When people talk about a person who has beliefs, this is perceived as a positive characteristic. But what if our beliefs and traditional view of events have back side that prevents us from clearly understanding the processes taking place in the world?

Konstantin Smygin, founder of MakeRight book hacking service, talks about Stephen D. Levitt and Stephen J. Dubner's acclaimed book "Freaky Thought".

“Thinking like a freak” means finding out-of-the-box solutions, avoiding common psychological traps, and looking at current events from a perspective that is usually inaccessible to a blinkered mind.

Few people are able to "think like a freak", and here's why:

1. Studies show that even the smartest people are looking for evidence in the world around them to support their point of view, and are not ready to accept new information that runs counter to their ideas about the world. Our consciousness distorts and adjusts the surrounding reality.
2. In addition, people are greatly influenced by their environment, the environment in which they live. It is easier for a person as a social animal to agree with the existing order of things than to question it, causing the wrath of fellow tribesmen. The authors call this phenomenon “transferring the thought process to someone else.”
3. The third reason also follows from the peculiarities of human nature: “people have no time to think about how they think. What’s more, they don’t spend much time thinking at all.”

In their book, Levitt and Dubner argue for the need for more people to think "like freaks." That is, more productive, inventive and rational.

The power of "I don't know" and the sickness of the experts

Most people find it shameful to show their ignorance and appear ignorant. In their opinion, it is better to try to look like an expert in something that you do not understand at all. In this situation, electronic methods of communication are only at hand. On the other hand, the unwillingness to admit one's ignorance and incompetence means that the human mind is closed to learning and real knowledge.

Recent studies (eg by Philip Tetlock) have shown that experts predict the future only slightly more accurately than "arbitrarily choosing a dart-throwing chimpanzee." The accuracy of their forecasts is only about 47.4%. This is equivalent to predicting at random, with the only difference that it will cost you nothing, while forecasters will charge a lot of money for their services.

Interestingly, researcher Philip Tetlock characterizes the worst predictors as overconfident - even if their forecast does not come true.

Nevertheless, people continue to listen to forecasts or give in to the temptation to predict. Why? This is due to the fact that (given the extremely intricate causal relationships of our world) few people remember failed predictions. But if the prediction comes true, then the person who made it can gain the fame of a prophet or receive a large reward.

How to confess ignorance?

The authors urge not to be shy to admit their ignorance. In order not to put yourself in a stupid position, say something that is unpleasant for you and end with the phrase: "... but maybe I can find out." Most likely, people will respond positively to such frankness, especially if you return to them with the necessary information.

Go to the root!

Causal relationships are complex, confusing, and not obvious. However, most people continue to think and explain the causes of certain phenomena according to patterns formed for them.

To see real reasons events, you need to go beyond the prevailing ideas.

1. What is the cause of poverty and hunger? On the one hand, it is the lack of money and food. On the other hand, food supplies and material aid starving countries do not change anything. The problem is in an unworkable economy, when those in power think, first of all, about satisfying their own needs.
2. Why are there so many wars in Africa? Of course, there are many reasons, but the main one is in the colonial division of Africa by Europeans in the 19th century. Europeans divided territories simply by looking at a map (which is why the borders between African countries are often perfectly straight lines). As a result, friendly African tribes could end up on opposite sides of the border, and warring tribes could be in the same country.
3. Why is heart disease more common among blacks in the US? It was found that slave owners selected slaves for the salinity of their sweat. Since salt retains moisture, a slave with saltier sweat was more likely to survive during the exhausting sea ​​travel to the New World and not die of dehydration. Salt sensitivity is hereditary, and studies show that African Americans are 50% more likely to have hypertension than whites (and blacks in other countries), and as a result, a higher risk of heart problems.
4. Until the 1980s, it was believed that stomach ulcers were caused by stress and spicy food. Barry Marshall proved that the cause of ulcers (which can later lead to cancer) is the bacterium Helicobacter pylori. To overcome the resistance of the medical community, which did not take Marshall's hypothesis seriously, he performed a heroic deed - he drank a liquid containing bacteria, after which he developed symptoms of gastritis.

Think like a child

Freakout often involves the ability to think like a child. The authors note that this is one of better ways search non-standard solutions and generating ideas. Children are curious and ask questions that adults are afraid to ask. Being open-minded is a huge advantage for someone who wants to get to the bottom of things.

Because big problems usually consist of many small tasks, it is quite reasonable to start by turning your attention to one of them. The advantage here is also the fact that a small task is easier to translate into reality.

The main life principle of a freak

If you want to think like a freak, then the authors advise you to always use real incentives that act on people.

There are many incentives - monetary, social, moral. The ability to recognize and apply them is a whole science, because different stimuli act in certain cases and with certain people.

Determining the incentive that will affect a particular person is not easy. People usually do not admit to what they can be addicted to, and the authors do not recommend taking anyone's word on this issue.

There is another effect, the so-called cobra effect. It is connected with the fact that often manifestations of generosity cause a backlash. It got its name after the situation in which the English colonists found themselves in India. Deciding to reduce the snake population in Delhi, the colonists announced a monetary reward for each killed cobra. The result was the opposite - the Indians began to breed and grow cobras, receiving money for them, and when the awards were canceled, all cobras were released into the wild.

In addition, avoid incentives that look like thinly disguised attempts at manipulation. People feel good about them.

The use of incentives is useful in another way as well. Often the one who cheats or lies reacts to them in a particular way. Based on this, the authors derive a principle that they call "teach your garden to weed itself." The point is that you need to foresee the situation in which a person with evil intentions will reveal himself.

As an example, the authors give known history about King Solomon. Once two women with a child came to his court, each of whom claimed that the child was hers. Solomon announced to them that he had decided to cut the child and give each mother a half. This helped to figure out the real mother, who said in horror that it would be better for her child to get another, but she would live. The impostor agreed to kill the child.

How do you convince people who don't want to be convinced?

It is extremely stupid to pass off your proposal as ideal - it always alarms people, simply because this does not happen. So that a person does not feel a catch, tell yourself about weak points your offer.

But to convince someone is a difficult task due to psychological effects. If a person's beliefs (which often happens) are based on stereotypes and herd thinking, convincing him using logic and common sense is a waste of time. It is better to work not on the logic of proofs, but on their effectiveness.

Another trick is to acknowledge strengths arguments of the opponent, which will help to give significance to their own arguments.

In addition, in no case should you cross the line, hang labels and slide down to insults. This will immediately deprive you of all positions. Best Strategy persuasion - telling stories. Stories grab attention and take you to another level of understanding, making your arguments and ideas better understood.

Retreat Benefits

It is important not to fall into a common mental trap - if we have already invested time and money in something, then we continue to invest money and time in these projects, even when they do not bring anything useful. This is called the sunk cost fallacy. Thus, by stepping back from the unprofitable Concorde development project in time, the governments of France and Great Britain would be able to save their budgets from billions of dollars in expenses.

We are afraid to stop because it will be an admission of our mistake. As a result, we are forced to continue a hopeless business. But, as noted earlier, thinking like a freak involves not being afraid to admit your own mistakes.

An effective way to avoid sunk cost mistakes is to remind yourself of them. Always look for alternative ways and solutions to a given situation.

Ask yourself: “What would I do now, using the same time, money and resources?”.

Russian philologist Dmitry Nikolaevich Ushakov in his explanatory dictionary gives such a definition of the concept of "method" - a way, a way, a technique theoretical research or the practical implementation of something (D. N. Ushakov, 2000).

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

In mathematics, there is no general rules, allowing to solve any non-standard task, since such tasks are to some extent unique. A non-standard task in most cases is perceived as "a challenge to the intellect, and gives rise to the need to realize oneself in overcoming obstacles, in developing creative abilities" .

Consider several methods for solving non-standard problems:

• · algebraic;
• · arithmetic;
• enumeration method;
• method of reasoning;
• practical;
• the method of guessing.

Algebraic method problem solving develops creative abilities, the ability to generalize, forms abstract thinking and has such advantages as brevity of writing and reasoning when drawing up equations, saves time.

In order to solve the problem by the algebraic method, it is necessary:

• to analyze the problem in order to choose the main unknown and identify the relationship between the quantities, as well as expressing these dependencies in mathematical language in the form of two algebraic expressions;
• find the basis for connecting these expressions with the sign "=" and make an equation;
• find solutions to the resulting equation, organize a check of the solution of the equation.

All these stages of solving the problem are logically interconnected. For example, we mention the search for a basis for connecting two algebraic expressions with an equal sign as a special stage, but it is clear that at the previous stage, these expressions are not formed arbitrarily, but taking into account the possibility of connecting them with the “=” sign.

Both the identification of dependencies between quantities and the translation of these dependencies into mathematical language require intense analytical and synthetic mental activity. Success in this activity depends, in particular, on whether students know what relationships these quantities can have in general, and whether they understand the real meaning of these relationships (for example, relationships expressed in the terms “later by ...”, “older by ... times " etc.). Further, an understanding is required of what kind of mathematical action or property of the action or what connection (dependence) between the components and the result of the action can describe one or another specific relationship.

Let us give an example of solving a non-standard problem by the algebraic method.

A task. The fisherman caught a fish. When asked: “What is its mass?”, He replied: “The mass of the tail is 1 kg, the mass of the head is the same as the mass of the tail and half of the body. And the mass of the body is the same as the mass of the head and tail together. What is the mass of the fish?

Let x kg be the mass of the body; then (1+1/2x) kg is the mass of the head. Since, by condition, the mass of the body is equal to the sum of the masses of the head and tail, we compose and solve the equation:

x = 1 + 1/2x + 1,

4 kg is the mass of the body, then 1+1/2 4=3 (kg) is the mass of the head and 3+4+1=8 (kg) is the mass of the whole fish;

Answer: 8 kg.

Arithmetic Method solutions also require a lot of mental stress, which has a positive effect on the development of mental abilities, mathematical intuition, on the formation of the ability to foresee a real life situation.

Consider an example of solving a non-standard problem by an arithmetic method:

A task. Two fishermen were asked, "How many fish are in your baskets?"

“In my basket is half of what he has in the basket, and 10 more,” the first answered. “And I have as many in my basket as he has, and even 20,” the second one calculated. We counted, and now you count.

Let's build a diagram for the problem. Let the first segment of the diagram denote the number of fish the first fisherman has. The second segment denotes the number of fish from the second fisherman.

Due to the fact that a modern person needs to have an idea about the main methods of data analysis and probabilistic patterns that play an important role in science, technology and economics, elements of combinatorics, probability theory and mathematical statistics are introduced into the school mathematics course, which are convenient to understand using enumeration method.

The inclusion of combinatorial problems in the course of mathematics has a positive impact on the development of schoolchildren. “Targeted learning to solve combinatorial problems contributes to the development of such a quality of mathematical thinking as variability. Under the variability of thinking, we mean the direction of the student's mental activity to search for various solutions to the problem in the case when there are no special instructions for this.

Combinatorial problems can be solved by various methods. Conventionally, these methods can be divided into "formal" and "informal". With the “formal” solution method, you need to determine the nature of the choice, select the appropriate formula or combinatorial rule (there are sum and product rules), substitute numbers and calculate the result. The result is the amount options, but the variants themselves are not formed in this case.

With the “informal” method of solving, the very process of drawing up comes to the fore. various options. And the main thing is not how much, but what options can be obtained. Such methods include enumeration method. This method is also available junior schoolchildren, and allows to accumulate experience in the practical solution of combinatorial problems, which serves as the basis for the introduction of combinatorial principles and formulas in the future. In addition, in life a person has to not only determine the number of possible options, but also directly compose all these options, and, having mastered the methods of systematic enumeration, this can be done more rationally.

Tasks are divided into three groups according to the complexity of enumeration:

• one . Tasks in which you need to make a complete enumeration of all possible options.
• 2. Tasks in which it is impractical to use the full enumeration technique and it is necessary to immediately exclude some options without considering them (that is, to carry out an abbreviated enumeration).
• 3. Tasks in which the enumeration operation is performed several times and in relation to various kinds of objects.

Here are the relevant examples of tasks:

A task. Placing the signs "+" and "-" between the given numbers 9 ... 2 ... 4, make up all possible expressions.

There is a full list of options:

• a) two characters in the expression can be the same, then we get:
  • 9 + 2 + 4 or 9 - 2 - 4;
• b) two signs can be different, then we get:
  • 9 + 2 - 4 or 9 - 2 + 4.

A task. The teacher says that he drew 4 figures in a row: large and small squares, large and small circles so that the circle is in the first place and the figures of the same shape do not stand side by side, and invites the students to guess the sequence in which these figures are arranged.

There are 24 different arrangements of these figures in total. And it is not advisable to compose them all, and then choose those that correspond to this condition, therefore, an abbreviated enumeration is carried out.

A large circle can be in the first place, then a small one can only be in third place, while large and small squares can be placed in two ways - in second and fourth place.

A similar reasoning is carried out if the first place is a small circle, and two options are also compiled.

A task. Three partners of the same firm keep securities in a safe with 3 locks. The companions want to distribute the keys to the locks among themselves so that the safe can only be opened in the presence of at least two companions, but not one. How can I do that?

First, all possible cases of key distribution are enumerated. Each companion can be given one key, or two different keys, or three.

Let's assume that each companion has three different keys. Then the safe can be opened by one companion, and this does not meet the condition.

Let's assume that each companion has one key. Then if two of them come, they won't be able to open the safe.

Let's give each companion two different keys. The first - 1 and 2 keys, the second - 1 and 3 keys, the third - 2 and 3 keys. Let's check when any two companions come to see if they can open the safe.

The first and second companions can come, they will have all the keys (1 and 2, 1 and 3). The first and third companions can come, they will also have all the keys (1 and 2, 2 and 3). Finally, the second and third companions can come, they will also have all the keys (1 and 3, 2 and 3).

Thus, to find the answer in this problem, you need to perform the iteration operation several times.

When selecting combinatorial problems, one should pay attention to the subject and form of presentation of these problems. It is desirable that the tasks do not look artificial, but are understandable and interesting to children, cause them to positive emotions. You can use practical material from life to draw up tasks.

There are other problems that can be solved by enumeration.

As an example, let's solve the problem: “Marquis Karabas was 31 years old, and his young energetic Puss in Boots was 3 years old, when the events known from the fairy tale took place. How many years have passed since then, if now the Cat is three times younger than its owner? The enumeration of options is represented by a table.

Age of the Marquis of Carabas and Puss in Boots

14 - 3 = 11 (years)

Answer: 11 years have passed.

At the same time, the student, as it were, experiments, observes, compares facts and, on the basis of particular conclusions, makes certain general conclusions. In the process of these observations, his real-practical experience is enriched. This is precisely the practical value of enumeration problems. In this case, the word "enumeration" is used in the sense of analyzing all possible cases that satisfy the conditions of the problem, showing that there can be no other solutions.

This problem can also be solved by an algebraic method.

Let the Cat be x years old, then the Marquis is 3x, based on the condition of the problem, we will compose the equation:

• 3x - x \u003d 28,
• 2x = 28,

The cat is now 14 years old, then 14 - 3 = 11 (years) passed.

Answer: 11 years have passed.

reasoning method can be used to solve mathematical sophisms.

The mistakes made in sophism usually come down to the following: performing "forbidden" actions, using erroneous drawings, incorrect word usage, inaccurate wording, "illegal" generalizations, incorrect applications of theorems.

To reveal sophism means to point out an error in reasoning, based on which the external appearance of proof was created.

Analysis of sophisms, first of all, develops logical thinking, instills the skills of correct thinking. To detect an error in sophism means to recognize it, and the awareness of an error prevents it from being repeated in other mathematical reasoning. In addition to the criticality of mathematical thinking, this type of non-standard tasks reveals the flexibility of thinking. Will the student be able to “break out of the grip” of this path, which at first glance is strictly logical, break the chain of inferences at the very link that is erroneous and makes all further reasoning erroneous?

The analysis of sophisms also helps the conscious assimilation of the material being studied, develops observation and a critical attitude towards what is being studied.

a) Here, for example, is a sophism with an incorrect application of the theorem.

Let us prove that 2 2 = 5.

Let's take the following obvious equality as the initial ratio: 4: 4 = 5: 5 (1)

We take out of brackets the common factor in the left and right parts, we get:

4 (1: 1) = 5 (1: 1) (2)

The numbers in brackets are equal, so 4 = 5 or 2 2 = 5.

In the reasoning, when passing from equality (1) to equality (2), an illusion of plausibility is created on the basis of a false analogy with distributive property multiplication with respect to addition.

b) Sophism using "illegal" generalizations.

There are two families - Ivanovs and Petrovs. Each consists of 3 people - father, mother and son. Ivanov's father does not know Petrov's father. Ivanov's mother does not know Petrova's mother. The only son of the Ivanovs does not know the only son of the Petrovs. Conclusion: not a single member of the Ivanov family knows a single member of the Petrov family. Is this true?

If a member of the Ivanov family does not know a member of the Petrov family equal in marital status, this does not mean that he does not know the whole family. For example, Ivanov's father may know Petrov's mother and son.

The reasoning method can also be used to solve logical tasks. Logical tasks are usually understood as those tasks that are solved using only logical operations. Sometimes their solution requires lengthy reasoning, the necessary direction of which cannot be foreseen in advance.

A task. They say that Tortila gave the golden key to Pinocchio not as simply as A. N. Tolstoy said, but in a completely different way. She brought out three boxes: red, blue and green. On the red box it was written: “Here lies a golden key”, and on the blue one - “The green box is empty”, and on the green one - “Here sits a snake”. Tortila read the inscriptions and said: “Indeed, there is a golden key in one box, a snake in the other, and the third is empty, but all the inscriptions are wrong. If you guess which box contains the golden key, it's yours." Where is the golden key?

Since all the inscriptions on the boxes are incorrect, the red box does not contain a golden key, the green box is not empty and there is no snake in it, which means that the key is in the green box, the snake is in the red one, and the blue one is empty.

When solving logical problems, logical thinking is activated, and this is the ability to deduce consequences from premises, which is essential for the successful mastery of mathematics.

A rebus is a riddle, but a riddle is not quite an ordinary one. Words and numbers in mathematical puzzles are depicted using drawings, asterisks, numbers and various signs. To read what is encrypted in the rebus, you must correctly name all the objects depicted and understand which sign depicts what. People used puzzles even when they couldn't write. They composed their letters from objects. For example, the leaders of one tribe once sent a bird, a mouse, a frog and five arrows instead of a letter to their neighbors. This meant: “Can you fly like birds and hide in the ground like mice, jump through swamps like frogs? If you don't know how, then don't try to fight us. We will bombard you with arrows as soon as you enter our country.”

Judging by the first letter of the sum 1), D = 1 or 2.

Suppose that D = 1. Then, Y? 5. Y \u003d 5 is excluded, because P cannot be equal to 0. Y? 6, because 6 + 6 = 12, i.e. P = 2. But such a value of P is not suitable for further verification. Likewise, U? 7.

Suppose Y = 8. Then, P = 6, A = 2, K = 5, D = 1.

A magic (magic) square is a square in which the sum of the numbers vertically, horizontally and diagonally is the same.

A task. Arrange the numbers from 1 to 9 so that vertically, horizontally and diagonally you get the same sum of numbers, equal to 15.

Although there are no general rules for solving non-standard problems (which is why these problems are called non-standard), we have tried to give a number of general guidelines - recommendations that should be followed when solving non-standard problems of various types.

Each non-standard task is original and unique in its solution. In this regard, the developed methodology for teaching search activities when solving non-standard tasks does not form skills for solving non-standard tasks, we can only talk about developing certain skills:

• ability to understand the task, highlight the main (supporting) words;
• the ability to identify the condition and question, known and unknown in the problem;
• the ability to find a connection between the data and the desired, that is, to analyze the text of the problem, the result of which is the choice of an arithmetic operation or a logical operation to solve a non-standard problem;
• the ability to record the progress of the solution and the answer to the problem;
• The ability to carry out extra work over the task;
• the ability to select useful information contained in the problem itself, in the process of solving it, to systematize this information, correlating it with existing knowledge.

Non-standard tasks develop spatial thinking, which is expressed in the ability to recreate in the mind the spatial images of objects and perform operations on them. Spatial thinking is manifested when solving problems like: “On top of the edge of a round cake, 5 dots of cream were placed at the same distance from each other. Cuts were made through all pairs of points. How many pieces of cake did you get in total?

practical method can be considered for non-standard division problems.

A task. The stick needs to be cut into 6 pieces. How many cuts will be required?

Solution: Cuts will need 5.

When studying non-standard division problems, you need to understand: in order to cut a segment into P parts, you should make a (P - 1) cut. This fact must be established with children inductively, and then used in solving problems.

A task. In a three-meter bar - 300 cm. It must be cut into bars 50 cm long each. How many cuts do you need to make?

Solution: We get 6 bars 300: 50 = 6 (bars)

We argue as follows: to divide the bar in half, that is, into two parts, you need to make 1 cut, into 3 parts - 2 cuts, and so on, into 6 parts - 5 cuts.

So, you need to make 6 - 1 = 5 (cuts).

Answer: 5 cuts.

So, one of the main motives that encourage students to study is interest in the subject. Interest is an active cognitive orientation of a person to a particular object, phenomenon and activity, created with a positive emotional attitude to them. One of the means of developing interest in mathematics is non-standard tasks. A non-standard task is understood as such tasks for which there are no general rules and regulations in the course of mathematics that determine the exact program for their solution. Solving such problems allows students to actively engage in learning activities. There are various classifications of problems and methods for their solution. The most commonly used are algebraic, arithmetic, practical methods and the method of enumeration, reasoning and conjecture.

Borodich

Irina Sergeevna

Textbook for a teacher on an elective course of mathematics for grade 11 (physical and mathematical profile)

« Non-standard methods solving problems in mathematics "

Introduction. AT modern conditions meaningful modernization of education, a continuum of problems arises that has social and personal characteristics and inhibits positive changes.

Mathematical education in the secondary system general education occupies one of the leading places, which is determined unconditionally practical significance mathematics, its capabilities in the development and formation of human thinking, its contribution to the creation of ideas about scientific methods knowledge of reality.

According to PIZA research, the level of mathematical competence of students in Russia remains very low, although we are used to being proud of the achievements of academic science.

The most important problem of today's mathematical education is the lack of development of formal-operational structures of the intellect (logical thinking) and the low motivation for theoretical intellectual activity in most schoolchildren.

On the other hand, authoritarian methods of pedagogy, which did not contribute to the development of intelligence in children, and collective methods of work, which reduced interest in mathematical science, led to this deficit.

Therefore, the most important aspect of today's education is the individualization of the educational process in the study of mathematics and tutor support by teachers of the development of the child's intellect.

Relevance. Course on non-standard solution methods math problems is relevant, first of all, because it makes education more open, expanding the intellectual capabilities of high school students. Secondly, this course provides more fluency in mathematical tools as part of the final certification. On the other hand, mathematics, being an oversubject area of ​​knowledge, contributes to the development of logical thinking, intellect in general and communication skills that contribute to the self-realization of the individual. The course is also relevant in connection with the expansion of the applied application of mathematical calculus in other areas of knowledge.

The course will help students assess their needs, opportunities and make informed choices about their future life path.

Starting work in mathematics with younger teenagers, in grades 6-7, as part of dividing the subject into two sections, I conduct a mathematical ability analysis test, differentiating the results obtained to form task packages: for students with a low level of creativity - developing packages, with an average level of creativity - tasks of increased complexity, with high level- creative tasks. Evaluating the effect of activities, I repeat this testing in grades 8-9 and 10-11. The result showed that such differentiation contributes to a more intensive and harmonious development of students.

The purpose of profile education, as one of the areas of modernization of mathematical education, is to provide an in-depth study of the subject and prepare students for continuing education.

The course "Non-standard methods for solving problems in mathematics" involves the study of such issues that are not included in general course secondary school mathematics, but are necessary for its further study, with the final certification in the form of the Unified State Examination. The appearance of problems solved by non-standard methods in exams is far from accidental, because with their help, the technique of mastering formulas is checked elementary mathematics, ways to solve equations and inequalities, the ability to build a logical chain of reasoning, the level of logical thinking of students and their mathematical culture.

The solution of problems of this type is not given due attention in the school curriculum, most students (not in physical and mathematical specialized groups) either do not cope with such problems at all, or give cumbersome calculations. The reason for this is the lack of a system of tasks on this topic in school textbooks. In this regard, there was a need to develop and conduct an elective course for students in grades 11 of the physical and mathematical profile.

The variety of non-standard tasks covers the entire course of school mathematics, therefore, the possession of methods for solving them can be considered a criterion for knowledge of the main sections of school mathematics, the level of mathematical and logical thinking.

The study of non-standard methods for solving mathematical problems provides excellent material for this educational and research work.

The course will allow students to systematize, expand and strengthen knowledge, prepare for further study of mathematics, learn to solve various problems of varying complexity.

The course will help the teacher to best prepare students for mathematical olympiads, passing the exam and exams for admission to universities.

Novelty. The course is innovative, as it contributes to a deeper development mathematical science in senior classes, both in specialized groups and at the basic level. The novelty is the construction of a course on methods for solving mathematical problems and methods for implementing mathematical knowledge. The course is a kind of simulator in preparation for the final certification and professional choice of mathematical specialties.

Literature review. This course is intended for students of the 11th grade of the physical and mathematical profile. Content educational material corresponds to the goals and objectives of profile education. At the beginning of the elective course, a diagnostics of mathematical creativity was carried out. Methodologically, I rely in the theoretical part on the works of V.P. Suprun "Mathematics for high school students: Non-standard methods for solving problems in mathematics" and Olekhnik S. N. Non-standard methods for solving equations and inequalities: .

The main objective of the course:

Creation of conditions for the development of logical thinking, mathematical culture and intuition of students by solving problems of increased complexity by non-traditional methods;

Course objectives:

to form students' competence in solving non-standard tasks;

the study of the course involves the formation of students' interest in the subject, the development of their mathematical abilities, preparation for the exam and further education at the university;

to develop research and cognitive activity of students;

creation of conditions for self-realization of students in the process learning activities.

develop the ability to independently acquire and apply knowledge.

The general principles for selecting course content are:

Consistency

Integrity

Scientific.

Accessibility, according to the psychological and age characteristics of students in specialized classes.

The course contains the material necessary to achieve the planned goals. This course is a source that expands and deepens learning, provides the integration of the necessary information for the formation of mathematical thinking, logic and the study of related disciplines.

The place of this course is determined by the need to prepare for professional activity, takes into account the interests and professional inclinations of high school students, which allows you to get a higher final result.

Course concept.

When studying the high school mathematics course at the basic level, the study of sections continues: "Algebra", "Functions", "Equations and Inequalities", "Geometry", "Elements of Logic, Combinatorics, Statistics and Probability Theory", the line "Beginnings of Mathematical Analysis" is introduced .

In the course of mastering the content of mathematical education, students master a variety of different ways activities, acquire and improve experience:

building and researching mathematical models for describing and solving applied problems, problems from related disciplines;

implementation and self-compilation of algorithmic prescriptions and instructions on mathematical material; performing practical calculations; the use of mathematical formulas and self-compilation of formulas based on the generalization of particular cases and experiment;

independent work with sources of information, generalization and systematization of the information received, its integration into personal experience;

conducting evidence-based reasoning, logical substantiation of conclusions, distinguishing between proven and unproven statements, reasoned and emotionally convincing judgments;

independent and collective activities, inclusion of their results in the results of the work of the group, correlation of their opinion with the opinion of other members of the educational team and the opinion of authoritative sources.

In the profile course, the content of education develops in the following directions:

systematization of information about the number; formation of representations of numerical sets, as a way of constructing a new mathematical apparatus necessary for solving problems of the surrounding world and internal problems of mathematics; improvement of computing techniques;

development and improvement of the technique of algebraic transformations, solving equations, inequalities, systems;

systematization and expansion of information about functions, improvement graphic skills; familiarity with the basic ideas and methods of mathematical analysis in a volume that allows you to explore elementary functions and solve the simplest geometric, physical and other applied problems;

development of ideas about probabilistic and statistical patterns in the world around;

improvement mathematical development to a level that allows you to freely apply the studied facts and methods in solving problems from various sections of the course, as well as use them in non-standard situations;

formation of the ability to build and explore the simplest mathematical models when solving applied problems, problems from related disciplines, deepening knowledge about the features of the application mathematical methods to the study of processes and phenomena in nature and society.

in the course of studying mathematics in the profile course of high school, students continue to master a variety of activities, acquire and improve experience:

conducting evidence-based reasoning, logical substantiation of conclusions, using various languages ​​of mathematics for illustration, interpretation, argumentation and proof;

solving a wide class of problems from various sections of the course, search and creative activities in solving problems of increased complexity and non-standard problems;

planning and implementation of algorithmic activities: implementation and self-compilation of algorithmic prescriptions and instructions on mathematical material; use and self-compilation of formulas based on the generalization of particular cases and experimental results; performing practical calculations;

construction and research of mathematical models for the description and solution of applied problems, problems from related disciplines and real life; checking and evaluating the results of their work, correlating them with the task, with personal life experience;

independent work with sources of information, analysis, generalization and systematization of the information received, its integration into personal experience.

Russian schools begin a phased transition to federal state educational standards of the second generation of general education (hereinafter referred to as GEF), the main mission of which is to improve the quality of education. A feature of the 2011/2012 academic year is the introduction of the GEF of primary general education in primary school and consistent preparation for the introduction of the GEF of basic general education. Therefore, already now it is necessary to understand its theoretical and methodological basis, structure and content.

The Federal State Educational Standard will be provided with state guarantees that educational results will be achieved in a certain information and educational environment, which is made up of: teaching staff, material, technical, financial, economic, information support.

Although the content of mathematical education is presented in the form of traditional substantive sections: "Arithmetic", "Algebra", "Geometry", "Mathematical Analysis", "Probability and Statistics", at the same time, acquaintance with the history of mathematics and mastering the following general mathematical concepts and methods is assumed:

definitions and initial (undefined) concepts, proofs, axioms and theorems, hypotheses and refutations, counterexamples, typical errors in reasoning;

direct and inverse theorem, existence and uniqueness of an object, necessary and sufficient condition for the validity of a statement, proof by contradiction, method of mathematical induction;

mathematical model, mathematics and problems of physics, chemistry, biology, economics, geography, linguistics, sociology, etc.

Based on the above positions, non-standard methods for solving problems in mathematics are a tool for the formation of both mathematical thinking and mathematical competencies, i.e. willingness to apply non-standard methods in solving theoretical and applied mathematical calculations.

At the same time, mathematical models of certain processes of nature and technology require mathematical processing, not always in traditional ways.

Such approaches to the application and use of mathematics contribute to the formation through personal actions of personal (self-improvement and self-respect), metasubject (formation of goals, objectives, processes for their solution) and subject results.

Non-standard approaches to the development of mathematics, as a super-subject area, makes education open, and the educational environment developing.

Topics of abstract, research and design works:

History of mathematics

Renaissance mathematicians

Number as a basic concept of mathematics

Reading and writing natural numbers

The relation of consciousness to matter: mathematics and objective reality

Mathematical intuition

The numbers that changed the world

Bernoulli

Irrational equations

The use of graphs in solving equations

The course is intended for students of the 11th physical and mathematical class.

The volume of hours is 33 hours (1 hour per week).

The course is divided into modules, three hours each, united by the topic of problem solving.

Educational and thematic plan

Topics and sections

Total hours

Including

Forms of holding

Introduction

Personal

Mini lecture

1. Method of functional substitution

Regulatory

Seminar, training

2. Method of trigonometric substitution

Cognitive, personal and regulatory

Seminar, training

3. Methods based on the application of numerical inequalities

Regulatory and communicative

Seminar, training

4. Methods based on the use of monotonicity of functions

Regulatory and communicative

Seminar, training

5. Methods for solving functional equations

Seminar, training

6. Methods based on the use of vectors

personal and regulatory

Seminar, training

7. Combined Methods

Cognitive, personal and regulatory, communicative

Critical Thinking Technology

8. Methods based on the use of limited functions

Regulatory and communicative

Seminar, training

9. Methods for solving symmetric systems of equations

Regulatory

Seminar, training

10. Methods for solving equations containing integer or fractional parts of a number

Regulatory

Seminar, training

FINAL lesson

Communicative

Lesson - conference (defense of design, research and abstract works)

Introduction: 1 hour (1 - theoretical)

The value of mathematics as a science and in human life. Applied value. The beauty of non-standard ways of solving problems. Distribution of topics of design, research and abstract works.

1. Method of functional substitution: 3 hours (1 hour - seminar; 2 hours - training)

Functional substitution method. New variable , its application. Irrational equations. Systems of equations. Equations like x 2 +(ah) 2 2 =s. Return equations. A number of other equations, the solution of which requires the introduction of a new variable.

2. Trigonometric substitution method: 3 hours (1 hour - seminar; 2 hours - training)

Trigonometric substitution method. Replacing an unknown variableX trigonometric function:x= orx=. Irrational equations. Rational equations. exponential equations. Systems of equations.

3. Methods based on the application of numerical inequalities: 3 hours (1 hour - seminar; 2 hours - training)

Methods based on the application of numerical inequalities. Cauchy's inequality. Bernoulli's inequality. Cauchy-Bunyakovsky inequality.

4. Methods based on the use of the monotony of functions: 3 hours (1 hour - seminar; 2 hours - training)

Methods based on the use of monotonicity of functions. An equation of the form f (x) \u003d g (x). Investigation of functions for monotonicity.

5. Methods for solving functional equations: 3 hours (1 hour - seminar; 2 hours - training)

Methods for solving functional equations. Equations of the form f (f (…(f (x))…))=x . Equations of the form f (g (x)) \u003d f (h (x)).

6. Methods based on the use of vectors: 3 hours (1 hour workshop; 2 hours training)

Methods based on the use of vectors. A vector in 3D space. The length of the vector. Sum and difference of two vectors. Collinear vectors. Triangle Inequality.

7. Combined methods: 3 hours (1 hour seminar; 2 hours training)

Combined methods. Tasks with parameters. Irrational equations. Logarithmic Equations. Equations and inequalities containing a module. Systems of equations. Proof of inequalities.

8. Methods based on the use of limited functions: 3 hours (1 hour - seminar; 2 hours - training)

Methods based on the use of restricted functions. trigonometric functions. Inverse trigonometric functions. Functions containing modulus, degree, root with even degree.

9. Methods for solving symmetric systems of equations: 3 hours (1 hour - seminar; 2 hours - training)

Methods for solving symmetric systems of equations. Systems of equations with symmetric occurrence of terms or factors.

10. Methods for solving equations containing integer or fractional parts of a number: 3 hours (1 hour - seminar; 2 hours - training)

Methods for solving equations containing integer and fractional parts of a number. whole part real number. The fractional part of a real number.

11. Final lesson: 2 hours (Lesson - conference (defense of design, research and abstract works))

Implementation of the formation of universal educational activities in the framework of the implementation of the Federal State Educational Standards of the II generation to profile level high school

PERSONAL RESULTS

Evaluate situations and actions(value settings, moral orientation)

Make a choice in relation to actions, forming attitudes towards socially approved and moral models of behavior, resolving moral contradictions on the basis of:

Universal values ​​and Russian values, including philanthropy, respect for work, culture;

The importance of fulfilling age-related social roles (“son”, “daughter”, the role of a “good student”), the importance of studying and learning new things;

The importance of caring for human health and nature;

The importance of developing the spiritual potential of the individual (distinguishing between "beautiful" and "ugly", the need for "beautiful" and the denial of "ugly", craving for self-knowledge and self-improvement);

The importance of education, a healthy lifestyle, the beauty of nature and creativity.

Predict assessments of the same situations from the standpoint of different people who differ in nationality, worldview, position in society, etc. (tolerant thinking and behavior)

Learn to notice and recognize the discrepancies between your actions and your declared positions, views, opinions.

Explain the meaning of your assessments, motives, goals

(personal self-reflection, ability for self-development, motivation for knowledge, study)

REMEMBER

Explain positive and negative assessments, including ambiguous actions, from the standpoint of universal and Russian civic values.

Explain differences in assessments of the same situation, act different people(including himself) as representatives of different worldviews, different groups of society.

Own social choice and choice of behavior patterns.

SELF-AWARENESS

Explain to yourself:

Positive "I am a concept"

- “what is good in me and what is bad” (personal qualities, character traits), “what I want” (goals, motives), “what I can” (results).

Self-determination in life values(in words)and act in accordance with them, being responsible for their actions(personal position, Russian and civil identity)

SELF-DETERMINATION

To recognize oneself as a citizen of Russia and a valuable part of the many-sided changing world, including

Explain what connects you:

with family, with family

with your family, friends, classmates

with fellow countrymen

with your country

with all people

with nature

Explain what connects you with the history, culture, fate of your people and all of Russia;

Feel a sense of pride in your people, your homeland, empathize with them in joys and troubles and show these feelings in good deeds;

Defend (within their capabilities) humane, equal, civil democratic orders and prevent their violation;

Look for your position in the variety of social and worldview positions, aesthetic and cultural preferences;

Strive for mutual understanding with representatives of other cultures, worldviews, peoples and countries, based on mutual interest and respect;

Respect other opinions, history and culture of other peoples and countries, not allow them to be insulted, ridiculed;

Carry out good deeds that are useful to other people, to your country, including giving up some of your desires for their sake.

Determination of one's place in the world of nature and the world of culture;

Form a non-conflict model of behavior that contributes to non-violent and equal overcoming of the conflict.

Make a conscious choice of a behavior model in ambiguous situations, based on:

Culture, people, worldview, to which you feel your involvement,

Basic Russian civic values,

Universal, humanistic values, including the value of peaceful good-neighborly relations between people of different cultures, positions, worldviews,

Known and simple generally accepted rules of "kind", "safe", "beautiful", "correct" behavior,

Empathy in the joys and troubles of "one's own": relatives, friends, classmates,

Empathy for the feelings of other people who are not like you, responsiveness to the troubles of all living beings.

To form adequate self-esteem and responsibility for the actions performed and loved ones.

REGULATORY UUD

Determine and formulate the purpose of the activity, draw up an action plan to solve the problem (task)

To determine the purpose of educational activity and the goal-setting of learning independently, to look for means of its implementation.

Find and formulate the main educational problem and idea, first together with the teacher, and then, independently, choose the topic of the project with the help of the teacher and independently.

Make a plan for completing tasks, solving problems of a creative and exploratory nature, completing a project together with the teacher.

To master the basics of research and project activities through educational and extracurricular work.

Take action to implement the plan

When working on a project, plan its stages with a view to implementation and, if necessary, adjust the stages of its implementation.

Learn to work with information, using it in the implementation of plans and solving educational and research problems (reference literature, complex devices, ICT tools).

Compare the result of your activity with the goal and evaluate it

In dialogue with the teacher, learn to develop assessment criteria and determine the degree of success in the performance of one's work and the work of everyone, based on the existing criteria, improve assessment criteria and use them in the course of assessment and self-assessment.

During the presentation of the project, learn to evaluate its results.

Understand the reasons for your failure and find ways out of this situation.

Extract information, navigate in your knowledge system and realize the need for new knowledge, make a preliminary selection of information sources to search for new knowledge, obtain new knowledge (information) from various sources and in various ways

Independently assume what information is needed to solve the subject educational problem, which consists of several steps.

Independently select the necessary dictionaries, encyclopedias, reference books, electronic disks for solving subject educational problems.

COGNITIVE UUD

Compare and select information obtained from various sources (dictionaries, encyclopedias, reference books, electronic disks, the Internet).

Form your own position in the world of information

Process information to obtain the desired result, including the creation of a new product

Perform universal logical actions:

Perform analysis (feature extraction),

To produce a synthesis (composing a whole from parts, including with independent completion),

Choose the basis for comparison, seriation, classification of objects,

Predict the expected result of solving educational problems,

Establish analogies and cause-and-effect relationships,

Build a logical chain of reasoning

Relate objects to known concepts.

Create models with highlighting the essential characteristics of the object and presenting them in a spatial-graphic or sign-symbolic form, transform models in order to identify the general laws that define this subject area.

Use information in project activities under the guidance of a teacher-adviser.

Convert information from one form to another and choose the most convenient form for yourself

Present information in the form of tables, diagrams, basic notes, including using ICT tools.

Write a simple and complex text plan.

Be able to convey content in a compressed, selective or expanded form.

COMMUNICATIVE UUD

Communicate your position to others, mastering the techniques of monologue and dialogic speech

To master effective speech activity by means of the native language and its emotional component.

To formulate their thoughts in oral and written speech, taking into account their educational and life speech situations including the use of ICT.

If necessary, defend your point of view, arguing it. Learn to back up arguments with facts.

Learn to be critical of your own opinion.

Understand other positions (views, interests)

Listen to others, try to take a different point of view, be ready to change your point of view.

Analyze the text being studied, while doing:

Comparing it with their own position on this issue (problem);

Proofread all types of textual information (factual, subtextual, conceptual).

To reflect on one's own attitude to the idea of ​​the work;

Negotiate with people, coordinating their interests and views with them, in order to do something together

Organize educational interaction in a group (distribute roles, negotiate with each other, etc.).

Accept other people's opinions in the group.

Anticipate (predict) the consequences of collective decisions.

Didactic support

The course has the character of deepening the study of mathematics in specialized groups and in preparation for competitions and olympiads. The course involves an additional analysis of the most complex methods for solving mathematical problems and equations. At the same time, the course is based mainly on two forms of activity: seminars and trainings. At the seminars, which have the character of tutorials, the theoretical aspects of mathematical science are considered. The purpose of the study is to master non-standard methods for solving complex mathematical problems. At the same time, due to the complexity and ambiguity of the methods, students in the training mode develop logical thinking and mathematical competencies.

Classes are built with the active participation of students who: track solutions, form critical thinking and adequate assessment and self-esteem. At the same time, all universal learning activities are formed and, as a result, key educational competencies:

analytical - activity,

predictive,

information,

communicative

reflexive.

All classes are built according to the plan developed by me in the process of practice.

when getting acquainted with new ways of solving - the work of a teacher with a demonstration of examples;

when improving;

training sessions;

individual work;

analysis of ready-made solutions;

independent work with tests;

In the classroom, various forms and methods of working with students are used:

Seminars, mini-lectures, round tables, master classes, trainings, individual and small group work.

Teaching methods are determined by the objectives of the course, aimed at the formation of mathematical abilities of students and basic competencies in the subject.

AT thematic planning the practical part is allocated, which is implemented on the knowledge of students obtained during the course of theoretical training.

At the end of each section, intermediate control is expected in the form of training tests and other active methods.

The effectiveness of the course is determined during the final lesson-conference, built on the defense of research, design and abstract work.

The course material is built taking into account the use of active teaching methods, and the rational distribution of sections of the program will allow you to obtain high-quality knowledge and achieve the planned results. The course is provided with the educational and methodological complex necessary for its implementation.

In the process of studying this course, it is supposed to use various methods of enhancing the cognitive activity of schoolchildren, as well as various forms organization of their independent work.

The result of mastering the course program is the presentation by schoolchildren of creative individual and group works at the final lesson.

Technologies used: technology for the development of critical thinking, problem technology, technology for solving research problems (TRIZ), information and communication technology.

Literature for the teacher:

Azarov AI Mathematics for high school students: Functional and graphic methods for solving examination problems /A. I. Azarov, S. A. Barvenov.- Minsk: Aversev, 2004.

Epifanova T. N. Finding extremal values ​​of functions by various

ways / T. N. Epifanova Mathematics at school. - No. 4. – 2000.

Mukhametzyanova F.S. methodologist of the Department of Physics and Mathematics Education of UIPCPRO, Honored Teacher of the Russian Federation Features of teaching subject"Mathematics" in the 2011-2012 academic year. (24.02.2009).

Olekhnik S. N. Non-standard methods for solving equations and inequalities: A Handbook / S. N. Olekhnik, M. K. Potapov, P. I. Pasichenko. - M.: Publishing house of Moscow State University, 1991.

Potapov, M. K. Reasoning with numerical values ​​in solving systems of equations / M. K. Potapov, A. V. Shevkin / Mathematics at school. - Number 3. – 2005.

Approximate basic educational program of an educational institution

correspondent of the Russian Academy of Education A. M. Kondakov, academician of the Russian Academy of Education L. P. Kezina)

V. P. Suprun. Mathematics for high school students. Tasks of increased complexity. - Minsk: "Aversev", 2002.

Literature for students:

Suprun V. P. Non-standard methods for solving problems in mathematics / Suprun V. P. - Minsk: Polymya, 2000.

Algebra and mathematical analysis. Grade 10: Textbook for schools and classes with in-depth study of mathematics / N. Ya. Vilenkin, O. S. Ivashev-Musatov, S. I. Shvartsburd. - M.: Mnemosyne, 2006

Algebra and mathematical analysis. Grade 11: Textbook for schools and classes with in-depth study of mathematics / N. Ya. Vilenkin, O. S. Ivashev-Musatov, S. I. Shvartsburd. - M.: Mnemosyne, 2006

The text of the work is placed without images and formulas.
Full version work is available in the "Files of work" tab in PDF format

Introduction

Mathematics education received at school is an essential component of general education and common culture modern man. Almost everything that surrounds a modern person is all connected in one way or another with mathematics. And recent advances in physics, technology and information technology leave no doubt that things will remain the same in the future. Therefore, the solution of many practical problems is reduced to solving various kinds equations.

Equations in the school course of algebra occupy a leading place. More time is devoted to their study than to any other topic of the school mathematics course. The strength of the theory of equations is that it not only has theoretical significance for the knowledge of natural laws, but also serves specific practical purposes.

Relevance of the topic is that in the lessons of algebra, geometry, physics, we very often meet with the solution of quadratic equations. Most problems about spatial forms and quantitative relations real world is reduced to solving various types of equations. By mastering the ways of solving them, people find answers to various questions from science and technology (transport, Agriculture, industry, communications, etc.). Therefore, each student should be able to correctly and rationally solve quadratic equations, this can also be useful to me when solving more challenging tasks, including in grade 9, as well as 10 and 11 and when passing exams.

Target: Learn standard and non standard ways solutions of quadratic equations

Tasks

1. Outline the most well-known methods for solving equations
2. Outline non-standard ways of solving equations
3. Draw a conclusion

Object of study: quadratic equations

Subject of study: ways to solve quadratic equations

Research methods:

• Theoretical: study of literature on the research topic;
• Analysis: information obtained in the study of literature; results obtained by solving quadratic equations in various ways.
• Comparison of methods for the rationality of their use in solving quadratic equations.

Chapter 1. Quadratic equations and standard solutions

1.1 Definition quadratic equation

quadratic equation is called an equation of the form ax 2 + bx + c= 0, where X- variable , a, b and With- some numbers, and, a≠ 0.

Numbers a, b and With - coefficients of the quadratic equation. Number a is called the first coefficient, the number b- second coefficient and number c- free member.

Complete quadratic equation is a quadratic equation in which all three terms are present i.e. the coefficients in and c are non-zero.

Incomplete quadratic equation is an equation in which at least one of the coefficients in or, c is equal to zero.

Definition 3. The root of the quadratic equation Oh 2 + bX + With= 0 is any value of the variable x for which square trinomial Oh 2 + bX+ With goes to zero.

Definition 4. Solving a quadratic equation means finding all of its

roots or establish that there are no roots.

Example: - 7 x + 3 =0

In each of the equations of the form a + bx + c= 0, where a≠ 0, the highest power of the variable x- square. Hence the name: quadratic equation.

A quadratic equation in which the coefficient at X 2 equals 1, called reduced quadratic equation.

Example

X 2 - 11x+ 30=0, X 2 -8x= 0.

1.2 Standard methods for solving quadratic equations

Solving quadratic equations by squaring a binomial

Solution of a quadratic equation in which both coefficients of the unknowns and the free term are nonzero. This method of solving a quadratic equation is called the selection of the square of the binomial.

Factoring the left side of the equation.

Let's solve the equation x 2 + 10x - 24 = 0. Let's factorize the left side:

x 2 + 10x - 24 \u003d x 2 + 12x - 2x - 24 \u003d x (x + 12) - 2 (x + 12) \u003d (x + 12) (x - 2).

Therefore, the equation can be rewritten as: (x + 12)(x - 2) = 0

A product of factors is zero if at least one of its factors is zero.

Answer: -12; 2.

Solving a quadratic equation using a formula.

Quadratic discriminantax 2 + bx + c\u003d 0 expression b 2 - 4ac \u003d D - by the sign of which one judges the presence of real roots in this equation.

Possible cases depending on the value of D:

1. If a D>0, then the equation has two roots.
2. If a D= 0, then the equation has one root: x =
3. If a D< 0, then the equation has no roots.

Solving equations using the Vieta theorem.

Theorem: The sum of the roots of the given quadratic equation is equal to the second coefficient, taken from opposite sign, and the product of the roots is equal to the free term.

The given quadratic equation has the form:

x 2 + bx + c= 0.

We denote the second coefficient by the letter p, and the free term by the letter q:

x 2 + px + q= 0, then

x 1 + x 2 \u003d - p; x 1 x 2 = q

Chapter 2

2.1. Solution using the properties of the coefficients of the quadratic equation

Properties of the coefficients of a quadratic equation is such a way to solve quadratic equations that will help you quickly and verbally find the roots of the equation:

ax 2 + bx + c= 0

1. If aa+b+c= 0, thenx 1 = 1, x 2 =

Example. Consider the equation x 2 +3x - 4= 0.

a+ b + c = 0, then x 1 = 1, x 2 =

1+3+(-4) = 0, then x 1 = 1, x 2 = = - 4

Let's check the obtained roots by finding the discriminant:

D=b2- 4ac= 3 2 - 4 1 (-4) = 9+16= 25

x 1 = = = = = - 4

Therefore, if +b+c= 0, then x 1 = 1, x 2 =

1. If ab= a + c , thenx 1 = -1, x 2 =

x 2+ 4X+1 = 0, a=3, b=4, c=1

If a b=a + c, then x 1 = -1, x 2 = , then 4 = 3 + 1

Equation roots: x 1 = -1, x 2 =

So the roots of this equation are -1 and. Let's check this by finding the discriminant:

D=b2- 4ac= 4 2 - 4 3 1 = 16 - 12 = 4

x 1 = = = = = - 1

Consequently, b=a + c, then x 1 = -1, x 2 =

2.2. The method of "transfer"

With this method, the coefficient a is multiplied by the free term, as if “thrown” to it, which is why it is called transfer method. This method is used when it is easy to find the roots of an equation using Vieta's theorem and, most importantly, when the discriminant is an exact square.

If a a± b+c≠0, then the transfer technique is used:

3x 2 +4x+ 1=0; 3+4+1 ≠ 0

Applying the method of "transfer" we get:

X 2 + 4x+3= 0

Thus, using the Vieta theorem, we obtain the roots of the equation:

x 1 \u003d - 3, x 2 \u003d -1.

However, the roots of the equation must be divided by 3 (the number that was "thrown"):

So, we get the roots: x 1 \u003d -1, x 2 \u003d.

Answer: ; - one

2.3. Solution using the regularity of coefficients

1. If the equationax 2 + bx + c= 0, coefficientb= (a 2 +1), and the coefficientc = a, then its roots are x 1 = - a, x 2 =

ax2+(a 2 + 1)∙ x + a = 0

Example. Consider Equation 3 x 2 +10x+3 = 0.

Thus, the roots of the equation: x 1 = -3 , x 2 =

D=b2- 4ac= 10 2 - 4 3 3 = 100 - 36 = 64

x 1 = = = = = - 3

x 2 = = = = = ; Therefore, x 1 = - a, x 2 =

1. If the equationax 2 - bx + c= 0, coefficientb= (a 2 +1), and the coefficientc = a, then its roots are x 1 = a, x 2 =

Thus, the equation to be solved should look like

ax 2-(a 2 + 1)∙ x+ a= 0

Example. Consider Equation 3 x 2 - 10x+3 = 0.

, x 2 =

Let's check this solution using the discriminant:

D=b2- 4ac= 10 2 - 4 3 3 = 100 - 36 = 64

a, x 2 =

1. If the equationax 2 + bx - c= 0, coefficientb= (a 2 -1), and coefficientc = a, then its roots are x 1 = - a, x 2 =

Thus, the equation to be solved should look like

ax2+(and 2 - 1)∙ x - a = 0

Example. Consider Equation 3 x 2 + 8x - 3 = 0..

So the roots of the equation are: x 1 = - 3, x 2 =

Let's check this solution using the discriminant:

D=b2- 4ac= 8 2 + 4 3 3 = 64 + 36 = 100

x 1 = = = = = - 3

x 2 = = = = =; Therefore, x 1 = - a, x 2 =

1. If the equationax 2-bx-c= 0, coefficientb= (a 2 -1), and coefficientc = a, then its roots are x 1 = a, x 2 =

Thus, the equation to be solved should look like

ax 2-(and 2 - 1)∙ x - a = 0

Example. Consider Equation 3 x 2 - 8x - 3 = 0..

Thus, the roots of the equation: x 1 \u003d 3 , x 2 = -

Let's check this solution using the discriminant:

D=b2- 4ac= 8 2 + 4 3 3 = 64 + 36 = 100

x 2 = = = = = 3; Therefore, x 1 = a, x 2 = -

2.4. Solution with a compass and straightedge

I propose the following method for finding the roots of a quadratic equation ah 2+bx + c = 0 using a compass and a ruler (Fig. 6).

Let us assume that the desired circle intersects the axis

abscissa in points B(x 1; 0) and D(x 2; 0), where x 1 and x 2- roots of the equation ah 2+bx + c = 0, and passes through the points

A(0; 1) and C(0;c/ a) on the y-axis. Then, by the secant theorem, we have OB . OD = OA . OC, where OC = = =

The center of the circle is at the point of intersection of the perpendiculars SF and SK, restored at the midpoints of the chords AC and BD, that's why

1) construct the points S (the center of the circle) and A(0; 1) ;

2) draw a circle with a radius SA;

3) the abscissas of the points of intersection of this circle with the axis Oh are the roots of the original quadratic equation.

In this case, three cases are possible.

1) The radius of the circle is greater than the ordinate of the center (AS > SK, orR > a + c/2 a) , the circle intersects the x-axis at two points (Fig. 7a) B(x 1; 0) and D(x 2; 0), where x 1 and x 2- roots of the quadratic equation ah 2+bx + c = 0.

2) The radius of the circle is equal to the ordinate of the center (AS = SB, orR = a + c/2 a) , the circle touches the Ox axis (Fig. 8b) at the point B(x 1; 0), where x 1 is the root of the quadratic equation.

3) The radius of the circle is less than the ordinate of the center AS< S, R<

the circle has no common points with the abscissa axis (Fig. 7c), in this case the equation has no solution.

a)AS>SB, R> b) AS=SB, R= in) AS

Two Solutions x 1 andx 2 One Solution x 1 There is no decision

Example.

Let's solve the equation x 2 - 2x - 3 = 0(Fig. 8).

Solution. Determine the coordinates of the point of the center of the circle by the formulas:

x = - = - = 1,

y = = = -1

Let's draw a circle of radius SA, where A (0; 1).

Answer: x 1 = - 1; x 2 = 3.

2.5. Geometric method for solving quadratic equations.

In ancient times, when geometry was more developed than algebra, quadratic equations were solved not algebraically, but geometrically. I will give an example that has become famous from the "Algebra" of al-Khwarizmi.

Examples.

1) Solve the equation x 2 + 10x = 39.

In the original, this problem is formulated as follows: “A square and ten roots are equal to 39” (Fig. 9).

Solution. Consider a square with side x, rectangles are built on its sides so that the other side of each of them is 2.5, therefore, the area of ​​\u200b\u200beach is 2.5x. The resulting figure is then supplemented to a new square ABCD, completing four equal squares in the corners, the side of each of them is 2.5, and the area is 6.25.

Square S square ABCD can be represented as the sum of the areas:

original square x 2, four rectangles (4. 2.5x = 10x) and four attached squares (6,25. 4 = 25) , i.e. S = x 2 + 10x + 25. Replacing

x 2 + 10x number 39 , we get that S = 39 + 25 = 64 , whence it follows that the side of the square ABCD, i.e. line segment AB = 8. For the desired side X the original square we get:

x = 8 - 2 - 2 = 3

2) But, for example, how the ancient Greeks solved the equation y 2 + 6y - 16 = 0.

Solution shown in Figure 10. where

y 2 + 6y = 16, or y 2 + 6y + 9 = 16 + 9.

Solution. Expressions y 2 + 6y + 9 and 16 + 9 geometrically represent

the same square, and the original equation y 2 + 6y - 16 + 9 - 9 = 0 is the same equation. From where we get that y + 3 = ± 5, or y 1 = 2, y 2 = - 8(rice. .

fig.10

3) Solve geometric equation y 2 - 6y - 16 = 0.

Transforming the equation, we get

y 2 - 6y \u003d 16.

In Figure 11 we find the "images" of the expression y 2 - 6y, those. from the area of ​​a square with side y subtract twice the area of ​​a square with side equal to 3 . So, if the expression y 2 - 6y add 9 , then we get the area of ​​a square with a side y - 3. Replacing the expression y 2 - 6y its equal number 16,

we get: (y - 3) 2 \u003d 16 + 9, those. y - 3 = ± √25, or y - 3 = ± 5, where y 1 = 8 and y 2 = - 2.

Conclusion

In the course of my research work, I believe that I coped with the set goal and tasks, I managed to generalize and systematize the studied material on the above topic.

It should be noted that each method of solving quadratic equations is unique in its own way. Some solutions help save time, which is important when solving tasks on tests and exams. When working on the topic, I set the task to find out which methods are standard and which are non-standard.

So, standard methods(used more often when solving quadratic equations):

• Solution by squaring the binomial
• Factoring the Left Side
• Solving quadratic equations by formula
• Solution using Vieta's theorem
• Graphical solution of equations

Non-standard methods:

• Properties of the coefficients of a quadratic equation
• Solution by transferring coefficients
• Solution using the regularity of coefficients
• Solving quadratic equations using a compass and straightedge.
• Investigation of the equation on intervals of the real axis
• Geometric way

It should be noted that each method has its own characteristics and limits of application.

Solving equations using Vieta's theorem

A fairly easy way, it makes it possible to immediately see the roots of the equation, while only integer roots are easily found.

Solution of equations by transfer method

For the minimum number of actions, you can find the roots of the equation, it is used in conjunction with the method of the Vieta theorem, while it is also easy to find only integer roots.

Properties of the coefficients of a quadratic equation

Affordable method for verbally finding the roots of a quadratic equation, but only suitable for some equations

Graphical solution of a quadratic equation

A visual way to solve a quadratic equation, however, errors may occur when plotting

Solving quadratic equations with a compass and straightedge

A visual way to solve a quadratic equation, but errors can also occur

Geometric way of solving quadratic equations

A visual way, similar to the way to select a full square

Solving equations in different ways, I came to the conclusion that knowing a set of methods for solving quadratic equations, you can solve any equation offered in the learning process.

At the same time, it should be noted that one of the more rational ways to solve quadratic equations is the method of “transferring” the coefficient. However, the most universal way can be considered the standard way of solving equations using a formula, because this method allows you to solve any quadratic equation, although sometimes for a longer time. Also, such solution methods as the “transfer” method, the property of coefficients and the Vieta theorem help to save time, which is very important when solving tasks in exams and tests.

I think that my work will be of interest to students in grades 9-11, as well as those who want to learn how to solve quadratic equations rationally and prepare well for final exams. It will also be of interest to teachers of mathematics, by considering the history of quadratic equations and systematizing ways to solve them.

Bibliography

1. Glazer, G.I. History of mathematics at school / G.I. Glaser.-M.: Enlightenment, 1982 - 340s.
2. Gusev, V.A. Maths. Reference materials / V.A. Gusev, A.G. Mordkovich - M.: Enlightenment, 1988, 372p.
3. Kovaleva G. I., Konkina E. V. "A functional method for solving equations and inequalities", 2014
4. Kulagin E. D. "300 competitive tasks in mathematics", 2013
5. Potapov M. K. “Equations and inequalities. Non-standard solution methods, M. Drofa, 2012
6. .Barvenov S. A "Methods for solving algebraic equations", M. "Aversev", 2006
7. Suprun V.P. "Non-standard methods for solving problems in mathematics" - Minsk "Polymya", 2010
8. Shabunin M.I. "Manual in mathematics for university applicants", 2005.
9. Bashmakov M.I. Algebra: textbook. for 8 cells. general education institutions. - M.: Enlightenment, 2004. - 287p.
10. Shatalova S. Lesson - workshop on the topic "Quadratic Equations". - 2004.

Non-standard ways to solve quadratic equations

9th grade student

Work manager:

Firsova Daria Evgenievna

mathematic teacher

It is often more useful for a student of algebra to solve the same problem in three different ways than to solve three or four problems. By solving one problem in different ways, you can find out by comparison which one is shorter and more efficient. This is how experience is developed.

W.U. Sawyer (English mathematician of the 20th century)

Objective

Learn all the existing ways to solve a quadratic equation. Learn how to use these.

Tasks

• Understand what is called a quadratic equation.
• Find out what types of quadratic equations exist.
• Find information about how to solve a quadratic equation and study it.

Relevance of the topic: People have been studying quadratic equations since ancient times. I wanted to know the history of the development of quadratic equations.

School textbooks do not provide complete information about quadratic equations and how to solve them.

An object: Quadratic equations.

Subject: Methods for solving these equations.

Research methods: analytical.

Hypothesis - if I can realize the goal and tasks set by me while studying this topic, then I will accordingly go to the implementation of pre-profile training in the field of mathematical education.

Research methods:

• Work with educational and popular science literature.
• Observation, comparison, analysis.
• Problem solving.

Expected results: In the course of studying this work, I will really be able to assess my intellectual potential and, accordingly, in the future decide on the profile of training, create a project product on the topic under study in the form of a computer presentation, studying this issue will allow me to compensate for the lack of knowledge on the designated topic.

I consider my work promising, since in the future both students can use this material to improve mathematical literacy, and teachers in optional classes

Quadratic Equations in Ancient Babylon

The need to solve equations not only of the first, but also of the second degree, even in ancient times, was caused by the need to solve problems related to finding the areas of land and earthworks of a military nature., as well as with the development of astronomy and mathematics itself. The Babylonians knew how to solve quadratic equations around 2000 BC. Applying modern algebraic notation, we can say that in their cuneiform texts there are, in addition to incomplete ones, such, for example, complete quadratic equations:

The rule for solving these equations, set forth in the Babylonian texts, coincides with the modern one, but it is not known how the Babylonians came to this rule. Almost all the cuneiform texts found so far give only problems with solutions stated in the form of recipes, with no indication of how they were found. Despite the high level of development of algebra in Babylonia, the concept of a negative number and general methods for solving quadratic equations are absent in cuneiform texts.

How Diophantus compiled and solved

quadratic equations

THE EQUATION:

"Find two numbers knowing that their sum is 20 and their product is 96"

Diophantus argues as follows: from the condition of the problem it follows that the desired numbers not are equal, because if they were equal, then their product would not be 96, but 100. Thus, one of them will be more than half of their sum, i.e. 10+X , the other is smaller, i.e. 10-X .

Difference between them 2 X

From here X=2 . One of the desired numbers is 12, the other is 8. Solution X = -2 for Diophantus does not exist, since Greek mathematics knew only positive numbers.

0 One of the problems of the famous Indian mathematician of the 12th century Bhaskara A flock of frisky monkeys After eating to their heart's content, they had fun. Part eight of them in a square I had fun in the clearing. And twelve along the lianas ... They began to jump hanging ... How many monkeys were You tell me, in this flock ?. The equation corresponding to the problem: Baskara writes under the form: Padded the left side to a square," width="640"

Quadratic equations in India

Problems on quadratic equations are also found in the astronomical treatise Aryabhattam, compiled in 499 by the Indian mathematician and astronomer Aryabhatta. Another Indian scientist, Brahmagupta, outlined a general rule for solving quadratic equations reduced to a single canonical form: ax ² +bx=c, a0

One of the problems of the famous Indian mathematician of the XII century Bhaskara

Frisky flock of monkeys

Eating well, having fun.

Them squared part eight

Having fun in the meadow.

And twelve in vines ...

They began to jump hanging ...

How many monkeys were

You tell me, in this flock?.

The equation corresponding to the problem is:

Baskara writes under the guise:

Complement the left side to a square,

Quadratic Equations in Ancient Asia

X 2 +10 x = 39

Here is how the Central Asian scientist al-Khwarizmi solved this equation:

He wrote: "The rule is this:

double the number of roots x=2x ·5

get five in this problem, 5

multiply by this equal to him, there will be twenty-five, 5 5=25

add that to thirty nine, 25+39

will be sixty four 64

take the root out of it, there will be eight, 8

and subtract from this half the number of roots, i.e. five, 8-5

will remain 3

this will be the root of the square you were looking for."

What about the second root? The second root was not found, since negative numbers were not known.

Quadratic equations in Europe XIII-XVII centuries.

The general rule for solving quadratic equations reduced to a single canonical form x2 + in + c = 0 was formulated in Europe only in 1544 Mr. Stiefel.

Formulas for solving quadratic equations in Europe were first stated in 1202 by an Italian mathematician

Leonard Fibonacci.

Vieta has a general derivation of the formula for solving a quadratic equation, but Vieta recognized only positive roots. Only in the 17th century thanks to the work Descartes, Newton and other scientists the method of solving quadratic equations takes on a modern form

About Vieta's theorem

A theorem expressing the relationship between the coefficients of a quadratic equation and its roots, bearing the name of Vieta, was formulated by him for the first time in 1591. As follows: “If B + D times A-A is equal to BD, then A is equal to B and equal to D” .

To understand Vieta, it should be remembered that A, like any vowel, meant the unknown (our x), while the vowels B, D are coefficients for the unknown.

In the language of modern algebra, Vieta's formulation above means :

If the given quadratic equation x 2 +px+q=0 has real roots, then their sum is equal to -p, and the product is q, that is x 1 + x 2 =-p, x 1 x 2 = q

(the sum of the roots of the given quadratic equation is equal to the second coefficient, taken with the opposite sign, and the product of the roots is equal to the free term).

• Factoring the left side of the equation
• Vieta's theorem
• Applying Quadratic Coefficient Properties
• The solution of quadratic equations by the method of "transferring" the highest coefficient
• Full square selection method
• Graphical way to solve quadratic equations
• Solving quadratic equations with a compass and straightedge
• Solving quadratic equations using a nomogram
• Geometric way of solving quadratic equations

Factorization method

bring the general quadratic equation to the form:

A(x) B(x)=0,

where A(x) and B(x) are polynomials with respect to x.

Target:

Ways:

• Taking the common factor out of brackets;
• Using abbreviated multiplication formulas;
• grouping method.

Example:

: X 2 + 10x - 24 = 0

Let's factorize the left side of the equation:

X 2 + 10x - 24 = x 2 + 12x - 2x - 24 \u003d x (x + 12) - 2 (x + 12) \u003d \u003d (x + 12) (x - 2);

(x + 12) (x - 2) = 0;

x + 12 = 0 or x - 2 = 0;

X 1 = -12 x 2 = 2 ;

The numbers - 12 and 2 are the roots of this equation.

Answer: x 1 = -12; X 2 = 2.

Solving equations using Vieta's theorem

x 1 and X 2 are the roots of the equation

For example :

X 2 + 3X – 10 = 0

X 1 ·X 2 = - 10, so the roots have different

signs

X 1 + X 2 = - 3, means greater in modulus

root - negative

By selection we find the roots: X 1 = – 5, Х 2 = 2

Properties of the coefficients of a quadratic equation

Let the quadratic equation ax be given 2 + bx + c = 0

If a a + b + c = 0 (i.e. the sum of the coefficients

equation is zero), then X 1 = 1 , X 2 = c/a

If a a - b + c = 0 , or b = a + c , then X 1 = – 1 , X 2 = – s/a .

Example :

137x 2 + 20x – 157 = 0.

a = 137, b = 20, c = -157.

a + b + c = 137 + 20 – 157 =0.

x 1 = 1,

Answer: 1;

0, according to the theorem converse to the Vieta theorem, we obtain the roots: 5; 6, then we return to the roots of the original equation: 2.5; 3." width="640"

Solving equations using the "transfer" method

Roots of quadratic equations ax 2 + bx + c = 0 and y 2 + by + ac = 0 related by the ratio : x = y/a .

Consider the quadratic equation ax ² + bx + c = 0 , where a ≠ 0. Multiplying both sides by a , we get the equation a²x² + abx + ac = 0. Let ah = y , where X = u/a; then we come to the equation y² + bu + ac = 0 , which is equivalent to the given one. its roots at 1 and at 2 find with the help of Vieta's theorem. Finally we get X 1 =y 1 /a and X 2 =y 2 /a .

Solve the equation: 2x 2 - 11x +15 = 0.

Let's transfer the coefficient 2 to the free term

at 2 - 11y + 30 = 0. D0, according to the theorem, the inverse of Vieta's theorem, we get the roots: 5; 6, then we return to the roots of the original equation: 2.5; 3.

Full square selection method

X 2 + 6x - 7 = 0

Let's select a full square on the left side. To do this, we write the expression X 2 + 6x in the following form:

X 2 + 6x = x 2 + 2 x 3

In the resulting expression, the first term is the square of the number X, and the second is the double product X on the 3 , so to get a full square, you need to add 3 2 , because

X 2 + 2 x 3 + 3 2 = (x + 3) 2

We now transform the left side of the equation X 2 + 6x - 7 = 0, adding to it and subtracting 3 2 , we have:

X 2 + 6x - 7 = x 2 + 2 x 3 + 3 2 – 3 2 – 7 =

= (x + 3) 2 – 9 – 7 = (x + 3) 2 – 16

Thus, this equation can be written as follows:

(x + 3) 2 –16 = 0 , i.e. (x + 3) 2 = 16 .

Consequently, x + 3 - 4 = 0 or x + 3 + 4 = 0

X 1 = 1 X 2 = -7

Answer: -7; one.

Graphical way to solve a quadratic equation

Without using formulas, a quadratic equation can be solved graphically

way. Let's solve the equation

To do this, we will build two graphs:

The abscissas of the intersection points of the graphs will be the roots of the equation.

If the graphs intersect at two points, then the equation has two roots.

If the graphs intersect at one point, then the equation has one root.

If the graphs do not intersect, then the equation has no roots.

Answer:

Solving quadratic equations with

compasses and rulers

1. Let's choose a coordinate system.

2. Let's build points S (-b/ 2 a; a+c/ 2 a) is the center of the circle and BUT( 0; 1 ) .

3. Draw a circle with a radius SA .

Abscissas points of intersection of the circle with the x-axis are roots given quadratic equation.

x 1

x 2

Solving quadratic equations using a nomogram

This is an old and undeservedly forgotten way to solve quadratic equations, placed on p. 83 "Four-digit mathematical tables" Bradis V.M.

For the equation

nomogram gives roots

Table XXII. Nomogram for Equation Solving

This nomogram allows, without solving the quadratic equation, to determine the roots of the equation by its coefficients.

Geometric way of solving quadratic equations

An example that has become famous is from the "Algebra" of al-Khwarizmi: X 2 + 10x = 39. In the original, this problem is formulated as follows: "The square and ten roots are equal to 39."

S=x 2 + 10 x + 25 (X 2 + 10 x = 39 )

S= 39 + 25 = 64 , whence follows,

what is the side of the square ABCD ,

those. line segment AB = 8 .

x = 8 - 2,5 - 2,5 = 3

Based on the survey, it was found that:

• The most difficult were the following methods:

Factoring the left side of the equation,

Full square selection method.

• Rational solution methods:

Solution of quadratic equations by formula;

Solving equations using Vieta's theorem

• Has no practical application

Geometric way of solving quadratic equations.

• Never heard of methods before:

Application of the properties of the coefficients of a quadratic equation;

With the help of a nomogram;

Solving quadratic equations with a compass and straightedge;

The method of "transfer" (this method aroused interest among the students).

Conclusion

• these decision methods deserve attention, since they are not all reflected in school mathematics textbooks;

• mastering these techniques will help students save time and solve equations efficiently;

• the need for a quick solution is due to the use of a test system of entrance exams;

THANKS PER ATTENTION!