The process of training students in the eighth grade elective classes in mathematics

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Content

Introduction

Chapter 1. General organization and holding of elective courses in mathematics

1. History and development of extracurricular activities in mathematics

2. Features of extracurricular activities and their objectives

3. Selection of content, the choice of methods and forms of extracurricular activities in the eighth form

4. Psychological and physiological characterization of teenagers

Chapter 2. Elaboration of an optional course "options in Geometry"

1. Analysis of school textbooks on geometry set federal

2. Elaboration of an optional course "options in Geometry"

3. Thematic planning optional course "options in Geometry"

Conclusion

Bibliography



Introduction

Even at the turn of the XIX-XX centuries educational community has concluded that the teaching of a comprehensive school-wide program on the subject is more successful if its complement group sessions, intended only for the public. In the development of group activities were taken into account needs and interests of students, the real potential teachers, quantitative and age of students. The main purpose of extracurricular group activities is to develop and maintain students' interest in a particular subject with his in-depth study.

The second half of the XX century is characterized by rapid growth of scientific knowledge and practical human activity. Therefore, the mathematical education was considered to be a means to improve the level of training of future professionals both in natural science and the humanities. In 1967-68, in the curricula of secondary schools were included extracurricular activities. The methodology of mathematics is currently considered the first phase the introduction of electives in mathematics at school, which were introduced in order to deepen and broaden the scientific and theoretical knowledge, the development of mathematical thinking, the disclosure of the application of mathematics in practice. The second stage in the development of extracurricular activities began in 1980 and was associated with the transition to the new high school program in mathematics. The third stage began with the Congress workers in public education, which was held in Moscow in December 1988. The reform envisages further development of all forms of differentiation, including electives.

In 1990 he published a new program of electives. The main purpose of the program is to improve knowledge on the basic course received in class, learning the decision more difficult and diverse tasks. The methodology of the elective courses in mathematics was discovered in the works of RS Cherkasov, Carpenter, AA, SI Shvartsburd and others, and the foundations of the organization and conduct of extracurricular activities were presented in the works of MP Kashin, Monakhova VM, VV Firsov, and many others.

A distinctive feature of the present stage of development of an optional form of education is that the teacher can not stick to subjects under the sections and to be creative, making its program of extracurricular activities. With this approach, the teacher is a big responsibility, as in the preparation of an optional course he must consider features a selection of the content of the material, forms and methods of extracurricular activities, psycho-pedagogical features of a particular class, the interests and wishes of students, high school core areas. Therefore, to date research related to the development of the content and methodology of the electives, are relevant.

The object of research is the process of training students in the eighth grade elective classes in mathematics.

The subject of the study - to build a system of extracurricular activities in mathematics for eighth-grade students.

The aim is to develop a diploma elective course "parameters in the geometry" and the development of methods for teaching his eighth-grade students.

To achieve this goal it was necessary to address the following research objectives:

- To analyze the methodological, pedagogical and psychological literature on the subject of research paper;

- The role and place of extracurricular activities in the teaching of mathematics in the school;

- To select the contents of an optional course "parameters in the geometry";

- To make a psychological and educational characteristics of eighth-grade students;

- Develop a plan for elective "parameters in the geometry" and notes specific occupations;

The work used different methods: the study and analysis of the methodological and psychological literature on the subject of work, conversation.

Thesis consists of an introduction, two chapters, the results of the experiments, conclusions, bibliography and appendices.

In the introduction, the urgency of the study, are its main characteristics.

The first chapter deals with the general issues of the organization of the course on mathematics. It covers the history of the origin and formation of school electives, the purpose of the elective courses, depending on the orientation of the profile of the senior classes, psycho-pedagogical characteristics adolescents features a selection of the content, forms and methods of extracurricular activities.

The second chapter presents the developed elective course on "parameters in the geometry" for eighth-grade students, which consists of eight lessons.

From the methodological point of view the study of the optional course "parameters in the geometry" contributes to the development of spatial, logical and creative thinking and mathematical abilities, education for sustainable interest in mathematics, the arts and the study of the environment.

Optional geometry creative course



Chapter 1. General organization and holding of elective courses in mathematics

1. History and development of extracurricular activities in mathematics

In the early twentieth century in the secondary school were created extracurricular group lessons in mathematics in order to develop and maintain students' interest in the subject with his in-depth study.

Group lessons meant only for those who wish, in their design takes into account the interests and needs of students, the real potential teachers, quantitative and age of students.

1 All-Russian Congress of Teachers of Mathematics, which was held in St. Petersburg in 1913, opened a new stage in the development of mathematics education of the Russian school. Congress acted on such famous teachers as Kulishev AR Lebedintsev KF, KA Posse and many others. Among the issues discussed most acute problem stood out the possibility of convergence of the mathematics in high school with a math course in higher education.

KA Posse noted that the most appropriate solution would be to create two courses in mathematics in secondary schools: general and special. The general course is compulsory for all, and a special - designed for students who wish to enroll in universities in the Physics and Mathematics and tie his career with math.

At this congress, in its report Lermontov VV also suggested that the in-depth study of mathematics optional gifted children in order to implement the principles of individualization and differentiation of instruction.

At the end of all presentations was proposed task: develop a detailed methodology for the teaching of mathematics in high school, which would retain the character and general education math would allow specialization in high school, taking into account the individual characteristics of students.

The ministry listened to the opinion of teachers, and in 1915. Major problems were solved school education reform. The students an opportunity, depending on their abilities and interests to choose one of the areas that have emerged in high school: a mathematical, natural, human-classical, neo-humanitarian.

In 1957. At the meeting of the Academy of Pedagogical Sciences of the report by Professor N. Goncharov He proposed in grades 8-10 to enter office with a predominance of subjects of Physics and Mathematics, and the technical, biological and agronomic, socio-economic and humanitarian cycle. This would create real conditions for the development of individual aptitudes and abilities of students, a conscious choice of profession, the best preparation for classes, to study at the university level.

In 1966, November 10 was issued government decree "On measures to further improve the work of the secondary school." It noted that the level of the educational activities of the school does not meet the new, more stringent requirements for the quality of student learning, and does not meet the needs of society in dire need of highly qualified personnel. Among the measures to deal with the backlog has been offered such an important form of education for the school as electives. Under electives understood course, students studied at will to deepen and broaden the scientific and theoretical knowledge. Elective classes were introduced in order to improve knowledge of physics and mathematics, the natural sciences and humanities for the development of varied interests and abilities of students.

Thus, the optional subject were a form of differentiation of instruction of the individual's aptitudes and abilities of students.

The introduction of the school practice of extracurricular activities helped to solve a number of pressing challenges facing the school.

- Extracurricular activities are as important as the lessons of the mandatory program;

- Extracurricular Activities have a clear place in the schedule for each school;

- Extracurricular activities have helped thousands of students to determine their own life and career, to make the right career choice;

- Extracurricular activities help teachers raise the standard of teaching at a high theoretical level;

- Work for the teachers of extracurricular activities was paid along with the lesson (this positive moral and material effect contributed to raising the profile of extracurricular activities).

In the practice of the School Extracurricular Activities included, from the academic year 1967-1968. The methodology to be considered the beginning of the first of three stages of the introduction of electives in mathematics at school.

The first course was entitled "Additional chapters and problems of mathematics" and "Special Courses". Their programs have been published in the journal "Mathematics in school."

Some topics, such as "The method of coordinates", "geometric transformation", "derivatives", "Integral" and others after successful testing in optional classes have been included in the basic math course.

Over the next decade (1966-1975 he) Optional course in mathematics was interpreted as "elective in a single nationwide program" ("One optional course"). In 1975, it issued the "Regulations on the optional subjects in the secondary schools of the RSFSR." By virtue of the provision of optional courses teaching prescribes a number of features, essentially brings him to teach required courses.

Over time, there were significant deficiencies "One elective course":

-Teachers were not prepared for the elective classes due to lack of the necessary teaching aids, methods of elective classes and lack of training for most software issues;

-Rigid quantitative standards for the cohort of students (minimum of 15 people);

-Ban on reading course for non-parallel classes;

-A mandatory program for all schools on an optional course;

-Displacement elective course circles.

Thus "One Optional course" with its rigid system of prohibitions and almost normal time limit method has not brought results in the development and popularity of elective classes.

To 1980. Had completed the transition to the new high school program in mathematics, optional course was replaced. He gave the teachers more opportunities in the self-selection of programs for elective courses. A second phase of the introduction of extracurricular activities at school.

New Optional courses include three sections:

1 Selected issues of Mathematics (8 11klassy);

2 Mathematics in Applications (10 11klassy);

Three algorithms and programming (9-11klassy).

The main purpose of the program was bringing the material being studied in class to a logical conclusion and finding its relationship to science and its applications.

The study of "Mathematics in Applications" aims at developing the knowledge and skills obtained in the study of the basic course of mathematics, the application of that knowledge to solve practical problems in applied mathematical problems, and related disciplines.

The objectives of the course "Algorithms and Programming" was to examine the students the elements of computer programming.

The beginning of the third phase of the introduction of elective classes in mathematics at school can be considered as the Congress of National Education, which was held in Moscow in December 1988. It adopted a concept of secondary education, the focus of which was declared a broad differentiation of instruction. The reform envisages further development of all forms of differentiation, including optional, the main purpose of which is the possibility of an in-depth study of specific subjects, including mathematics.

In 1990. Published a new program of optional courses, which provide for a grade 7.

The main purpose of this program is to improve the knowledge on the basic course, obtained in the classroom, teaching solve more difficult problems. In high school, the deepening of the basic course is systematic and prepares students for further education and to pass the college entrance exams.

Elective course covers the following topics:

-For a textbook of mathematics (7-9);

And mathematical mosaic (7-9);

And preparatory electives (10-11).

A distinctive feature of the present stage of development of the Optional forms of training is that the teacher can not stick to subjects under the sections and to be creative, making its program of elective classes. With this approach, the teacher is a big responsibility, as in the preparation of the Optional course he must consider features a selection of the content, forms and methods of training, psychological and pedagogical features of a particular class, the interests and desires of students and a high school core areas.

2. Features of extracurricular activities and their objectives

Requirements for the training program, designed to sredneuspevayuschego student. But in primary school pupils are allocated as hard learning a mandatory learning outcomes and pupils show interest and ability to learn.

All this leads to the need for a differentiated approach to learning. But even with this approach, the time frame does not allow the lesson finally eliminate gaps in knowledge slow learners and most fully reveal the possibilities and can improve knowledge. To the aid of class work with students.

Under the extracurricular work refers optional systematic studies of students with the teacher after school. There are two types of extracurricular activities in mathematics:

1) additional classes for slow learners, whose main aim is to eliminate the gaps in knowledge on the course of mathematics;

2) class work with pupils show interest and ability to learn mathematics.

There are various forms of extra-curricular activities: mathematical circles, evening or matinee performances, tours, contests, quizzes, contests, a week or a month of mathematics, mathematics fun clubs, after-school reading of popular scientific literature, summer jobs, and others.

The objectives of extracurricular activities for mathematics are:

- The awakening and development of the students' interest in mathematics;

- Expansion of knowledge of students on the program material;

- The development of mathematical skills and culture of mathematical thinking;

- Improved knowledge of students about the importance of mathematics in engineering and the life;

- Education sense of collectivism.

Optional - training course, students studied at will to deepen and broaden the scientific theoretical knowledge. In contrast to the mathematical circle and other forms of extracurricular activities, extracurricular activities are provided to the eighth grade. If in 5-7 classes of interest in students is poor, and the goal is to develop extracurricular activities for pupils in mathematics through entertaining tasks and games, 8-9 grades students consciously choose electives in mathematics in order to deepen the knowledge gained in the classroom, learn new, learn how to solve difficult problems.

Extracurricular activities, like study sessions required course must be based on government programs. These programs are defined by mathematical elective subjects and recorded the time allowed for the consideration of a topic. This determines the amount of knowledge and skills achieved by students during the passage of each topic.

However, in accordance with its own possibilities, possibilities of his students, the teacher can choose any of the elective courses recommended by the Ministry of Education courses. Programs include a variety of variation of elective courses. Therefore, each teacher can vary to some extent the content of the course, not going beyond the elective programs. The above is even more true of the special courses in mathematics, which generally involves a sequence of instruction in a particular subject for a long time.

Extracurricular activities are optional for students. Them schoolchildren, who chose this option on at will.

The condition of an optional choice imposes certain requirements on the system of extracurricular activities dictating its limitations related to both the content and methodology of these studies.

First, the various classes of electives should be possible independent of each other. Only in high school, students who have already formed a relatively stable interest in mathematics, the possibility of setting specific courses, designed more than a year. It is desirable that such courses were of an applied nature, giving students the opportunity of career counseling in the field of mathematics and its applications.

Second, the content and methodology of extracurricular activities should involve students.

This is ensured by the inclusion in the program of curricular topics of general education and a great practical value. Study of these can significantly improve students' mathematical development, which is the main task of mathematical electives.

To elective courses in mathematics to be effective, they need to be organized, where there is:

1) highly qualified teachers or other specialists who are able to teach at a high level of scientific methodology;

2) at least 15 students who wish to study this elective course.

If the school has classes with a small occupancy, the groups of students for extracurricular activities are to be fitted on the parallels of the students or related classes (8-9 classes, grades 10-11).

Record students to extracurricular activities is voluntary, in accordance with their interests. Requirements for students participating in the optional course, the same as with any school subject: compulsory attendance, homework, discipline, discipline in school.

Math teacher is solely responsible for the quality of extracurricular activities, extracurricular activities to the calendar and paid teacher.

Choice of electives students made freely, in accordance with their interests. Requirements for a student participating in the optional course, the same as with any school subject: compulsory attendance, homework and other assignments, etc.

The school extracurricular activities in mathematics are introduced with specific goals:

- School Enrichment;

- The development of mathematical thinking;

- The formation of the active cognitive interest in the subject;

- The development of spatial imagination;

- Promoting career guidance students;

- The study of the history of science.

Unlike other forms of extracurricular activities in mathematics elective course provides students a large amount of scientific and theoretical knowledge, develop skills, to shape the worldview characterizes meaningful connection with the history of science. Optional includes students in various forms of self-employment through the use of heuristics in the classroom, problem, partially-search methods, combining mathematical rigor with mathematical presentation of mathematical beauty and entertaining, has great potential in building a culture of thinking students.

3. Selection of content, the choice of methods and forms of extracurricular activities in the eighth form

The methodology of the elective courses must not copy the method of ordinary lesson. Since in this case becomes a more elective classes in mathematics.

Specificity of extracurricular activities is shown not to use any special methods of teaching, and in non-traditional combination of content selection of educational material, the choice of methods and forms of education. Composition of students in optional classes allows for more efficient use of heuristic teaching method, which consists in self-disclosure of students new content.

The methodology produce psychological and pedagogical features that should be considered in the selection of the content of the training material, methods and forms of elective courses in mathematics.

1 Ratio of extracurricular activities with the basic math course.

In the methodological literature distinguish theoretical plan electives and electives, consisting of a large number of diverse mathematical topics. First develop and substantially complement the mandatory course in mathematics at the geometric, algebraic, function-theoretic or combinatorial and probabilistic direction, the latter provide a more complete, although approximate, representation of the current state of science, mathematics. But each of them develop some of the ideas, concepts, methods known to school children in the curriculum. It follows that the extracurricular activities need to be balanced with the basic math course. This ratio allows for greater efficiency, since students do not need to enter into the circle of the basic concepts of themes familiar with its terminology and symbols. Depth study of the main topics of the course provides an opportunity to summarize and organize the results of mandatory training allows them to solve more difficult problems.

2 content integrity and compliance with the forms and methods of teaching content, aims and objectives of education.

Elective course can not cover all the major areas of modern science, on the integrity of the content of an optional course determined by the goals and objectives of the course, the inner relationship of content, examination of the main concepts, laws and practices. This allows students to concentrate in one direction increases the availability of the material allows for a short period of time allotted for electives to achieve greater efficiency and quality of education.

Specificity of electives - they are not necessary. And because the work has to choose the most attractive form of presentation of new material, Teaching methods: lectures, practical work, reports of students, field trips, seminars - discussion ...

The use of particular methods of teaching in optional classes directly related to the content of the material being studied. Different content requires different forms and methods. If the teacher must inform the student a lot of concepts and information, proofs of theorems - he resorts to lectures. If the new material provides the facts should be known to the students, the teacher is talking, and material having predominantly practical in nature, can be registered in the system tasks, which are used to solve the problem or search training methods.

3 corresponds to the content of the educational and developmental goals of education.

The use of historical material in optional classes is an educational and educational. The history of mathematics, information on the life and work of the great scientists promote cognitive, creative and moral development of pupils.

Classes are developmental in nature, if they are organized with the widespread use of creative teaching methods - the problem, research, heuristic, self-study students. These classes support the sustained interest in mathematics, raising a desire for knowledge, the desire not only to learn new material, but also to explore the new.

Elective course "Options in geometry" provides more in-depth solution of geometrical problems with parameters than in the school course. This will not only develop spatial thinking, but also interest in a stable geometry, since the tasks are designed more not even on the calculation or the evidence, but rather on the imagination and ingenuity.

4 The contents of the training and methodological support.

The teacher must have the necessary scientific and popular literature and manuals on the studied material.

The content must be accompanied by elective visual aids and technical facilities in an amount sufficient for successful learning. Extracurricular activities should be interesting and exciting for students, as it helps the most easy to understand the different ideas and methods of mathematical sciences, techniques of creative activity.

5.Napravlennost elective courses in mathematics at the side of the application of mathematics.

The study of all of the course should provide a discussion of the importance that they have in various areas of science and industry.

The purpose of the teacher - students interested in his subject, learn how to apply their knowledge in practice, to bring them independence and curiosity.

These features of selecting the content of forms and methods of extracurricular activities to address the educational, pedagogical and developing learning objectives, possible to identify the main phase of the optional course "parameters in the geometry":

- Review the literature that have included training manuals, articles from popular science magazines and books on the subject of an optional course;

- To select the contents of an optional course that meets the mentioned features and goals of education;

- To distribute content on occupations;

- To develop a program of each session, indicating the content of the material considered in class, forms and methods of classes, homework.

4. Psychological and physiological characterization of teenagers

The behavior of adolescents are determined by specific social circumstances, and, above all, the change of the child in society, when a teenager becomes subjectively new relationship with the adult world, which is the new content of his consciousness, forming a psychological formation of this age, as self-consciousness.

A characteristic feature is the self-manifestation of the adolescent abilities and needs to know yourself as a person, with its specific qualities. This generates a teenager desire for self-assertion, self-expression and self-development. It is promoted and the new circumstances that distinguish life from a teenager living children of primary school age. First of all, it's high demands to the teenager from adults, friends, public opinion, which is determined not so much progress in student learning, and other traits of his personality, attitudes, abilities, character. All this creates motivations teenager turn to the analysis itself and to compare themselves with others. So, he has gradually established value orientation, formed relatively stable patterns of behavior, which, in contrast to the sample of children of primary school age are not so much as the image of a particular person, but in the specific requirements that adolescents show to people and to himself .

At a certain stage of development of the former space occupied by a child in the system surrounding human relationships, it is recognized as not corresponding to his or her abilities, and he is trying to change it. There is an open conflict between the way of life and its possibilities, has already identified this lifestyle. Under the restructured its operations.

The first signs of puberty (in boys 12-13 years old, girls 10-12 years) results in a restriction in blood supply, which is reflected not only in the muscles, but also in other organs, including the brain. Thus, adolescents in this age group have reduced physical activity and overall endurance, their intellectual activity temporarily reduced.

Later, in the third stage of pubertal development (13-15 years, 12-14 years old boys and girls) the volume rate of flow is increased and therefore there has been some increase in physical and intellectual capabilities.

Inherent teenager at this stage adaptation, categorical judgments, the desire by all means seem to adults, while flaunting his supposed independence, underscores the marginal nature of the adolescent stage of socialization. Abrupt changes in this period, the body and the psyche teen make him irritable and easily vulnerable. He is trying to form their own views of the world system, but much more is not thought through, it is based only on casual observation, and the teenager is quite easy to change their views or under the influence of new experiences, either in the course of a further more detailed consideration.

Teens at this time reflects the impulsive, emotional, sensitive, negativism, critical mind, perfectionism, daydreaming. On the one hand, failed social positions, undigested social roles, and the other - the desire to take on independent issues. This stage of age is often called "the crisis." The crisis of adolescence due to the gap between the incipient development of the inner world of the child and those relations with the outside world, which were formed in the previous phase.

At this age, young people often fall performance. There are conflicts with others, including those with older, followed by a painful and agonizing experience.

This age is different fundamental shift caused rearrangement previously established psychological structures. It lays the foundation of conscious behavior, emerges the general direction in the formation of moral ideas and attitudes. Roll up and die earlier for adolescents, but heavily based on the formation of new manifestations of positive factors - is increasing its independence, much more diverse it becomes meaningful relationships with other people, adults, he is actively developing other people's social position, there is soul-searching.

As a teenager, greatly expands the amount of the child's activity, qualitatively changes its character. Significant changes are taking place in the intellectual activities of children. Increases the desire to make complex, requiring creative tension activities.

By adolescence, the man has quite mature thinking, the ability to analyze certain aspects of reality, the ability to understand their complex contradictions. Teens tend to understand the logic of events, refusing to take anything for granted, require proof system. The main feature of the intellectual 10-16 year old boy is growing every year the ability to think abstractly. When activated abstract thinking young visual components of thinking does not regress, do not disappear, but are retained and developed, while continuing to play a significant role in the overall structure of thinking. An important feature of this age is the formation of an active, independent and creative thinking of children.

On adolescents is characterized not only by a large amount and stability, but also a specific selectivity. In this time of growing deliberate attention. Selective, targeted, is analyzing and perception. With the strong propensity for romantic, adolescent imagination becomes more realistic and critical nature. They are more sober assessment of their capabilities.

Adolescents greatly increases the amount of memory, and at the expense not only help you remember the material, but his logical thinking. Memory teenager, like attention, gradually takes on the character of organized, regulated and controlled processes.

In connection with the doctrine, maturity, the accumulation of experience and, therefore, promote the general, the psychological development of the children to the beginning of adolescence formed new and broader interests, there are different interests and show a tendency to take a different, more independent position.



Chapter 2. Elaboration of an optional course "options in Geometry"

1. Analysis of school textbooks on geometry set federal

extracurricular optional course mathematics

The optional course "parameters in the geometry" I looked at three textbooks:

1 Geometry 7.9 Atanasyan LS and etc.

2 Geometry 11.7 AV Pogorelov

3 Geometry 7.9 Sharygin IF

None of these books does not contain information on the introductory theoretical parameter in geometry, that is, there is no definition, comparison with the algebraic, and therefore do not pay special attention to the problems with the parameters.

1. Textbook on geometry LS Atanasyan etc. for schools is based on the axiomatic approach. Addressing the students, the author writes that the geometry is a continuation of Mathematics, "in mathematics lessons you have already seen some geometric figures, and imagine what a point, line, segment, ray, angle, as they can be positioned relative to each other. You familiar with such figures as a triangle, rectangle, circle, etc., you know how the segments are measured with a ruler ... and how to measure angles with a protractor. But all this - only the first geometric information. Now you have to broaden and deepen your knowledge of geometric shapes. You will meet new shapes and with many important and interesting features that you already know the figures. You will learn how to use the properties of geometric shapes in practice. " Thus, the author himself defines the practical part of the study of geometry to a greater extent than the theoretical. Hence the abundance of problems on the measurement and calculation.

The textbook Atanasyan LS And others are the following tasks, which require the use of geometric parameters:

Grade 7

Chapter 1 - Basic geometric information.

§ 1 and Direct segment.

Task number 3. Spend three straight so that the any two of them intersect. Label the points of intersection of these lines. How get points? Consider all the possible cases.

§ 4 The measurement intervals.

Objective number 32. Points A, B and C are collinear. It is known that AB = BC = 12cm 13.5cm. What can be the length of the segment AC?

Objective number 33. The points B, D and M are collinear. It is known that BD = 7cm MD = 16cm. What can be the distance BM?

Additional tasks:

Objective number 80. It is known that AOB = 35 °, 50 ° = BOC. Find the angle AOC. For each of the possible cases make a drawing with a ruler and a protractor.

Objective number 81. Hk = 120 ° angle, and the angle hm = 150 °. Find the angle km. For each of the possible cases do the drawing.

Grade 8.

Chapter 5 - The quadrangle.

§ 2 parallelogram and trapezoid.

Task number 374. Bisector A parallelogram ABCD intersects the side BC at point C. Find the perimeter of the parallelogram, if VC = 15cm COP = 9cm.

Task number 375. Find the perimeter of the parallelogram if the bisector of one of the angles of a parallelogram divides the side into segments 7cm and 14cm.

§ 3 rectangle, rhombus, square.

Task number 401. Find the perimeter of rectangle ABCD, if bisector A shares: a) side BC at 45.6 cm intervals and 7.85 cm, and b) on the DC side sections 2.7 and 4.5 dm dm.

Chapter 8 - The Circle.

§ 2 The central and inscribed angles.

Task number 655. Central angle of 30 degrees to the AOB more inscribed angle, based on the arc AB. Locate each of these angles.

Task number 656. Chord AB subtends an arc equal to 115o, and chord AC - an arc of 43o. Find the angle BAC.

2. Textbook on geometry A. Pogorelov for schools is also built on the axiomatic approach. But unlike the textbook on geometry LS Atanasyan etc. there more problems on the evidence, and there are four tasks that require the use of geometric parameters:

Grade 7

§ 4 of the angles of a triangle.

Objective number 25. One of the angles of an isosceles triangle is equal to 70 degrees. Find other corners. How many solutions is the problem?

Grade 8

§ 6 quadrangle.

Objective number 32. In an isosceles right triangle inscribed in a rectangle so that the two vertices are on the hypotenuse and the other two - the other two sides. What are the sides of the rectangle, if you know that they are both 5-2, and the hypotenuse of a triangle is 45 cm?

§ 7 Pythagorean theorem.

Objective number 4. The two sides of a right triangle are equal to 3m. and 4m. Find a third party. (Two cases)

§ 8 Cartesian coordinates.

Objective number 27. Find the center of the circle on the x-axis, if it is known that the circle passes through the point (1, 4) and the radius of the circle is 5.

3. A new textbook on geometry for schools implementing the author, clearly - the empirical concept of building school geometry course. It is expressed primarily in the rejection of the axiomatic approach. More attention than traditional textbooks given to methods for solving geometric problems. System tasks differentiated by level of complexity. The author writes in the introduction: "Geometry is not mathematics. Anyway, it's not the math, which until now you had to deal with. Geometry is a subject for those who like to dream, to draw and look at the pictures, who know how to observe, observe and draw conclusions. Geometry extremely important and interesting subject, and anyone can find it on the corner of the soul. " Of this approach follows the relative abundance of tasks "to choose from", ie with geometric parameters. The textbook contains Sharygina following tasks:

Grade 7.

§ 2.1 The geometry of the straight line.

Objective number 8. c) On the line are the points A, B, C and D. Find the length of the segment with endpoints at the midpoints of AB and CD, if AC = 5, BD = 7.

Objective number 19. The point B lies on the segment AC, AB = 2, BC = 1. Point to the line AB all points M for which AM + BM = CM.

§ 2.2. Basic properties of a straight line on a plane.

Objective number one. How many parts are separated by a plane two lines?

§ 2.3 The flat corners.

Objective number 7. b) What can be equal to the angle of the AOC, if the angle AOB = 161o, 172o angle BOC =?

Objective number 9. What can be the same angle AOD, if the angle AOB = BOC = angle and the angle COD =, where: a) = 34o, 33o =, = 32 °, b) = 78o, = 79o, = 83o, = 132o, = 161o, = 141o?

§ 2.4 of plane curves, polygons, circles.

Objective number 1.b) How many points on the line can not cross the border of the quadrangle? (We assume that the line does not pass through the vertices)

§ 3.3 Inequalities in a triangle. Touch the circle with lines and circles.

Objective number 19. In the plane, there are two circles. What is the radius of the circle tangent to the given circles and having a center on a line passing through the centers, if the radii of these circles and the distance between their centers, respectively: a) 1,3,5 b) 5,2,1 c) 3 , 4.5? How many solutions is the problem?

Objective number 22. At the vertices of the triangle centers are located three mutually tangent circles. The radius of the circles, if the triangle are 5,6,7. How many solutions is the problem?

§ 4.4 On solving geometric problems.

Objective number 4. Are points on the line ABC and D, at what AV = 2, CD = 3. Segments AC and BD are the diameters of the two circles. Find the distance between the centers of the circles.

Objective number 7. Through a point on the line and draw a line p and q. It is known that the angle between the lines a and p is equal to 2 °, and the angle between the lines a and q is equal to 80 °. What is the angle between the lines p and q?

Objective number 11. Through the vertices A and C of the triangle ABC draw a line perpendicular to the bisector of the angle ABC intersects the line CB and BA at the points K and M. Find AB if BM = 8, RC = 1.

8 Class.

§ 5.1 Parallel lines in the plane.

Task number 3. On the plane shows several polygons. Sum of the angles of the polygon is 540o. which and how many polygons are shown. (Check all possible)?

Objective number 11 b) Find an isosceles triangle if one of its angles equal to 80 °.

Objective number 16 b) Find the angles of an isosceles triangle if one of its exterior angles is 100 °.

Objective number 22. Find the angle of the triangle ABC, if you know that a bisector divides the triangle into two isosceles triangles.

Objective number 25. Angle ABC =. What is the angle MRC, if the line is parallel RC VA, direct parallel PM Sun

§ 5.2 Measuring the angles associated with the circle.

Objective number 6. What can equal inscribed angle, based on the chord equal to the radius of the circle?

Objective number 14. Diagonals of the quadrilateral ABCD, the top of which are located on a circle intersect at M, the angle is 80 ° AVM. lines AB and CD intersect at point K, at what angle AKD is 20 °, while direct Su and DA - at the point N, ANV angle is 40 °. Find the angle of the quadrilateral ABCD. How many solutions is the problem?

§ 6.1 The parallelogram, rectangle, rhombus, square.

Objective number 15. On the parallelogram ABCD is known that the angle ABD is 40 ° and that the center of the circle circumscribed about the triangle ABC and CDA, lie on the diagonal BD. Find the angle DBC.

Objective number 18. Of the parallelogram with a line intersecting the two opposite sides, cut a diamond. The remaining parallelogram in the same way again cut diamond. And again from the remaining parallelogram again cut diamond. The result is a parallelogram with sides 1 and 2. Find the source side of the parallelogram.

§ 6.3 of similar triangles. The similarity of triangles.

Objective number 31. a) Two circles with diameters of 3 and 5 relate to each other at point A. The line passing through A intersects the smaller circle again at point B, and more - at C. Find the chords AB and AC, if the aircraft anyway.

§ 8.1 The remarkable point of the triangle.

Objective number 10. In the triangle ABC is equal to the angle A, H - The intersection of heights. What can be equal to the angle of BHC?

§ 8.7 The tasks to be repeated.

Objective number 15. In a circle of radius held chord AB is equal to 2. Let M - a point on the circle, different from A and B. What can be equal to the angle AMB?

The textbooks are 10, 4 and 23 tasks, respectively, and the course can be offered to students undergoing basic geometry course Textbook Pogorelov AV and Atanasyan LS, since these books are not just little problems, but also in the text each task required his attention is drawn to the fact that the decision will be few. Therefore, if there is no such link, the student to think and even in the course of solving the problem that may have other solutions, and, standing on the ground, do not completely solve the problem.

2. Elaboration of an optional course "options in geometry." Explanatory note

Currently in math gets more distant from her branch. Without knowledge of mathematics can not be understanding of any modern technology, natural phenomena, analysis of social political economic and other changes. Shows the influence of mathematics education in the humanities, as in the process of mathematical activity students learn the necessary methods such as analysis and synthesis, induction and deduction, classification and analogy. In the course of developing creative problem solving and theoretical thinking, and the proof of the facts produced by the ability to formulate, justify, logical reasoning.

Elective classes in mathematics broaden and deepen their knowledge in the chosen subject. This elective course is designed for 8 classes of secondary school.

The main goal of the course is:

-Expanding horizons;

To increase the level of mathematical training;

-Formation of interest in the subject;

-Development of creative thinking;

-The development of spatial thinking;

Self-development and cultural identity.

Requirements for the mathematical preparation of students:

Will know that after learning of the optional course "options in geometry" students will know what the geometric parameters, how to resolve the problem with the parameters in the geometry.

After that will be able to study the optional course "options in geometry", students will be able to solve problems with geometric parameters.

§ 3. Thematic planning elective course

Lesson number Subject lesson the number of school hours

1 Introduction to the parameters in the geometry, the simplest tasks. 2

2 Decision of construction problems in the subject two triangles

3 Solving problems on the circle 2

4 Solving the topic two quads

5 Solution of the topic two quads

6 Solving problems in the subject of the circle and two tons of Pythagoras

7 Decision problems on the Pythagorean theorem 2

8 Decision Dido problem 2

Course Structure

Classes are held on the first lecture, in which the concept is given a parameter, the parameter in the geometry, talk about the differences and similarities between the geometric and algebraic parameters, as well as to show presentations in PowerPoint, which is a simple example of the geometric parameters (three possible cases of the location of the three points on the line ).

At the second session begins direct problem solving, and thus there is an active repeat the course grade 7. This is important, since the repetition rate at the beginning of the year is given very little time (Zhokhov VI Kartasheva GD, Krainev LB, SM, Sahakian Approximate planning educational material and tests in math, grades 5-11), and the extra hours allow students to quickly get down to work, and remember learned.

From the fourth session of the problem are solved using the material of the eighth grade. During employment is widely used IDE "POWERPOINT". It built drawings to solve problems, and some tasks directly in the environment and addressed. This alleviates the teacher, as many drawings and some of them are quite complex, so it is difficult to play on the board. Can also be used for self-learners.

Complete the course a solution of Dido, which was mentioned in the first introductory lesson. This problem can be solved only by the example of rectangles for the groups that are the main course textbook Pogorelov AV Other examples are considered not appropriate, because the students are not familiar with the formula for the area of ??triangles, quadrangles, etc.

For those groups who are the main course textbook Atanasyan LS Dido problem and others can be viewed on the examples of triangles and quadrangles.

The course content

1 lesson (introductory lecture)

The term "parameter" in Greek means "to measure." It is usually used in combination with other mathematical terms, for example, the parameter of the equation, the parameter of inequality, function parameters, etc. By the problem with the parameters of a problem, in which the technical and logical flow solutions and form of the result depends on the members of the condition variables, the values ??of which are not defined explicitly, but must be known.

The-variable whose value is to distinguish one element of a set of other elements of the set.

Under the geometrical parameter we mean any element or elements of a geometric figure on the size, location, or the relative position of which depends on the solution of the problem, its existence or amount.

An example of one of the first problems with the parameter is the famous Dido problem. In the IX century. BC Phoenician princess Dido, fleeing the persecution of his brother, went to the west along the coast of the Mediterranean Sea to look for a refuge, it has attracted a place on the coast of the Gulf of Tunis. Dido led negotiations with the local leader Yarbom the sale of land. She asked a very small area - "as much as you can surround bovine skin." Dido persuaded Yarba, and the transaction took place. Dido then cut up by the bull's skin into small ribbons, tied them together and surrounded by a large area on which the fortress and the city of Carthage. This legend is found in the poem "The Aeneid," the Roman poet Virgil, Publius Maron, and also in his treatise "On the isoperimetric figures" Zenodora ancient Greeks, who lived between the III. BC BC and the beginning of

The task of finding among all closed curves with a given perimeter that which covers the maximum area, called the problem of Dido.

What in this problem is the? Formulate the problem in this form of Dido, "in which the figure F, for a given perimeter, the area will be the greatest?". In this case, the parameters are the non-numeric data, and the figure, with different values ??of this parameter, that is, with different shapes problem will have different solutions.

Mathematics operates strictly defined concepts, and the world around us at every step there are continuous uncertainties, contingencies. "If there is rain, the holiday" Day of Knowledge "is a program, and if the rain will not be, - Program B". Can we consider the condition "will be - will not go rain" as a parameter? Or mathematics needed only numeric values? For algebra - it's natural. But geometry includes not only the numerical ratio between shapes or elements shapes and geometric. Consequently, for the geometry parameters can be the classic "algebraic" parameters, and very specific "geometric" parameters.

In this course we will consider the problems with geometrical parameters, and some of them you have met in the past year:

1. Points A, B and C are collinear. It is known that AB = BC = 12cm 13.5cm. What can be the length of the segment AC?

Of the three points on the line is one and only one lies between the other two. Since we are talking about three points, each of which may lie between the other two, so we have three different cases.

A negative number is not a solution, since the length is a positive chislo.Otvet: 22.5 cm or 1.5 cm