Analysis of typical errors when solving problems in a school mathematics course: equations, trigonometry, planimetry. Quadratic inequalities

Consider an example where there is a negative coefficient in front of “x 2” in the quadratic inequality.

Introduction… ……………………………………………………… 3

1. Classification of errors with examples…………………………… .…… …5

1.1. Classification by types of tasks…… ……………………… … ……….5

1.2. Classification by types of transformations……………………………10

2. Tests………………………………………………….… .………………….12

3. Protocols of decisions……………… ……….….………………… ………… 18

3.1. Protocols of incorrect decisions……………………………………… 18

3.2. Answers (protocols of correct decisions)………………………………….34

3.3. Errors made in decisions…………………………………… 51

Appendix……………………….…………………………………………… 53

Literature……………………………………………………………………………….56

INTRODUCTION

“You learn from mistakes,” says folk wisdom. But in order to learn a lesson from a negative experience, you first need to see the mistake. Unfortunately, a student is often unable to detect it when solving a particular problem. As a result, the idea arose to conduct a study, the purpose of which was to identify typical mistakes made by students, as well as classify them as completely as possible.

As part of this study, a large set of problems from the April testing options, tests and written assignments for entrance exams at Omsk State University, various manuals and collections of tasks for applicants to universities were reviewed and solved, and materials from the correspondence school at the Omsk State University Faculty of Philosophy were carefully studied. The obtained data was subjected to detailed analysis, while much attention was paid to the logic of decisions. Based on these data, the most frequently made mistakes, that is, typical ones, were identified.

Based on the results of this analysis, an attempt was made to systematize typical errors and classify them by types of transformations and types of problems, among which the following were considered: quadratic inequalities, systems of inequalities, fractional rational equations, equations with a modulus, irrational equations, systems of equations, motion problems, work problems and labor productivity, trigonometric equations , systems trigonometric equations, planimetry.

The classification is accompanied by an illustration in the form of incorrect decision protocols, which makes it possible to help schoolchildren develop the ability to check and control themselves, critically evaluate their activities, find errors and ways to eliminate them.

The next stage was working with tests. For each problem, five possible answers were proposed, of which one was correct and the other four were incorrect, but they were not taken at random, but corresponded to a solution in which a specific error, standard for problems of this type, was made. This provides a basis for predicting the degree of “severity” of an error and the development of basic mental operations (analysis, synthesis, comparison, generalization). The tests have the following structure:

Error codes are divided into three types: OK - the correct answer, a digital code - an error from the classification by type of task, a letter code - an error from the classification by type of transformation. Their decoding can be found in Chapter 1. Classification of errors with examples.

Next, tasks were proposed to find an error in the solution. These materials were used when working with students of the correspondence school at the NOF Omsk State University, as well as in advanced training courses for teachers in Omsk and the Omsk region, conducted by the NOF Omsk State University.

In the future, based on the work done, it is possible to create a system for monitoring and assessing the level of knowledge and skills of the test taker. It becomes possible to identify problem areas in work, record successful methods and techniques, and analyze what content of training is appropriate to expand. But for these methods to be most effective, student interest is required. For this purpose, I, together with Chubrik A.V. and a small software product was developed that generates incorrect solutions of linear and quadratic equations(theoretical basis and algorithms - me and Chuubrik A.V., assistance in implementation - student MP-803 Filimonov M.V.). Working with this program gives the student the opportunity to act as a teacher whose student is the computer.

The results obtained can serve as the beginning of a more serious study, which in the near and long term will be able to make the necessary adjustments to the mathematics teaching system.

1. CLASSIFICATION OF ERRORS WITH EXAMPLES

1.1. Classification by task types

1. Algebraic equations and inequalities.

1.1. Quadratic inequalities. Systems of inequalities:

1.1.1. The roots of the quadratic trinomial were found incorrectly: Vieta's theorem and the formula for finding the roots were incorrectly used;

1.1.2. The graph of a quadratic trinomial is shown incorrectly;

1.1.3. The values ​​of the argument at which the inequality is satisfied are incorrectly defined;

1.1.4. Division by an expression containing an unknown quantity;

1.1.5. In systems of inequalities, the intersection of solutions to all inequalities is incorrectly taken;

1.1.6. Ends of intervals are incorrectly included or not included in the final answer;

1.1.7. Rounding.

1.2. Fractional rational equations:

1.2.1. The ODZ is incorrectly indicated or not indicated: it is not taken into account that the denominator of the fraction should not be equal to zero;

ODZ: .

1.2.2. When receiving a response, DZ is not taken into account;

Sections: Mathematics

Class: 9

A mandatory learning outcome is the ability to solve inequalities of the form:

ax 2 + bx+ c ><0

based on a schematic graph of a quadratic function.

Most often, students make mistakes when solving quadratic inequalities with a negative first coefficient. In such cases, the textbook suggests replacing the inequality with an equivalent one with a positive coefficient at x 2 (example No. 3). It is important that students understand that they need to “forget” about the original inequality; to solve the problem, they need to draw a parabola with branches pointing upward. One can argue differently.

Let's say we need to solve the inequality:

–x 2 + 2x –5<0

First, let's find out whether the graph of the function y=-x 2 +2x-5 intersects the OX axis. To do this, let's solve the equation:

The equation has no roots, therefore, the graph of the function y=-x 2 +2x-5 is located entirely below the X-axis and the inequality -x 2 +2x-5<0 выполняется при любых значения Х. Необходимо показать учащимся оба способа решения и разрешить пользоваться любым из них.

The ability to solve is developed on No. 111 and No. 119. It is imperative to consider the following inequalities x 2 +5>0, -x 2 -3≤0; 3x 2 >0 etc.

Of course, when solving such inequalities, you can use a parabola. However, strong students should give answers immediately without resorting to drawing. In this case, it is necessary to require explanations, for example: x 2 ≥0 and x 2 +7>0 for any values ​​of x. Depending on the level of preparation of the class, you can limit yourself to these numbers or use No. 120 No. 121. In them it is necessary to perform simple identical transformations, so here the material covered will be repeated. These rooms are designed for strong students. If a good result is achieved and solving quadratic inequalities does not cause any problems, then you can ask students to solve a system of inequalities in which one or both inequalities are quadratic (exercise 193, 194).

It is interesting not only to solve quadratic inequalities, but also where else this solution can be applied: to find the domain of definition of a function of studying a quadratic equation with parameters (exercise 122-124). For the most advanced students, you can consider quadratic inequalities with parameters of the form:

Ax 2 +Bx+C>0 (≥0)

Ax 2 +Bx+C<0 (≤0)

Where A,B,C are expressions depending on the parameters, A≠0,x are unknown.

Inequality Ax 2 +Bx+C>0

It is studied according to the following schemes:

1)If A=0, then we have linear inequality Bx+C>0

2) If A≠0 and discriminant D>0, then we can factor the square trinomial and obtain the inequality

A(x-x1) (x-x2)>0

x 1 and x 2 are the roots of the equation Ax 2 +Bx+C=0

3)If A≠0 and D<0 то если A>0 the solution will be the set of real numbers R; at A<0 решений нет.

The remaining inequalities can be studied similarly.

Can be used to solve quadratic inequalities, hence the property of the quadratic trinomial

1)If A>0 and D<0 то Ax2+Bx+C>0- for all x.

2)If A<0 и D<0 то Ax2+Bx+C<0 при всех x.

When solving a quadratic inequality, it is more convenient to use a schematic representation of the graph of the function y=Ax2+Bx+C

Example: For all parameter values, solve the inequality

X 2 +2(b+1)x+b 2 >0

D=4(b+1) 2 -4b 2 =4b 2 +8b+4-4b 2

1) D<0 т.е. 2b+1<0

The coefficient in front of x 2 is 1>0, then the inequality is satisfied for all x, i.e. X є R

2) D=0 => 2b+1=0

Then x 2 +x+¼>0

x є(-∞;-½) U (-½;∞)

3) D>0 =>2b+1>0

The roots of a square trinomial are:

X 1 =-b-1-√2b+1

X 2 =-b-1+√2b+1

The inequality takes the form

(x-x 1) (x-x 2)>0

Using the interval method we get

x є(-∞;x 1) U (x 2 ;∞)

For independent decision give the following inequality

As a result of solving inequalities, the student must understand that in order to solve inequalities of the second degree, it is proposed to abandon excessive detail in the method of constructing a graph, from finding the coordinates of the vertices of the parabola, observing the scale, and one can limit oneself to drawing a sketch of the graph of a quadratic function.

At the senior level, solving quadratic inequalities is practically not an independent task, but acts as a component of solving another equation or inequality (logarithmic, exponential, trigonometric). Therefore, it is necessary to teach students how to solve quadratic inequalities fluently. You can refer to three theorems borrowed from the textbook by A.A. Kiseleva.

Theorem 1. Let a square trinomial ax 2 +bx+c be given, where a>0, having 2 different real roots (D>0).

Then: 1) For all values ​​of the variable x less than the lesser root and greater than the greater root, the square trinomial is positive

2) For values ​​of x between the square roots, the trinomial is negative.

Theorem 2. Let a square trinomial ax 2 +bx+c be given, where a>0 having 2 identical real roots (D=0). Then for all values ​​of x different from the roots of the square trinomial, the square trinomial is positive.

Theorem 3. Let a square trinomial ax 2 +bx+c be given where a>0 having no real roots (D<0).Тогда при всех значениях x квадратный трехчлен положителен. Доказательство этих теорем приводить не надо.

For example: the inequality should be solved:

D=1+288=289>0

The solution is

X≤-4/3 and x≥3/2

Answer (-∞; -4/3] U