Practical work on the topic of inverse trigonometric functions. "inverse trigonometric functions" - Document

The final work on the topic "Inverse trigonometric functions. Problems containing inverse trigonometric functions" was completed in advanced training courses.

Contains brief theoretical material, detailed examples and tasks for independent solution for each section.

The work is addressed to high school students and teachers.

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GRADUATION WORK

TOPIC:

“INVERSE TRIGONOMETRIC FUNCTIONS.

PROBLEMS CONTAINING INVERSE TRIGONOMETRIC FUNCTIONS"

Performed:

mathematic teacher

Municipal educational institution secondary school No. 5, Lermontov

GORBACHENKO V.I.

Pyatigorsk 2011

INVERSE TRIGONOMETRIC FUNCTIONS.

PROBLEMS CONTAINING INVERSE TRIGONOMETRIC FUNCTIONS

1. BRIEF THEORETICAL INFORMATION

1.1. Solutions of the simplest equations containing inverse trigonometric functions:

Table 1.

The equation	Solution

1.2. Solving simple inequalities involving inverse trigonometric functions

Table 2.

Inequality	Solution

1.3. Some identities for inverse trigonometric functions

From the definition of inverse trigonometric functions, the identities follow

, (1)

, (2)

, (3)

, (4)

Moreover, the identities

, (5)

, (6)

, (7)

, (8)

Identities relating unlike inverse trigonometric functions

(9)

(10)

2. EQUATIONS CONTAINING INVERSE TRIGONOMETRIC FUNCTIONS

2.1. Equations of the form etc.

Such equations are reduced to rational equations by substitution.

Example.

Solution.

Replacement ( ) reduces the equation to a quadratic equation whose roots.

Root 3 does not satisfy the condition.

Then we get the reverse substitution

Answer .

Tasks.

2.2. Equations of the form, Where - rational function.

To solve equations of this type it is necessary to put, solve the equation of the simplest formand do the reverse substitution.

Example.

Solution .

Let . Then

Answer . .

Tasks .

2.3. Equations containing either different arc functions or arc functions of different arguments.

If the equation includes expressions containing different arc functions, or these arc functions depend on different arguments, then the reduction of such equations to their algebraic consequence is usually carried out by calculating some trigonometric function on both sides of the equation. The resulting foreign roots are separated by inspection. If tangent or cotangent is chosen as a direct function, then solutions included in the domain of definition of these functions may be lost. Therefore, before calculating the value of the tangent or cotangent from both sides of the equation, you should make sure that there are no roots of the original equation among the points not included in the domain of definition of these functions.

Example.

Solution .

Let's reschedule to the right side and calculate the value of the sine from both sides of the equation

As a result of transformations we get

The roots of this equation

Let's check

When we have

Thus, is the root of the equation.

Substituting , note that the left side of the resulting relationship is positive, and the right side is negative. Thus,- extraneous root of the equation.

Answer. .

Tasks.

2.4. Equations containing inverse trigonometric functions of one argument.

Such equations can be reduced to the simplest using basic identities (1) – (10).

Example.

Solution.

Answer.

Tasks.

3. INEQUALITIES CONTAINING INVERSE TRIGONOMETRIC FUNCTIONS

3.1. The simplest inequalities.

The solution to the simplest inequalities is based on the application of the formulas in Table 2.

Example.

Solution.

Because , then the solution to the inequality is the interval.

Answer .

Tasks.

3.2. Inequalities of the form, - some rational function.

Inequalities of the form, is some rational function, and- one of the inverse trigonometric functions is solved in two stages - first, the inequality with respect to the unknown is solved, and then the simplest inequality containing the inverse trigonometric function.

Example.

Solution.

Let it be then

Solutions to inequalities

Returning to the original unknown, we find that the original inequality can be reduced to two simplest ones

Combining these solutions, we obtain solutions to the original inequality

Answer .

Tasks.

3.3. Inequalities containing either opposite arc functions or arc functions of different arguments.

It is convenient to solve inequalities connecting the values ​​of various inverse trigonometric functions or the values ​​of one trigonometric function calculated from different arguments by calculating the values ​​of some trigonometric function from both sides of the inequalities. It should be remembered that the resulting inequality will be equivalent to the original one only if the set of values ​​of the right and left sides of the original inequality belong to the same monotonicity interval of this trigonometric function.

Example.

Solution.

Multiple Valid Valuesincluded in the inequality:. At . Therefore, the valuesare not solutions to the inequality.

At both the right side and the left side of the inequality have values ​​belonging to the interval. Because in betweenthe sine function increases monotonically, then whenthe original inequality is equivalent

Solving the last inequality

Crossing with a gap, we get a solution

Answer.

Comment. Can be solved using

Tasks.

3.4. Inequality of the form, Where - one of the inverse trigonometric functions,- rational function.

Such inequalities are solved using the substitutionand reduction to the simplest inequality in Table 2.

Example.

Solution.

Let it be then

Let's do the reverse substitution and get the system

Answer .

Tasks.

Sections: Mathematics

Inverse trigonometric functions are widely used in mathematical analysis.

Problems related to inverse trigonometric functions often cause significant difficulties for high school students. This is due, first of all, to the fact that current textbooks and teaching aids do not pay too much attention to such problems, and if students still somehow cope with problems of calculating the values ​​of inverse trigonometric functions, then the equations and inequalities containing these functions often leave them stumped. The latter is not surprising, since practically no textbook (including textbooks for classes with in-depth study of mathematics) sets out a method for solving even the simplest equations and inequalities of this kind. The proposed program is devoted to methods for solving equations and inequalities and transforming expressions containing inverse trigonometric functions.

It will be useful for teachers working in high schools - both general education and mathematics, as well as for students interested in mathematics.

This course expands the basic mathematics course and provides an opportunity to get acquainted with interesting questions in mathematics. The issues covered in the course go beyond the compulsory mathematics course. At the same time, they are closely related to the main course. Therefore, this elective course will contribute to the improvement and development of students’ mathematical knowledge and skills.

When conducting classes, traditional forms, such as lectures and seminars, should be used, but in the first place it is necessary to bring such organizational forms as discussion, debate, presentations, and writing abstracts.

Options for final certification may be the following: testing, tests, writing essays on topics proposed by the teacher; individual assignments in which it is necessary to conduct independent research, thematic tests.

The goals of the course are to create conditions for the implementation of specialized training; formation of an integral system of mathematical knowledge and a basis for continuing mathematical education in universities of various profiles.

Course objectives:

• expand the scope of students' mathematical knowledge;
• expand students' understanding of inverse trigonometric functions;
• generalize the basic methods for solving equations and inequalities containing inverse trigonometric functions;
• consider methods for constructing graphs of inverse trigonometric functions.

Requirements for the level of preparation of students.

• Students should know:
  – definition of inverse trigonometric functions, their properties;
  – basic formulas;
  – methods for solving equations and inequalities containing inverse trigonometric functions;
  – methods for constructing function graphs: y=arcsinx, y= arccosx, y=arctgx, y=arcctgx.
• Students must be able to:
  – apply the properties and basic formulas of inverse trigonometric functions;
  – solve simple equations and inequalities;
  – perform transformations of expressions containing inverse trigonometric functions;
  – apply various methods for solving equations and inequalities;
  – solve equations and inequalities with parameters containing inverse trigonometric functions;
  – build graphs of inverse trigonometric functions.

The thematic course planning provided is approximate. The teacher can vary the number of hours allocated to the study of individual topics, taking into account the level of preparation of the students.

Thematic planning

	Subject	Number of hours	Forms of educational activities
	Inverse trigonometric functions and their properties. Values ​​of inverse trigonometric functions.		Independent work with educational literature, seminar lesson.
	Graphs of inverse trigonometric functions.		Practical work.
	Converting expressions containing inverse trigonometric functions.		Analysis and analysis of solutions. Testing.
	Solving simple trigonometric equations and inequalities.		Seminar lesson.
	Methods for solving equations and inequalities containing inverse trigonometric functions.		Analysis and analysis of solutions. Dispute. Test.
	Solving equations and inequalities containing parameters.		Analysis and analysis of solutions. Discussion.
	Generalizing repetition		Development and protection of the project.
	Final control of the course.		Test. Defense of the abstract.

“Inverse trigonometric functions, their graphs. Values ​​of inverse trigonometric functions."

Definition of inverse trigonometric functions, their properties. Finding the values ​​of inverse trigonometric functions.

"Graphs of inverse trigonometric functions."

Functionsy= arcsinx, y= arccosx, y= arctgx, y= arcctgx,their graphs.

"Transformation of expressions containing inverse trigonometric functions."

Calculating the values ​​of trigonometric functions from the values ​​of inverse trigonometric functions. Checking the validity of equalities containing inverse trigonometric functions. Simplifying Expressions Containing Imagescomplex trigonometric functions» .

"Solving the simplest trigonometric equations and inequalities containing inverse trigonometric functions."

Equations:arcsinx=a,arccosx=a,arctgx=a,arcctgx=a.
Inequalities:arcsinx>ah,arccosx>ah,arctgx>ah,arcctgx>ah,arcsinx<а, arccosx<а, arctgx<а, arcctgx<а.

"Methods for solving equations and inequalities containing inverse trigonometric functions."

Equations and inequalities whose left and right sides are the same inverse trigonometric functions. Equations and inequalities whose left and right sides are opposite inverse trigonometric functions. Variable replacement. Use of monotonicity and boundedness of inverse trigonometric functions.

"Solving equations and inequalities containing parameters."

Methods for solving equations and inequalities containing parameters.

"Generalization repetition."

Solving equations and inequalities of different levels.

Final control of the course (2 hours).

Control work can be presented in the formtests in several versions and different levels of difficulty. Defense of abstracts on given topics.

Literature for students:

1. Kramor V.S., Mikhailov P.A. Trigonometric functions. – M.: Education, 1983.
2. Litvinenko V. N., Mordkovich A. G. Workshop on solving mathematical problems. – M.: Education, 1984.
3. Tsypkin A. G., Pinsky A. I. Reference manual on methods for solving problems for secondary school. – M.: Nauka, 1983.
4. CD disk 1C: Tutor.Mathematics. 1 part.
5. Internet resources: Collection of abstracts.

Literature for teachers:

1. Ershov V., Raichmist R.B. Plotting function graphs. – M.: Education, 1984.
2. Vasilyeva V. A., Kudrina T. D., Molodozhnikova R. N. Methodological manual on mathematics for applicants to universities. – M.: MAI, 1992.
3. Ershova A.P., Goloborodko V.V. Algebra. Start of analysis. – M.: ILEKSA, 2003.
4. Collection of problems in mathematics for competitive exams in colleges and universities / Ed. M. I. Scanavi. – M.: Higher School, 2003.
5. Magazines "Mathematics at school".

Municipal educational institution gymnasium No. 2

Mathematic teacher

Gabrielyan Zhasmena Artushovna

Explanatory note.

The proposed program of the elective subject is developed for students of specialized (10-11th) classes of physics and mathematics and is designed for 17 hours; of which 9 hours are allocated for studying theoretical material, 8 hours are allocated for practical classes. At the end of studying this academic subject, students complete a test work consisting of theoretical and practical parts. The program is intended for students who have chosen a specialty where mathematics plays the role of the main apparatus, a specific means for studying the laws of the surrounding world and issues related to economic activity.

Purpose of the item: generalization and systematization, expansion and deepening of knowledge of the general education program in mathematics on the topic “Inverse trigonometric functions”, acquisition of practical skills in performing tasks with inverse trigonometric functions, increasing the level of mathematical training of schoolchildren.

Subject Objectives:

Develop the thinking and creative abilities of students;

To introduce students to the application of theoretical knowledge in solving competitive and olympiad problems;

Involve students in independent work;

Teach students to work with reference and scientific literature;

To teach how to prepare a test paper using computer technology;

Promote the development of students' algorithmic thinking;

To promote the formation of cognitive interest in mathematics.

Requirements for the level of mastery of educational material.

As a result of studying the program of the elective subject “Inverse trigonometric functions”, students:

must know : definitions of inverse trigonometric functions; basic properties and formulas of inverse trigonometric functions; methods for solving equations and inequalities containing inverse trigonometric functions;

must be able to : apply definitions, properties of inverse trigonometric functions when solving competitive and olympiad problems; read and construct graphs of functions whose analytical expression contains the concepts of arcsine, arccosine, arctangent; solve equations, inequalities, systems of equations and inequalities containing arcsine, arccosine, arctangent.

Inverse function. Graph of the inverse function. Definitions of inverse trigonometric functions: y=arcsinx, y=arccosx, y=arctgx, y=arcctgx.

Values ​​of functions y=arcsinx and y=arccosx at points

Values ​​of the function y=arctgx at points Finding the numerical values ​​y=arctgx, y=arcsinx, y=arccosx using computer technology.

Domain of definition, set of values, monotonicity of functions y=arcsinx, y=arccosx, y=arctgx, continuity, boundedness, maximum and minimum values, extrema.

Graphs of the functions y=arcsinx, y=arсosх, y=arctgх and functions related to them. Identities for inverse trigonometric functions. Transformations of expressions containing inverse trigonometric functions. Values ​​of basic trigonometric functions from their inverses. Equations and inequalities, systems of equations and systems of inequalities containing inverse trigonometric functions. Derivatives and antiderivatives of inverse trigonometric functions. Study of functions containing inverse trigonometric functions and construction of their graphs.

Thematic planning of course lessons

"Inverse trigonometric functions"

	Lesson topic	Number of hours
	Inverse function. Graph of an inverse function
	Definition of functions inverse to basic trigonometric functions: y=arcsinx, y=arccosx, y=arctgx, y=arcctgx
	Values ​​of functions y=arcsinx, y=arccosx, y=arctgx, y=arcctgx at given points
	Finding the numerical values ​​of arcsine, arccosine and arctangent using computer technology
	Properties of the functions y=arcsinx, y=arccosx, y=arctgx
	Graphs of functions y=arcsinx, y=arccosx, y=arctgx
	Basic relationships between inverse trigonometric functions
	Calculating the values ​​of trigonometric functions from the values ​​of inverse trigonometric functions
	Proof of identities on a set containing inverse trigonometric functions
	Converting expressions containing inverse trigonometric functions
	Solving equations containing inverse trigonometric functions
	Solving systems of equations containing inverse trigonometric functions
	Solving inequalities involving inverse trigonometric functions
	Solving systems of inequalities containing inverse trigonometric functions
	Derivatives and antiderivatives of inverse trigonometric functions
	Study of functions containing inverse trigonometric functions and plotting their graphs
	Test work

Literature

1. Veresova E.E., Denisova N.S., Polyakova T.P. Workshop on solving mathematical problems. - Moscow “Enlightenment”, 1979.

2. Ishkhanovich Yu.A. Introduction to modern mathematics. Moscow “Science”, 1965

3. Kushchenko V.S. Collection of competition problems in mathematics. Moscow “Enlightenment”, 1979

4. Nikolsky S.M. Elements of mathematical analysis. Moscow "Science", 1989

5. Pontryagin L.S. Mathematical analysis for schoolchildren. Moscow "Science", 1983

6. Tsypkin A.G. Handbook of Mathematics. Moscow "Science", 1983

7. Tsypkin A.G., Pinsky A.I. A reference guide on methods for solving problems in mathematics. Moscow “Science”, 1984

reverse functions table 3 Argument Function sin  cos ... , then you should use the properties of the corresponding reversetrigonometricfunctions, then: When a = 1; ...

Preparation for the Unified State Exam in mathematics

Experiment

Lesson 9. Inverse trigonometric functions.

Practice

Lesson summary

We will mainly need the ability to work with arc functions when solving trigonometric equations and inequalities.

The tasks that we will now consider are divided into two types: calculating the values ​​of inverse trigonometric functions and their transformations using basic properties.

Calculation of values ​​of arc functions

Let's start by calculating the values ​​of the arc functions.

Task No. 1. Calculate.

As we see, all the arguments of arc functions are positive and tabular, which means that we can restore the value of angles from the first part of the table of values ​​of trigonometric functions for angles from to . This range of angles is included in the range of values ​​of each of the arc functions, so we simply use the table, find the value of the trigonometric function in it and restore which angle it corresponds to.

A)

b)

V)

G)

Answer. .

Task No. 2. Calculate

.

In this example we already see negative arguments. A typical mistake in this case is to simply remove the minus from under the function and simply reduce the task to the previous one. However, this cannot be done in all cases. Let us remember how in the theoretical part of the lesson we discussed the parity of all arc functions. The odd ones are arcsine and arctangent, i.e., the minus is taken out of them, and arccosine and arccotangent are functions of a general form; to simplify the minus in the argument, they have special formulas. After calculation, to avoid errors, we check that the result is within the range of values.

When the function arguments are simplified to positive form, we write out the corresponding angle values ​​from the table.

The question may arise: why not write down the value of the angle corresponding, for example, directly from the table? Firstly, because the table before is harder to remember than before, and secondly, because there are no negative sine values ​​​​in it, and negative tangent values ​​​​will give the wrong angle according to the table. It is better to have a universal approach to a solution than to get confused by many different approaches.

Task No. 3. Calculate.

a) A typical mistake in this case is to start taking out a minus and simplify something. The first thing to notice is that the arcsine argument is not in the scope of

Therefore, this entry has no meaning, and the arcsine cannot be calculated.

b) The standard mistake in this case is that they confuse the values ​​of the argument and the function and give the answer. This is not true! Of course, the thought arises that in the table the cosine corresponds to the value , but in this case, what is confused is that arc functions are calculated not from angles, but from the values ​​of trigonometric functions. That is, not .

In addition, since we have found out what exactly the argument of the arc cosine is, it is necessary to check that it is included in the domain of definition. To do this, let us remember that , i.e., which means arccosine does not make sense and cannot be calculated.

By the way, for example, the expression makes sense, because , but since the value of the cosine equal is not tabular, it is impossible to calculate the arc cosine using the table.

Answer. The expressions don't make sense.

In this example, we do not consider arctangent and arccotangent, since their domain of definition is not limited and the function values ​​will be for any arguments.

Task No. 4. Calculate .

In essence, the task comes down to the very first one, we just need to separately calculate the values ​​of the two functions, and then substitute them into the original expression.

The arctangent argument is tabular and the result belongs to the range of values.

The arccosine argument is not tabular, but this should not scare us, because no matter what the arccosine is equal to, its value when multiplied by zero will result in zero. There is one important note left: it is necessary to check whether the arccosine argument belongs to the domain of definition, since if this is not the case, then the entire expression will not make sense, regardless of the fact that it contains multiplication by zero. But, therefore, we can say that it makes sense and we get zero in the answer.

Let us give another example in which it is necessary to be able to calculate one arc function, knowing the value of another.

Problem #5. Calculate if it is known that .

It may seem that it is necessary to first calculate the value of x from the indicated equation, and then substitute it into the desired expression, i.e., into the inverse tangent, but this is not necessary.

Let us remember the formula by which these functions are related to each other:

And let’s express from it what we need:

To be sure, you can check that the result lies in the arc cotangent range.

Transformations of arc functions using their basic properties

Now let's move on to a series of tasks in which we will have to use transformations of arc functions using their basic properties.

Problem #6. Calculate .

To solve, we will use the basic properties of the indicated arc functions, only making sure to check the corresponding restrictions.

A)

b) .

Answer. A) ; b) .

Problem No. 7. Calculate.

A typical mistake in this case is to immediately write 4 in response. As we indicated in the previous example, to use the basic properties of arc functions, it is necessary to check the corresponding restrictions on their argument. We are dealing with the property:

at

But . The main thing at this stage of the decision is not to think that the specified expression does not make sense and cannot be calculated. After all, we can reduce the four, which is the argument of the tangent, by subtracting the period of the tangent, and this will not affect the value of the expression. Having done these steps, we will have a chance to reduce the argument so that it falls within the specified range.

Because since, therefore, , because .

Problem No. 8. Calculate.

In the above example, we are dealing with an expression that is similar to the basic property of the arcsine, but only it contains cofunctions. It must be reduced to the form sine from arcsine or cosine from arccosine. Since it is easier to convert direct trigonometric functions than inverse ones, let’s move from sine to cosine using the “trigonometric unit” formula.

As we already know:

In our case, in the role. Let us first calculate for convenience .

Before substituting it into the formula, let’s find out its sign, i.e., the sign of the original sine. We must calculate the sine from the arc cosine value, whatever this value is, we know that it lies in the range. This range corresponds to the angles of the first and second quarters, in which the sine is positive (check this yourself using a trigonometric circle).

In today's practical lesson we looked at the calculation and transformation of expressions containing inverse trigonometric functions

Strengthen the material with exercise equipment

Trainer 1 Trainer 2 Trainer 3 Trainer 4 Trainer 5

Target:

Assignment: Create a test “Inverse trigonometric functions”

Internet resources

Delivery date - according to the technical specifications

Independent work No. 14 (2 hours)

On the topic: “Stretching and compression along coordinate axes”

Target: systematization and consolidation of the acquired theoretical knowledge and practical skills of students;

Assignment: Abstract on the topic: “Extension and compression along coordinate axes”

Literature: A.G. Mordkovich “Algebra and the beginnings of mathematical analysis” 10th grade

Internet resources

Delivery date - according to the technical specifications

Independent work No. 15 (1 hour)

On the topic: “Stretching and compression along coordinate axes”

Target: formation of independent thinking, ability for self-development, self-improvement and self-realization

Assignment: presentation: “Extension and compression along coordinate axes”

Literature: A.G. Mordkovich “Algebra and the beginnings of mathematical analysis” 10th grade

Internet resources

Delivery date - according to the technical specifications

Independent work No. 16 (2 hours)

On the topic: “Inverse trigonometric functions, their properties and graphs”

Target: systematization and consolidation of acquired theoretical knowledge and practical skills of students

Task completion form: research work.

Literature: A.G. Mordkovich “Algebra and the beginnings of mathematical analysis” 10th grade

Internet resources

Delivery date - according to the technical specifications

Independent work No. 18 (6 hours)

On topic: “Half argument formulas”

Goal: deepening and expanding theoretical knowledge

Assignment: Write a message on the topic “Formulas of half an argument.” Create a reference table for trigonometry formulas

Literature: A.G. Mordkovich “Algebra and the beginnings of mathematical analysis” 10th grade

Internet resources

Delivery date - according to the technical specifications

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The list of bibliographic sources is presented under the heading “Literature”. The list of references must include all sources used: information about books (monographs, textbooks, manuals, reference books, etc.) must contain: the author’s surname and initials, book title, place of publication, publisher, year of publication. If there are three or more authors, it is allowed to indicate the surname and initials of only the first of them with the words “etc.” The name of the place of publication must be given in full in the nominative case: abbreviation of the name of only two cities is allowed: Moscow (M.) and St. Petersburg (SPb.). The cited bibliographic sources should be sorted in alphabetical order in ascending order. The list must consist of at least three sources.

Each new part of the work, new chapter, new paragraph begins on the next page.

The application is drawn up on separate sheets, each application has a serial number and a thematic heading. The inscription “Appendix” 1 (2.3...) is placed in the upper right corner. The application title is formatted as a paragraph title.

The volume of work is at least 10 sheets of pages printed on a computer (typewriter); table of contents, bibliography and appendices are not included in the specified number of pages.

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Three types of font are used in the work: 1 - to highlight chapter titles, headings “Table of Contents”, “Literature”, “Introduction”, “Conclusion”; 2 - to highlight paragraph titles; 3 - for text

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The first slide contains:

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The second slide indicates the content of the work, which is best presented in the form of hyperlinks (for interactivity of the presentation).

The last slide contains a list of literature used in accordance with the requirements, Internet resources are listed last.

Slide design
Style	8 it is necessary to maintain a uniform design style; 8 you need to avoid styles that will distract from the presentation itself; 8 auxiliary information (control buttons) should not prevail over the main information (text, pictures)
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Animation effects	8 you need to use the capabilities of computer animation to present information on a slide; 8 you should not overuse various animation effects; animation effects should not distract attention from the content of the information on the slide
Presentation of information
Contents of information	8 short words and sentences should be used; 8 verb tenses should be the same everywhere; 8 you should use a minimum of prepositions, adverbs, adjectives; 8 headlines should grab the audience's attention
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Ways to highlight information	You should use: 8 frames, borders, shading 8 different font colors, shading, arrows 8 pictures, diagrams, charts to illustrate the most important facts
Amount of information	8, you should not fill one slide with too much information: people can remember no more than three facts, conclusions, and definitions at a time. 8, the greatest effectiveness is achieved when key points are reflected one at a time on each individual slide.
Types of slides	To ensure variety, you should use different types of slides: with text, with tables, with diagrams.

During the work, students:

Review and study the necessary material, both in lectures and in additional sources of information;

Make a list of words separately according to directions;

Make up questions for selected words;

Check the spelling of the text and compliance with the numbering;

Create a finished crossword puzzle.

General requirements for composing crossword puzzles:

The presence of “blanks” (unfilled cells) in the crossword puzzle grid is not allowed;

Random letter combinations and intersections are not allowed;

The hidden words must be nouns in the nominative singular case;

Two-letter words must have two intersections;

Three-letter words must have at least two intersections;

Abbreviations (ZiL, etc.), abbreviations (orphanage, etc.) are not allowed;

All texts must be written legibly, preferably printed.

Design requirements:

The crossword puzzle design must be clear;

All crossword grids must be completed in two copies:

1st copy - with filled words;

2nd copy - only with position numbers.

Answers are published separately. The answers are intended to check the correctness of the crossword puzzle solution and provide an opportunity to familiarize yourself with the correct answers to the unsolved positions of the conditions, which helps solve one of the main tasks of solving crossword puzzles - increasing erudition and increasing vocabulary.

Criteria for evaluating completed crossword puzzles:

1. Clarity of presentation of the material, completeness of the topic research;

2. Originality of the crossword puzzle;

3. Practical significance of the work;

4. The level of stylistic presentation of the material, the absence of stylistic errors;

5. Level of work design, presence or absence of grammatical and punctuation errors;

6. The number of questions in the crossword puzzle, their correct presentation.

In order for practical classes to bring maximum benefit, it is necessary to remember that the exercise and solution of situational problems are carried out on the basis of material read in lectures and are associated, as a rule, with a detailed analysis of individual issues of the lecture course. It should be emphasized that only after mastering the lecture material from a certain point of view (namely, from the one from which it is presented in the lectures) will it be reinforced in practical classes, both as a result of discussion and analysis of the lecture material, and by solving situational problems. Under these conditions, the student will not only master the material well, but will also learn to apply it in practice, and will also receive an additional incentive (and this is very important) to actively study the lecture.

When solving assigned problems independently, you need to justify each stage of action based on the theoretical principles of the course. If a student sees several ways to solve a problem (task), then he needs to compare them and choose the most rational one. It is useful to draw up a brief plan for solving the problem (task) before starting to solve the problems. The solution to problematic problems or examples should be presented in detail, accompanied by comments, diagrams, drawings and drawings, and instructions for implementation.

It should be remembered that the solution to each educational problem should be brought to the final logical answer required by the condition, and, if possible, with a conclusion. The obtained result should be verified in ways that arise from the essence of this task.

· The main terms of the test task must be clearly and explicitly defined.

· Test tasks must be pragmatically correct and designed to assess the level of educational achievements of students in a specific area of ​​knowledge.

· Test tasks should be formulated in the form of condensed short judgments.

· You should avoid test items that require the test taker to make detailed conclusions about the requirements of the test items.

· When constructing test situations, you can use various forms of their presentation, as well as graphic and multimedia components in order to rationally present the content of educational material.

The number of words in a test task should not exceed 10-12, unless this distorts the conceptual structure of the test situation. The main thing is a clear and explicit reflection of the content of a fragment of the subject area.

The average time a student spends on a test task should not exceed 1.5 minutes.