Non-standard tasks as a means of developing logical thinking. Non-standard tasks and their types Non-standard tasks

NON-STANDARD TASKS IN MATHEMATICS LESSONS

Teacher primary classes Shamalova S. V.

Each generation of people makes its own demands on school. An ancient Roman proverb says: “We study not for school, but for life.” The meaning of this proverb is still relevant today. Modern society dictates to the education system an order to educate an individual who is ready to live in constantly changing conditions, to continue education, and who is capable of learning throughout his life.

Among the spiritual abilities of man, there is one that has been the subject of close attention of scientists for many centuries 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 learning, in everyday life.

Any activity of a worker, schoolchild and scientist is inseparable from mental work. In any real matter, it is necessary to rack your brains, to stretch your mind, that is, in the language of science, you need to carry out a mental action, intellectual work. It is known that a problem can be solved or not solved, one person will cope with it quickly, the other thinks for a long time. There are tasks that are feasible even for a child, and some have been worked on by entire teams of scientists for years. This means there is the ability to think. Some are better at it, others worse. What kind of skill is this? In what ways does it arise? How to buy it?

No one will argue that every teacher should develop the logical thinking of students. This is stated in the methodological literature, in explanatory notes to curriculum. However, we teachers do not always know how to do this. This often leads to the fact that the development of logical thinking is largely spontaneous, so the majority of students, even high school students, do not master the initial techniques 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 targeted logical development of students in the early stages of education. Most modern manuals for preschoolers and primary schoolchildren contain a set of various tasks, focusing on such techniques 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 thinking techniques even in high school, and these techniques need to be taught to younger students.

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

In the process of solving educational problems, children develop the ability to be distracted from unimportant details. This action is given to younger schoolchildren with no less difficulty than highlighting the essential. Younger schoolchildren, as a result of studying at school, when it is necessary to regularly complete tasks without fail, learn to manage 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 what is available, and to carry out simple proofs using their life experience. Logic exercises gradually become more complex.

I also use non-standard developmental logical tasks in my practice. There is a significant variety 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 developmental tasks: tasks for intelligence, tasks for ingenuity, tasks with a “twist”. In all their diversity, we can distinguish into a special class such tasks, which are called tasks - traps, provoking tasks. The conditions of such tasks contain various kinds of references, instructions, hints that encourage the choice of an erroneous solution path or an incorrect answer. I will give examples of such tasks.

Problems that impose one, very definite answer.

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

Tasks that encourage you to make an incorrect choice of 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 heavier luggage?

Tasks whose conditions push you to perform some action with given numbers, whereas there is no need to perform this action at all.

The Mercedes car traveled 100 km. How many kilometers did each of its wheels travel?

Petya once said to his friends: “The day before yesterday I was 9 years old, and next year I will turn 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 was studying, one replied: “Vadim is studying Chinese, Sergei is not studying Chinese, and Mikhail is not studying Arabic.” Subsequently, it turned out that only one statement in this statement is true. What language is each of them studying?

The shorties from Flower City planted a watermelon. Watering it requires exactly 1 liter of water. They only have two empty 3 liter cans. And 5 l. How to use these cans. Collect exactly 1 liter from the river. water?

How many years did Ilya Muromets sit on the stove? It is known that if he had stayed in prison 2 more times, his age would have been the largest two-digit number.

Baron Munchausen counted the number of magical hairs in the beard of old man Hottabych. 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:V first of all , tasks should be introduced into the learning process in a certain system with a gradual increase in complexity, since an impossible task will have little effect on the development of students;V o secondly , it is necessary to provide students with maximum independence when searching for solutions to problems, give them the opportunity to go to the end along the wrong path in order to make sure of the mistake, return to the beginning and look for another, correct solution path;Thirdly , you need to help students understand some 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 provide simple proofs. The exercises themselves are entertaining in nature, so they contribute to the emergence of children’s interest 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 the 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 as tasks gradually became more complex. For this purpose I selectedtasks to identify patterns, dependencies and formulate generalizationswith 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, logic exercises should always be included in oral work in class. For example:

Find the result using this equality:

3+5=8

3+6=

3+7=

3+8=

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

2+3*2x3

4+4*3x4

4+5*4x5

5+6*5x6

Continue the series of numbers.

3. 5, 7, 9, 11…

1, 4, 7, 10…

Come up with something for everyone this example similar example.

12+6=18

16-4=12

What do the numbers on each line have in common?

12 24 20 22

30 37 13 83

Numbers given:

23 74 41 14

40 17 60 50

Which number is the odd one in each line?

In elementary school math lessons, I often use counting stick exercises. These are problems of a geometric nature, since during the solution, as a rule, there is transfiguration, the transformation of some figures into others, 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 involved in an active search for a solution, while striving for the final goal, the required modification of the figure.

Exercises with counting sticks can be combined into 3 groups: tasks on composing a given figure from a certain number of sticks; problems for changing figures, to solve which you need to remove or add a specified number of sticks; tasks, the solution of which consists in rearranging sticks in order to modify, transform a given figure.

Exercises with counting sticks.

Tasks on making figures from a certain number of sticks.

Make two different squares using 7 sticks.

Problems involving changing a figure, where you need to remove or add a specified number of sticks.

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

Problems involving rearranging sticks for the purpose of transformation.

Arrange two sticks to make 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 teaching of logical concepts does not give effect.

Solving non-standard problems develops in students the ability to make assumptions, check their accuracy, and justify them logically. Speaking for the purpose of evidence contributes to the development of speech, the development of skills to draw conclusions, and build conclusions. In the process of using these exercises in lessons 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.

The collection presents materials on developing students' skills in solving non-standard problems. The ability to solve non-standard problems, that is, those for which the solution algorithm is not known in advance, is an important component of school education. How to teach schoolchildren to solve non-standard problems? About one of possible options such training - a constant competition for solving problems was described on the pages of the Mathematics supplement (No. 28-29, 38-40/96). The set of tasks offered to your attention can also be used in extracurricular activities. The material was prepared at the request of teachers in the city of Kostroma.

Problem solving skills are the most important (and easiest to control) component of students' mathematical development. It's about not about standard tasks (exercises), but about tasks non-standard, the solution algorithm for which is not known in advance (the boundary between these types of problems is arbitrary, and what is non-standard for a sixth-grader may be familiar to a seventh-grader! The 150 problems proposed below (a direct continuation of non-standard problems for fifth-graders) are intended to annual competition in 6th grade. These tasks can also be used in extracurricular activities.

Comments on tasks

All tasks can be divided into three groups:

1.Challenges for ingenuity. Solving such problems, as a rule, does not require deep knowledge; all that is needed is intelligence and the desire to overcome the difficulties encountered on the way to a solution. 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 to consolidate the material. From time to time, it is necessary to solve problems designed solely to consolidate the learned ideas. Note that it is advisable to check the degree of assimilation of new material some time after studying it.

3.Tasks for propaedeutics of new ideas. Problems 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. For example, calculating 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 word problems in this set will prepare them for studying the topics: Systems linear equations", "Uniform movement", etc. As experience shows, the longer the material is studied, the easier it is to learn.

About problem solving

Let us note the fundamentally important points:

1. We provide “purely arithmetic” solutions to word problems where possible, even if students can easily solve them using equations. This is explained by the fact that reproducing material in verbal form requires significantly greater logical effort and therefore most effectively develops students’ thinking. 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 students’ minds it is connected with other material, therefore, as a rule, we refer to already solved problems (such links are typed in italics).

3. Problems are useful to solve different ways(a positive mark is given for any solution method). Therefore, for all word problems except arithmetic is being considered algebraic solution (equation). The teacher is recommended to conduct comparative analysis proposed solutions.

Problem conditions

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

1.2. If Anya walks to school and takes the bus back, then she spends a total of 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 Anya spend on the road if she walks to and from school?

1.3. Potatoes fell in price 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 a chess knight from the lower left corner of the board to the upper right corner, visiting each square exactly once? If 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 30 minutes, and my brother - 40 minutes. How many minutes will it take me to catch up with my brother if he left the house 5 minutes before me?

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

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

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

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

3.3. Given 173 numbers, 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 he read 0.3 of the rest and 20 more pages, and on the third day he read 0.75 of the new remainder and the last 30 pages. How many pages are in the book?

3.5. A painted cube with an edge of 10 cm was sawn into cubes with an edge of 1 cm. How many of them would there be cubes with one colored edge? With two painted edges?

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

4.2. The car is traveling at a speed of 60 km/h. How much do you need to increase your speed to cover a kilometer one minute faster?

4.3. One square has been added to the tic-tac-toe board (see picture). How should the first player play to ensure he wins?

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

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

▼ 5.1. They paid 5,000 rubles for the book. And there remains to pay as much as there would be left to pay if they paid for it as much as there was left to pay. How much does the 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, you will find out my age 13 years ago.” How old is your 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. 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, 6 sticks – 3 cm, 5 sticks – 4 cm. Is it possible to make a square from this set, using all the sticks, without breaking them or stacking one on top of the other?

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

6.2. Three runners A , B And IN competed in the 100 m race. When A reached the end of the race B lagged behind him by 10 m, When B reached the finish line IN lagged behind him by 10 m. How many meters lagged behind IN from A , When A finished?

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

6.4 . Watermelon balances out the melon and beets. The melon balances out the cabbage and beets. Two watermelons weigh the same as three heads of cabbage. How many times is a melon heavier than a beet?

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, floating along the river, lost his hat under a bridge. After 15 minutes he noticed it was missing, returned and caught the hat 1 km from the bridge. What is the speed of the river flow?

7.3. It is known that one of the coins is counterfeit and is lighter than the others. In how many weighings on a cup scale without weights can you determine which coin is counterfeit?

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

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

▼ 8.1. 47 boxers compete in the Olympic system (loser is eliminated). How many fights must be fought to determine the winner?

8.2. Apple and cherry trees grow in the garden. If you take all the cherries and all the apple trees, then there will be an equal number of both trees, 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 won which place, Kolya replied: “Neither the first nor the fourth.” Borya said: “Second,” and Vova noted 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 whose length is 9 cm and width 4 cm into two equal parts so that they can be folded into a square.

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

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

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

9.3. What number ends in the sum 135x + 31y + 56x+y, if x and y – integers?

9.4. Five boys Andrey, Borya, Volodya, Gena and Dima are of 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 are together. How old is Borya? Who else's age can be determined?

9.5. The chessboard has two squares cut off: the lower left and the upper right. Is it possible to cover such a chessboard with 2x1 domino “bones”?

▼ 10.1. Is it possible from the numbers 1,2,3,…. 11.12 create 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 director of the plant usually arrives in the city by train at 8 o'clock. Exactly 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 walked to the plant. 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 . There are 140 kg of flour in two bags. If you transfer 1/8 of the flour contained in the first bag from the first bag into the second, then there will be equal amounts of flour in both bags. How much flour was initially in each bag?

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

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

▼ 11.1. Continue the series of numbers: 10, 8, 11, 9, 12, 10 until the eighth number. By what rule is it compiled?

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

11.3. 2100?

11.4. Pupils in two sixth grades 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 ​​each cell is 1 sq. cm).

▼ 12.1. The moisture content of freshly cut grass is 60%, and that of hay is 15%. How much hay will be produced 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 the round-robin tournament if a total of 190 games were played?

12.4. What digit does the number 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 rubles. After the fare reduction, the number of passengers increased by 50%, and revenue increased by 25%. How much did the ticket cost after the reduction?

13.2. The ship takes 5 days from Nizhny Novgorod to Astrakhan, and 7 days back. How long will the rafts take to travel from Nizhny Novgorod to Astrakhan?

13.3. Yura I borrowed the book for 3 days. On the first day he read half the book, on the second - a third of the remaining pages, and the number of pages read on the third day was equal to half the pages read on the first two days. Did you have time? 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, another by tram, and the third by trolleybus. One day after school Alyosha I went to accompany my friend to the bus stop. When a trolleybus passed by them, a third friend shouted from the window: “ Borya, You forgot your notebook at school!” What type of transport does everyone use to go home?

13.5. I'm doubled now more years What was it like for you 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. The number given is 2001. 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, the tire burst. He walked the rest of the way and spent 2 times more time on it than riding a bicycle. How many times faster was the cyclist traveling than he was walking?

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

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

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

▼ 15.1. The length of the rectangular section was increased by 35%, and the width was reduced by 14%. By what percentage did the area of ​​the plot 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 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 matter 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'm not a criminal. 2. Jones too.

Jones: 1, This is not Brown. 2. This is Smith.

Lived: 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. The clock shows 19:15. What is the angle between the minute and hour hands?

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

16.2. There are less than 50 students in the class. For the test, one seventh of the students received a grade of “5”, the third received a grade of “4”, and half received a grade of “3”. The rest received a "2". How many such works were there?

16.3. Two cyclists left the points at the same time A And IN towards each other and met 70 km from A. Continuing to move at the same speeds, they reached their final destinations and, after resting for an equal amount of time, returned back. The second meeting took place 90 km from IN. Find the distance from A before IN.

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.” Then the little one ran up Xiecutting and said that dad is 40 years older than him. How many years Vanya?

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. 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 rose twice as slow as the first, but descended twice as fast. Which fly will crawl back first?

17.4. 25 boxes of apples of three varieties were brought to the store, and each box contained apples of one variety. Is it possible to find 9 boxes 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 coincide. What is the intended number?

18.2. At the ball, each gentleman danced with three ladies, and each lady with three gentlemen. Prove that at the ball the number of ladies 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 this area?

18.5. Find all triangles whose side lengths are integer 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. Out of 75 identical-looking rings, one is different in weight from the others. How can you determine in two weighings on a cup scale whether this ring is lighter or heavier than the others?

19.3. The plane flew from A to B at first at a speed of 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 plane along the entire route was 200 km/h. Determine the distance from A to V.

19.4. The policeman turned around at the sound of breaking glass and saw four teenagers running away from a broken display case. 5 minutes later they were at the police station. Andrey stated that the glass was broken Victor, Victor claimed he was guilty Sergey.Sergey assured that Victor lies, but Yuri insisted that it was not he who did 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. Having walked the path, they sat down to rest. “How much longer to go?” - one asked the other. “We have 12 km more to go than we have already covered,” was the answer. What is the distance between city and village?

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

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

20.4. There are 10 watermelons and 8 melons in a dark room (melons and watermelons are indistinguishable to the 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 class was 8 times greater than those absent. Today 2 more students did not come and it turned out that 20% of the students present in the class were absent. How many students are in the class?

21.3. What is more 3200 or 2300?

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

21.5. In the middle of the square-shaped plot there is a flower bed, which also has the shape of a square. The area of ​​the plot is 100 m2. The side of the flowerbed is half the size of the side of the plot. 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 scraps. 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 among them there are at least two red and three blue?

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 one,

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

8. From the number ...5960, cross out one hundred digits so that the resulting number is the largest.

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

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

23.3. From point A to point IN two beetles crawl and return. The first beetle crawled in both directions at the same speed. The second one crawled in IN 1.5 times faster, and back 1.5 times slower than the first one. Which beetle has returned to A earlier?

23.4. Which number is greater: 2,379∙23 or 2,378∙23?

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

a) increase the side of the square by 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 A V IN drove 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 sawed into meter-long logs. Which logs are more profitable to cut?

▼ 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 (weigh 10 g each), and one contains fake coins (weigh 11 g each). With one weighing on an electronic scale, you can 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. We bought several identical books and identical albums. They paid 10 rubles for the books. 56 kopecks How many books were bought if the price of one book is more than a ruble higher than the price of an album, and 6 more books were bought than albums.

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

26.2. Ten teams are participating in a football tournament. Prove that for any given 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 B, and then back. Its own speed is constant. When will the plane fly the entire distance faster: in the absence of wind or in the wind constantly blowing in the direction from A to B?

26.4. The numbers 100 and 90 are divided by one and the same number. In the first case, the remainder was 4, and in the second, 18. What number was the division done by?

26.5. Six transparent flasks with water are arranged in two parallel rows of 3 flasks each. In Fig. 1, three front flasks are visible, and in Fig. 2 – two right side ones. Through the transparent walls of the flasks, the water levels in each visible flask and in all the flasks behind them are visible. Determine what order the flasks are in and what the water level is in each of them.

▼ 27.1. On the first day, the mowing team 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). Just by weighing you can determine which bag contains counterfeit coins.

27.3. A box contains 10 red, 8 blue and 4 yellow pencils. Pencils are taken from the box in the dark. What is the smallest number of pencils that must be taken so that among them there will certainly be: 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 knows the solution to the equation xy 8+ x 8y = 1995 in natural numbers. Prove that Vasya is 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 this: Grisha fires 5 shots and for each hit on the target he gets the right to fire 2 more shots. In total, Grisha fired 17 shots. How many times did he hit the target?

28.2. A piece 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. The rider galloped for the first half of the journey at a speed of 20 km/h, and for the second half at a speed of 12 km/h. Find the average speed of the rider.

28.4. There are 4 watermelons of different weights. Using cup scales without weights, how can you arrange them in ascending order of mass in no more than five weighings?

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

▼ 29.1. Prime A number 1?

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

29.3. Couriers move evenly, but at different speeds, from A V IN towards each other. After the meeting, to arrive at their destination, one needed to spend another 16 hours, and the other - 9 hours. How long does it take each of them to travel the entire path from A to B?

29.4. What is greater, 3111 or 1714?

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

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

▼ 30.1. Is it possible to represent the number 203 as the sum of several terms, the product of which also equals 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 passes.

30.3. Of the four externally identical parts, one differs in mass from the other three, but it is unknown whether its mass is greater or less. How to identify this part by two weighings on cup scales 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 there be? Draw 4 straight lines with the same condition. How many parts are there now?

SOLUTIONS TO PROBLEMS

1.1. By checking we are convinced: if the number is multiplied by 9, the result will be Question to 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 there one way by bus in 15 minutes. If Anya walks to school and takes the bus back, then she spends a total of 1.5 hours on the road, which means she gets there on foot one way in 1 hour 15 minutes. If Anya walks to and from school, then she spends 2 hours 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 the previously purchased potatoes, and buy another 1/4 of the potatoes with the remaining 20%, which is 25%. 4

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

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

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

No wonder that entertaining mathematics has become entertainment “for of all times and peoples." To solve such problems, no special knowledge is required - one guess is enough, which, however, is sometimes more difficult to find than methodically solving a standard school problem.

Solving a fun arithmetic problem.
For 3 – 5 grades

How many dragons?

2-headed and 7-headed dragons gathered for a rally.
At the very beginning of the meeting, the Dragon King, the 7-headed Dragon, counted everyone gathered by their heads.

He looked around his crowned middle head and saw 25 heads.
The king was pleased with the results of the calculations and thanked everyone present for their attendance at the meeting.

How many dragons came to the rally?

(a) 7; (b) 8; 9; (d) 10; (e) 11;
Solution:

Let us subtract 6 heads belonging to him from the 25 heads counted by the Dragon King.

There will be 19 goals left. All remaining Dragons cannot be two-headed (19 is an odd number).

There can only be 1 7-headed Dragon (if 2, then for two-headed Dragons there will be an odd number of heads left. And for three Dragons there are not enough heads: (7 · 3 = 21 > 19).

Subtract 7 heads of this single Dragon from 19 heads and get the total number of heads belonging to two-headed Dragons.

Therefore, 2-headed Dragons:
(19 - 7) / 2 = 6 Dragons.

Total: 6 +1 +1 (King) = 8 Dragons.

Correct answer:b = 8 Dragons

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Solving a fun math problem

For 4 - 8 grades

How many wins?

Nikita and Alexander are playing chess.
Before the start of the game they agreed

that the winner of the game will receive 5 points, the loser will receive no points, and each player will receive 2 points if the game ends in a draw.

They played 13 games and got 60 points together.
Alexander received three times more points for those games that he won than for those that were drawn.

How many victories did Nikita win?

(a) 1; (b) 2; 3; (d) 4; (e) 5;
Correct answer: (b) 2 victories (Nikita won)

Solution.

Each draw game gives 4 points, and each win gives 5 points.
If all the games ended in a draw, the boys would score 4 · 13 = 52 points.
But they scored 60 points.

It follows that 8 games ended with someone winning.
And 13 - 5 = 5 games ended in a draw.

Alexander scored 5 · 2 = 10 points in 5 draw games, which means that if he won, he scored 30 points, that is, he won 6 games.
Then Nikita won (8-6=2) 2 games.

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Solving a fun arithmetic problem

For 4 - 8 grades

How many days without food?
The Martian interplanetary spacecraft arrived on a visit to Earth.
Martians eat at most once a day, either in the morning, at noon, or in the evening.

But they only eat when they feel hungry. They can go without food for several days.
During the Martians' stay on Earth, they ate 7 times.
We also know that they went without food 7 times in the morning, 6 times at noon and 7 times in the evening.
How many days did the Martians spend without food during their visit?

(a) 0 days; (b) 1 day; 2 days; (d) 3 days; (e) 4 days; (a) 5 days;
Correct answer: 2 days (the Martians spent without food)

Solution.
The Martians ate for 7 days, once a day, and the number of days they ate lunch was one more number days when they had breakfast or dinner.

Based on these data, it is possible to create a food intake schedule for Martians. This is the probable picture.

The aliens had lunch on the first day, had dinner on the second day, had breakfast on the third, had lunch on the fourth, had dinner on the fifth, had breakfast on the sixth, and had lunch on the seventh.

That is, the Martians ate breakfast for 2 days, and spent 7 days without breakfast, ate dinner 2 times, and spent 7 days without dinner, ate lunch 3 times, and lived without lunch for 6 days.

So 7 + 2 = 9 and 6 + 3 = 9 days. This means they lived on Earth for 9 days, and 2 of them went without food (9 - 7 = 2).

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Solving an entertaining non-standard problem

For 4 - 8 grades

How much time?
The cyclist and the pedestrian left point A at the same time and headed to point B at a constant speed.
The cyclist arrived at point B and immediately set off on the way back and met the Pedestrian an hour later from the moment they left point A.
Here the Cyclist turned around again and they both began to move in the direction of point B.

When the cyclist reached point B, he turned back again and met the Pedestrian again 40 minutes after their first meeting.
What is the sum of the digits of a number expressing the time (in minutes) required for a Pedestrian to get from point A to point B?
(a) 2; (b) 14; 12; (d) 7; (e)9.
Correct answer: e) 9 (the sum of the digits of the number is 180 minutes - this is how long the Pedestrian travels from A to B)

Everything becomes clear if you draw a drawing.
Let's find the difference between the two paths of the Cyclist (one path is from A to the first meeting (solid green line), the second path is from the first meeting to the second (dashed green line)).

We find that this difference is exactly equal to the distance from point A to the second meeting.
A pedestrian covers this distance in 100 minutes, and a cyclist travels in 60 minutes - 40 minutes = 20 minutes. This means the cyclist travels 5 times faster.

Let us denote the distance from point A to the point at which 1 meeting occurred as one part, and the Cyclist’s path to the 1st meeting as 5 parts.

Together, by the time of their first meeting, they had covered double the distance between points A and B, i.e. 5 + 1 = 6 parts.

Therefore, from A to B there are 3 parts. After the first meeting, the pedestrian will only have to walk 2 more parts to point B.

He will cover the entire distance in 3 hours or 180 minutes, since he covers 1 part in 1 hour.

Tests and questionnaires 3rd grade.

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

“The students of the first class bought 12 carnations, and the students of the second...”, “The master made 20 parts during the shift, and his student...”.

Everything in the text is important; And characters, and their actions, and numerical characteristics. When working with a mathematical model of a problem (a 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 the text of a mathematical problem is an important result and an important condition for the student’s overall development. And this needs to be done not only in mathematics lessons, but also in reading and fine arts lessons. Some problems make good subjects for drawings. And any task - good topic for retelling. And if there are theater lessons in the class, then some mathematical problems can be dramatized. Of course, all these techniques: retelling, drawing, dramatization - can also take place in the mathematics lessons themselves. So, working on the texts of mathematical problems is an important element of the child’s overall development, an element of developmental education.

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

about the number of elements of a certain set;

about movement, its speed, path and time;

about price and cost;

about work, its time, volume and productivity.

The four topics listed are standard. It is believed that the ability to solve problems on these topics can teach one to solve problems in general. Unfortunately, it is not. Good students who can solve practically

any problem from a textbook on the listed topics, they are often unable to understand the conditions of a problem on another topic.

The way out is not to limit yourself to any topic of word problems, but to solve non-standard problems, that is, problems whose topics are not in themselves the object of study. After all, we don’t limit the plots of stories in reading lessons!

Non-routine problems need to be solved in class every day. They can be found in mathematics textbooks for grades 5-6 and in magazines " Primary School", "Mathematics at school" and even "Quantum".

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

Among the problems there are completely similar ones in mathematical terms. If children see this, great. The teacher can show this 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 connections between tasks with a similar plot.

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

The concept of “non-standard task” is used by many methodologists. Thus, Yu. M. Kolyagin explains this concept as follows: “Under non-standard is understood task, upon presentation of which students do not know in advance either how to solve it or how educational material the decision is based."

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

Non-standard tasks 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 identify 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 presupposes a research character. However, if solving a problem in mathematics for one student is non-standard, since he is unfamiliar with methods for solving problems of this type, then for another, solving the problem occurs in a standard way, since he has already solved such problems and more than one. The same problem in mathematics in the 5th grade is non-standard, but in the 6th grade it is ordinary, and not even of increased complexity.

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

Based on an analysis of the theory and practice of using non-standard problems in teaching mathematics, it is possible to establish their general and specific role. Non-standard tasks:

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

Non-standard tasks:

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

Solving non-standard problems activates students' activities. Students learn to compare, classify, generalize, analyze, and this contributes to a more durable and 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 assignments, 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 can easily and quickly cope with the main part of independent work in class, or for those who wish to do so 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 problems, but B.A. Kordemsky identifies the following types of such tasks:

• · Problems related to the school mathematics course, but of increased difficulty - such as problems of mathematical olympiads. Intended mainly for schoolchildren with a definite interest in mathematics; thematically, these tasks are usually related to one or another specific section of the school curriculum. The exercises related here deepen the educational material, complement and generalize individual provisions of the school course, expand mathematical horizons, and develop skills in solving difficult problems.
• · Problems such as mathematical entertainment. They are not directly related to the school curriculum and, as a rule, do not require extensive mathematical training. This does not mean, however, that the second category of tasks includes only light exercises. There are problems with very difficult solutions and problems for which no solution has yet been obtained. “Unconventional problems, presented in an exciting way, bring an emotional element to mental exercises. Not associated with the need to always apply memorized rules and techniques to solve them, they require the mobilization of all accumulated knowledge, teach people to search for original, non-standard solutions, enrich the art of solving with beautiful examples, and make one admire the power of the mind.”

This type of task includes:

various number puzzles (“... examples in which all or some numbers are replaced by asterisks or letters. The same letters replace the same numbers, different letters - different numbers.”) and puzzles for ingenuity;

logical problems, the solution of which does not require calculations, but is based on building a chain of precise reasoning;

tasks whose solution is based on a combination of mathematical development and practical ingenuity: weighing and transfusion under difficult conditions;

mathematical sophisms are a deliberate, false conclusion that has the appearance of being correct. (Sophism is proof of a false statement, and the error in the proof is skillfully disguised. Sophistry translated from Greek means a clever 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 searching for relationships between given objects, processes or phenomena;
• · problems that are insoluble or cannot be solved by means of a school course at a given level of knowledge of students;
• tasks that require:

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

implementation of practical demonstration, abstraction from certain properties of an object, process, phenomenon or specification of one or another aspect of a given phenomenon;

establishing cause-and-effect relationships between given objects, processes or phenomena;

constructing analytically or synthetically cause-and-effect chains with subsequent analysis of the resulting options;

correct implementation of a sequence of certain actions, avoiding “trap” errors;

making a 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 single classification of non-standard tasks. There are several of them, but the author of the work used in the study the classification proposed by I.V. Egorchenko.