As mentioned above, the main goal of using a mathematical game in extracurricular classes about mathematics is to develop students’ sustainable cognitive interest in the subject through the variety of mathematical games used.
We can also highlight the following purposes for using mathematical games:
o Development of thinking;
o Deepening theoretical knowledge;
o Self-determination in the world of hobbies and professions;
o Organization of free time;
o Communication with peers;
o Fostering cooperation and collectivism;
o Acquisition of new knowledge, skills and abilities;
o Formation of adequate self-esteem;
o Development of strong-willed qualities;
o Knowledge control;
o Motivation educational activities and etc.
Mathematical games are designed to solve the following problems.
Educational:
b Promote students’ solid assimilation of educational material;
b Contribute to broadening the horizons of students, etc.
Educational:
b To develop creative thinking in students;
b To promote the practical application of skills acquired in lessons and extracurricular activities;
b Promote the development of imagination, fantasy, creative abilities, etc.
Educational:
b Contribute to the education of a self-developing and self-realizing personality;
b To develop moral views and beliefs;
b Contribute to the development of independence and will in work, etc.
Mathematical games serve various functions.
1. During a mathematical game, gaming, educational and work activities occur simultaneously. Indeed, the game brings together what is not comparable in life and separates what is considered one.
2. A mathematical game requires the student to know the subject. After all, without knowing how to solve problems, solve, decipher and unravel, a student will not be able to participate in the game.
3. In games, students learn to plan their work, evaluate the results of not only someone else’s, but also their own activities, be smart when solving problems, take a creative approach to any task, use and select the right material.
4. The results of the games show schoolchildren their level of preparedness and training. Mathematical games help students improve themselves and thereby stimulate their cognitive activity and increase their interest in the subject.
5. While participating in mathematical games, students not only receive new information, but also gain experience in collecting the necessary information and applying it correctly.
There are a number of requirements for game forms of extracurricular activities.
Participants in a mathematical game must have certain knowledge requirements. In particular, to play, you need to know. This requirement gives the game an educational character.
The rules of the game should be such that students show a desire to participate in it. That's why games should be developed taking into account the age characteristics of children, the interests they show at a given age, their development and existing knowledge.
Mathematical games should be designed taking into account individual characteristics students, taking into account different groups of students: weak, strong; active, passive, etc. They must be such that each type of student can express themselves in the game, show their abilities, capabilities, independence, perseverance, ingenuity, and experience a sense of satisfaction and success.
When developing a game easier game options should be provided, assignments for weak students and, conversely, a more difficult option for strong students. For very weak students, games are being developed where you don’t need to think, but only need ingenuity. In this way, it is possible to attract more students to attend extracurricular mathematics classes and thereby contribute to the development of their cognitive interest.
Mathematical games should be designed taking into account the subject and its material. They should be varied. The variety of types of mathematical games will help increase the effectiveness of extracurricular work in mathematics and serve as an additional source of systematic and solid knowledge.
Thus, a mathematical game as a form of extracurricular work in mathematics has its own goals, objectives and functions. Compliance with all the requirements for mathematical games will allow you to achieve good results to attract more students to extracurricular work in mathematics, to develop their cognitive interest in it. Not only strong students will become more interested in the subject, but weak students will also begin to show their activity in learning.
Learning is easier, more fun and much more effective now thanks to new technologies and the development of online methods! Fun math games are a great way to turn difficult-to-learn material into fun. Mathematics games can make even a pure humanist not only understand, but also love counting - and all this without any effort! And most importantly, there is no coercion: the puzzles and virtual lessons are so interesting that even careless students will study with great pleasure.
Fun lessons
The first, and most obvious, form of online entertainment dedicated to studying is a virtual classroom in which a favorite character acts as a teacher.
Dasha the Pathfinder also in her programs likes to draw children’s attention to how important it is to know and be able to do everything, and now, standing at the board, she is more convincing than ever! Exercises on addition, subtraction, multiplication and division are accompanied by funny pictures depicting Dasha's adventures, and at the end the student will receive a grade corresponding to his knowledge. Be careful: to solve the examples, the student must already be familiar with negative numbers!
But Sofia the Wonderful Mathematics Game has prepared a test especially for girls in which you need to choose for each problem whether the solution is correct. Testing yourself is very simple: the answer counter, depending on the result, increases by one immediately after the choice is made. The test, which was compiled by the beautiful Barbie, is organized according to the same exact principle. Such mathematical games teach you not only to count without errors, but also to think quickly, because the time to answer is limited!
And if you need to train a certain mathematical operation - for example, to improve your addition or division skills - then you should go to the White Cat for help. The fluffy purr is a strict teacher. It requires you to have time to correctly solve the task and choose the required answer from the four presented to choose from in a limited time.
Numbers and life
Solving examples is good way learn to fold quickly, but it often seems that this activity is useless and will not be useful in the future. How could it not be useful if in our world you can’t take a single step without mathematics, and adventure games about it only prove this!
The crew involved in tank combat is forced to constantly think about complex problems, especially when it comes to shooting themselves or calculating how to avoid enemy shells. This process is presented in a simplified form by the game Mathematics on Tank, which you can play on this page. A wrong decision will lead to an explosion and death of personnel, and only a player who knows how to count will help save him from the inevitable!
In the games, the student will have to solve math problems to get candy, deal with bees, or deliver pizza to the right table. Without arithmetic, an arrow in a tournament will not reach its target, and space rockets will not take off. However, it’s useful to know that without solving special problems (only much more complex than those taught in second grade!) the rocket really won’t take off - but that’s a completely different story...
Logachev Alexey Evgenievich, mathematics teacher, Municipal Educational Institution Secondary Secondary School No. 7, Dmitrov [email protected]
Mathematical game as a form of extracurricular work in mathematics
Abstract. The article is devoted to the description of mathematical games as one of the forms of extracurricular work in mathematics. It provides an analysis of the concept of “mathematical game”; various classifications of games are given, the need to include mathematical games in the process of teaching mathematics is substantiated. The rules of the most popular of them are given. Key words: additional mathematical education for schoolchildren, mathematical competitions, problem solving, forms of education and development of schoolchildren, development of interest in the subject. Section: (01) pedagogy; history of pedagogy and education; theory and methods of teaching and education (by subject areas).
Mathematical games as a form of extracurricular activity play a huge role in the development of students’ cognitive interest. The game has a noticeable impact on the activities of students. The gaming motive is for them a reinforcement of the cognitive motive, promotes the activity of mental activity, increases concentration, perseverance, efficiency, interest, and creates conditions for the emergence of the joy of success, satisfaction, and a sense of teamwork. While playing, children are carried away and do not notice that they are learning. The gaming motive is equally effective for all categories of students, both strong and average, and weak. Children eagerly take part in mathematical games of various nature and form. A mathematical game is very different from a regular lesson, and therefore arouses the interest of most students and the desire to participate in it. It should also be noted that many forms of extracurricular work in mathematics may contain game elements, and vice versa, some forms of extracurricular work may be part of a mathematical game. The introduction of game elements into extracurricular activities destroys the intellectual passivity of students, which occurs in students after prolonged mental work in the classroom. The mathematical game is massive in scope and cognitive, active, creative in relation to the activities of students. The main goal of using a mathematical game is to develop sustainable cognitive interest in students through the variety of applications of mathematical games. A mathematical game is one of the forms of extracurricular work in mathematics. It is used in the system of extracurricular activities to develop children's interest in the subject, acquire new knowledge, abilities, skills, and deepen existing knowledge. Play, along with learning and work, is one of the main types of human activity, an amazing phenomenon of our existence. What is meant by the word game? The term “game” has many meanings; in widespread use, the boundaries between play and non-game are extremely blurred. As D.B. Elkonin and S.A. Shkakov rightly emphasized, the words “game” and “play” are used in a variety of senses: entertainment, performance of a piece of music or roles in a play. The main function of the game is recreation and entertainment. This property is precisely what distinguishes a game from a non-game. The phenomenon of children's play has been studied quite widely and comprehensively by researchers, both in domestic developments and abroad. A game, according to many scientific psychologists, is a type of developmental activity, a form of mastering social experience, one of complex human abilities. Russian psychologist A.N. Leontiev considers play to be the leading type of child activity, with the development of which the main changes in the psyche of children occur, preparing the transition to a new, highest degree of their development. By having fun and playing, a child finds himself and becomes aware of himself as a person. A game, in particular a mathematical one, is extremely informative and “tells” a lot about the child himself. It helps the child find himself in a group of comrades, in the whole society, humanity, in the universe. In pedagogy, games include a wide variety of actions and forms of children’s activities. Play is an activity, firstly, that is subjectively significant, enjoyable, independent and voluntary, secondly, it has an analogue in reality, but is distinguished by its non-utilitarian and literal reproduction, thirdly, it arises spontaneously or is created artificially for the development of any functions or qualities of the individual, consolidating achievements or relieving tension. A mandatory characteristic feature of all games is special emotional condition, against the background and with the participation of which they take place. A.S. Makarenko believed that “the game should constantly replenish knowledge, be a means of comprehensive development of the child, his abilities, evoke positive emotions, replenish the life of the children’s team with interesting content.” The following definition can be given. games. Game is an activity that imitates real life, having clear rules and limited duration. But, despite the differences in approaches to defining the essence of a game, its purpose, all researchers agree on one thing: a game, including a mathematical one, is a way of developing a personality, enriching it life experience. Therefore, the game is used as a means, form and method of teaching and education. There are many classifications and types of games. If we classify the game by subject area, we can single out a mathematical game. A mathematical game in the field of activity is, first of all, an intellectual game, that is, a game where success is achieved mainly due to a person’s thinking abilities, his mind, and his existing knowledge in mathematics. A mathematical game helps to consolidate and expand the knowledge and skills provided for in the school curriculum and skills.It is highly recommended for use in after-school activities and evenings. But these games should not be perceived by children as a process of deliberate learning, as this would destroy the very essence of the game. The nature of the game is such that in the absence of absolute voluntariness, it ceases to be a game. modern school a mathematical game is used in the following cases: as an independent technology for mastering a concept, topic or even a section of an academic subject; as an element of a broader technology; as a lesson or part of it; as a technology for extracurricular activities. A mathematical game included in a lesson, and simple gaming activities during the learning process have a noticeable impact on the activities of students. The gaming motive is for them a real reinforcement of the cognitive motive, helps to create additional conditions for the active mental activity of students, increases concentration, perseverance, performance, creates additional conditions for the emergence of the joy of success, satisfaction, and a sense of teamwork. A mathematical game, and indeed any game in educational process, has characteristic features. On the one hand, the conditional nature of the game, the presence of a plot or conditions, the presence of objects and actions used with the help of which the game problem is solved. On the other hand, freedom of choice, improvisation in external and internal activities allow game participants to receive new information, new knowledge, and be enriched with new sensory experience and experience of mental and practical activity. Through the game, the real feelings and thoughts of the game participants, their positive attitude, real actions, creativity, it is possible to successfully solve educational problems, namely, the formation of positive motivation in educational activities, a sense of success, interest, activity, the need for communication, the desire to achieve a better result, to surpass oneself, to improve one’s skills. Thus, among the forms of extracurricular work, we can highlight the mathematical game as the most vibrant and attractive for students. Games and gaming forms are included in extracurricular activities not only to entertain students, but also to interest them in mathematics, to arouse their desire to overcome difficulties, and to acquire new knowledge in the subject. A mathematical game successfully combines gaming and educational motives, and in such a way play activity There is a gradual transition from gaming motives to educational motives. Mathematical games are designed to solve the following problems. 1. Educational: to promote students’ strong assimilation of educational material; to help broaden students’ horizons, etc. 2. Developmental: to develop creative thinking in students; to promote the practical application of skills and skills acquired in lessons and extracurricular activities; to promote the development of imagination, fantasy, creative abilities, etc. 3. Educational: to contribute to the education of a self-developing and self-realizing personality; to foster moral views and beliefs; to promote independence and will in work, etc. Mathematical games perform various functions.1.During a mathematical game, gaming, educational and work activities occur simultaneously. Indeed, the game brings together what is not comparable in life and separates what is considered one. 2. A mathematical game requires the student to know the subject. After all, without knowing how to solve problems, solve, decipher and unravel, a student will not be able to participate in the game. 3. In games, students learn to plan their work, evaluate the results of not only someone else’s, but also their own activities, be smart when solving problems, and take a creative approach to any task , use and select the right material. 4. The results of the games show schoolchildren their level of preparedness and training. Mathematical games help in the self-improvement of students and, thereby, stimulate their cognitive activity, increasing interest in the subject. 5. While participating in mathematical games, students not only receive new information, but also gain experience in collecting the necessary information and applying it correctly. To game forms There are a number of requirements for extracurricular activities. Participants in a mathematical game must have certain knowledge requirements. In particular, to play you need to know. This requirement gives the game an educational character. The rules of the game must be such that students show a desire to participate in it. Therefore, games should be developed taking into account the age characteristics of children, the interests they show at a given age, their development and existing knowledge. Mathematical games should be developed taking into account the individual characteristics of students, taking into account different groups of students: weak, strong; active, passive, etc. They must be such that each type of student can express themselves in the game, show their abilities, capabilities, independence, perseverance, ingenuity, and experience a sense of satisfaction and success. When developing a game, it is necessary to provide easier versions of the game and tasks for weak students and, conversely, a more difficult version for strong students. For very weak students, games are being developed where you don’t need to think, but only need ingenuity. In this way, more students can be attracted to attend extracurricular activities in mathematics and thereby help them develop their cognitive interest. Math games should be designed keeping the subject and its material in mind. They should be varied. The variety of types of mathematical games will help increase the efficiency of extracurricular work in mathematics and serve as an additional source of systematic and lasting knowledge. Thus, a mathematical game as a form of extracurricular work in mathematics has its own goals, objectives and functions. Compliance with all the requirements for mathematical games will allow one to achieve good results in attracting more students to extracurricular work in mathematics and developing their cognitive interest in it. Not only strong students will show more interest in the subject, but also weak students will begin to show their activity in learning. The typification of mathematical games can be as follows: board games; mathematical mini-games; quizzes; station games; mathematical competitions; KVNs; travel games; mathematical labyrinths; Mathematical carousel"; battles. Some of the above types of games can be included in other, larger mathematical games, as one of their stages. Now let's look at a few examples.
Mathematical biathlon is a problem-solving competition (can be individual or team). The winner is the one who showed best time. Problems are solved at three firing lines (“Prone”, “Kneeling”, “Stand”). Sometimes they add a fourth milestone “On the run” to solve controversial issues; At this point, no additional ammunition is issued. At the beginning of the game, all participants are located at the first firing line. After the presenter’s signal, the participants receive 5 cartridge problems and begin to solve them. If the participant believes that all problems have been solved, then he presents their solutions to the judge. If any of the tasks are solved incorrectly, the participant receives additional tasks and cartridges (no more than three at each milestone). The next firing line is considered completed successfully (without penalty time) if the participant managed to close all five targets (each correctly solved task of this line closes one of his targets), perhaps with the help of additional tasks and cartridges. Otherwise, each uncovered target of the next firing line is punishable by 10 minutes of penalty time. The participant moves to the next firing line (receives the next series of five rounds of ammunition) immediately after closing five targets of the previous line or after penalty time has been accrued. The game ends for the participant if a) the time allotted for the competition has expired, or b) the participant has left the last shooting line. Result The participant's time is the sum of the time spent passing all the firing lines (net time) and the accrued penalty time. The participant’s net time is recorded by the judge at the moment of passing the last milestone. “Prone” 1. Place 4x12+18:6+3 brackets in the record so that you get the lowest possible result. 2. 15 identical balls can be folded into a triangle, but cannot be folded into a square; one ball is missing. How many balls, not exceeding 50, can be used to form both a triangle and a square? 3. How many zeros does the product 1·2·3·4·…·105 end with? 4. It takes 1 gram of paint to paint a 2x2x2 cube. How much paint will be needed to color a 6x6x6 cube? 5. What angle do the hour and minute hands make at twenty minutes past one? “From the knee” 1. First digit three-digit number is equal to 4. If you move it to the end, you get a number that is 3/4 of the original. Find the original number. 2. There are 20 gloves lying in disarray in a box: 5 pairs of black and 5 pairs of brown. What is the smallest number of gloves that must be taken without looking, so that from them one can surely choose two pairs of gloves of the same color? 3. If I want to buy 4 pencils, then 3 rubles will not be enough for me, and if I buy 3 pencils, then I will have 6 rubles left. How much money do I have? 4. An electrician must repair a garland of four light bulbs connected in series, one of which has burned out. It takes 10 seconds to unscrew any lamp from a garland, and 10 seconds to screw it in too. The time spent on other activities is negligible. In what minimum time can an electrician be guaranteed to repair a garland if he has a spare lamp? 5. Find two two-digit prime numbers obtained from each other by rearranging the digits, the difference of which perfect square. "Stand" 1. Average age eleven players on the football team are 22 years old. During the match, one of the players was sent off for roughness. The average age of the remaining players on the field became 21 years old. How old is the removed football player? 2. At exactly noon, a 15-meter pillar casts a 10-meter shadow. What is the height of the tree that is casting a 15-meter shadow at the same moment?3. What percentage are there more fingers than hands (Each hand has 5 fingers). 4. The equation XI = I is laid out from 7 matches. How can one rearrange one match in it so that it becomes true? 5. Four spies eat 4 secret packages in 4 minutes. How many spies do you need to invite so that they eat 20 secret packages in 8 minutes? “On the run”1. It is known that in January there are 4 Mondays and 4 Fridays. What day of the week was January 1st? 2. From the numbers 21, 19, 30, 25, 3, 12, 9, 15, 6, 27, choose three whose sum is 50. 3. Winnie the Pooh was given a barrel of honey weighing 7 kg on his birthday. When Winnie the Pooh ate half of the honey, the barrel with the remaining honey began to have a mass of 4 kg. How many kilograms of honey were originally in the barrel? 4. 15 trees were planted in one row at a distance of 5 m from each other. What is the distance between the outermost trees? 5. By what percentage will the area of the rectangle change if its length is increased by 20% and its width is decreased by 10%?
Mathematical game "Dots" "Dots" ("Cities") a game on checkered paper for two people. Opponents take turns placing one point at the intersection of the sheet lines (point) in a cell, each with its own color. The first move of each player takes place in the central part of the field. Subsequent moves can be to any point, as long as it is not in an encircled area. There is no option to skip a move. When creating a continuous line (vertically, horizontally, diagonally) closed line an area is formed. If there are enemy points inside it (and there may be points that are not occupied by anyone else’s points), then this is considered an encirclement area in which it is further prohibited for any player to place a point. If there are no opponent's points, then the area is free and points can be placed in it. When an opponent's point appears in a free area, the free area will be considered an encirclement area, provided that the opponent's point was not the final one in his environment. Points that fall into the environment area do not further participate in the formation of lines for the environment. Points placed on the edge of the field are not surrounded. The game ends when there are no empty spaces left, by mutual agreement of the players, or when one of the players refuses to make a move, stopping the game. If player A stops the game, then his opponent is given a fixed time, during which he will place points alone, further surrounding the free points of player A. After this time, the game ends automatically. Victory is determined by counting the surrounded points (the player who surrounded larger number opponent's points) or by mutual agreement of the players.
Links to sources 1. Gorev P.M. Developmental mathematics lessons in 56th grade high school// Concept. 2012. No. 10 (October). ART 12132. 0.6 p.l. URL: http://www.covenok.ru/koncept/2012/12132.htm.2. Elkonin D.B. Psychology of the game. M.: Pedagogy, 1978.304 p. 3. Sidenko A. Game approach to teaching // Public education. 2000. No. 8.S. 134136.4.Game in the pedagogical process. Novosibirsk, 1989.5. Makarenko A.S. About family upbringing. M.: Uchpedgiz, 1955.6. Minsky E.M. From play to knowledge. M: Education, 1979.192 p. 7. Dyshinsky E.A. Mathematical club toy library. 1972.142 p.8. Technology of gaming activity / L.A. Baykova, L.K. Terenkina, O.V. Eremkina. Ryazan: Publishing house RGPU, 1994. 120 p.
Alexey Logatchev, mathematics teacher of secondary school No. 7, [email protected] game as a form of extracurricular activities in mathematicsAbstract.The article describes the mathematical games as a form of extracurricular activities in mathematics. It provides an analysis of the concept of "mathematical game", there are different classifications of games rationale for the inclusion of mathematical games in the processof learning mathematics. Rules are the most popular ones.Key words: additional mathematics education students, math competitions, problem solving, learning and development form pupils develop interest in the subject.
Mathematical game as a form of extracurricular activities in mathematics within the framework of the implementation of the Federal State Educational Standard
Today, there are various forms of extracurricular activities in mathematics with students. These include:
Mathematical circle;
School math evening;
Mathematical Olympiad;
Math game;
School Math Print;
Mathematical excursion;
Mathematical abstracts and essays;
Mathematical Conference;
Extracurricular reading of mathematical literature, etc.
Obviously, the forms of conducting these classes and the techniques used in these classes must satisfy a number of requirements.
Firstly, they must differ from the forms of conducting lessons and other compulsory events. This is important because extracurricular activities are voluntary and usually take place after school. Therefore, in order to interest students in the subject and attract them to extracurricular activities, it is necessary to conduct it in an unusual form.
Secondly, these forms of extracurricular activities should be varied. After all, in order to maintain the interest of students, you need to constantly surprise them and diversify their activities.
Thirdly, the forms of extracurricular activities should be designed for different categories of students. Extracurricular activities should attract and be carried out not only for students interested in mathematics and gifted students, but for students who do not show interest in the subject. Perhaps, thanks to the correctly chosen form of extracurricular work, designed to interest and captivate students, such students will begin to pay more attention to mathematics.
And finally, fourthly, these forms should be selected taking into account the age characteristics of the children for whom it is carried out extracurricular activity .
Failure to comply with these basic requirements may result in few or no students attending extracurricular mathematics classes. Students study mathematics only in lessons, where they do not have the opportunity to experience and realize the attractive aspects of mathematics, its potential for improving mental abilities, and to fall in love with the subject. Therefore, when organizing extracurricular activities, it is important not only to think about its content, but also, of course, about the methodology and form.
Game forms of classes or mathematical games are activities imbued with game elements, competitions containing game situations.
A mathematical game as a form of extracurricular activity plays a huge role in the development of cognitive interest in students. The game has a noticeable impact on the activities of students. The gaming motive is for them a reinforcement of the cognitive motive, promotes the activity of mental activity, increases concentration, perseverance, efficiency, interest, and creates conditions for the emergence of the joy of success, satisfaction, and a sense of teamwork. While playing, children are carried away and do not notice that they are learning. The gaming motive is equally effective for all categories of students, both strong and average, and weak. Children eagerly take part in mathematical games of various nature and form. A mathematical game is very different from a regular lesson, and therefore arouses the interest of most students and the desire to participate in it. It should also be noted that many forms of extracurricular work in mathematics may contain game elements, and vice versa, some forms of extracurricular work may be part of a mathematical game. The introduction of game elements into extracurricular activities destroys the intellectual passivity of students, which occurs in students after prolonged mental work in class.
A mathematical game as a form of extracurricular work in mathematics is massive in scope and cognitive, active, and creative in relation to the activities of students.
The main goal of using a mathematical game is to develop sustainable cognitive interest in students through the variety of applications of mathematical games.
Thus, among the forms of extracurricular work, a mathematical game can be distinguished as the most vibrant and attractive for students. Games and gaming forms are included in extracurricular activities not only to entertain students, but also to interest them in mathematics, to arouse their desire to overcome difficulties, and to acquire new knowledge in the subject. A mathematical game successfully combines gaming and educational motives, and in such gaming activities there gradually occurs a transition from gaming motives to educational motives.
Mathematical games as a means of developing cognitive interest in mathematics
Organizational stages of a mathematical game
In order to conduct a mathematical game, and its results would be positive, it is necessary to carry out a series of consistent actions to organize it. There are a number of stages involved in organizing a mathematical game. Each stage, as part of a single whole, includes a certain logic of actions of the teacher and students.
First stage - Thispreliminary work . At this stage, the game itself is selected, goals are set, and a program for its implementation is developed. The choice of a game and its content primarily depends on which children it will be played for, their age, intellectual development, interests, communication levels, etc. The content of the game must correspond to the goals set; the timing of the game and its duration are also of great importance. At the same time, the place and time of the game are specified, and the necessary equipment is prepared. At this stage, the game is also offered to children. The proposal can be oral or written, and may include a brief and precise explanation of the rules and techniques of action. The main task of offering a mathematical game is to arouse students' interest in it.
Second phase – preparatory . Depending on a particular type of game, this stage may differ in time and content. But still they have common features. During the preparatory stage, students become familiar with the rules of the game, and a psychological mood for the game occurs. The teacher organizes the children. The preparatory stage of the game can take place either immediately before the game itself, or begin well in advance of the game itself. In this case, students are warned about what type of tasks will be in the game, what the rules of the game are, what they need to prepare (assemble a team, prepare homework, performance, etc.). If the game is based on any academic section of the subject of mathematics, then schoolchildren will be able to repeat it and come to the game prepared. Thanks to this stage, children become interested in the game in advance and participate in it with great pleasure, receiving positive emotions and a sense of satisfaction, which contributes to the development of their cognitive interest.
Third stage – this is immediatethe game itself , implementation of the program in activities, implementation of functions by each participant in the game. The content of this stage depends on what kind of game is being played.
Fourth stage - ThisThe final stage orstage of summing up the game . This stage is mandatory, because without it the game will be incomplete, incomplete, and will lose its meaning. As a rule, at this stage the winners are determined and they are awarded. It also sums up the general results of the game: how the game went, whether the students liked it, whether similar games should still be held, etc.
The presence of all these stages, their clear thoughtfulness makes the game holistic, complete, the game produces the greatest positive effect on students, the goal is achieved - to interest schoolchildren in mathematics.
Requirements for selecting tasks
Any mathematical game presupposes the presence of problems that the schoolchildren participating in the game must solve. What are the requirements for their selection? U different types games they are different.
If you takemath mini games , then the tasks included in them can be on any topic school curriculum, and unusual tasks, original, with a fascinating formulation. Most often they are of the same type, based on the application of formulas, rules, theorems, differing only in the level of complexity.
Questions for the quiz should have easily visible content, not cumbersome, not requiring any significant calculations or notes, and for the most part accessible to solution in the mind. Typical problems, usually solved in class, are not interesting for a quiz. In addition to problems, you can include various questions on mathematics in the quiz. There are usually 6-12 tasks and questions in a quiz; quizzes can be devoted to a single topic.
INgames by station , the tasks at each station should be of the same type; it is possible to use tasks not only on knowledge of the material of the subject of mathematics, but also tasks that do not require deep mathematical knowledge (for example, sing as many songs as possible, the text of which contains numbers). The set of tasks at each stage depends on the form in which it is carried out and what mini-game is used.
To the tasksmath competitions AndKVNov the following requirements are presented: they must be original, with a simple and captivating formulation; solving problems should not be cumbersome, require long calculations, and may involve several solutions; should be of different levels of complexity and contain material not only from the school mathematics curriculum.
Fortravel games easy problems are selected that can be solved by students, mainly based on program material, and do not require large calculations. You can use entertaining tasks.
If the game is planned to be played for weak students who do not show interest in mathematics, then it is best to choose tasks that do not require good knowledge of the subject, tasks that test intelligence, or simple tasks that are not at all difficult.
You can also include tasks of a historical nature in the games, on knowledge of some unusual facts from the history of mathematics, of practical significance.
INlabyrinths Typically, tasks are used to test knowledge of the material in any section of the school mathematics course. The difficulty of such tasks increases as you move through the maze: the closer you get to the end, the more more difficult task. It is possible to carry out a maze using problems of historical content and problems on knowledge of material not included in the school mathematics course. Tasks that require ingenuity and innovative thinking can also be used in mazes.
IN"mathematical carousel" Andmath battles Usually, problems of increased difficulty are used, which require deep knowledge of the material and innovative thinking, since quite a lot of time is allocated for solving them and mainly only strong students participate in such games. In some mathematical battles, the tasks may not be difficult, but sometimes they are simply entertaining, just to test your wits (for example, tasks for captains).
It is possible to use tasks to consolidate or deepen the material studied. Such tasks can attract strong students and arouse their interest. Children, trying to solve them, will strive to gain new knowledge that is not yet known to them.
Taking into account all the requirements, age and type of students, you can develop a game that will be interesting to all participants. During lessons, children solve quite a lot of problems, all of them are the same and not interesting. When they come to a math game, they will see that solving problems is not at all boring, they are not so complex or, on the contrary, monotonous, that problems can have unusual and interesting formulations, and no less interesting solutions. By solving problems of practical importance, they realize the full significance of mathematics as a science. In its turn game uniform, in which problem solving will take place, will make the whole event not at all educational, but entertaining character and the children will not notice that they are learning.
Requirements for conducting a mathematical game
Compliance with all the requirements for conducting a mathematical game contributes to the fact that the extracurricular mathematics event will be held at a high level, children will like it, and all the goals will be achieved.
During the game, the teacher should play a leading role in its implementation. . The teacher must maintain order during the game. Deviation from the rules, tolerance of minor pranks or discipline can ultimately lead to disruption of the lesson. A mathematical game will not only not be useful, it will cause harm.
The teacher is also the organizer of the game.The game must be clearly organized, all its stages highlighted, The success of the game depends on this. This requirement should be given the most serious importance and kept in mind when holding a game, especially a mass one. Keeping the stages clear will prevent the game from turning into a chaotic, incomprehensible sequence of actions. A clear organization of the game also assumes that all handouts and equipment necessary for carrying out one or another stage of the game will be used at the right time and there will be no technical delays in the game.
When playing a math gameit is important to ensure that schoolchildren remain interested in the game . In the absence of interest or its fading, in no casechildren should not be forced to play , since in this case it loses its voluntariness, teaching and developmental significance, the most valuable thing - its emotional beginning - falls out of the gaming activity. If interest in the game is lost, the teacher should take actions leading to a change in the situation. This can be achieved through emotional speech, a friendly environment, and support for those lagging behind.
Very importantplay expressively . If the teacher talks to the children dryly, indifferently, and monotonously, then the children are indifferent to the game and begin to get distracted. In such cases, it can be difficult to maintain their interest, to maintain the desire to listen, watch, and participate in the game. Often, this does not succeed at all, and then the children do not receive any benefit from the game, it only causes them fatigue. A negative attitude towards mathematical games and mathematics in general arises.
The teacher himself must be involved in the game to a certain extent. , be its participant, otherwise its leadership and influence will not be natural enough. He must initiate the creative work of students and skillfully introduce them to the game.
Students must understand the meaning and content of the entire game what is happening now and what to do next. All rules of the game must be explained to the participants. This happens mainly on preparatory stage. Mathematical content should be understandable to schoolchildren. All obstacles must be overcomethe proposed tasks must be solved by the students themselves , and not the teacher or his assistant. Otherwise, the game will not generate interest and will be played formally.
All participants in the game must actively participate in it , busy with business. Long waits for their turn to join the game reduce children's interest in this game.Easy and difficult competitions should alternate . In terms of content itmust be pedagogical and depend on the age and outlook of the participants . During the gameStudents must be mathematically competent in their reasoning , mathematical speech must be correct.
During the gamecontrol over the results must be ensured , on the part of the entire team of students or selected individuals. Accounting of results must be open, clear and fair. Errors in accounting for ambiguities in the accounting organization itself lead to unfair conclusions about the winners, and, consequently, to dissatisfaction among the participants in the game.
The game should not include even the slightest possibility of risk , threatening children's health . Availability necessary equipment , which must be safe, convenient, suitable and hygienic. It is very important thatDuring the game the dignity of the participants was not humiliated .
Anythe game must be successful . The result can be a win, a loss, a draw. Only a completed game, with a summary, can play a positive role and make a favorable impression on students.
Interesting game, which gives children pleasure, has a positive impact on subsequent mathematical games and their attendance. When conducting mathematical gamesfun and learning must be combined so that they do not interfere, but rather help each other.
The mathematical side of the game content should always be clearly highlighted . Only then will the game fulfill its role in the mathematical development of children and foster interest in mathematics.
These are all the basic requirements for playing a mathematical game.
CITY CLASSICAL LYCEUM
ABSTRACT
Math games and puzzles
Prepared by:
Petrov A. A.,
10B class (physics-mat)
Kemerovo - 1999
Math games and puzzles are very popular, as are all games. And the more complex game is not always the more interesting. Often millions of people play the simplest games with undying interest, and it is these games that are most valued, they are the ones that go down in the history of mathematics and glorify their creators.
The closest thing to mathematics are puzzles, but many puzzles were formed from games that once existed (and some still exist). Most of these fundamental games were invented by ancient Greek mathematicians.
Recently, attention has been paid to mathematical games mainly for finding winning strategies, which has been greatly influenced by the spread of programming: creating an algorithm by which a computer could play a game is often more difficult and more interesting than learning how to play it yourself, while You delve deeper into the essence of the game, after which you can win almost anyone at it.
Games
The simplest mathematical games are often used as tasks in which you need to find a winning strategy, or transfer one situation to another. Sometimes the problems are very simple when they are solved by known methods such as invariant and coloring, but there are also very simple, but still unsolved problems associated with mathematical games.
An example would be popular game Tic-tac-toe on an endless field (renju). As is known, with the correct strategy of both players it is infinite, but no one knows the winning strategy. Currently, many algorithms for this game have been invented, based primarily on enumerating various options and analyzing the game for the next few moves, which are very close to a winning strategy, but only when implemented on a computer - a person practically cannot follow them. There are simple techniques of this game that players use, but attentiveness is most often decisive.
Game of nim and other similar games
There are several games in which two players A and B, guided by certain rules, take turns taking out a certain number of chips from one or more piles - the one who takes the last chip wins. The simplest such game is a game with one pile of chips, and to make a move in it means to take any number of chips from the pile from 1 to m inclusive. Many similar games can be studied using the Sprague-Grundy number G(C). An empty position O containing no tokens corresponds to G(O)=0. Let us denote a combination of heaps consisting respectively of x, y, ... chips as C=(x, y, ...) and assume that valid moves transform C into other combinations: D, E, ... Then G(C) is the smallest non-negative number, different from G(D),G(E), ... This allows us to determine G(C) by induction for any combination of C allowed by the rules of the game. So, in the mentioned problem G(x)=x mod (m+1).
If G(C)>0, then the player making the next move, say player A, can secure a win if he manages to move to a “safe” combination S with G(S)=0. Indeed, by the definition of G(S) in this case, either S is an empty position, and then A has already won, or B, with the next move, must move to the “dangerous” position U with G(U)>0 - and then everything repeats again. Such a game ends with A's victory after a finite number of moves.
Similar games include him. There is an arbitrary number of piles of chips, and players take turns choosing one pile and removing any number of chips from it (but at least one is required).
A more general case is represented by the game Moore, which can also be called k-nim. Its rules are the same as in a regular nim (1-nim), but here you are allowed to take chips from any number of piles not exceeding k.
Another similar game - Skittles. In it, the chips are laid out in a row, and with each move, one chip or two adjacent ones are removed. In this case, the series can be split into two smaller series. The one who takes the last chip wins. A generalized variation of this game is known as the game Wythoff .
There is an interesting variation of the game called nim “star neem”. It is quite simple, but the strategy in it is not immediately visible. This game is played on the star-shaped figure shown in Fig. 1, left. Place one chip on each of the nine points of the star. Players A and B take turns taking turns, removing at each move either one or two pieces connected by a straight line. The one who removes the last chip wins.
Player B, when playing star nim, has a winning strategy that uses the symmetry of the game board (in general, the winning strategies of many mathematical games are based on this). Let's imagine that the straight segments connecting the vertices of the star are threads. Then the entire configuration can be turned into a circle, topologically equivalent to a thread star. If A removes one token from the circle, then B removes two tokens from the opposite section of the circle. If A takes two chips, then B removes one chip from the opposite section of the circle. In both cases, two groups of three tiles remain on the circle. Whatever tile (or tiles) A takes from one group, B takes the corresponding tile (or tiles) from the other group. It is clear that the last chip will go to player B.
Other math games
In the late 60s, J. Leuthwaite from the Scottish city of Thurso invented a wonderful game with a skillfully hidden strategy of “paired moves”, providing the second player with a certain win. On a board measuring 5 * 5 square cells, 13 black and 12 white chips are placed in a checkerboard pattern, after which any of the black chips, for example, standing on the central field, is removed (Fig. 2, left).
Player A moves with white chips, player B with black ones. Moves are made vertically and horizontally. The loser is the player who is the first to fail to make the next move. If the board is colored like a chessboard, it will become clear that each piece moves from its field to a field of a different color and that not a single piece can be made to move twice. Therefore, the game for each player cannot last more than 12 moves. But it can end even earlier with a win for any player, unless B follows a rational strategy.
A rational strategy for player B is to mentally imagine the entire matrix (except for the empty square) being covered with twelve non-overlapping dominoes. How exactly they are laid out on the board does not matter. In Fig. 2, on the right, shows one way to cover a board with dominoes. Whatever move A makes, B simply moves to the domino he just left. A. With such a strategy, B always has a move after A's next move, so Obviously wins in 12 or fewer moves.
Leuthwaite's game can be played not only with pieces on the board, but also with square tiles or cubes moved inside a flat box, on the bottom of which a matrix is drawn. Let us now assume that the rules of the game are amended to allow any player at any time to move any number (from 1 to 4) of chips standing on the same horizontal or vertical, if the first and last chips in the horizontal or vertical of his choice are “his” color. Here is an excellent example of how a trivial (at first glance) change in the rule leads to a sharp complication of game analysis. Leithwaite was unable to find a winning strategy for either player in this variation of the game.
Most of the games we reviewed had a winning strategy, but this does not mean that almost all such games have one. There are many games in which a winning strategy has not yet been invented, and there are many that do not have one at all.
Puzzles
There are a variety of mathematical puzzles: rotational (Rubik's cube), “Magic rings”, “Games with a hole” (tag), lattice and many others. We will look at just a few of them.
Rotation puzzles
Rotational puzzles are called puzzles, the essence of which is to rotate the rows of cubes (and not only cubes) of which they are composed.
The most famous puzzle of our time - the Rubik's cube - began its victorious march around the world in 1978, when mathematicians first became acquainted with it at the International Mathematical Congress in Helsinki. Only a few cubes were taken away by mathematicians from the congress, but this became the initial impetus for the avalanche of spreading the toy throughout the world.
Almost everyone can solve one side of a Rubik's Cube, but solving it completely often requires serious thought. When you assemble the first face (or first layer), you don't have to worry about the rest, but when you have to swap the last few cubes, it's very easy to mess everything up and start over.
The Rubik's Cube is a rotational puzzle. distinctive feature which is that it’s easy to confuse them, but not everyone knows how to quickly assemble them. When we get confused, we act haphazardly and try to ruin everything at once; when assembling, it is too difficult to cover the whole picture at once; it is more convenient for us to move methodically, step by step, first installing one piece, adjusting the second to it, etc. As we build the correct picture our freedom of action is limited, because what we have achieved must be preserved in subsequent steps. And closer to the end of the assembly, the next advances are no longer possible without sacrifices - we are forced to temporarily give up what we have won in order to return it at a profit. Here specially designed operations are already required, we can call them “local” or “minimal”, which make the smallest changes in the arrangement of the puzzle elements, for example, rearranging two or three elements or turning them over. Moreover, “minimal” does not mean “small” - they usually consist of quite large number moves.
Let's consider an algorithm for solving rotational puzzles using the example of a Rubik's cube.
Formulas for operations in the “Rubik’s cube”
When using “minimal” operations, a natural question arises: how to systematize or formulate them so that they are convenient to use when solving a cube. First of all, before using one or another already developed operation, you should somehow designate the faces of the cube relative to which they should be carried out. Their standard names are: front, rear, left, right, top, bottom. And the designations are respectively: F, T, L, P, V, N. Any formula of operations can be performed by rotating the side or central faces of the cube. One rotation of a face clockwise is designated in the same way as the face itself (F, T, etc.). If the edge is turned counterclockwise, then the sign '(Ф', Т', etc.) is assigned to the designation of this action. It is clear that two clockwise turns are identical to two counter-clockwise turns, and therefore they are designated the same way: sign 2. (F 2, T 2, etc.). Using this notation system, only rotations of the side faces can be formulated; for the central ones, the notation is shown in Figure 3.
Below is a list of the most common “minimal” operations used when solving a Rubik's cube. It should be noted that these are only universal combinations, and to create a more advanced algorithm for solving the cube, it is necessary to develop more “global” operations, which are quite difficult for a person to remember, but in general reduce the number of actions required to solve the cube from each specific position.
First layer
Operation “ladder” (elevator) 2:NLN 'L ’
Two ladders 1:NLN'L'N'F'NF
Only two combinations are performed with the top edge rotated between them:
(PSN) 4
(F 'PFP ’) 2
(PF 'P 'F) 2
“Games with a hole”
Before the invention of the Rubik's cube, for many people, their acquaintance with puzzles began with “tag” - this is how the famous game “15” is often called.
The history of games with a hole begins with tag - puzzles in which chips move around the playing field due to the fact that one of the places on the field is free. The “tags” have many relatives, which form a whole section of these puzzles.
The game “15” was invented in the 70s of the 19th century by the famous American puzzle inventor Samuel Loyd. The time of appearance of his toy and the well-known Rubik's cube is separated by exactly one hundred years. It is curious that both inventors were the same age when they came up with their famous puzzles - a little over thirty. Before “tag”, no other puzzle had enjoyed such success.
The great Mark Twain, a contemporary of Loyd's and a witness to the general excitement around the game of 15, included in his satirical story "The American Challenger" an account of a message purportedly transmitted by the Associated Press, which stated that "it has become fashionable in the last few weeks a new puzzle toy... and that from the Atlantic to the Pacific the entire population of the United States has stopped working and is engaged only in this toy; that in connection with this all business life in the country has come to a standstill, for judges, lawyers, burglars, priests, thieves, traders, workers, murderers, women, children, infants - in a word, everyone is busy from morning to night with one single highly intellectual and complicated matter... that fun and joy left the people - they were replaced by concern, thoughtfulness, anxiety, everyone's faces were long, despair and wrinkles appeared on them - traces of the years lived and the difficulties experienced, and with them sadder signs , indicating mental inferiority and incipient insanity; that in eight cities factories are working day and night, and yet the demand for the puzzle has not yet been met.”
Soon after its birth, the box with the numbers 15 on the lid crossed the ocean, quickly spread to all European countries and acquired a new name “taken”. The inventor was lucky enough to find that elusive measure of complexity when the puzzle was solved without difficulty by almost everyone and at the same time required a certain intelligence, thanks to which everyone could enjoy the consciousness of their high intellectual level.
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By placing this ad, Loyd knew that he was not risking anything, since he was offering an impossible task. This problem also played a cruel joke on the inventor when he tried to patent his game - he was told that he could not patent a game that did not have a solution.
The secret of the game “15”
It is not always possible to transfer a puzzle from one state to another - such transitions are prohibited in which certain conservation laws are violated. There is such a law in the game “15”. To explain it, let's mentally fill the empty space with a chip with the number 16. Then each move - a shift of a chip - will consist in the fact that this chip changes places with chip 16. An operation in which some two chips (not necessarily adjacent!) change in some places, let's call it that - exchange; mathematical term for such operations - transposition. It is obvious that from any arrangement of 16 chips it is possible to obtain the correct position in no more than 15 exchanges - let’s denote it S 0 - and in general any other arrangement. During these exchanges, it is not prohibited to remove chips from the box. For example, you can first put chip 1 in its place, exchanging it with the chip that occupies this place, then put chip 2 in its place in the same way, etc. We will exchange chips 15 and 16 last - and both will be placed correctly at once . Of course, it is possible that along the way some chips will automatically fall into place and you won’t have to touch them, but the number of exchanges will be less than 15. You can arrange the chips using the same system, but in a different order, say 16, 15 , 14, .... or completely different, and then the number of exchanges may be different. However, no matter how you choose a sequence of exchanges that transforms one given arrangement of chips into another, the parity of the number of exchanges in this sequence will always be the same.
We will prove this very important and non-obvious point below. It allows us to give the following definition: the arrangement is called even, if it can be turned into a correct position by an even number of exchanges, and odd otherwise. In mathematics they usually say not “arrangement”, but “permutation”; We will return to this later. The correct arrangement S 0 is always even, and the Lloyd trap L odd. But why don't they translate into each other?
As mentioned above, each move in the game “15” can be considered as an exchange of a chip with one of the neighboring ones. Consequently, with each move, the parity of the arrangement of 16 chips changes: if before the move the arrangement could be arranged in N exchanges, then after it - in N+1 exchanges (taking this move back), and the numbers N and N+1 have different parities. In both arrangements of the classic Loyd problem, the hole (or piece 16) is located in the same way. If we were able to translate one arrangement into another, then chip 16 would have to make as many moves up as down, and as many moves to the right as to the left, otherwise it would not come back. So we would do even number moves, and since the parity of the arrangement changes with each move, it would be the same at the beginning and at the end. But the positions S 0 and L, as we have seen, have different parities.
We have looked at only a small part of the wonderful puzzles that mathematicians of different times have come up with, but if they ever invent a more popular puzzle than, for example, the game “15”, then the famous Rubik’s cube is probably the most famous!
Bibliography
1. Ya. I. Perelman “Entertaining mathematics”
2. Martin Gardner "Time Travel". – Moscow, “Mir”, 1990
3. W. Ball, G. Coxeter “Mathematical essays and entertainment.” – Moscow, “Mir”, 1986
4. V. N. Dubrovsky, A. T. Kalinin “Mathematical puzzles.” – Moscow, “Knowledge”, 1990
5. “Mathematical Flower Garden” (compiler and editor D. A. Klarner). – Moscow, “Mir”, 1983
