The parameter \(a\) is a number that can take any value from \(\mathbb(R)\) .

To study an equation/inequality for all values ​​of a parameter means to indicate at what values ​​of the parameter which particular solution a given equation/inequality has.

Examples:

1) the equation \(ax=2\) for all \(a\ne 0\) has a unique solution \(x=\dfrac 2a\), and for \(a=0\) it has no solutions (since then the equation takes the form \(0=2\) ).

2) the equation \(ax=0\) for all \(a\ne 0\) has a unique solution \(x=0\), and for \(a=0\) it has infinitely many solutions, i.e. \(x\in \mathbb(R)\) (since then the equation takes the form \(0=0\) ).

notice, that

I) both sides of the equation cannot be divided by an expression containing a parameter (\(f(a)\) ) if this expression can be equal to zero. But two cases can be considered:
the first when \(f(a)\ne0\) , in which case we can divide both sides of the equality by \(f(a)\) ;
the second case is when \(f(a)=0\) , and in this case we can check each value of \(a\) separately (see example 1, 2).

II) both sides of the inequality cannot be divided by an expression containing a parameter if the sign of this expression is unknown. But three cases can be considered:
the first, when \(f(a)>0\) , and in this case we can divide both sides of the inequality by \(f(a)\) ;
second, when \(f(a)<0\) , и в этом случае при делении обеих частей неравенства на \(f(a)\) мы обязаны поменять знак неравенства на противоположный;
the third is when \(f(a)=0\) , in which case we can check each value of \(a\) individually.

Example:

3) the inequality \(ax>3\) for \(a>0\) has a solution \(x>\dfrac3a\), for \(a<0\) имеет решение \(x<\dfrac3a\) , а при \(a=0\) не имеет решений, т.к. принимает вид \(0>3\) .

Task 1 #1220

Task level: Easier than the Unified State Exam

Solve the equation \(ax+3=0\)

The equation can be rewritten as \(ax=-3\) . Let's consider two cases:

1) \(a=0\) . In this case, the left side is equal to \(0\) , but the right side is not, therefore, the equation has no roots.

2) \(a\ne 0\) . Then \(x=-\dfrac(3)(a)\) .

Answer:

\(a=0 \Rightarrow x\in \varnothing; \\ a\ne 0 \Rightarrow x=-\dfrac(3)(a)\).

Task 2 #1221

Task level: Easier than the Unified State Exam

Solve the equation \(ax+a^2=0\) for all values ​​of the parameter \(a\) .

The equation can be rewritten as \(ax=-a^2\) . Let's consider two cases:

1) \(a=0\) . In this case, the left and right sides are equal to \(0\), therefore, the equation is true for any values ​​of the variable \(x\) .

2) \(a\ne 0\) . Then \(x=-a\) .

Answer:

\(a=0 \Rightarrow x\in \mathbb(R); \\ a\ne 0 \Rightarrow x=-a\).

Task 3 #1222

Task level: Easier than the Unified State Exam

Solve the inequality \(2ax+5\cos\dfrac(\pi)(3)\geqslant 0\) for all values ​​of the parameter \(a\) .

The inequality can be rewritten as \(ax\geqslant -\dfrac(5)(4)\) . Let's consider three cases:

1) \(a=0\) . Then the inequality takes the form \(0\geqslant -\dfrac(5)(4)\) , which is true for any values ​​of the variable \(x\) .

2) \(a>0\) . Then, when dividing both sides of the inequality by \(a\), the sign of the inequality will not change, therefore, \(x\geqslant -\dfrac(5)(4a)\) .

3)\(a<0\) . Тогда при делении на \(a\) обеих частей неравенства знак неравенства изменится, следовательно, \(x\leqslant -\dfrac{5}{4a}\) .

Answer:

\(a=0 \Rightarrow x\in \mathbb(R); \\ a>0 \Rightarrow x\geqslant -\dfrac(5)(4a); \\ a<0 \Rightarrow x\leqslant -\dfrac{5}{4a}\) .

Task 4 #1223

Task level: Easier than the Unified State Exam

Solve the inequality \(a(x^2-6) \geqslant (2-3a^2)x\) for all values ​​of the parameter \(a\) .

Let's transform the inequality to the form: \(ax^2+(3a^2-2)x-6a \geqslant 0\). Let's consider two cases:

1) \(a=0\) . In this case, the inequality becomes linear and takes the form: \(-2x \geqslant 0 \Rightarrow x\leqslant 0\).

2) \(a\ne 0\) . Then the inequality is quadratic. Let's find the discriminant:

\(D=9a^4-12a^2+4+24a^2=(3a^2+2)^2\).

Because \(a^2 \geqslant 0 \Rightarrow D>0\) for any parameter values.

Therefore, the equation \(ax^2+(3a^2-2)x-6a = 0\) always has two roots \(x_1=-3a, x_2=\dfrac(2)(a)\) . Thus, the inequality will take the form:

\[(ax-2)(x+3a) \geqslant 0\]

If \(a>0\) , then \(x_1 \(x\in (-\infty; -3a]\cup \big[\dfrac(2)(a); +\infty)\).

If \(a<0\) , то \(x_1>x_2\) and the branches of the parabola \(y=(ax-2)(x+3a)\) are directed downward, which means that the solution is \(x\in \big[\dfrac(2)(a); -3a]\).

Answer:

\(a=0 \Rightarrow x\leqslant 0; \\ a>0 \Rightarrow x\in (-\infty; -3a]\cup \big[\dfrac(2)(a); +\infty); \ \a<0 \Rightarrow x\in \big[\dfrac{2}{a}; -3a\big]\) .

Task 5 #1851

Task level: Easier than the Unified State Exam

For what \(a\) is the set of solutions to the inequality \((a^2-3a+2)x -a+2\geqslant 0\) contains half-interval \(\).

Answer:

\(a\in (-\infty;\dfrac(4)(3)\big]\cup

Let's consider two cases:

1) \(a+1=0 \Rightarrow a=-1\) . In this case, the equation \((*)\) is equivalent to \(3=0\) , that is, it has no solutions.

Then the whole system is equivalent \(\begin(cases) x\geqslant 2\\ x=2 \end(cases) \Leftrightarrow x=2\)

2) \(a+1\ne 0 \Rightarrow a\ne -1\). In this case, the system is equivalent to: \[\begin(cases) x\geqslant -2a\\ \left[ \begin(gathered) \begin(aligned) &x_1=-2a \\ &x_2=\dfrac3(a+1) \end(aligned) \end( gathered) \right. \end(cases)\]

This system will have one solution if \(x_2\leqslant -2a\) , and two solutions if \(x_2>-2a\) :

2.1) \(\dfrac3(a+1)\leqslant -2a \Rightarrow a<-1 \Rightarrow \) we have one root \(x=-2a\) .

2.2) \(\dfrac3(a+1)>-2a \Rightarrow a>-1 \Rightarrow \) we have two roots \(x_1=-2a, x_2=\dfrac3(a+1)\) .

Answer:

\(a\in(-\infty;-1) \Rightarrow x=-2a\\ a=-1 \Rightarrow x=2\\ a\in(-1;+\infty) \Rightarrow x\in\( -2a;\frac3(a+1)\)\)

As statistics show, many graduates consider finding solutions to problems with a parameter the most difficult thing when preparing for the Unified State Exam 2019 in mathematics. What is this connected with? The fact is that often problems with a parameter require the use of research methods of solution, that is, when calculating the correct answer, you will need not just to apply formulas, but also to find those parametric values ​​at which a certain condition for the roots is satisfied. At the same time, sometimes there is no need to look for the roots themselves.

Nevertheless, all students who are preparing to take the Unified State Exam must cope with solving tasks with parameters. Similar tasks are encountered regularly in certification tests. The Shkolkovo educational portal will help you fill gaps in knowledge and learn how to quickly find solutions to tasks with a parameter in the Unified State Examination in mathematics. Our experts have prepared and presented in an accessible form all the basic theoretical and practical material on this topic. With the Shkolkovo portal, solving problems for selecting a parameter will be easy for you and will not entail any difficulties.

Basic moments

It is important to understand that there is simply no single algorithm for solving parameter selection problems. The methods for finding the correct answer may vary. Solving a mathematical problem with a parameter in the Unified State Examination means finding what the variable is equal to at a certain value of the parameter. If the original equation and inequality can be simplified, this should be done first. In some problems, you can use standard solution methods for this, as if the parameter were an ordinary number.

Have you already read the theoretical material on this topic? To fully assimilate the information when preparing for the Unified State Exam in mathematics, we recommend that you practice completing tasks with a parameter; For each exercise we have provided a complete analysis of the solution and the correct answer. In the corresponding section you will find both simple and more complex tasks. Students can practice solving exercises with parameters, modeled after tasks in the Unified State Exam, online, while in Moscow or any other city in Russia.

A person who knows how to solve problems with parameters knows the theory perfectly and knows how to apply it not mechanically, but with logic. He “understands” the function, “feels” it, considers it his friend or at least a good acquaintance, and does not just know about its existence.

What is an equation with a parameter? Let the equation f (x; a) = 0 be given. If the task is to find all such pairs (x; a) that satisfy this equation, then it is considered as an equation with two equal variables x and a. But we can pose another problem, assuming the variables are unequal. The fact is that if you give the variable a any fixed value, then f (x; a) = 0 turns into an equation with one variable x, and the solutions to this equation naturally depend on the chosen value of a.

The main difficulty associated with solving equations (and especially inequalities) with a parameter is the following: - for some values ​​of the parameter, the equation has no solutions; -with others – has infinitely many solutions; - in the third case, it is solved using the same formulas; - with the fourth – it is solved using other formulas. - If the equation f (x; a) = 0 needs to be solved with respect to the variable X, and a is understood as an arbitrary real number, then the equation is called an equation with parameter a.

Solving an equation with a parameter f (x; a) = 0 means solving a family of equations resulting from the equation f (x; a) = 0 for any real values ​​of the parameter. An equation with a parameter is, in fact, a short representation of an infinite family of equations. Each of the equations of the family is obtained from a given equation with a parameter for a specific value of the parameter. Therefore, the problem of solving an equation with a parameter can be formulated as follows:

It is impossible to write down every equation from an infinite family of equations, but nevertheless, every equation from an infinite family must be solved. This can be done, for example, by dividing the set of all parameter values ​​into subsets according to some appropriate criterion, and then solving the given equation on each of these subsets. Solving Linear Equations

To divide the set of parameter values ​​into subsets, it is useful to use those parameter values ​​at which or when passing through which a qualitative change in the equation occurs. Such parameter values ​​can be called control or special. The art of solving an equation with parameters is precisely to be able to find the control values ​​of the parameter.

Type 1. Equations, inequalities, their systems that must be solved either for any parameter value or for parameter values ​​belonging to a predetermined set. This type of problem is basic when mastering the topic “Problems with parameters”, since the invested work predetermines success in solving problems of all other basic types.

Type 2. Equations, inequalities, their systems, for which it is necessary to determine the number of solutions depending on the value of the parameter (parameters). When solving problems of this type, there is no need either to solve given equations, inequalities, or their systems, or to provide these solutions; In most cases, such unnecessary work is a tactical mistake that leads to unnecessary waste of time. But sometimes a direct solution is the only reasonable way to get the answer when solving a type 2 problem.

Type 3. Equations, inequalities, their systems, for which it is required to find all those parameter values ​​for which the specified equations, inequalities, and their systems have a given number of solutions (in particular, they do not have or have an infinite number of solutions). Problems of type 3 are in some sense the inverse of problems of type 2.

Type 4. Equations, inequalities, their systems and sets, for which, for the required values ​​of the parameter, the set of solutions satisfies the specified conditions in the domain of definition. For example, find the parameter values ​​at which: 1) the equation is satisfied for any value of the variable from a given interval; 2) the set of solutions to the first equation is a subset of the set of solutions to the second equation, etc.

Basic methods (methods) for solving problems with a parameter. Method I (analytical). The analytical method of solving problems with a parameter is the most difficult method, requiring high literacy and the greatest effort to master it. Method II (graphical). Depending on the problem (with variable x and parameter a), graphs are considered either in the Oxy coordinate plane or in the Oxy coordinate plane. Method III (decision regarding parameter). When solving in this way, the variables x and a are assumed to be equal, and the variable with respect to which the analytical solution is considered simpler is selected. After natural simplifications, we return to the original meaning of the variables x and a and complete the solution.

Example 1. Find the values ​​of the parameter a for which the equation a(2a + 3)x + a 2 = a 2 x + 3a has a single negative root. Solution. This equation is equivalent to the following:. If a(a + 3) 0, that is, a 0, a –3, then the equation has a single root x =. X

Example 2: Solve the equation. Solution. Since the denominator of the fraction cannot be equal to zero, we have (b – 1)(x + 3) 0, that is, b 1, x –3. Multiplying both sides of the equation by (b – 1)(x + 3) 0, we obtain the equation: This equation is linear with respect to the variable x. For 4b – 9 = 0, that is b = 2.25, the equation takes the form: For 4b – 9 0, that is b 2.25, the root of the equation is x =. Now we need to check whether there are any values ​​of b for which the found value of x is equal to –3. Thus, for b 1, b 2.25, b –0.4, the equation has a single root x =. Answer: for b 1, b 2.25, b –0.4 root x = for b = 2.25, b = –0.4 there are no solutions; when b = 1 the equation does not make sense.

Types of problems 2 and 3 are distinguished by the fact that when solving them, it is not necessary to obtain an explicit solution, but only to find those parameter values ​​at which this solution satisfies certain conditions. Examples of such conditions for a solution are the following: there is a solution; there is no solution; there is only one solution; there is a positive solution; there are exactly k solutions; there is a solution belonging to the specified interval. In these cases, the graphical method of solving problems with parameters turns out to be very useful.

We can distinguish two types of application of the graphical method when solving the equation f (x) = f (a): On the Oxy plane, the graph y = f (x) and the family of graphs y = f (a) are considered. This also includes problems solved using a “bundle of lines”. This method turns out to be convenient in problems with two unknowns and one parameter. On the Ox plane (which is also called the phase plane), graphs are considered in which x is the argument and a is the value of the function. This method is usually used in problems that involve only one unknown and one parameter (or can be reduced to such).

Example 1. For what values ​​of the parameter a does the equation 3x 4 + 4x 3 – 12x 2 = a have at least three roots? Solution. Let's construct graphs of the functions f (x) = 3x 4 + 4x 3 – 12x 2 and f (x) = a in one coordinate system. We have: f "(x) = 12x x 2 – 24x = 12x(x + 2)(x – 1), f "(x) = 0 at x = –2 (minimum point), at x = 0 (maximum point ) and at x = 1 (maximum point). Let's find the values ​​of the function at the extremum points: f (–2) = –32, f (0) = 0, f (1) = –5. We construct a schematic graph of the function taking into account the extremum points. The graphical model allows us to answer the question posed: the equation 3x 4 + 4x 3 – 12x 2 = a has at least three roots if –5

Example 2. How many roots does the equation have for different values ​​of the parameter a? Solution. The answer to the question posed is related to the number of intersection points of the graph of the semicircle y = and the straight line y = x + a. A straight line that is tangent has the formula y = x +. The given equation has no roots at a; has one root at –2

Example 3. How many solutions does the equation |x + 2| = ax + 1 depending on the parameter a? Solution. You can plot graphs y = |x + 2| and y = ax + 1. But we will do it differently. At x = 0 (21) there are no solutions. Divide the equation by x: and consider two cases: 1) x > –2 or x = 2 2) 2) x –2 or x = 2 2) 2) x

An example of using a “bundle of lines” on a plane. Find the values ​​of the parameter a for which the equation |3x + 3| = ax + 5 has a unique solution. Solution. Equation |3x + 3| = ax + 5 is equivalent to the following system: The equation y – 5 = a(x – 0) defines on the plane a pencil of lines with center A (0; 5). Let's draw straight lines from a bunch of straight lines that will be parallel to the sides of the corner, which is the graph of y = |3x + 3|. These lines l and l 1 intersect the graph y = |3x + 3| at one point. The equations of these lines are y = 3x + 5 and y = –3x + 5. In addition, any line from the pencil located between these lines will also intersect the graph y = |3x + 3| at one point. This means that the required values ​​of the parameter [–3; 3].

Algorithm for solving equations using the phase plane: 1. Find the domain of definition of the equation. 2. Express the parameter a as a function of x. 3. In the xOa coordinate system, we construct a graph of the function a = f(x) for those values ​​of x that are included in the domain of definition of this equation. 4. Find the intersection points of the straight line a = c, where c є (-; +) with the graph of the function a = f (x). If the line a = c intersects the graph a = f(x), then we determine the abscissas of the intersection points. To do this, it is enough to solve the equation a = f(x) for x. 5.Write down the answer.

An example of solving an inequality using the “phase plane”. Solve the inequality x. Solution: By equivalent transition Now on the Ox plane we will construct graphs of functions Points of intersection of the parabola and the straight line x 2 – 2x = –2x x = 0. The condition a –2x is automatically satisfied at a x 2 – 2x Thus, in the left half-plane (x

Diploma

Research skills can be divided into general and specific. The general research skills, the formation and development of which occurs in the process of solving problems with parameters, include: the ability to see behind a given equation with a parameter various classes of equations, characterized by the common presence of the number and type of roots; ability to use analytical and graphic-analytical methods....

Equations and inequalities with a parameter as a means of developing the research skills of students in grades 7-9 (essay, coursework, diploma, test)

Graduate work

Pabout the topic: Equations and inequalities with a parameter as a means of forming research skills of students in grades 7 - 9

The development of creative thinking abilities is impossible outside of problem situations, therefore non-standard tasks are of particular importance in learning. These also include tasks containing a parameter. The mathematical content of these problems does not go beyond the scope of the program, however, solving them, as a rule, causes difficulties for students.

Before the reform of school mathematics education in the 60s, the school curriculum and textbooks had special sections: the study of linear and quadratic equations, the study of systems of linear equations. Where the task was to study equations, inequalities and systems depending on any conditions or parameters.

The program does not currently contain specific references to studies or parameters in equations or inequalities. But they are precisely one of the effective means of mathematics that help solve the problem of forming an intellectual personality set by the program. To eliminate this contradiction, it became necessary to create an elective course on the topic “Equations and inequalities with parameters.” This is precisely what determines the relevance of this work.

Equations and inequalities with parameters are excellent material for real research work, but the school curriculum does not include problems with parameters as a separate topic.

Solving most of the problems in a school mathematics course is aimed at developing in schoolchildren such qualities as mastery of rules and algorithms of action in accordance with current programs, and the ability to conduct basic research.

Research in science means the study of an object in order to identify the patterns of its occurrence, development, and transformation. In the research process, accumulated experience, existing knowledge, as well as methods and methods of studying objects are used. The result of the research should be the acquisition of new knowledge. In the process of educational research, the knowledge and experience accumulated by the student in the study of mathematical objects are synthesized.

When applied to parametric equations and inequalities, the following research skills can be distinguished:

1) The ability to express through a parameter the conditions for a given parametric equation to belong to a particular class of equations;

2) The ability to determine the type of equation and indicate the type of coefficients depending on the parameters;

3) The ability to express through parameters, the conditions for the presence of solutions to a parametric equation;

4) In the case of the presence of roots (solutions), be able to express the conditions for the presence of a particular number of roots (solutions);

5) The ability to express the roots of parametric equations (solutions to inequalities) through parameters.

The developmental nature of equations and inequalities with parameters is determined by their ability to implement many types of mental activity of students:

Development of certain thinking algorithms, Ability to determine the presence and number of roots (in an equation, system);

Solving families of equations that are a consequence of this;

Expressing one variable in terms of another;

Finding the domain of definition of an equation;

Repetition of a large volume of formulas when solving;

Knowledge of appropriate solution methods;

Wide use of verbal and graphic argumentation;

Development of graphic culture of students;

All of the above allows us to talk about the need to study equations and inequalities with parameters in the school mathematics course.

At present, the class of problems with parameters has not yet been clearly methodically worked out. The relevance of choosing the topic of the elective course “Quadratic equations and inequalities with a parameter” is determined by the importance of the topic “Quadratic trinomial and its properties” in the school mathematics course and, at the same time, by the lack of time to consider problems related to the study of a quadratic trinomial containing a parameter.

In our work, we want to show that parameter problems should not be a difficult addition to the main material being studied, which only capable children can master, but can and should be used in a general education school, which will enrich learning with new methods and ideas and help students develop their thinking.

The purpose of the work is to study the place of equations and inequalities with parameters in the algebra course for grades 7–9, to develop an elective course “Quadratic equations and inequalities with a parameter” and methodological recommendations for its implementation.

The object of the study is the process of teaching mathematics in grades 7–9 of a secondary school.

The subject of the research is the content, forms, methods and means of solving equations and inequalities with parameters in a secondary school, ensuring the development of an elective course “Quadratic equations and inequalities with a parameter.”

The research hypothesis is that this elective course will help provide a more in-depth study of the content of the mathematics section “Equations and Inequalities with Parameters”, eliminate discrepancies in the requirements in mathematics for the preparation of school graduates and university applicants, and expand opportunities for the development of mental activity students, if in the process of studying it the following will be used:

· consideration of graphical techniques for solving quadratic equations and inequalities with a parameter using schoolchildren’s work with educational literature;

· solving problems on the study of a quadratic trinomial containing a parameter, using self-control of schoolchildren and mutual control;

· tables for summarizing the material on the topics “Sign of the roots of a square trinomial”, “location of a parabola relative to the abscissa axis”;

· use of various methods for assessing learning outcomes and a cumulative point system;

· studying all the topics of the course, giving the student the opportunity to independently find a way to solve the problem.

In accordance with the purpose, object, subject and hypothesis of the study, the following research objectives are put forward:

· consider general provisions for the study of equations and inequalities with parameters in grades 7–9;

· develop an elective course in algebra “Quadratic equations and inequalities with a parameter” and a methodology for its implementation.

The following methods were used during the study:

· literature analysis;

· analysis of experience in developing elective courses.

Chapter 1. Psychological and pedagogical features studying Topics « Equations and inequalities with parameters" in the course of algebra 7−9 class

§ 1. Age-related, physiological and psychological characteristicsbenefits of schoolchildren in grades 7–9

Middle school age (adolescence) is characterized by rapid growth and development of the whole organism. There is an intensive growth of the body in length (in boys there is an increase of 6–10 centimeters per year, and in girls up to 6–8 centimeters). Ossification of the skeleton continues, bones acquire elasticity and hardness, and muscle strength increases. However, the development of internal organs occurs unevenly, the growth of blood vessels lags behind the growth of the heart, which can cause disruption of the rhythm of its activity and increased heart rate. The pulmonary apparatus develops, breathing becomes rapid at this age. The volume of the brain approaches that of an adult human brain. The control of the cerebral cortex over instincts and emotions improves. However, excitation processes still prevail over inhibition processes. The increased activity of associative fibers begins.

At this age, puberty occurs. The activity of the endocrine glands, in particular the sex glands, increases. Secondary sexual characteristics appear. The teenager’s body exhibits greater fatigue due to dramatic changes in it. A teenager’s perception is more focused, organized and planned than that of a younger schoolchild. The teenager’s attitude towards the observed object is of decisive importance. Attention is voluntary, selective. A teenager can focus on interesting material for a long time. Memorization of concepts, directly related to comprehension, analysis and systematization of information, comes to the fore. Adolescence is characterized by critical thinking. Students of this age are characterized by greater demands on the information provided. The ability for abstract thinking improves. The expression of emotions in teenagers is often quite violent. Anger is especially strong. This age is quite characterized by stubbornness, selfishness, withdrawal into oneself, the severity of emotions, and conflicts with others. These manifestations allowed teachers and psychologists to talk about the crisis of adolescence. The formation of identity requires a person to rethink his connections with others, his place among other people. During adolescence, intensive moral and social formation of personality occurs. The process of formation of moral ideals and moral beliefs is underway. They often have an unstable, contradictory character.

The communication of teenagers with adults differs significantly from the communication of younger schoolchildren. Teenagers often do not consider adults as possible partners for free communication; they perceive adults as a source of organization and support for their lives, and the organizational function of adults is perceived by adolescents most often as only restrictive and regulating.

The number of questions addressed to teachers is reduced. The questions asked relate, first of all, to the organization and content of the life activities of adolescents in cases in which they cannot do without the relevant information and instructions from adults. The number of ethical issues is reduced. Compared to the previous age, the authority of the teacher as a bearer of social norms and a possible assistant in solving complex life problems is significantly reduced.

§ 2. Age characteristics of educational activities

Teaching is the main activity for a teenager. The educational activity of a teenager has its own difficulties and contradictions, but there are also advantages that a teacher can and should rely on. The great advantage of a teenager is his readiness for all types of educational activities, which make him an adult in his own eyes. He is attracted by independent forms of organizing lessons in the classroom, complex educational material, and the opportunity to independently build his cognitive activity outside of school. However, the teenager does not know how to realize this readiness, since he does not know how to carry out new forms of educational activity.

A teenager reacts emotionally to a new academic subject, and for some this reaction disappears quite quickly. Often their general interest in learning and school also decreases. As psychological research shows, the main reason lies in the lack of development of learning skills in students, which does not make it possible to satisfy the current need of age - the need for self-affirmation.

One of the ways to increase the effectiveness of learning is the purposeful formation of learning motives. This is directly related to the satisfaction of the prevailing needs of age. One of these needs is cognitive. When it is satisfied, he develops stable cognitive interests, which determine his positive attitude towards academic subjects. Teenagers are very attracted by the opportunity to expand, enrich their knowledge, penetrate into the essence of the phenomena being studied, and establish cause-and-effect relationships. They experience great emotional satisfaction from research activities. Failure to satisfy cognitive needs and cognitive interests causes not only a state of boredom and indifference, but sometimes a sharply negative attitude towards “uninteresting subjects.” In this case, both the content and the process, methods, and techniques of acquiring knowledge are equally important.

The interests of adolescents differ in the direction of their cognitive activity. Some students prefer descriptive material, they are attracted by individual facts, others strive to understand the essence of the phenomena being studied, explain them from the point of view of theory, others are more active in using knowledge in practical activities, others - to creative, research activities. 15]

Along with cognitive interests, an understanding of the significance of knowledge is essential for a positive attitude of adolescents towards learning. It is very important for them to realize and comprehend the vital significance of knowledge and, above all, its significance for personal development. A teenager likes many educational subjects because they meet his needs as a comprehensively developed person. Beliefs and interests, merging together, create an increased emotional tone in adolescents and determine their active attitude towards learning.

If a teenager does not see the vital importance of knowledge, then he may develop negative beliefs and a negative attitude towards existing academic subjects. Of significant importance when teenagers have a negative attitude towards learning is their awareness and experience of failure in mastering certain academic subjects. Fear of failure, fear of defeat sometimes leads teenagers to look for plausible reasons not to go to school or leave class. The emotional well-being of a teenager largely depends on the assessment of his educational activities by adults. Often the meaning of assessment for a teenager is the desire to achieve success in the educational process and thereby gain confidence in their abilities and capabilities. This is due to such a dominant need of age as the need to realize and evaluate oneself as a person, one’s strengths and weaknesses. Research shows that it is during adolescence that self-esteem plays a dominant role. It is very important for the emotional well-being of a teenager that assessment and self-esteem coincide. Otherwise, internal and sometimes external conflict arises.

In middle grades, students begin to study and master the basics of science. Students will have to master a large amount of knowledge. The material to be mastered, on the one hand, requires a higher level of educational, cognitive and mental activity than before, and on the other hand, is aimed at their development. Students must master the system of scientific concepts and terms, therefore new academic subjects make new demands on the methods of acquiring knowledge and are aimed at developing higher-level intelligence - theoretical, formal, reflective thinking. This kind of thinking is typical for adolescence, but it begins to develop in younger teenagers.

What is new in the development of a teenager’s thinking lies in his attitude to intellectual tasks as those that require their preliminary mental solution. The ability to operate with hypotheses in solving intellectual problems is a teenager’s most important acquisition in analyzing reality. Conjectural thinking is a distinctive tool of scientific reasoning, which is why it is called reflective thinking. Although the assimilation of scientific concepts at school in itself creates a number of objective conditions for the formation of theoretical thinking in schoolchildren, however, it is not formed in everyone: different students may have different levels and quality of its actual formation.

Theoretical thinking can be formed not only by mastering school knowledge. Speech becomes controlled and manageable, and in some personally significant situations, adolescents especially strive to speak beautifully and correctly. In the process and as a result of the assimilation of scientific concepts, new content of thinking, new forms of intellectual activity are created. A significant indicator of inadequate assimilation of theoretical knowledge is the inability of a teenager to solve problems that require the use of this knowledge.

The central place begins to be occupied by the analysis of the content of the material, its originality and internal logic. Some teenagers are characterized by flexibility in choosing ways to learn, others prefer one method, and some try to organize and logically process any material. The ability to logically process material often develops spontaneously in adolescents. Not only academic performance, depth and strength of knowledge, but also the possibility of further development of the teenager’s intelligence and abilities depend on this.

§ 3. Organization of educational activitiescharacteristics of schoolchildren in grades 7–9

Organizing the educational activities of adolescents is the most important and complex task. A middle school student is quite capable of understanding the arguments of a teacher or parent and agreeing with reasonable arguments. However, due to the peculiarities of thinking characteristic of this age, a teenager will no longer be satisfied with the process of communicating information in a ready-made, complete form. He will want to check their reliability, to make sure that his judgments are correct. Disputes with teachers, parents, and friends are a characteristic feature of this age. Their important role is that they allow you to exchange opinions on a topic, check the truth of your views and generally accepted views, and express yourself. In particular, in teaching, the introduction of problem-based tasks has a great effect. The foundations of this approach to teaching were developed back in the 60s and 70s of the 20th century by domestic teachers. The basis of all actions in the problem-based approach is the awareness of the lack of knowledge to solve specific problems and the resolution of contradictions. In modern conditions, this approach should be implemented in the context of the level of achievements of modern science and the tasks of socialization of students.

It is important to encourage independent thinking, the student expressing his own point of view, the ability to compare, find common and distinctive features, highlight the main thing, establish cause-and-effect relationships, and draw conclusions.

For a teenager, interesting and fascinating information that stimulates his imagination and makes him think will be of great importance. A good effect is achieved by periodically changing types of activities - not only in class, but also when preparing homework. A variety of types of work can become a very effective means of increasing attention and an important way to prevent general physical fatigue, associated both with the educational load and with the general process of radical restructuring of the body during puberty. 20]

Before studying the relevant sections of the school curriculum, students often already have certain everyday ideas and concepts that allow them to navigate fairly well in everyday practice. This circumstance, in cases where their attention is not specifically drawn to the connection of the knowledge they acquire with practical life, deprives many students of the need to acquire and assimilate new knowledge, since the latter has no practical meaning for them.

Moral ideals and moral beliefs of adolescents are formed under the influence of numerous factors, in particular, strengthening the educational potential of learning. In solving complex life problems, more attention should be paid to indirect methods of influencing the consciousness of adolescents: not presenting a ready-made moral truth, but leading to it, and not expressing categorical judgments that adolescents can perceive with hostility.

§ 4. Educational research in the system of basic requirements for the content of mathematical education and the level of students’ preparation

Equations and inequalities with parameters are excellent material for real research work. But the school curriculum does not include problems with parameters as a separate topic.

Let us analyze various sections of the educational standard of Russian schools from the point of view of identifying issues related to learning to solve problems with parameters.

Studying the program material allows primary school students to “get an initial understanding of a problem with parameters that can be reduced to linear and quadratic” and learn how to construct graphs of functions and explore the location of these graphs in the coordinate plane depending on the values ​​of the parameters included in the formula.

The line "function" does not mention the word "parameter" but says that students have the opportunity to "systematize and develop knowledge about function; develop a graphic culture, learn to “read” graphs fluently, reflect the properties of a function on a graph.”

Having analyzed school textbooks on algebra by such groups of authors as: Alimov Sh. A. et al., Makarychev Yu. N. et al., Mordkovich A. G. et al., we come to the conclusion that problems with parameters in these textbooks are given little attention. In textbooks for 7th grade there are several examples on studying the question of the number of roots of a linear equation, on studying the dependence of the location of the graph of a linear function y = kh and y = kh + b depending on the values ​​of k. In textbooks for grades 8–9, in sections such as “Problems for extracurricular work” or “Repetition exercises”, 2–3 tasks are given for the study of roots in quadratic and biquadratic equations with parameters, the location of the graph of a quadratic function depending on the values ​​of the parameters.

In the mathematics program for schools and classes with in-depth study, the explanatory note says “the section “Requirements for the mathematical preparation of students” sets the approximate amount of knowledge, skills and abilities that schoolchildren must master. This scope, of course, includes those knowledge, abilities and skills, the mandatory acquisition of which by all students is provided for by the requirements of the general education school program; however, a different, higher quality of their formation is proposed. Students must acquire the ability to solve problems of a higher level of complexity than the required level of complexity, accurately and competently formulate the theoretical principles they have studied and present their own reasoning when solving problems...”

Let's analyze some textbooks for students with advanced study of mathematics.

The formulation of such problems and their solutions do not go beyond the scope of the school curriculum, but the difficulties that students encounter are explained, firstly, by the presence of a parameter, and secondly, by the branching of the solution and answers. However, the practice of solving problems with parameters is useful for developing and strengthening the ability for independent logical thinking and for enriching mathematical culture.

In general education classes at school, as a rule, negligible attention is paid to such tasks. Since solving equations and inequalities with parameters is, perhaps, the most difficult section of a course in elementary mathematics, it is hardly advisable to teach solving such problems with parameters to the mass of schoolchildren, but strong students who show interest, inclination and ability in mathematics, who strive to act independently, teach It is certainly necessary to solve such problems. Therefore, along with such traditional content-methodological lines of the school mathematics course as functional, numerical, geometric, the line of equations and the line of identical transformations, the line of parameters must also take a certain position. The content of the material and the requirements for students on the topic “problems with parameters” should, of course, be determined by the level of mathematical preparation of the entire class as a whole and each individual.

The teacher must help meet the needs and requests of schoolchildren who show interest, aptitude and ability in the subject. On issues of interest to students, consultations, clubs, additional classes and electives can be organized. This fully applies to the issue of problems with parameters.

§ 5. Educational research in the structure of cognitive activity of schoolchildren

At the moment, the issue of preparing a student who strives to act independently, beyond the requirements of the teacher, who does not limit the scope of his interests and active research to the educational material offered to him, who knows how to present and argueably defend his solution to a particular problem, who knows how to specify or, conversely, generalize the result under consideration, identify cause-and-effect relationships, etc. In this regard, studies that analyze the fundamentals of the psychology of mathematical creativity in school-age children, examine the problem of managing the process of mental activity of students, forming and developing their skills to independently acquire knowledge, apply knowledge, replenish and systematize it, the problem of increasing the activity of cognitive activity of schoolchildren (L.S. Vygotsky, P. Ya. Krutetsky, N.A. Menchinskaya, S.L. Rubinstein, L.M. Friedman, etc.).

The research method of teaching includes two research methods: educational and scientific.

Solving a significant part of the problems of a school mathematics course presupposes that students have developed such qualities as mastery of the rules and algorithms of actions in accordance with current programs, and the ability to conduct basic research. Research in science means the study of an object in order to identify the patterns of its occurrence and development of transformation. In the research process, accumulated previous experience, existing knowledge, as well as methods and methods (techniques) of studying objects are used. The result of research should be the acquisition of new scientific knowledge.

When applied to the process of teaching mathematics in secondary school, it is important to note the following: the main components of educational research include the formulation of a research problem, awareness of its goals, preliminary analysis of available information on the issue under consideration, conditions and methods for solving problems close to the research problem, proposing and formulating the initial hypotheses, analysis and generalization of the results obtained during the study, verification of the initial hypothesis based on the obtained facts, final formulation of new results, patterns, properties, determination of the place of the found solution to the problem posed in the system of existing knowledge. The main place among the objects of educational research is occupied by those concepts and relations of the school mathematics course, in the process of studying which the patterns of their change and transformation, the conditions for their implementation, uniqueness, etc. are revealed.

Serious potential in the formation of such research skills as the ability to purposefully observe, compare, put forward, prove or disprove a hypothesis, the ability to generalize, etc., has tasks on constructing in a geometry course, equations and inequalities with parameters in an algebra course, the so-called dynamic problems, in the process of solving which students master the basic techniques of mental activity: analysis, synthesis (analysis through synthesis, synthesis through analysis), generalization, specification, etc., purposefully observes changing objects, puts forward and formulates a hypothesis regarding the properties of the objects under consideration, tests the put forward hypothesis, determines the place of the learned result in the system of previously acquired knowledge, its practical significance. The organization of educational research by the teacher is of decisive importance. Teaching methods of mental activity, the ability to carry out elements of research - these goals constantly attract the attention of the teacher, encouraging him to find answers to many methodological questions related to solving the problem under consideration.

Studying many issues of the program provides excellent opportunities to create a more holistic and complete picture associated with the consideration of a particular problem.

In the process of educational research, the knowledge and experience accumulated by the student in the study of mathematical objects are synthesized. Of decisive importance in organizing a student’s educational research is attracting his attention (first involuntary, and then voluntary), creating conditions for observation: ensuring deep awareness, the necessary attitude of the student to the work, the object of study ("https://site", 9).

In school mathematics teaching, there are two closely related levels of educational research: empirical and theoretical. The first is characterized by observation of individual facts, their classification, and the establishment of a logical connection between them, verifiable by experience. The theoretical level of educational research is different in that as a result the student formulates general mathematical laws, on the basis of which not only new facts, but also those obtained at the empirical level are more deeply interpreted.

Conducting educational research requires the student to use both specific methods, characteristic only for mathematics, and general ones; analysis, synthesis, induction, deduction, etc., used in the study of objects and phenomena of various school disciplines.

The organization of educational research by the teacher is of decisive importance. In application to the process of teaching mathematics in secondary school, it is important to note the following: the main components of educational research include the formulation of a research problem, awareness of its goals, preliminary analysis of available information on the issue under consideration, conditions and methods for solving problems close to the research problem, proposing and formulating the initial hypothesis, analysis and generalization of the results obtained during the study, verification of the initial hypothesis based on the facts obtained, the final formulation of new results, patterns, properties, determination of the place of the found solution to the problem posed in the system of existing knowledge. The main place among the objects of educational research is occupied by those concepts and relations of the school mathematics course, in the process of studying which the patterns of their change and transformation, the conditions for their implementation, uniqueness, etc. are revealed.

Suitable material for educational research is material related to the study of the functions studied in the algebra course. As an example, consider a linear function.

Assignment: Examine a linear function for even and odd. Hint: Consider the following cases:

2) a = 0 and b? 0;

3) a? 0 and b = 0;

4) a? 0 and b? 0.

As a result of the research, fill out the table, indicating the result obtained at the intersection of the corresponding row and column.

As a result of the solution, students should receive the following table:

	even and odd
	odd	neither even nor odd

Its symmetry evokes a feeling of satisfaction and confidence in the correctness of filling.

The formation of methods of mental activity plays a significant role both in the overall development of schoolchildren and in order to instill in them the skills of conducting educational research (in general or in fragments).

The result of educational research is subjectively new knowledge about the properties of the object (relationship) under consideration and their practical applications. These properties may or may not be included in a high school math curriculum. It is important to note that the novelty of the result of a student’s activity is determined both by the nature of the search for a way to carry out the activity, the method of activity itself, and the place of the result obtained in the knowledge system of that student.

The method of teaching mathematics using educational research is called research, regardless of whether the educational research scheme is implemented in full or in fragments.

When implementing each stage of educational research, elements of both performing and creative activity are necessarily present. This is most clearly observed in the case of a student independently conducting a particular study. Also, during educational research, some stages can be implemented by the teacher, others by the student himself. The level of independence depends on many factors, in particular, on the level of formation, the ability to observe a particular object (process), the ability to focus one’s attention on the same subject, sometimes for quite a long time, the ability to see a problem, clearly and unambiguously formulate, the ability to find and use suitable (sometimes unexpected) associations, the ability to concentrately analyze existing knowledge in order to select the necessary information, etc.

It is also impossible to overestimate the influence of a student’s imagination, intuition, inspiration, ability (and perhaps talent or genius) on the success of his research activities.

§ 6 . Research in the system of teaching methods

More than a dozen fundamental studies have been devoted to teaching methods, on which the considerable success of the work of the teacher and the school as a whole depends. And, despite this, the problem of teaching methods, both in teaching theory and in pedagogical practice, remains very relevant. The concept of teaching method is quite complex. This is due to the exceptional complexity of the process that this category is intended to reflect. Many authors consider the teaching method to be a way of organizing the educational and cognitive activities of students.

The word “method” is of Greek origin and translated into Russian means research, method. “Method - in the most general sense - is a way of achieving a goal, a certain way of ordering activity.” It is obvious that in the learning process the method acts as a connection between the activities of the teacher and students to achieve certain educational goals. From this point of view, each teaching method organically includes the teaching work of the teacher (presentation, explanation of the material being studied) and the organization of active educational and cognitive activity of students. Thus, the concept of teaching method reflects:

1. Methods of teaching work of the teacher and methods of educational work of students in their interrelation.

2. The specifics of their work to achieve various learning goals. Thus, teaching methods are ways of joint activity between teacher and students aimed at solving learning problems, that is, didactic tasks.

That is, teaching methods should be understood as the methods of the teacher’s teaching work and the organization of educational and cognitive activities of students to solve various didactic tasks aimed at mastering the material being studied. One of the acute problems of modern didactics is the problem of classifying teaching methods. Currently there is no single point of view on this issue. Due to the fact that different authors base the division of teaching methods into groups and subgroups on different criteria, there are a number of classifications. But in the 20s in Soviet pedagogy there was a struggle against the methods of scholastic teaching and mechanical rote learning that flourished in the old school and a search was made for methods that would ensure conscious, active and creative acquisition of knowledge by students. It was in those years that teacher B.V. Vieviatsky developed the position that there can only be two methods in teaching: the research method and the method of ready-made knowledge. The method of ready-made knowledge, naturally, was criticized. The research method, the essence of which boiled down to the fact that students supposedly should learn everything on the basis of observation and analysis of the phenomena being studied, independently approaching the necessary conclusions, was recognized as the most important teaching method. The same research method in the classroom may not be applied to all topics.

Also, the essence of this method is that the teacher breaks down a problematic problem into subproblems, and students carry out individual steps to find its solution. Each step involves creative activity, but there is no holistic solution to the problem yet. During research, students master the methods of scientific knowledge and develop experience in research activities. The activity of students trained using this method is to master the techniques of independently posing problems, finding ways to solve them, research tasks, posing and developing problems that teachers present to them.

It can also be noted that psychology establishes some patterns with developmental psychology. Before you start working with students using methods, you need to thoroughly study the methods of studying their developmental psychology. Familiarity with these methods can be of practical benefit directly to the organizers of this process, since these methods are suitable not only for one’s own scientific research, but also for organizing an in-depth study of children for practical educational purposes. An individual approach to training and education assumes good knowledge and understanding of the individual psychological characteristics of students and the uniqueness of their personality. Consequently, the teacher needs to master the ability to study students, to see not a gray, homogeneous student mass, but a collective in which everyone represents something special, individual, and unique. Such study is the task of every teacher, but it still needs to be properly organized.

One of the main methods of organization is the observation method. Of course, the psyche cannot be observed directly. This method involves indirect knowledge of the individual characteristics of the human psyche through the study of his behavior. That is, here it is necessary to judge the student by individual characteristics (actions, deeds, speech, appearance, etc.), the mental state of the student (processes of perception, memory, thinking, imagination, etc.), and by his personality traits , temperament, character. All this is necessary for the student with whom the teacher works using the research method of teaching when performing some tasks.

Solving a significant part of the problems of a school mathematics course presupposes that students have developed such qualities as mastery of rules and algorithms of action in accordance with current programs, and the ability to conduct basic research. Research in science means the study of an object to identify the patterns of its occurrence, development, and transformation. In the research process, accumulated previous experience, existing knowledge, as well as methods and methods (techniques) of studying objects are used. The result of the research should be the acquisition of new scientific knowledge. Teaching methods of mental activity, the ability to carry out elements of research - these goals constantly attract the attention of the teacher, encouraging him to find answers to many methodological questions related to solving the problem under consideration. Studying many issues of the program provides excellent opportunities to create a more holistic and complete picture associated with the consideration of a particular task. The research method of teaching mathematics naturally fits into the classification of teaching methods depending on the nature of the students’ activities and the degree of their cognitive independence. To successfully organize a student’s research activity, the teacher must understand and take into account both his personal qualities and the procedural features of this type of activity, as well as the student’s level of proficiency in the course material studied. It is impossible to overestimate the influence of a student’s imagination, intuition, inspiration, and ability on the success of his research activities.

The forms of tasks in the research method can be different. These can be tasks that can be quickly solved in class and at home, or tasks that require an entire lesson. Most research assignments should be small search assignments that require completion of all or most steps of the research process. Their complete solution will ensure that the research method fulfills its functions. The stages of the research process are as follows:

1 Purposeful observation and comparison of facts and phenomena.

Identification of unclear phenomena to be investigated.

Preliminary analysis of available information on the issue under consideration.

4. Proposition and formulation of a hypothesis.

5. Construction of a research plan.

Implementation of the plan, clarifying the connections of the phenomenon being studied with others.

Formulation of new results, patterns, properties, determination of the place of the found solution to the assigned research in the system of existing knowledge.

Checking the solution found.

Practical conclusions about the possible application of new knowledge.

§ 7 . Ability to research in systemswe have special knowledge

Skill is the conscious application of the student’s knowledge and skills to perform complex actions in various conditions, i.e., to solve relevant problems, because the execution of each complex action acts for the student as a solution to the problem.

Research skills can be divided into general and specific. The general research skills, the formation and development of which occurs in the process of solving problems with parameters, include: the ability to see behind a given equation with a parameter various classes of equations, characterized by the common presence of the number and type of roots; ability to use analytical and graphic-analytical methods.

Special research skills include skills that are formed and developed in the process of solving a specific class of problems.

When solving linear equations containing a parameter, the following special skills are formed:

§ The ability to identify special parameter values ​​at which a given linear equation has:

Single root;

An infinite number of roots;

3) Has no roots;

Ability to interpret the answer in the language of the original task. Special research skills, the formation and development of which occurs in the process of solving linear inequalities containing a parameter, include:

§ The ability to see the coefficient of the unknown and the free term as a function of the parameter;

§ The ability to identify special parameter values ​​at which a given linear inequality has as a solution:

1) Interval;

2) Has no solutions;

§ Ability to interpret the answer in the language of the original task. Special research skills, the formation and development of which occurs in the process of solving quadratic equations containing a parameter, include:

§ The ability to identify a special value of a parameter at which the leading coefficient becomes zero, i.e. the equation becomes linear and to find a solution to the resulting equation for the identified special values ​​of the parameter;

§ Ability to solve the question of the presence and number of roots of a given quadratic equation depending on the sign of the discriminant;

§ Ability to express the roots of a quadratic equation (if available) through a parameter;

Among the special research skills, the formation and development of which occurs in the process of solving fractional-rational equations containing a parameter that can be reduced to quadratic ones, include:

§ Ability to reduce a fractional rational equation containing a parameter to a quadratic equation containing a parameter.

Special research skills, the formation and development of which occurs in the process of solving quadratic inequalities containing a parameter, include:

§ The ability to identify a special value of a parameter at which the leading coefficient becomes zero, that is, the inequality becomes linear and to find many solutions to the resulting inequality for special values ​​of the parameter;

§ Ability to express the set of solutions to a quadratic inequality through a parameter.

Listed below are educational skills that translate into teaching and research, as well as research skills.

6−7 grade:

- quickly use old knowledge in the situation of acquiring new ones;

- freely transfer a complex of mental actions from one material to another, from one subject to another;

distribute the acquired knowledge to a large set of objects;

combine the process of “collapse” and “unfoldment” of knowledge;

purposefully summarize the ideas of the text by highlighting the main thoughts in its segments and parts;

systematize and classify information;

— compare information on systems of characteristics, highlighting similarities and differences;

- be able to connect symbolic language with written and oral speech;

— analyze and plan methods for future work;

“connect” quickly and freely the components of new knowledge;

be able to succinctly present the main thoughts and facts of the text;

- obtain new knowledge by moving from system-forming knowledge to the specific with the help of diagrams, tables, notes, etc.;

use various forms of recording during a lengthy listening process;

choose optimal solutions;

prove or disprove using interrelated techniques;

- use various types of analysis and synthesis;

- consider the problem from different points of view;

— express a judgment in the form of an algorithm of thoughts.

Mathematical education in the processes of formation of thinking or mental development of students should be and is given a special place, because the means of teaching mathematics most effectively influence many of the basic components of the holistic personality and, above all, thinking.

Thus, special attention is paid to the development of the student’s thinking, since it is precisely this that is connected with all other mental functions: imagination, flexibility of mind, breadth and depth of thought, etc. Let us note that, when considering the development of thinking in the context of student-centered learning, one should remember that a necessary condition for the implementation of such development is the individualization of learning. It is this that ensures that the characteristics of the mental activity of students of various categories are taken into account.

The path to creativity is individual. At the same time, all students in the process of studying mathematics should feel its creative nature, become acquainted in the process of learning mathematics with some skills of creative activity that they will need in their future life and activities. To solve this complex problem, teaching mathematics must be structured so that the student often looks for new combinations, transforming things, phenomena, processes of reality, and looks for unknown connections between objects.

An excellent way to introduce students to creative activity when teaching mathematics is independent work in all its forms and manifestations. Very fundamental in this regard is the statement of Academician P. L. Kapitsa that independence is one of the most basic qualities of a creative personality, since the cultivation of creative abilities in a person is based on the development of independent thinking.

The level of preparedness of students and study groups for independent creative activity can be determined by answering the following questions:

How effectively can schoolchildren use notes, reference notes, and read diagrams and different types of tables?

Do students know how to objectively evaluate the proposed ideas when solving a problem problem by the teacher, and take into account the possibility of their application? 3) How quickly do schoolchildren move from one way of solving a problem to another? 4) Analyze the effectiveness of orienting students during the lesson to self-organization of independent work; 5) Explore students' ability to model and solve problems flexibly.

Chapter 2. Methodological analysis of the topic “Equations and inequalities with parameters” and development of an elective course “Quadratic equations and inequalities with a parameter”

§ 1. Role And place parametric equations And inequalities in the formation research skillth students

Despite the fact that the secondary school mathematics curriculum does not explicitly mention problems with parameters, it would be a mistake to say that the issue of solving problems with parameters is in no way addressed in the school mathematics course. Suffice it to recall the school equations: ax2+bx+c=0, y=khx, y=khx+b, ax=b, in which a, b, c, k are nothing more than parameters. But within the framework of the school course, attention is not focused on such a concept, the parameter, how it differs from the unknown.

Experience shows that problems with parameters are the most complex section of elementary mathematics in logical and technical terms, although from a formal point of view the mathematical content of such problems does not go beyond the limits of programs. This is caused by different points of view on the parameter. On the one hand, a parameter can be considered as a variable, which is considered a constant value when solving equations and inequalities; on the other hand, a parameter is a quantity whose numerical value is not given, but must be considered known, and the parameter can take arbitrary values, i.e. . the parameter, being a fixed but unknown number, has a dual nature. Firstly, the assumed knownness allows the parameter to be treated as a number, and secondly, the degree of freedom is limited by its unknownness.

In each of the descriptions of the nature of the parameters, there is uncertainty - at what stages of the solution the parameter can be considered as a constant and when it plays the role of a variable. All these contradictory characteristics of the parameter can cause a certain psychological barrier in students at the very beginning of their acquaintance.

In this regard, at the initial stage of getting to know the parameter, it is very useful to resort to a visual and graphical interpretation of the results obtained as often as possible. This not only allows students to overcome the natural uncertainty of the parameter, but also gives the teacher the opportunity, in parallel, as propaedeutics, to teach students to use graphical methods of proof when solving problems. We should also not forget that the use of at least schematic graphic illustrations in some cases helps to determine the direction of research, and sometimes allows us to immediately select the key to solving a problem. Indeed, for certain types of problems, even a primitive drawing, far from a real graph, makes it possible to avoid various types of errors and obtain an answer to an equation or inequality in a simpler way.

Solving mathematical problems in general is the most difficult part of schoolchildren’s activities when studying mathematics and this is explained by the fact that solving problems requires a fairly high level of development of intelligence of the highest level, i.e. theoretical, formal and reflective thinking, and such thinking, as already noted, still developing during adolescence.

Course work

Performer: Bugrov S K.

The study of many physical processes and geometric patterns often leads to solving problems with parameters. Some universities also include equations, inequalities and their systems in exam papers, which are often very complex and require a non-standard approach to solution. At school, this one of the most difficult sections of the school mathematics course is considered only in a few elective classes.

In preparing this work, I set the goal of a deeper study of this topic, identifying the most rational solution that quickly leads to an answer. In my opinion, the graphical method is a convenient and fast way to solve equations and inequalities with parameters.

My essay discusses frequently encountered types of equations, inequalities and their systems, and I hope that the knowledge I gained in the process of work will help me when passing school exams and when entering a university.

Inequality

¦(a, b, c, …, k, x)>j(a, b, c, …, k, x), (1)

where a, b, c, …, k are parameters, and x is a real variable, is called an inequality with one unknown containing parameters.

Any system of parameter values ​​a = a0, b = b0, c = c0, ..., k = k0, for some function

¦(a, b, c, …, k, x) and

j(a, b, c, …, k, x

make sense in the domain of real numbers, called a system of permissible parameter values.

is called a valid value of x if

¦(a, b, c, …, k, x) and

j(a, b, c, …, k, x

take valid values ​​for any admissible system of parameter values.

The set of all admissible values ​​of x is called the domain of definition of inequality (1).

A real number x0 is called a partial solution of inequality (1) if the inequality

¦(a, b, c, …, k, x0)>j(a, b, c, …, k, x0)

true for any system of permissible parameter values.

The set of all particular solutions to inequality (1) is called the general solution of this inequality.

Solving inequality (1) means indicating at what values ​​of the parameters a general solution exists and what it is.

Two inequalities

¦(a, b, c, …, k, x)>j(a, b, c, …, k, x) and (1)

z(a, b, c, …, k, x)>y(a, b, c, …, k, x) (2)

are called equivalent if they have the same general solutions for the same set of systems of admissible parameter values.

We find the domain of definition of this inequality.

We reduce the inequality to an equation.

We express a as a function of x.

In the xOa coordinate system, we construct graphs of the functions a =¦ (x) for those values ​​of x that are included in the domain of definition of this inequality.

We find sets of points that satisfy this inequality.

Let's explore the influence of the parameter on the result.

Let's find the abscissa of the intersection points of the graphs.

let's set a straight line a=const and shift it from -¥ to +¥

We write down the answer.

This is just one of the algorithms for solving inequalities with parameters using the xOa coordinate system. Other solution methods are also possible, using the standard xOy coordinate system.

§3. Examples

I. For all admissible values ​​of parameter a, solve the inequality

In the domain of definition of the parameter a, defined by the system of inequalities

this inequality is equivalent to the system of inequalities

If , then the solutions to the original inequality fill the interval.

II. At what values ​​of parameter a does the system have a solution?

Let's find the roots of the trinomial on the left side of the inequality -

(*)

The straight lines defined by equalities (*) divide the coordinate plane aOx into four regions, in each of which there is a square trinomial

maintains a constant sign. Equation (2) defines a circle of radius 2 centered at the origin. Then the solution to the original system will be the intersection of shaded

region with a circle, where , and the values ​​of and are found from the system

and the values ​​and are found from the system

Solving these systems, we obtain that

III. Solve inequality depending on the values ​​of parameter a.

Finding the range of acceptable values ​​–

Let's construct a graph of the function in the xOy coordinate system.

when the inequality has no solutions.

at for solution x satisfies the relation , Where

Answer: Solutions to the inequality exist when

Where , and when solving ; when deciding .

IV. Solve inequality

Finding ODZ or discontinuity lines (asymptotes)

Let's find the equations of functions whose graphs need to be constructed in the UCS; why let's move on to equality:

Let's factorize the numerator.

because That

Let us divide both sides of the equality by . But it is a solution: the left side of the equation is equal to the right side and is equal to zero at .

3. We build graphs of functions in the UCS xOa

and number the resulting areas (the axes do not play a role). This resulted in nine regions.

4. We are looking for which of the areas is suitable for this inequality, for which we take a point from the area and substitute it into the inequality.

For clarity, let's make a table.

		inequality:

5. Find the intersection points of the graphs

6. Let's set the straight line a=const and shift it from -¥ to +¥.

at

there are no solutions

at

Bibliography

Dalinger V. A. “Geometry helps algebra.” Publishing house “School - Press”. Moscow 1996

Dalinger V. A. “Everything to ensure success in final and entrance exams in mathematics.” Publishing house of Omsk Pedagogical University. Omsk 1995

Okunev A. A. “Graphical solution of equations with parameters.” Publishing house “School - Press”. Moscow 1986

Pismensky D. T. “Mathematics for high school students.” Publishing house “Iris”. Moscow 1996

Yastribinetsky G. A. “Equations and inequalities containing parameters.” Publishing house “Prosveshcheniye”. Moscow 1972

G. Korn and T. Korn “Handbook of Mathematics.” Publishing house “Science” physical and mathematical literature. Moscow 1977

Amelkin V.V. and Rabtsevich V.L. “Problems with parameters”. Publishing house “Asar”. Moscow 1996

Municipal autonomous educational institution "Lyceum No. 1" of Novtroitsk

Research

Methods for solving equations and inequalities with a parameter

Math modeling

Completed:

student 11 A class MOAU

"Lyceum No. 1"

Supervisor:

higher education teacher

Novotroitsk

Introduction. 3

Parameter. 5

Methods for solving trigonometric equations with a parameter. 9

Methods for solving exponential and logarithmic equations and inequalities with a parameter. 17

Methods for solving systems of equations and inequalities. 22

Conclusion. 31

List of used literature... 32

Introduction

Equations with a parameter cause great difficulty for students in grades 9-11. This is due to the fact that solving such equations requires not only knowledge of the properties of functions and equations, the ability to perform algebraic transformations, but also high logical culture and research techniques.

Difficulties when studying this type of equations are associated with their following features:

· abundance of formulas and methods used to solve equations of this type;

· the ability to solve the same equation containing a parameter in different ways.

Relevance topics is determined by the insufficient content of problems on this topic in the textbook “Algebra 11th grade”.

The importance of this topic is determined by the need to be able to solve such equations with parameters both when passing the Unified State Exam and during entrance exams to higher educational institutions.

Object of study: tasks with parameters.

The purpose of this work:

Identify, justify and clearly demonstrate methods for solving all types of equations with parameters;

Solve equations with parameters;

Deepen theoretical knowledge of solving equations with parameters;

To achieve this goal, it is necessary to solve the following tasks:

1. Define the concepts of an equation with parameters;

2. Show ways to solve equations with parameters.

The dignity of my work is as follows: algorithms for solving equations with parameters are indicated; problems are often found in various exams and olympiads. The work will help students pass the Unified State Exam.

My actions:

1. Select and study literature;

2. Solve selected problems;

Parameter

There are several definitions parameter:

- Parameter - this is a quantity included in formulas and expressions, the value of which is constant within the limits of the problem under consideration, but in another task changes its values ​​(- “Explanatory Dictionary of Mathematical Terms”).

- Variables a, b, c, …, k, which are considered constant when solving an equation or inequality are called parameters, and the equation (inequality) itself is called an equation (inequality) containing parameters (- “Mathematics Tutor”, Rostov-on-Don “Phoenix” 1997).

The solution to most equations containing a parameter comes down to quadratic equations with parameter. Therefore, in order to learn how to solve exponential, logarithmic, trigonometric equations and systems of equations with a parameter, you must first acquire solving skills quadratic equations with parameter.

Equation of the form ax2 + bx+ c=0 , where x is an unknown, a, b, c are expressions that depend only on the parameters, a¹0 is called quadratic equation relative to x. We will consider only those parameter values ​​for which a, b, c are valid.

Parameter control values

To solve quadratic equations with a parameter, it is necessary to find parameter control values.

Parameter control values– those values ​​at which it turns to 0:

The leading coefficient in an equation or inequality;

Denominators in fractions;

Discriminant of a quadratic binomial.

General scheme for solving equations reducible to quadratic equations with a parameter.

General scheme for solving equations reducible to quadratic equations with a parameter:

1. Indicate and exclude all values ​​of the parameter and variable at which the equation becomes meaningless.

2. Multiply both sides of the equation by a common denominator that is not zero.

3. Convert the corollary equation to the form https://pandia.ru/text/80/147/images/image002_13.png" width="128" height="24 src="> - real numbers or functions of a parameter.

4. Solve the resulting equation, considering the cases:

A) ; b) https://pandia.ru/text/80/147/images/image005_6.png" width="19" height="27">.png" width="21" height="27">.png" height="75">х=2b+1

Since x must lie in the range from 1 to 6, then:
1) 1<2b+1<6

2) 1<2b – 1<6

https://pandia.ru/text/80/147/images/image009_4.png" width="47" height="41 src=">=2b+1

1) 1<2b+1<6

2) 1<2b – 1<6

https://pandia.ru/text/80/147/images/image010_2.png" width="18 height=98" height="98">

y(1)>0 y=1-4b+4b2– 1>0

y(6)> 0 y=36-24b+4b2– 1>0

xвО(1; 6) 1<-<6

bО(-∞; 0) È (1; +∞).

2) 4b2-24b+35>0

D=576 – 560=16=42>0

b1=https://pandia.ru/text/80/147/images/image016_2.png" width="47" height="41 src=">=2.5 bÎ(0.5; 3)

bÎ(-∞;2.5)È(3.5;+∞)
bО(1; 2.5)

Answer: the roots of the equation x2-4bх+4b2–1=0 lie in the interval from