Work program for an elective course in mathematics, grade 11

program of an elective course in mathematics in grade 11, an elective course in algebra (grade 11) on the topic

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work program for an elective course in mathematics

Type of program: elective course program in mathematics in grade 11 (secondary education).

Program status: working program of the elective course.

Purpose of the program:

• for students, the program ensures the implementation of their right to information about educational services, the right to choose educational services and the right to guarantee the quality of the services received;
• for teaching staff, the program determines priorities in the content of basic general education and promotes the integration and coordination of activities for the implementation of general education;

Category of students: 11th grade students

Duration of the program: 1 year.

Amount of teaching time: 34 hours.

Lesson schedule: 1 hour per week

Forms of control: final test, Unified State Exam.

Explanatory note

The work program of the elective course is compiled on the basis of the following regulatory documents:

1. The federal component of the state standard of basic general education in mathematics, approved by order of the Ministry of Education of Russia dated March 5, 2004 No. 1089.
2. Law of the Russian Federation “On Education” (Article 7, 9, 32).
3. Curriculum of the MBOU "Shemordan Lyceum" for the 2014 - 2015 academic year.

In teaching any discipline, you cannot teach everyone the same thing, in the same volume and content, first of all, due to different interests, and then due to abilities, characteristics of perception, and worldview. It is necessary to provide students with the opportunity to choose a discipline for deeper study.

The school curriculum in mathematics contains only the most necessary, maximally simplified knowledge. Practice shows a huge gap between the content of the school mathematics curriculum and the requirements that are imposed on applicants entering higher education institutions. It becomes difficult for our graduates to enter a university not only due to economic and socio-political conditions, but also due to the discrepancy between the knowledge of the graduate, who was faithfully taught according to the program, and the level of entrance exams to the university. Students in 11th grade, overloaded, are forced to attend additional paid courses (which are not available to everyone), and school teachers are forced to organize various kinds of additional classes for them. For the best results, this should be done not only in the last years of study, but much earlier.

The main goal of the proposed program is not only to prepare for the entrance exam, and to master a certain amount of knowledge, ready-made methods for solving non-standard problems, but also to teach how to think independently and creatively approach any problem.

In this regard, an elective course program in mathematics is being created.

The elective course “Practicum in solving problems in mathematics” is designed for 34 hours for 11th grade students. This course program will be able to attract the attention of students who are interested in mathematics, who will need it when studying, preparing for exams, in particular, the Unified State Exam. This course can be attended by students from a variety of backgrounds.

This course has applied and general educational significance; it contributes to the development of students’ logical thinking and systematization of knowledge in preparation for final exams. Various forms of organizing classes are used, such as lectures and seminars, group and individual activities of students. The result of the proposed course should be successful passing of the Unified State Exam and centralized testing.

Course objectives:

• Based on the correction of students’ basic mathematical knowledge for the course of grades 5–11, to improve the mathematical culture and creative abilities of students. Expanding and deepening the knowledge gained from studying an algebra course.
• Consolidation of theoretical knowledge; development of practical skills and abilities. Ability to apply acquired skills when solving non-standard problems in other disciplines.
• Creating conditions for the formation and development of students’ skills in analysis and systematization of previously acquired knowledge; preparation for the final certification in the form of the Unified State Exam.

Course objectives:

• Implementation of individualization of training; meeting the educational needs of schoolchildren in algebra. Formation of students' sustainable interest in the subject.
• Identifying and developing their mathematical abilities.
• Preparation for studying at a university.
• Ensuring that students master the most common techniques and methods for solving problems. Development of skills to independently analyze and solve problems based on a model and in an unfamiliar situation;
• Formation and development of analytical and logical thinking.
• Expanding students' mathematical understanding of certain topics included in entrance examination programs for other types of educational institutions.
• Development of communication and general academic skills of working in a group, independent work, ability to lead a discussion, give reasons for answers, etc.

Types of activities in the classroom:

teacher lecture, conversation, workshop, consultation, ICT technology, distance learning.

Course Features:

1. Brief study of the material.
2. Practical significance.
3. Non-traditional forms of studying the material.

Students’ skills and abilities developed by the elective course:

• ability to independently work with tables and reference literature;
• drawing up algorithms for solving typical problems;
• skills in solving trigonometric, exponential and logarithmic equations and inequalities;
• studies of elementary functions for solving problems of various types.

Subject content

Topic 1. Word problems (5 hours)

Logic and general approaches to solving word problems. The simplest word problems. Basic properties are directly and inversely proportional quantities. Percentages, rounding up, rounding down. Choosing the optimal option. Selecting an option from two possible options Selecting an option from three possible options Selecting an option from four possible options. Text problems on percentages, alloys and mixtures, on movement, on teamwork.

Topic 2. Trigonometry (5 hours) Calculating the values ​​of trigonometric expressions. Converting numerical trigonometric expressions. Transforming literal trigonometric expressions. Trigonometric equations and inequalities. The simplest trigonometric equations. Two methods for solving trigonometric equations: introducing a new variable and factoring. Homogeneous trigonometric equations.

Topic 3. Planimetry (5 hours)

Triangle. Parallelogram, rectangle, rhombus, square. Trapezoid. Circle and circle. A circle inscribed in a triangle and a circle circumscribed about a triangle. Polygon. Sum of angles of a convex polygon. Regular polygons. Incircle and circumcircle of a regular polygon. Coordinate plane. Vectors. Calculation of lengths and areas.

Problems related to angles. Multiconfiguration planimetric problems.

Topic 4. Stereometry (5 hours)

Prism, its bases, side ribs, height, lateral surface; straight prism; correct prism. Parallelepiped; cube; symmetry in a cube, in a parallelepiped. Pyramid, its base, lateral ribs, height, lateral surface; triangular pyramid; correct pyramid. Sections of a cube, prism, pyramid. An idea of ​​regular polyhedra (tetrahedron, cube, octahedron, dodecahedron and icosahedron).

The magnitude of the angle, the degree measure of the angle, the correspondence between the magnitude of the angle and the length of the arc of a circle. The angle between straight lines in space; the angle between a straight line and a plane, the angle between planes. Distance from a point to a line, from a point to a plane; the distance between parallel and intersecting lines, the distance between parallel planes. Surface area of ​​a composite polyhedron.

Topic 5. Derivative (5 hours)

The concept of the derivative of a function, the geometric meaning of the derivative. The physical meaning of the derivative, finding the speed for a process given by a formula or graph. Equation of a tangent to the graph of a function. Derivatives of sums, differences, products, quotients. Derivatives of basic elementary functions. The second derivative and its physical meaning. Research of functions. Application of the derivative to the study of functions and graphing. The largest and smallest value of functions. Examples of using the derivative to find the best solution in applied, including socio-economic, problems. Research of works and private ones. Study of trigonometric functions. Study of functions without the help of derivatives.

Topic 6. Typical tasks C1, C2, C3, C4, C5, C6 (8 hours)

Trigonometric equations: solution methods and selection of roots.

Arithmetic method. Algebraic method. Geometric method. Basic methods for solving trigonometric equations. Trigonometric equations, linear with respect to the simplest

trigonometric functions. Trigonometric equations reduced to algebraic equations by substitution. Factorization method. Combined equations.

Polyhedra: types of problems and methods for solving them.

Distances and angles. The distance between two points. Distance from a point to a line.

Distance from a point to a plane. The distance between intersecting lines.

The angle between two straight lines. The angle between a straight line and a plane. Angle between planes.

Areas and volumes. Surface area of ​​a polyhedron. Sectional area of ​​a polyhedron. Volume of a polyhedron.

Systems of inequalities with one variable.

Solving exponential and logarithmic inequalities. Exponential inequalities. Logarithmic inequalities. Mixed inequalities. Systems of inequalities.

Planimetric problems with ambiguity in the condition (multivariate problems)

Function and parameter. Explicitly specified functions. Applying function properties. Functions specified in implicit form. Solving problems in different ways.

Integer problems. Divisibility of integers. Decimal notation of a number. Comparisons. Expressions with numbers. Expressions with variables. Methods for solving equations and inequalities in integers.

Final lesson.

Calendar-thematic plan

Requirements for the level of mastery of the subject

Carrying out practical exercises is aimed at consolidating students’ theoretical knowledge and developing practical skills in the field of algebra, and successfully passing the Unified State Examination in mathematics.

• Students should know what percentages and compound interest are, the basic property of proportion.
• Know the scheme for solving linear, quadratic, fractional-rational, irrational equations.
• Know how to solve systems of equations.
• Know the definition of a parameter; examples of equations with a parameter; main types of problems with parameters; basic methods for solving problems with parameters. Know the definition of a linear equation and inequality with parameters. Algorithms for solving linear equations and inequalities with parameters graphically. Definition of quadratic equation and inequality with parameters. Algorithms for solving quadratic equations and inequalities with parameters graphically
• carry out identical transformations of irrational, exponential, logarithmic and trigonometric expressions.
• solve irrational, logarithmic and trigonometric equations and inequalities.
• solve systems of equations using studied methods.
• build graphs of elementary functions and carry out graph transformations using the studied methods.
• apply the apparatus of mathematical analysis to problem solving.
• apply basic geometry methods (design, transformations, vector, coordinate) to solve geometric problems.
• Be able to apply the above knowledge in practice.

Forms for monitoring student achievement levels and assessment criteria

1. Current control: practical work, independent work.
2. Thematic control: test.
3. Final control: final test.

Planned results

Studying this course gives students the opportunity to:

- repeat and systematize previously studied material in the school mathematics course;

— master basic problem solving techniques;

— master the skills of constructing and analyzing the proposed solution to the problem;

— master and use in practice the technique of passing the test;

— get acquainted with and use in practice non-standard methods for solving problems;

— increase the level of your mathematical culture, creative development, cognitive activity;

— get acquainted with the possibilities of using electronic learning tools, including Internet resources, in preparation for the final certification in the form of the Unified State Exam.

Educational and methodological support

1. Goldich V.A. Algebra. Solving equations and inequalities. - St. Petersburg: Litera, 2008

2. Gornshtein P.I., Polonsky V.B., Yakir M.S. Problems with parameters. — M.-Kharkov: “ILEXA”, “Gymnasium”, 2009

3. Sharygin I.F. Optional course in mathematics. Problem solving – M. – “Enlightenment” 2008

4. Codifier, specification of Unified State Examination tasks 2013 -2014.

Internet sources:

1. Open bank of Unified State Exam problems: https://mathege.ru
2. Online tests:
3. https://uztest.ru/exam?idexam=25
4. https://egeru.ru

https://reshuege.ru/

5. FIPI https://fipi.ru/

6. MIOO https://www.mioo.ru/ogl.php#

7. https://shpargalkaege.ru/

Changes and Notes Sheet

Application

Testing and measuring materials for the course

Word problems

1. New textbooks on geometry for 2-3 courses were brought to the university library, 280 pieces for each course. All books are the same size. The bookcase has 7 shelves, each shelf holds 30 textbooks. How many cabinets can be completely filled with new textbooks?
2. The patient is prescribed a medicine that needs to be taken 0.5 g 3 times a day for 21 days. One package contains 10 tablets of medicine, 0.5 g each. What is the smallest number of packages that will be enough for the entire course of treatment?
3. The wholesale price of the textbook is 170 rubles. The retail price is 20% higher than the wholesale price. What is the largest number of such textbooks that can be purchased at a retail price of 7,000 rubles?
4. Pavel Ivanovich bought an American car, the speedometer of which shows the speed in miles per hour. An American mile is equal to 1609 m. What is the speed of a car in kilometers per hour if the speedometer shows 65 miles per hour? Round your answer to a whole number.
5. In order to knit a sweater, the housewife needs 800 grams of red wool. You can buy red yarn for 80 rubles per 100 g, or you can buy undyed yarn for 50 rubles per 100 g and dye it. One packet of paint costs 20 rubles and is designed to dye 400 g of yarn. Which purchase option is cheaper? In response, write how many rubles this purchase will cost.
6. To make bookshelves, you need to order 48 identical glasses from one of three companies. The area of ​​each glass is 0.25. The table shows prices for glass, as well as for glass cutting and edge grinding. How many rubles will the cheapest order cost?

1. An independent expert laboratory determines the rating of household appliances based on a value coefficient equal to 0.01 of the average price, indicators of functionality, quality and design. Each of the indicators is rated as an integer from 0 to 4. The final rating is calculated using the formula

The table shows the average price and ratings of each indicator for several models of electric meat grinders. Determine the highest rating of the electric meat grinder models presented in the table.

1. Four shirts are 8% cheaper than a jacket. What percentage are five shirts more expensive than a jacket?
2. The motor ship travels along the river to its destination for 200 km and, after stopping, returns to its point of departure. Find the speed of the current if the speed of the ship in still water is 15 km/h, the stay lasts 10 hours, and the ship returns to the point of departure 40 hours after departure. Give your answer in km/h.
3. The first pipe passes 1 liter of water per minute less than the second. How many liters of water per minute does the second pipe carry if it fills a 110-liter tank 1 minute faster than the first pipe?

Trigonometry

1. Solve the equation. Write the smallest positive root in your answer.
2. Find if and .
3. Find if
4. Find the meaning of the expression
5. Find the meaning of the expression.
6. Find the meaning of the expression
7. Find the meaning of the expression.
8. Find the meaning of the expression
9. Find the meaning of the expression
10. Given an equation a) Solve the equation; b) Indicate the roots of the equation belonging to the segment

Planimetry

1. On checkered paper with squares measuring 1 cm by 1 cm, a triangle is depicted (see picture). Find its area in square centimeters.
2. The sides of the parallelogram are 9 and 15. The height dropped on the first side is 10. Find the height dropped on the second side of the parallelogram.
3. The diagonals of a quadrilateral are 4 and 5. Find the perimeter of a quadrilateral whose vertices are the midpoints of the sides of the given quadrilateral.
4. The midline and height of the trapezoid are 3 and 2, respectively. Find the area of ​​the trapezoid.
5. Find the area of ​​the ring bounded by concentric circles whose radii are equal to and .
6. The sides of an equilateral triangle are equal to 3. Find the scalar product of the vectors and .
7. The points O(0;, 0), A(6; 8), B(6; 2), C(0; 6) are the vertices of the quadrilateral. Find the abscissa of the point P of intersection of its diagonals.

8. In a triangle, . Find .

9. A quadrilateral is inscribed in a circle. The angle is equal to , the angle is equal to . Find the angle. Give your answer in degrees.

10. On a line containing the median of a right triangle with a right angle , a point is taken that is distant from the vertex at a distance equal to 4. Find the area of ​​the triangle if , .

Stereometry

1. In a regular triangular pyramid, the medians of the base intersect at the point. The area of ​​the triangle is 2; the volume of the pyramid is 6. Find the length of the segment.
2. Find the square of the distance between vertices C and A1 of a rectangular parallelepiped for which AB = 5, AD = 4, AA1=3.
3. Find the distance between the vertices of the polyhedron shown in the figure.
4. Find the surface area of ​​the polyhedron shown in the figure
5. If each edge of a cube is increased by 1, its surface area increases by 54. Find the edge of the cube.
6. Find the lateral surface area of ​​a regular hexagonal prism whose base side is 5 and height is 10.

1. Find the surface area of ​​a regular quadrangular pyramid whose base sides are 6 and whose height is 4.
2. The sides of the base of a regular hexagonal pyramid are equal to 10, the side edges are equal to 13. Find the lateral surface area of ​​this pyramid.
3. In a cube, find the cosine of the angle between the planes and.
4. The following edges are known in a rectangular parallelepiped: , , . Find the angle between planes ABC and .

Derivative

1. A material point moves rectilinearly according to the law (where x is the distance from the reference point in meters, t is the time in seconds measured from the beginning of the movement). Find its speed in (m/s) at time t = 6 s.
2. The line is tangent to the graph of the function. Find the abscissa of the tangent point.
3. The figure shows a graph of the function y=f(x), defined on the interval (−2; 12). Find the sum of the extremum points of the function f(x).

1. Find the smallest value of the function on the segment.
2. Find the minimum point of the function.
3. Find the smallest value of the function on the segment.
4. Find the greatest value of the function on the segment
5. Find the smallest value of the function on the segment.
6. Find the minimum point of the function.
7. Find the largest value of the function

Calculations and Conversions

1. Solve the equation. If an equation has more than one root, write down the smaller root in your answer.
2. Find the root of the equation
3. Find the root of the equation: If the equation has more than one root, indicate the smaller one.
4. Find if at .
5. Find the meaning of the expression
6. Find the value of the expression when
7. Find the meaning of the expression.
8. Find the value of the expression at .
9. Find the meaning of the expression
10. Find the roots of the equation: Write down the largest negative root in your answer.

Practice-oriented tasks

1. At temperature, the rail has a length of m. As the temperature increases, thermal expansion of the rail occurs, and its length, expressed in meters, changes according to the law, where is the coefficient of thermal expansion and is the temperature (in degrees Celsius). At what temperature will the rail lengthen by 3 mm? Express your answer in degrees Celsius.
2. According to Ohm's law, for a complete circuit, the current strength, measured in amperes, is equal to , where is the emf of the source (in volts), Ohm is its internal resistance, and is the circuit resistance (in Ohms). At what minimum circuit resistance will the current be no more than the short circuit current? (Express your answer in Ohms.)
3. The distance (in km) from an observer located at a height h m above the earth, expressed in kilometers, to the horizon line he observes is calculated by the formula , where km is the radius of the Earth. A person standing on the beach sees the horizon 4.8 km away. How many meters does a person need to climb for the distance to the horizon to increase to 6.4 kilometers?
4. During the decay of a radioactive isotope, its mass decreases according to the law, where is the initial mass of the isotope, (min) is the time elapsed from the initial moment, and is the half-life in minutes. In the laboratory, a substance was obtained that initially contained mg of the isotope, the half-life of which was min. How many minutes will it take for the mass of the isotope to be at least 5 mg?
5. A diving bell, containing at the initial moment of time a mole of air with a volume of liters, is slowly lowered to the bottom of the reservoir. In this case, isothermal compression of air occurs to a final volume. The work done by water when compressing air is determined by the expression (J), where is a constant and is the air temperature. What volume (in liters) will air occupy if 10,350 J of work was done when compressing the gas?
6. The independent agency intends to introduce a rating of online news publications based on assessments of information content, efficiency, objectivity of publications, as well as the quality of the site. Each individual indicator is rated by readers on a 5-point scale with integers from 1 to 5. Analysts who compile the rating formula believe that objectivity is valued three times, and the informativeness of publications is twice as valuable as the efficiency and quality of the site. Thus, the formula took the form What should the number be so that the publication with all the highest ratings would receive a rating of 1?
7. A ball is thrown at an angle to a flat horizontal surface of the earth. The flight time of the ball (in seconds) is determined by the formula. At what smallest angle (in degrees) will the flight time be at least 3 seconds if the ball is thrown with an initial speed of m/s? Consider that the acceleration of free fall is m/s.
8. The figure shows a graph of precipitation in Kaliningrad from February 4 to February 10, 1974. Days are plotted on the x-axis, and precipitation in mm is plotted on the y-axis.
9. The taxi company currently has 20 cars available: 10 black, 2 yellow and 8 green. One of the cars, which happened to be closest to the customer, left when called. Find the probability that a green taxi will come to her.
10. In a random experiment, a symmetrical coin is tossed three times. Find the probability of getting heads exactly twice.

Stereometry

1. The height of the cone is 6, and the diameter of the base is 16. Find the generatrix of the cone.

2. The lateral surface area of ​​the cylinder is 21, and the diameter of the base is 7. Find the height of the cylinder.

3. A sphere is circumscribed around the cone (the sphere contains the circumference of the base of the cone and its vertex). The center of the sphere is at the center of the base of the cone. The generatrix of the cone is . Find the radius of the sphere.

4. The ball is inscribed in a cylinder. The surface area of ​​the sphere is 111. Find the total surface area of ​​the cylinder.

5. A vessel in the shape of a regular triangular prism was filled with 2300 water and the part was immersed in water. At the same time, the water level rose from 25 cm to 27 cm. Find the volume of the part. Express your answer in .

6. A regular quadrangular prism is circumscribed about a cylinder whose base radius and height are equal to 1. Find the lateral surface area of ​​the prism.

7. The base of a right triangular prism is a right triangle with legs 6 and 8, the side edge is 5. Find the volume of the prism.

8. Find the height of a regular triangular pyramid whose base sides are equal to 2 and whose volume is equal to . 9. A cone is inscribed in a ball. The radius of the base of the cone is equal to the radius of the ball. The volume of the cone is 6. Find the volume of the sphere.

10. A ball is inscribed in a cube with edge 3. Find the volume of this sphere divided by .

Typical tasks C1, C2, C3, C4, C5, C6

1. C1 Solve the equation.
2. C1 a) Solve the equation. b) Indicate the roots of this equation belonging to the interval
3. C 2 In a regular tetrahedron, find the angle between the height of the tetrahedron and the median of the side face.
4. C 2 Given a regular triangular pyramid DABC with vertex D. The side of the base of the pyramid is equal to , the height is equal to . Find the distance from the midpoint of the side edge BD to the straight line MT, where points M and T are the midpoints of the ribs AC and AB, respectively.
5. C 3 Solve the system of inequalities
6. C 3 Solve the system of inequalities
7. C4 Find the length of the segment of the common tangent to two circles, enclosed between the points of contact, if the radii of the circles are 23 and 7, and the distance between the centers of the circles is 34.
8. Given triangle ABC. Point E on line AC is chosen so that triangle ABE, whose area is 14, is isosceles with base AE and height BD. Find the area of ​​triangle ABC if it is known that and .
9. C5 Find all values ​​of a, for each of which the set of solutions to the inequality is a segment.
10. Find all values ​​of a, for each of which the function has more than two extremum points.
11. Before each of the numbers 14, 15, . . ., 20 and 4, 5, . . ., 8, a plus or minus sign is arbitrarily placed, after which each of the resulting numbers of the second set is subtracted from each of the resulting numbers of the first set, and then all 35 obtained results are added. What is the smallest modulo and what is the largest sum that can be obtained in the end?
12. The sum of two natural numbers is 43, and their least common multiple is 120 times their greatest common divisor. Find these numbers.

Elective course “Solving non-standard problems in mathematics” for 11th grade students

Explanatory note This program is intended for students of the 11th physics and mathematics class. The content of the educational material corresponds to the goals and objectives of specialized training: The main goal of the course:
creating conditions for the development of logical thinking, mathematical culture and intuition of students by solving problems of increased complexity using non-traditional methods;
Course objectives:
• to develop skills in using non-traditional methods for solving problems;
• develop the ability to independently acquire and apply knowledge; • to form among students a sustainable interest in the subject for further independent activity in preparation for the Unified State Exam and competitive exams at universities; The relevance
of the elective course “Solving non-standard problems in mathematics” is determined by the fact that this course will help students assess their needs, capabilities and make an informed choice of their future life path.
The general principles for selecting program content are:

1. Systematicity 2. Integrity 3. Scientificity.

4. Accessibility, according to the psychological and age characteristics of students in specialized classes.
The program contains the material necessary to achieve the planned goals. This course is a source that expands and deepens the basic component, provides the integration of the necessary information for the formation of mathematical thinking, logic and the study of related disciplines. The program is modernized, based on the program of the author G.N. Kuznetsova. for secondary schools, lyceums and gymnasiums and supplemented by an educational and methodological complex of authors: A.S. Budakov, Yu.A. Gusman, A.O. Smirnov “Collection of methodological instructions and tasks for applicants.” The place of this course is determined by the need to prepare for professional activity, takes into account the interests and professional inclinations of high school students, which allows for a higher final result. The course lasts 68 hours with a regularity of 2 hours per week. While studying the course, students should know: • ways and techniques for solving non-standard problems; must be able to:

• solve problems of higher complexity than the required level; • express your own reasoning accurately and competently; • be able to use mathematical symbols; • apply rational calculation methods; • independently work with methodological literature.

During the classes, various forms and methods of working with students are used:

- when getting acquainted with new ways of solving - the teacher’s work with demonstration of examples; - when using traditional methods - frontal work of students; - individual work; — analysis of ready-made solutions; - independent work with tests.

Teaching methods are determined by course objectives aimed at developing students' mathematical abilities and core competencies in the subject.
In thematic planning, a practical part is highlighted, which is implemented on the knowledge of students acquired during the course of theoretical training. At the end of each section, intermediate control is expected in the form of cut-off and test tasks and other active methods. The effectiveness of the course is determined during the final test, with the subsequent recording of the elective course in the certificate of secondary education. The program material is structured taking into account the use of active learning methods, and the rational distribution of program sections will allow you to obtain high-quality knowledge and achieve the planned results. The program is provided with the educational and methodological complex necessary for its implementation. List of educational and methodological support:
1. Mathematics: a collection of methodological instructions and tasks for applicants to SPbSUAP. Part 1. Compiled by: A.S. Budakov, Yu.A. Gusman, A.O. Smirnov. St. Petersburg: SPBGUAP, 1999. 2. Mathematics: a collection of methodological instructions and tasks for SPBGUAP applicants. Part 2. Compiled by: A.S. Budakov, Yu.A. Gusman, A.O. Smirnov. St. Petersburg: SPBGUAP, 1999. 3. Mathematics: a collection of methodological instructions and tasks for SPBGUAP applicants. Part 3. Compiled by: A.S. Budakov, Yu.A. Gusman, A.O. Smirnov. St. Petersburg: SPBGUAP, 1999.

Additional literature:

1. Denishcheva L.O., Glazkov Yu.A.
"Educational and training materials for preparing for the Unified State Exam." M. Intellect Center, 2004. 2. Dorofeev G. And others. "Mathematics. A collection of tasks for preparing and conducting a written exam for a high school course.” M. Bustard, 2001. 3. Sahakyan S.M. "Grade 11. Exam on algebra and principles of analysis." Verboom - M. 2001. 4. “Collection of problems in mathematics (for applicants to universities).” Textbook - St. Petersburg, 2000. 5. “Collection of problems in mathematics for those entering colleges of higher education” / edited by Skanavi M.I. M. Higher school, 1988 6. Shadriv I.P. “Materials for preparing for the Unified State Exam in mathematics.” Chelyabinsk, 2002. 7. Shamshin V.M. "Thematic tests for preparation for the Unified State Exam in mathematics." Ed. 3rd. Rostov-on-Don - Phoenix, 2004. For the full text of the material, Elective course “Solving non-standard problems in mathematics” for 11th grade students, see the downloadable file
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