three Wednesdays fell on even numbers. What date is the second Sunday of this month?

10.5. After 7 washes, the length, width and thickness of the bar of soap have halved. How many of the same washes will last the remaining soap?

▼ 11.1. Continue the series of numbers: 10, 8, 11, 9, 12, 10 until the eighth number. What rule is it based on?

11.2. From home to school Yura left 5 minutes late Lena, but walked twice as fast as she did. How many minutes after leaving Yura catch up Lena?

11.3. 2100?

11.4. Two sixth-grade students bought 737 textbooks, and each bought the same number of textbooks. How many sixth graders were there, and how many textbooks did each of them buy?

11.5 . Find the area of ​​the triangle shown in the figure (the area of ​​\u200b\u200beach cell is 1 sq. cm).

▼ 12.1. Humidity of freshly cut grass is 60%, and hay is 15%. How much hay will be made from one ton of freshly cut grass?

12.2. Five students bought 100 notebooks. Kolya and Vasya bought 52 notebooks, Vasya and Yura– 43, Yura and Sasha - 34, Sasha and Seryozha– 30. How many notebooks did each of them buy?

12.3. How many chess players played in a round robin tournament if 190 games were played in total?

12.4. What digit does Z100 end with?

12.5. It is known that the lengths of the sides of a triangle are integers, with one side equal to 5 and the other 1. What is the length of the third side?

▼ 13.1. The ticket cost Rs. After the fare reduction, the number of passengers increased by 50%, while revenue increased by 25%. How much did the ticket cost after the reduction?

13.2. From Nizhny Novgorod to Astrakhan the ship goes 5 days, and back - 7 days. How long will the rafts sail from Nizhny Novgorod to Astrakhan?

13.3. Yura took the book for 3 days. On the first day he read half the book, on the second day he read a third of the remaining pages, and the number of pages read on the third day is equal to half the pages read in the first two days. Did you manage Yura read a book in 3 days?

13.4. Alyosha, Borya and Vitya study in the same class. One of them goes home from school by bus, the other by tram, the third by trolley bus. One day after class Alyosha went to see a friend to the bus stop. When a trolley bus passed them, a third friend shouted from the window: “ Borya, you forgot your notebook at school!” What mode of transport does everyone use to go home?

13.5. I am now twice as old as you were when I was as old as you are now. Now we have been together for 35 years. How old are each of you?

▼ 14.1. 2001 is given. It is known that the sum of any four of them is positive. Is it true that the sum of all numbers is positive?

14.2. When the cyclist passed the tracks, a tire burst. He walked the rest of the way and spent 2 times more time on it than on a bike ride. How many times did the cyclist ride faster than he walked?

14.3. There are two pan scales and weights weighing 1, 3, 9, 27 and 81 g. A load is placed on one scale pan, weights are allowed to be placed on both pans. Prove that the balance can be balanced if the mass of the load is: a) 13 g; b) 19 g; c) 23 g; d) 31

14.4. The student wrote on the board an example for multiplying two-digit numbers. Then he erased all the numbers and replaced them with letters: the same numbers - the same letters, and different - different. The result is an equality: . Prove that the student was wrong.

14.5. Among musicians every seventh is a chess player, and among chess players every ninth is a musician. Who is more: musicians or chess players? Why?

▼ 15.1. The length of the rectangular area is increased by 35% and the width is reduced by 14%. By what percent did the area change?

15.2. Calculate the sum of the digits of the number 109! Then they calculated the sum of the digits of the newly obtained number and so continued until a single-digit number was obtained. What is this number?

15.3. Three Fridays of a certain month fell on even dates. What day of the week was the 18th of this month?

15.4. The case is being sorted out Brown, Jones and Smith. One of them committed a crime. During the investigation, each of them made two statements:

Brown: 1. I am not a criminal. 2. Jones too.

Jones: 1, It's not Brown. 2. This is Smith.

Lives: 1. Criminal Brown. 2. It's not me.

It was found that one of them lied twice, another told the truth twice, and the third lied once and told the truth once. Who committed the crime?

15.5. On the clock 19 h 15 min. What is the angle between the minute and hour hands?

▼ 16.1. If the person in front of you in line was taller than the person behind the person in front of you, was the person in front of you taller than you?

16.2. There are less than 50 students in the class. For the control work, the seventh part of the students received a mark of "5", the third part - "4", and half - "3". The rest received "2". How many such jobs were there?

16.3. Two cyclists left the checkpoints at the same time BUT and AT towards each other and met 70 km from BUT. Continuing to move at the same speeds, they reached their final destinations and, having rested for an equal amount of time, returned back. The second meeting took place 90 km from AT. Find the distance from BUT before AT.

16.4. Is the number divisible 111…111 (999 units) by 37?

16.5. Divide the 18x8 rectangle into pieces so that the pieces can be folded into a square.

▼ 17.1. When Vanya asked how old he was, he thought and said: "I'm three times younger than dad, but three times older than Seryozha." A little one ran up Secutting and said that dad is 40 years older than him. How many years Van?

17.2. The cargo was delivered to three warehouses. 400 tons were delivered to the first and second warehouses, 300 tons to the second and third together, and 440 tons to the first and third warehouses. How many tons of cargo were delivered to each warehouse separately?

17.3. From the ceiling of the room, two flies crawled vertically down the wall. Having descended to the floor, they crawled back. The first fly crawled in both directions at the same speed, and the second, although it ascended twice as slowly as the first, nevertheless descended twice as fast. Which of the flies will crawl back first?

17.4. 25 boxes of apples of three varieties were brought to the store, and in each of the boxes there were apples of one variety. Can you find 9 crates of apples of the same variety?

17.5. Find two prime numbers whose sum and difference is also a prime number.

▼ 18.1. A three-digit number is conceived, in which one of the digits coincides with any of the numbers 543, 142 and 562, and the other two do not match. What is the intended number?

18.2. At the ball, each gentleman danced with three ladies, and each lady danced with three gentlemen. Prove that the number of ladies at the ball was equal to the number of gentlemen.

18.3. The school has 33 classes, 1150 students. Is there a class in this school with at least 35 students?

18.4. In one area of ​​the city, more than 94% of houses have more than 5 floors. What is the smallest number of houses possible in the area?

18.5. Find all triangles whose side lengths are whole numbers of centimeters and the length of each of them does not exceed 2 cm.

▼ 19.1. Prove that if the sum of two natural numbers is less than 13, then their product is at most 36.

19.2. Of the 75 identical rings, one differs in weight from the others. How can you tell if this ring is lighter or heavier than the others in two weighings on a balance pan?

19.3. The plane flew from A to B at first at 180 km/h, but when it had 320 km less to fly than it had already flown, it increased its speed to 250 km/h. It turned out that the average speed of the aircraft for the entire journey was 200 km/h. Determine the distance from BUT to V.

19.4. The policeman turned around at the sound of breaking glass and saw four teenagers running away from the broken shop window. In 5 minutes they were at the police station. Andrei said the glass was broken Victor, Victor claimed to be guilty Sergey.Sergey assured that Victor lies, and Yuri insisted he didn't do it. From further conversation it turned out that only one of the guys was telling the truth. Who broke the glass?

19.5. All natural numbers from 1 to 99 are written on the board. Which numbers are more on the board - even or odd?

▼ 20.1. Two peasants left the village for the city. After walking the path, they sat down to rest. "How much more to go?" one asked the other. “We have 12 km more to go than we have already done,” was the answer. What is the distance between the city and the countryside?

20.2. Prove that the number 7777 + 1 is not divisible by 5.

20.3. There are four children in the family, they are 5, 8, 13 and 15 years old. Children's name Anya, Borya, Vera and Galya. How old is each child if one of the girls goes to kindergarten, Anya older Bori and sum of years Ani and Faith is divisible by 3?

20.4. There are 10 watermelons and 8 melons in a dark room (melons and watermelons are not distinguishable by touch). How many fruits do you need to take so that there are at least two watermelons among them?

20.5. A rectangular school plot has a perimeter of 160 m. How will its area change if the length of each side is increased by 10 m?

▼ 21.1. Find the sum 1 + 5 + ... + 97 + 101.

21.2. Yesterday, the number of students present in the class was 8 times greater than those absent. Today, 2 more students did not come and it turned out that 20% of the number of students present in the class are missing. How many students are in the class?

21.3. What is more than 3200 or 2300?

21.4. How many diagonals does a thirty-quadruple have?

21.5. In the middle of the square-shaped area there is a flower bed, which also has the shape of a square. The plot area is 100 m2. The side of the flower bed is half the size of the side of the site. What is the area of ​​the flower bed?

▼ 22.1. Reduce the fraction

22.2. A piece of wire 102 cm long must be cut into pieces 15 and 12 cm long so that there are no trimmings. How to do it? How many solutions does the problem have?

22.3. The box contains 7 red and 5 blue pencils. Pencils are taken from the box in the dark. How many pencils do you need to take so that there are at least two red and three blue among them?

22.4. In one vessel 2a liters of water and the other is empty. Half of the water is poured from the 1st vessel into the 2nd,

then water is poured from the 2nd into the 1st, then water is poured from the 1st into the 2nd, etc. How many liters of water will be in the first vessel after 1995 transfusion?

8. Cross out one hundred digits from the number ... 5960 so that the resulting number is the largest.

▼ 23.1. First they drank cups of black coffee and topped it up with milk. Then they drank the cups and refilled it with milk. Then they drank another half a cup and again topped it up with milk. Finally, they drank the whole cup. What did you drink more: coffee or milk?

23.2. 3 was added to the three-digit number on the left and it increased by 9 times. What is this number?

23.3. From paragraph BUT to paragraph AT two beetles crawl and come back. The first beetle crawled in both directions at the same speed. The second crawled into AT 1.5 times faster and back 1.5 times slower than the first one. Which beetle is back in BUT before?

23.4. Which number is greater: 2379∙23 or 2378∙23?

23.5. The square area is 16 m2. What will be the area of ​​the square if:

a) increase the side of the square, 2 times?

b) increase the side of the square by 3 times?

C) increase the side of the square by 2 dm?

▼ 24.1. What number must be multiplied by to get a number that is written using only fives?

24.2. Is it true that the number 1 is the square of some natural number?

24.3. car from BUT in AT traveled at an average speed of 50 km/h and returned back at a speed of 30 km/h. What is his average speed?

24.4. Prove that any amount of a whole number of rubles greater than seven can be paid without change in banknotes of 3 and 5 rubles?

24.5. Two types of logs were brought to the plant: 6 and 7 m long. They need to be sawn into meter-long logs. What kind of logs are more profitable to saw?

▼ 25.1. The sum of several numbers is 1. Can the sum of their squares be less than 0.01?

25.2. There are 10 bags of coins. Nine bags contain real coins (weighing 10 g each), and one contains fake coins (weighing 11 g each). With one weighing on an electronic scale, determine which bag contains counterfeit coins.

25.3. Prove that the sum of any four consecutive natural numbers is not divisible by 4.

25.3. From the number ... 5960, cross out one hundred digits so that the resulting number is the smallest.

25.4. Bought several identical books and identical albums. Books were paid 10 rubles. 56 kop. How many books were bought if the price of one book is more than a ruble higher than the price of an album, and books were bought 6 more than albums.

▼ 26.1. Two opposite sides of the rectangle are increased by their part, and the other two are reduced by a part. How has the area of ​​the rectangle changed?

26.2. Ten teams participate in the football tournament. Prove that for any schedule of games there will always be two teams that have played the same number of matches.

26.3. An airplane flies in a straight line from city A to city B and then back. Its own speed is constant. When will the plane fly all the way faster: in the absence of wind or with the wind constantly blowing in the direction from A to B?

26.4. The numbers 100 and 90 are divided by the same number. In the first case, the remainder was 4, and in the second - 18. By what number was the division performed?

26.5. Six transparent flasks with water are placed in two parallel rows of 3 flasks in each. On fig. 1, three front flasks are visible, and in fig. 2 - two right side. Through the transparent walls of the flasks, the water levels in each visible flask and in all the flasks behind them are visible. Determine the order in which the flasks are and what level of water is in each of them.

▼ 27.1. On the first day, the team of mowers mowed half of the meadow and another 2 hectares, and on the second day, 25% of the remaining part and the last 6 hectares. Find the area of ​​the meadow.

27.2. There are 11 bags of coins. Ten bags contain real coins (weigh 10 g each), and one contains fake coins (weigh 11 g each). In one weighing, determine which bag contains counterfeit coins.

27.3. There are 10 red, 8 blue and 4 yellow pencils in a box. Pencils are taken from a drawer in the dark. What is the smallest number of pencils you need to take so that among them there are: a) at least 4 pencils of the same color? B) at least 6 pencils of the same color? C) at least 1 pencil of each color?

D) at least 6 blue pencils?

27.4. Vasya said that he knew the solution of the equation hu 8+ x 8y= 1995 in natural numbers. Prove that Vasya was wrong.

27.5. Draw such a polygon and a point inside it so that no side of the polygon is completely visible from this point (in Fig. 3, the side is not completely visible from point O AB).

▼ 28.1. Grisha and dad went to the shooting range. The agreement was as follows: Grisha makes 5 shots and for each hit on the target he gets the right to make 2 more shots. In total, Grisha fired 17 shots. How many times did he hit the target?

28.2. A sheet of paper was cut into 4 pieces, then some (perhaps all) of those pieces were also cut into 4 pieces, etc. Could the result be exactly 50 pieces of paper?

28.3. For the first half of the journey, the rider rode at a speed of 20 km/h, and the second half at a speed of 12 km/h. Find the rider's average speed.

28.4. There are 4 watermelons of various weights. How, using a pan balance without weights, in no more than five weighings, arrange them in ascending order of mass?

28.5. Prove that it is impossible to draw a line so that it intersects all sides of a 1001-gon (without passing through its vertices).

▼ 29.1. Prime number 1?

29.2. One bottle contains white wine and the other red wine. We put one drop of red wine into the white, and then from the resulting mixture we return one drop to the red wine. What is more - white wine in red or red wine in white?

29.3. Couriers uniformly, but with different speeds, move from BUT in AT towards each other. After the meeting, one had to spend another 16 hours to arrive at their destination, and the other - 9 hours. How long does it take each of them to go all the way from A to B?

29.4. What is greater than 3111 or 1714?

29.5. a) The sum of the sides of a square is 40 cm. What is the area of ​​a square?

b) Area of ​​a square 64. What is its perimeter?

▼ 30.1. Can the number 203 be represented as the sum of several terms whose product is also equal to 203?

30.2. One hundred cities are connected by airlines. Prove that among them there are two cities through which the same number of airlines pass.

30.3. Of the four outwardly identical parts, one differs in mass from the other three, but it is not known whether its mass is greater or less. How to reveal this detail by two weighings on a pan balance without weights?

30.4. What digit does the number end with

13 + 23 + … + 9993?

30.5. Draw 3 straight lines so that the notebook sheet is divided into the largest number of parts. How many parts will it take? Draw 4 straight lines with the same condition. How many parts are there now?

PROBLEM SOLUTIONS

1.1. By checking we are convinced: if the number is multiplied by 9, then the result will be Question for students: why should only the number 9 be “checked”?)

1.2. If Anya travels both ways by bus, then the whole journey takes her 30 minutes, therefore, she gets to one end by bus in 15 minutes. If Anya goes to school on foot and back by bus, then she spends 1.5 hours on the road, which means that she gets there on foot in 1 hour and 15 minutes. If Anya walks to and from school, then she spends 2 hours and 30 minutes on the road.

1.3. Since potatoes have fallen in price by 20%, now you need to spend 80% of the available money on all potatoes bought earlier, and buy another 1/4 of the potatoes for the remaining 20%, which is 25%. 4

1.4. The course of the solution is visible from the table:

	in step	1st step	2nd step	3rd by them	4th step	5th step

1.5. In order to go around all 64 cells of the chessboard, having visited each field exactly once. The knight must make 63 moves. With each move, the knight moves from a white field to a black one (or from a black field to a white one), therefore, after moves with even numbers, the knight will go to fields of the same color as the original one, and after “odd” moves, to fields with the opposite Colour. Therefore, on the 63rd move, the knight cannot get into the upper right corner of the board, since it is the same color as the upper right.