Approximate calculations in mathematics curriculum

5. development of independence and creativity of students working with scientific - popular literature in mathematics. Working with additional literature is essential to enhance the overall development of students, preparing students for further education and self-education, to practice creativity.

Experience shows that the basis of the group of students attending optional course, are students with strong math skills, personality, thinking that is obvious. Petrova ES in his article [26] raises the question of differentiation and individualization of learning, the combination of collective, group and individual training - cognitive activity. The author offers an optional first of all the participants offer different specific objectives, while trying to seek out a common way students in solving problems of this kind, noticing some patterns. The overall conclusion of the whole group of disciples do on the basis of studies of each, that is, the outcome of its research brings teamwork.

During the elective course important system issues and exercises offered to students. It is important to cover all the issues learned.

The manual [26] highlighted the methods used to wake up from the audience of school elective creative activity:

- Heuristic. Used at familiarizing students with new material: in the case of algebraic and geometric approaches have been studied, in the preparation of a new algorithm for pupils mathematical method, problem solving, student withdrawal of the new rule, formula, theorem proving, the introduction of new concepts.

- Problem. Create problem situations - a necessary condition for encouraging students to creative solutions. For example, an indication of problems with practical content, entertaining or historical character, causing the student desire to solve them.

It is important to note that the studied for elective factual material can be either known or unknown to students. This fact is worth noting. If the actual content material elective already partially known, it is impossible to go that route. It is useful to ask participants a few questions on the house: list what facts worth repeating before working with this material, identify the feasibility of interdisciplinary connections in the study of the issue, and its practical application, to try to identify the problems.

In the process of laboratory work the students get interested to get the result. It combines two approaches to creativity:

- Consideration of the emotional side of the issue;

- Consideration of the operating structure, the possibility of algorithmic step guide research student. In this case, the teacher should:

1) consider the amount of knowledge of students;

2) Identify ways to update them;

3) to think of new information reported by the students, the purpose of work, the progress of the work, the necessary equipment, design the course and results of the students in the Workbook, and conclusions.

4) Identify ways to verify that the work of the students;

5) Identify the possibility of individualization and differentiation of instruction.

- Work with the literature. This is implemented in the following ways. Electives, students receive individual tasks to compile and write the essay. For example, to describe some of the facts of history prove theorems. Selection of literature in accordance with the theme.

An important condition is to ensure active cognitive activity of students for each class.

Part of the problem is an optional course intended for the organization of teaching and research activities of students. To solve the problem of teaching and research is not enough of one or more bright ideas, it is natural, is divided into a number of smaller tasks - from simple, does not require any previous knowledge and available to any student, to complex problems. Mashevetsky GI [26] identifies the general characteristics of teaching and research objectives:

The main groups of the requirements include: motivational (statement of the problem should attract attention, be connected with the surrounding reality, have practical applications, the task must be of real benefit to decide), accessibility (statement of the problem must be formulated on the basis of the learned knowledge, the problem can be naturally divided into number of support tasks, from simple to very difficult); education and educational value of the task (task should give research skills and experiment, the ability to process and synthesize the results; task designed to develop cognitive activity and imagination, creativity, students).

Petrova ES in the manual [26] raises the question of how to promote and censure students involved in the elective. The rating system of knowledge is now actually dvuhballnaya:'' 4'' and'' 5''. "Scoring system" for the tasks brings to class spirit of competition, game elements, interests of students.

Often carried division elective students into subgroups. The separation is carried out by different principles, depending on the didactic and educational goals of: different level of students' knowledge, interests of students, the possibility of mutual control, different extracurricular inform students, student relationships, different abilities.

When checking assignments every student use: mutual testing, group testing, a collective review.

These students are taught how to verify critical of the performance of tasks themselves and friends. Ability to work collectively gives the students of the scientific debate, the discussion of research.

Collective work of the group is carried out in the following forms:

- Validation of an algorithm for solving a new mathematical method, the production of individual students;

- Discussion of the correct and concrete solutions control tasks;

- Identification of possible errors in the study of individual students a particular topic, specify how to address them;

- Organization of the discussions on the proposed teacher theme.

Thus, in the elective course material is presented in portions, it is conducted in accordance with a plan that does not undergo major changes, there is some consistency in the presentation of the material. In addition, there are two points of view. First: a study on elective same material as in the classroom, but more deeply unconventional decision tasks, tasks of high difficulty. The goal is to maintain the interest in the subject, to enable students to the subject. Two: The course includes a study of some of the material and creative research. Elective course, means to perform creative works. Creative work can be based on a new, learned to elective. This new order is a source of creative works. Key - the question arose, a problem.

When designing a course on "Approximate calculation", you try to connect the two points of view. The course is the development of'd learned, and problems that require creative exploration.

2. The goals, objectives, structure, optional course

Development of an optional course took place in several stages.

In the first phase of school textbooks have been isolated problems in which knowledge can be useful for approximate calculations. Also had to think of other purposes for which the necessary knowledge and approximate calculations. The task list is given in Annex 3 to this work.

In the second phase was analyzed the logic of introducing the concept of accuracy of approximation, which is given in [15]. Here we distinguish the original elements cell concept approximation accuracy:

- The exact value of x;

- The approximate value of a;

- A measure of the deviation (absolute error) the approximate value of the exact.

Developed for the course, we used the first stage of development of the concept of cells, while in [15] are two.

Step1: a change in any of the elements of the cell. For example, if the approximate value is given as a changing, then:

It was assumed that in the course will be tested research objectives, in order to determine: whether the tasks for the students to study the subject, can we identify the problem, which will be for the students creative challenge? Therefore, in the third stage, we have identified possible topics of research tasks:

1. Error of the sum and difference. The accumulation of rounding errors in the preliminary.

2. Accuracy of the work.

3. Approximate solution of the equations.

The objectives of the elective course include:

- Increased representation of students;

- Creating conditions for independent academic research.

The possibility of the course at 7, 8 classes due not age students, as defined by features of the school curriculum. From the analysis of school textbooks have been allocated objects school mathematics associated with approximate calculations: periodic decimal fraction number ?, Plotting functions, root extract n-th root of the number, location of log and trigonometric functions that are necessary for the assimilation of the course and understanding of the root the degree and the rate is available for students 7 and 8 classes.

Designed elective course consists of two units.

The first block. Basic definitions

In the first section examines the basic concepts highlighted in the analysis of textbooks. At the beginning of each session devoted to the study of the new material, the students proposed a task which needs new knowledge. The basic concepts are introduced through the introduction of the concepts of logic topics "Approximate calculations" developed Kovaleva SA. [15] During the sessions, additional elements: true and significant figures.

The second block. Performing research assignments

In the second block proposed research tasks. In the study conducted by the experiments, measurements of objects. For the control of knowledge students invited to make their tasks, which use approximate calculations.

Students to the proposed problems, solution of which there is difficulty, the need for new knowledge. The complexity of tasks is increasing. In order to solve the following problems requires an understanding of the previous ones.

Optional course handouts accompany templates, memos, texts.

Templates are built graphics functions. Plotting functions - time-consuming process, moreover, is not the subject of the course, so there was a need to use them. In the end, it took little time to build. Students were given a memo to avoid confusion with the newly studied formulas, definitions, to organize knowledge. Under the text refers to a list of jobs to employment issues, the requirements for the execution of various tasks, the rules and regulations.

Program elective course

Block I: Theoretical part. Familiarity with basic definitions.

1. The concepts of exact, approximate values, the absolute error of approximation.

2. Communication source elements. VG and NG, range of variation. Finding accurate as the average.

3. Rounding. Rounding with precision.

4. Faithful and significant figures.

Block II: Implementation of training and research assignments.

5. Error of the sum and difference. The accumulation of rounding errors in the preliminary.

6. The relative error. Limit the absolute and relative errors.

7. Accuracy of the work.

8. Approximate solution of the equations.

- Selection;

- The method of successive approximations;

- Bisection of the interval;

(The choice of the most effective method, on the convergence of successive approximations).

3. Description of the course content

We give a brief description of the course content. More themes lessons, tasks, and comments are presented in Appendix 4 to this work.

Topic 1: "The concept of exact, approximate values. Absolute error of approximation. "

The concepts of: the exact value, approximate value, the absolute error.

In the early disciples proposed task, by the discussion that highlights key concepts. Key - the first task. Further prepared a diagram showing the relationship between concepts. Students should do the proposed scheme. Finally, the task of finding the absolute error - the task to perfect.

Topic 2: "Communication exact, approximate values and absolute error. VG and NG, range of variation. Finding accurate as average. "

Used concepts: the exact value, approximate value, the absolute error.

The concepts of: the upper and lower limits of the range of variation as accurately as average.

The study of relationships and dependencies between the exact and approximate values of the absolute error of approximation. Finding the answer to the question whether any of the three elements may be unknown. Familiarity with the upper and lower bound of the range variation is in the form of teamwork. Students can discover new knowledge. Getting to know how accurate the average is a problem.

Theme 3: "Rounding. Rounding to a given accuracy. "

The concepts of: Round, round with precision.

Need to understand the necessity of rounding systematize knowledge on the topic. The pupils learn a new way - the rule of even numbers. Generally they motivate and intriguing.

Theme 4: "Meet the faithful and significant figures."

The concepts of: correct digits significant figures.

Pupils are offered two records at first designating the same number. We need to find the difference. The difference is due meaningful and true numbers. In the future, this knowledge can be useful when writing a creative work on the topic.

Topic 5: "The error of the sum and difference. The accumulation of rounding errors in the preliminary. "

This is a research problem.

Here are several examples of students with multiple characters after the decimal point. In the task to find the sum and difference of up to a tenth of two ways, and then compare the results. Students are encouraged to discuss their methods of solution. As a result, selected two ways. In finding the value of the first method, you must first round up to the nearest tenth the terms, and then add / subtract. In finding the value of the second way first stacked / taken away, and then rounded to the nearest tenth.

Thus, we get a different answer. Question, why it happened. After analyzing each round, came to the conclusion that there was accumulation of error. This empirical study.

Topic 6: "The relative error. Limit the absolute and relative errors. "

The concepts of: the relative error limit is absolute and relative errors.

Students learn the concept of self-rationed. Concepts needed to solve research problems.

Theme 7: "The error of the product."

This is a research problem.

Students should enter the approximate formula for finding the maximum relative error of the work.

Topic 8: "The approximate solution of quadratic equations."

This is a research problem.

- In solving this problem was put on three issues:

- Which method: selection or successive approximations, the most effective?

- Which method: selection, successive approximation, halving the most effective?

- Is any equation can be solved by successive approximations? Which equations method works?

Thus, subjects 1, 2, 3 are used to logically introduce key concepts. Themes 4 and 6 introduce the concepts used in the research tasks. Subjects 5, 7, 8 - research tasks. They are aimed at identifying creative problem.

4. Testing course analysis of the results

Elective course was piloted at the Lyceum № 3 of Krasnoyarsk, in the 7th grade. In carrying out research tasks the children involved in the study. Some of the children were genuinely interested in receiving a response. But the most exciting for the students was the problem of the approximate determination of the root. In carrying out research tasks students applied to the obtained theoretical knowledge.

In the first stage, when the concepts were introduced, students were asked to make a similar task, they could not do it. Proposed tasks with the same plot. After having been worked out between the basic elements, the students were able to make a similar problem.



Chapter 3. Creative work as a form of further education students

1. Creativity and mathematical creativity

The concept of mathematical creativity.

In [5] states that under certain conditions, the educational process, may initiate, "start" the process of children's mathematical creativity. Initiation of such children's activity that promotes personal growth of students, the development of his mental faculties, for independent activity and planning is especially important for learning.

For a discussion of mathematical creativity is necessary to specify the concept of creativity.

AT Shumilin in his work [28] identifies the following important characteristics of creativity:

- Creativity is closely linked with cognitive activity. Creative act - an act of understanding the world. Mathematical creativity is a form of mastering mathematical knowledge.

- A necessary condition for the beginning of creative research (research) is to understand the problem, its formulation, the creative process - the process of solving the problem. In the process of creative problem statement is changing, it is specified, the solution is divided into a number of tasks. Different authors noted phasing, cycle in solving the problem, as is a history of creative exploration. Hypothesis or sample solution, even if it is not true, prepares the faithful representation of solutions, ie, the hypothesis - it is a means of solving problems.

- Creative work is original. In the creative process always creates a new thing (we have the new mathematical result), or apply new tools, methods, or new program of activities, but the result can be objectively known already, but individually, the new psychologically, that is achieved on its own. And in this case, too, talk about work. [24]

- Discussing the creation of a student with special notice of the presence of active personal position in relation to knowledge, personal interest in creativity, emotionally colored relationship to the test material (14).

The simplest structure of creative activity consists of two phases:

- Statement of the problem (the problem);

- Addressing (Tasks).

The book Shumilina AT [28, p. 47] identified four main stages of creation.

The first stage - awareness, production, formulation of the problem.

The second stage - the principle of finding the solution to the problem of non-standard tasks.

The third stage - the rationale and development of the principles found. Specification of the hypothesis, the development plan of the experimental hypothesis, the proof or disproof of it.

The fourth stage - the practical testing of the hypothesis.

The proposed structure reflects the situation with creative work, the steps you need to go when you write it.

Another important factor associated with the requirement to present the results of work in the form of text. Since the artwork should be submitted written for teachers and fellow students text, an important condition for its appearance - the presence of the students experience with the teacher about the subject of mathematics through written text.

In the scheme of AM Matyushkin allocated as special stages of the principle of finding solutions, its development and implementation. In [28, p. 32 - 51], he wrote: "... The first phase of any problem-solving process is described as the stage of" assimilation "of the problem. ... Problem situations, the main element of which is unknown new, something that should be disclosed to the proper execution of tasks to perform the desired action. "

In the second phase, "people looking for solutions for communications, had not had a direct relationship to the problem. At this stage revealed a new attitude, which leads to the "reverse engineering" problem, to identify a new mode of operation, to understand the solutions. "

The third stage - "the realization of the principle found", which boils down to the use of certain transactions related to the practice, to perform calculations, the justification of evidence. May identify new problems that will cause the search for new principles of implementation.

The fourth stage - "the final stage of the solution of the problem of the problem - check the correct decision."

Let us examine each of the steps in more detail. In the formulation of the problem occupies an important place justification urgency of the problem, because the belief in the relevance of the problem stimulates the search for solutions, the stability of the interest of the researcher. At the stage of the problem begins the process of sharpening contradictions.

The authors of [5] have identified the conditions under which a student can be considered a creative activity:

- He formulated the problem is solved by, the formulation of the problem the students acted as a solution to the challenges ahead;

- By itself, the result is valuable and meaningful to the student;

- The student is "involved" in dealing with the problem, emotionally going through the process of research.

In general, science develops hypotheses. But the attitude of "people - science" may be very different. There are at least two aspects:

- Consumer aspect - the person uses the results of science to solve practical problems of various types.

- Search aspect - is associated with the search for new laws to obtain new knowledge. This aspect of the research.

The hypothesis is a form of creative thinking and, therefore, should be regarded as a category of dialectical logic. Research works have shown that the content a search engine solution is to suggest hypotheses about how to address the problems and check the generation of hypotheses and their verification - the central mechanism of creativity.

Hypothesis - an assumption about how to resolve the conflict issue. Hypothesis can be either assumptions about the properties and structure of things (object system), allowing the contradiction problem or suggestion about the way activities are permitted by the latter. In the process of knowledge comes a point when addressing the hypothesis is necessary and inevitable when the movement of knowledge without putting forward the hypothesis is impossible. In this moment is the emergence of a problem situation, the problem.

It is important to emphasize that the strict wording of the problem is not always possible due to lack of information on the laws of the study (new) field of reality. "Perfect in every way of the problem, - says I. Mochalov in [21, p. 56] - suggests the presence of a researcher complete information about the object being studied. But then we would not have a problem! ".

A certain lack of rigor problem statement is inevitable in the early stages of the investigation. In the study, it is confirmed. And only in the works, which set out the solutions are already possible and requires a deep statement of problems and their rigorous formulation.

J. Bernal considered "finding problems" highest indicator of creativity. Solution and the content of creativity begins with finding the detection principle - the idea of the solution. (The idea - the basic idea behind the theoretical system, logical construction, a plan of action. Principle concept in the philosophy used to refer to the base, that is, what is the basis of a set of facts, knowledge, practice reveals the essence of their relationships, movement) .

Stage find the principle or idea of the solution is the culmination of creative search. The concept of "principle" is very close to the concept of an "idea" and is often used as a synonym.

The solution in the most general way is to open or create a new connection of things, such their compounds, conversion that would resolve the problem. Finding the solution is to find the support of certain new relationship (if open) or to create it (in the case of the invention).

It should be a distinction between the problem and the challenge. The problem appears as an issue for resolution, and the problem involves the question, and (data) of the solution. Rubinstein SL gives the following definition of "problem - it is always in their verbal, verbal formulation of the problem. She - a living testimony to the unity of thought and speech. " [28]

The essential difference between a problem and a challenge as V.E.Berkov shows [28], is that "the concept of the problem is related to a situation characterized by sufficient, means to achieve the goal of scientific knowledge, and the notion of the problem - their failure."

Demands placed on the formulation of the problem or task. This is due to the fact that it recorded the analysis of the problem situation, and that in the formulation already contain elements of the solution. The formulation of the problem (the problem) - an important stage in its understanding. In-depth analysis of problem situations, intelligent language challenges are essential to optimize creative research. Speech language - it is not an external factor to the thinking, but the process of it.

2. "The study of the rate of convergence of different methods for solving quadratic equations" - the theme of creative work

When writing a creative work of a preparatory step is to represent those creative works. At this stage, was proposed theme: "The problem of estimating the error in the solution of problems." The next stage: the stage of the exchange of texts and oral interview. Here are targeted student in materials, familiarity with the term, as the topic is not in the school curriculum. It was necessary to understand how an investigation of what may have problems. At this stage there is an accumulation of information on the studied subject. During the discovery of new areas to explore, so that changed the course of work.

The next stage - working with the results. Systematization, completion, execution drawings, design work in accordance with the plan.

On-site school participants to write creative work was proposed theme: "The approximate calculation. The accumulation of rounding errors in the preliminary. " This topic was chosen student of class 8 Gymnasium № 1.

During the joint work on the subject, we concluded that the subject is not creative. Note that, as in the creative task, and in the research necessary to conduct research, but also a creative task to interest the person, as opposed to research.

Theme "approximate calculations. The accumulation of rounding errors in the preliminary "was as follows. Proposed two equations, these two equations have approximately the same result. We had to figure out how it turns out, that is, to prove that these two equations are derived from a single quadratic equation. Also needed to provide the most accurate way and bring the rules following which did not occur to the accumulation of error or it would be minimal and with a certain degree of accuracy.

Working on the theme, it was proved that the two equations are derived from a quadratic equation, it was to identify the most accurate way. However, further study is not interested. Opened new areas to explore. The proposed theme expanded. In the future, we investigated the rate of convergence of the different methods for solving quadratic equations. The theme was restated: "Study the rate of convergence of different methods for solving quadratic equations."

Thus, the problem of our creative work is to study the rate of convergence of different methods for solving quadratic equations.

We have identified the following steps to solve it:

- Finding the roots of a quadratic equation in two ways and choosing the most effective way.

- Study of the method of bisection and the successive approximations.

- Application of the method of successive approximations to the different types of quadratic equations.

- Removing the conditions under which the quadratic equation can be solved by successive approximations.

We have highlighted the hypothesis:

1. Of equality and the equality of the form, gives a more accurate result.

2. The method of successive approximations can solve any quadratic equation.

In the first part of our work, we present theoretical information, the second part, we offer the course of our investigation.

The result of our creative work is that the equality of the form, is more accurate, and the method of successive approximations can solve quadratic equations, only those who obey the extracted condition.

Student's level of independence in this study are as follows. Apprentice proved that equality are derived from the same equation. Proved which of the equation is more accurate. After he decided to work only with this equation. How to find the roots of proposed supervisor, using the method to the equation a student yourself, chose the most efficient way, and decided to explore it. Equation, the root of which can not be found by the method of successive approximations was proposed supervisor. Next, student had built graph shown geometrically as the values are close to the root, and noticed that in the case of the possibility of finding the root of the method of successive approximations, the values tend to the point of intersection graphs, and in the other case, simply moving "on the square." Formulation conditions occurred with the manager.

Rejection of the original theme and the formulation of this can be attributed to the novelty of the material. Previously, nothing was known about the methods, moreover, are always excited when the root of one of the equations can not be found by the method of successive approximations. This creative work can be continued.

Creative work is protected in the school and the conference, and the speaker was awarded "the best paper and careful study."



Conclusion

As a result of this work were as follows:

1) found a link theoretical knowledge with the school program. Approximate solution of quadratic equations can lead students to the concept of approximate calculations, to open for them a new field of knowledge.

2) has been developed and tested an elective course aimed at the selection of tasks on approximate calculations, which were to study for the students.

3) has been developed and tested theme of creative work.

Further work on this topic can be continued. First, you should consider to develop an elective course as an independent, which was not only a means to highlight the research tasks. In - the second, possible revision of the identified research problems to those creative works. B - Third, thinking through possible links with electives items shown in the syllabus.



Literature

1. Algebra. Textbook. to Grade 8. Wed wk. / Ed. SA Telyakovskii. - M.: Education, 1988.

2. Algebra: The textbook for 8th grade middle school / Sh Ulama, JM Kolyagin etc. - M.: Education, 1991.

3. Ulama Sh Contraction mapping principle. Mathematics. Cybernetics. Subscription scientific - popular series. M.: Knowledge, 1983.

4. Ulama Sh Mathematics. Textbook for Grade 8. Wed wk. MA: Education 1992.

5. Aronov AM, Ermakov SV, OV Znamenskaya Training and education space in teaching development: Math Education: Monograph. - Krasnoyarsk, Krasnoyarsk State University, 2001.

6. NS Bakhvalov Numerical methods. Moscow: Nauka, 1973.

7. Shoes MI Algebra. Textbook for 11 class. - M.: Education, 1997.

8. Bradis VM Oral and written account. Aids calculations. / / Encyclopedia of elementary mathematics. Prince. 1. Arithmetic. Ed. PS Aleksandrov, Markushevich AI Khinchin AY / / Transl. ed. Technical - theoretical. Literature, 1951.

9. Vilenkin NY Mathematics. Tutorial for 5 cells. Wed wk. - M.: Education, 1994.

10. Vilenkin NY Algebra. A manual for schools and classes with intensive study of mathematics. - M.: Education, 1994.

11. Vygotsky MJ Handbook of elementary mathematics. Spb. St. - Petersburg Orchestra, 1994.

12. Zverkin GL Approximate calculations. / Encyclopedia for children. T. 11. Math / Heads. Ed. MD Aksenov. - Moscow: Avanta +, 1998.

13. Kaplan IA Practical classes in higher mathematics. Kharkov, Kharkov University, 1972.

14. Kashapov MM, Y. Tabakov Teaching creative problem solving in the teaching of psychology as a means of formulating the intellectual abilities of senior / / Pedagogy Development: Problems of modern childhood and school problems: Articles III scientific - practical conference. Krasnoyarsk, 1996. Part I.

15. SA Kovalev The place of approximate calculations in school mathematics. Thesis, 2001.

16. Kulanin ED, Lemeshko NN Shamshurin VL Microcalculators in Mathematics. - M.: High School, 1989.

17. Mathematics BES / Ch. Ed. Y. Prokhorov - 3rd ed. - Moscow: The Great Soviet Encyclopedia, 1998.

18. Mathematics: a textbook for Grade 5 high school / NY Vilenkin and others - Spb.: Lights, 1995.

19. Math: School Encyclopedia. M. Bustard, 1997.

20. Mordkovich AG Algebra. Grade 8.: Textbook for educational institutions. - Moscow: Mnemosyne, 1998.

21. Mochalov II Imaginary problems of science / / Problems of Philosophy. 1966. № 1.

22. Nurk ER, AE Telgmaa Mathematics. Grade 5: a textbook for secondary schools. - M. Bustard, 1995.

23. VD Stepanov Activation of extracurricular work in mathematics in high school. Book for teachers. Of work experience. M.: "Education", 1991.

24. Tabidze OI Value aspect of creativity / / Problems of Philosophy. 1981. № 6.

25. A manual for schools and classes with intensive study of mathematics. Ed. Vilenkin NY - M.: Education, 1994.

26. Elective course in mathematics in high school. Interuniversity scientific collection. Ed. Count.: Petrov ES, PhD. ped. Science, VI Sukhorukov, PhD. f. - M. Science. Saratov: Saratov State Pedagogical Institute. K. Fedin, 1989.

27. Tsikh AK Introduction into specialty "Mathematics": studies. Manual / Krasnoyarsk Krasnoyarsk. State. University Press, 1997.

28. Shumilin AT Problems of the theory of creation. Monograph - M.: High School, 1989.

29. Encyclopedic Dictionary of the young mathematician. Moscow: Pedagogy, 1989.



Appendixs

Appendix 1

Rounding rules

Rule 1: If the first digit to the left of the discarded is less than 5, the remaining decimal places do not change.

Rule 2: If the first digit to the left of the discarded more than 5, then the last remaining digit is added 1.

Rule 3: If the first on the left of the discarded digits is 5, and the rest of the discarded digits are zero, then the last of the remaining digits are added 1.

Rule 4: If the first on the left of the discarded digits is 5 and all other discarded digits are zero, then the last remaining digit is stored, if it is even, and is increased by 1 if it is odd (usually even number).



Appendix 2

Tasks

The concept of exact, approximate values. Absolute error of approximation.

1) How many integers divisible by 7 contains

c) 1 to 10 b) from 1 to 20) from 1 to 50 g) from 1 to 1,000,000?

Is it possible to give an exact figure?

2) Describe how to find answers

3) Why do I get different results?

4) It is known that 1 contains 142,857 to 1,000,000 digits. How much do you make a mistake?

5) Introduction of terminology.

6) Build a relationship diagram concepts.

7) State the definition of the exact value, the approximate value of the absolute error.

8) Let a - the approximate value of x. Find the approximation error if:

a) x = 5.346 and = 5.3 b) x = 15.9 a = 16) x = 4.82 and = 4.9

9) At the upper end of thermometers liquid is between the 21 and 22oC. As an approximate temperature take the number 21.5 which is the absolute error of approximation.

Approximation to the required degree of accuracy.

10) Construct the graphs of y = x, y = x-2, y = x-4.

11) Find the values graphically and solving the equation.

12) Up to what level have found the roots of the graph?

13) Introduction to the approximate values of lack and excess.

14) The approximate value of x = 2.4, the absolute error is less than 0.1. Find a gap which is all the exact value of x.

15) Let x = 5,8 ± 0,2 whether the exact value to be equal to:

a) 5.9 b) 6.001 c) 5.81

16) Is the number 4 is the approximate value of the fraction to within 4.3 to 0.5? To 0.1?

17), the approximate value of x is equal to the arithmetic mean approximation to the lack and excess

18) Prove that the number 0.43 is an approximate value of the fraction 13/30 to 0.01.

19) Indicate the approximate value of x is equal to the arithmetic mean approximation to the lack and excess

Rounding.

20) Solve the equation x * a = 6.3, a = 0.428571 ...

21) How can I solve the equation? (Use common fractions, take a finite number).

22) The rule of rounding. (Read)

23) On the day of the census number of residents equal to 57,328, but the number of residents in the city is constantly changing (arrival, departure ...) how many people in the city?

24) to round the number sequence to thousandths, hundredths, tenths, ones, tens, hundreds, thousands, a) 3285.05384 b) 6377.00753 c) 1234.5336

25) as the decimal point to 0.1 the number of: a) 13/8 b) 17/25) 39/129

26) The number of p--» 3,141592652 is the ratio of the circumference to its diameter. Round this number up to millions, thousandths, hundredths.

Methods for finding the result. Compare them.

27) Find the value of the expression with the required accuracy (precision is three decimal places)

28) Find the root of an equation with a given degree of accuracy (three decimal places) 5x2 + x-3 = 4

29) Consider your ways. Explain them.

30) Find the solution of the equation x = 1 +1 / x. What method to quickly obtain results?



Appendix 3

Assignments for elective courses, jobs, comments.

Topic 1: "The concept of exact, approximate values. Absolute error of approximation. "

(40 min)

Objectives: to introduce the concept of the exact, approximate values and absolute error in the logic.

Methods:

o heuristics.

Forms:

o Seminar;

o Lecture.

Means:

o Entries on the board;

o Speaking.

Introduction (3 minutes): Several sessions will be devoted to the approximate calculation. We will study the basic facilities and then formulate questions to answer that you want to do creative work.

Defining the form of comments

Problem 1: How many integers divisible by 7 contains

a) 1 to 10 b) from 1 to 20) from 1 to 50 g) from 1 to 1,000,000?

Is it possible to give an exact figure?

Describe how to find answers

Why do I get different results?

It is known that 1 contains 142,857 to 1,000,000 digits. How much do you make a mistake? Independent work of students (5 min), after frontal discussion (10 min). The discussion should define the terms:

The exact value of x;

The approximate value of a1, ..., An.

Approach - replacement of some other mathematical objects, in some sense close to the original.

Error - the task accuracy of the approximate number.

A. Section Represented e.

e = | x - a |

Task 2: Make Schemes concepts x, a, e. frontal discussion

(10 min.)

Task 3: Let a - the approximate value of x. Find the approximation error if:

a) x = 5.346 and = 5.3

b) x = 15.9 a = 16

h) x = 4.82 and = 4.9 Independent work of students (10 min) training exercise in finding AP

Topic 2: "Communication exact, approximate values and absolute error. VG and NG, range of variation. Finding accurate as average. "

(40 min)

Objectives:

Subject:

o Specify the exact relationship, the approximate values and absolute error.

o introduce the concept of HS and NG, the range of variation.

o Identify the way of finding the exact value as the average.

General culture:

o introduce a rule to speak.

Methods:

o Clarifying existing knowledge of oral sources.

Forms:

o Seminar;

o Lecture.

Means:

o Entries on the board;

o Speaking.

Unit 1: Working with a mathematical model. (20 min)

Defining the form of comments

Task 1: Remember the elements, whom he met at the last session. Indicate the relationship of the elements. Front poll.

Orally. (X - the exact value, a - the approximate value of x, e = | x-a | - absolute error.)

Task 2: The proposed tasks are known and unknown items. Can I find them? How? Point to model communication elements. (10 min)

Problem 1: Weight rolls should be 120 grams. When weighed two ready buns, giving you the 123gr. and 118gr. How wrong bakers?

Problem 2: It is known that the bun should weigh 120gr. When weighing the dough, use a circuit weights, which are insensitive to decrease or increase the weight by 5g. What value will be accurate, and what the approximate? Why? Independent solution (5 min), Frontal discussion (5 minutes).

Q: Have you considered two options: Unknown but also. Could there be another option? Come up with their tasks. Independent decision; Frontal discussion x unknown can not be.

Unit 2: Concepts and SH NG, range of variation. Finding accurate as the average. (20 min)

Problem 1:

a) On the interval mark a point 4/7.

B) The segment impose another segment, with marked divisions. Game.

(Teamwork)

10 min. (HS, NY, tolerance range).

Point 4/7 must be located between the points of 3/7 and 5/7.

Ie 3/7 - NY, 5/7 - HS;

(3/7, 5/7), the range of variation.

Task 2: Set the changing boundaries of x:

A) 20 Ј x Ј 22;

B) 3,7 Ј x Ј 4,1.

What is the average? Name three approximate value of x. Independent work of students. 10 min. The exact value of the average.

Summing up.

Theme 3: "Rounding. Rounding to a given accuracy. "

(40 min)

Objectives:

o Repeat rounding and rounding to the specified accuracy.

Methods:

o Elimination of private knowledge of the general;

o Clarifying existing knowledge from a written source.

Forms:

o Seminar.

Means:

o Entries on the board;

o Speaking;

o Handout.

Defining the form of comments

Task 1. (5 minutes).

Question 1: Answer the question, what it means to round?

Question 2: What is needed to round? Front poll. Versions children jot on the board:

round - to drop one or more digits, approximately represented by a finite number of bits,

Versions of children: to reduce record ...

Task 2: Answer the questions posed in the objectives. (5 minutes).

Problem 1: The diameter of the Earth is about 6400 km. So write in geography textbooks. What is the order of the numbers: if the distance is measured in tens, hundreds or thousands of miles away?

Task 2: Distance from Krasnoyarsk to Moscow 3312km. What is important for the traveler? frontal discussion

Task 3: Review the rounding rule. Are there any of these rules, the unknown, unfamiliar? Independent work of students (5 min.), Frontal discussion (5 minutes). Pupils given text:

Recall the rounding rule.

Rule 1: If the first digit to the left of the discarded is less than 5, the remaining decimal places do not change.

Rule 2: If the first digit to the left of the discarded more than 5, then the last remaining digit is added 1.

Rule 3: If the first on the left of the discarded digits is 5, and the rest of the discarded digits are zero, then the last of the remaining digits are added 1.

Rule 4: If the first on the left of the discarded digits is 5 and all other discarded digits are zero, then the last remaining digit is stored, if it is even, and is increased by 1 if it is odd

(Usually an even number).

Task 4: Go to one of the rules to build a model. Independent work of students (3 min.), Frontal discussion (5 minutes). In class students are usually interest the rule 4, since it is rare and is not offered in school textbooks.

We can construct the following model:

H-odd figure.

Task 5: Perform. (10 min)

1) round, knocking one digit

From what we round up? Up to what level?

2) Round off the number of consecutive thousandths, hundredths, tenths, ones, tens, hundreds, thousands:

1. 3285.0584;

2. 6377.00753;

3. 1234.5336.

3) Present as a decimal to the nearest 0.1 number:

1. 13/8;

2. 17/25;

3. 39/129. Independent work of students. Testing. Rounding with precision.

Summing up. (2 min.)

Theme 4: "Meet the faithful and significant figures."

(40 min)

Objectives:

o Introduce the concept of correct digits;

o introduce the concept of significant figures;

o Working with the faithful and significant figures.

Methods:

o Clarifying existing knowledge from a written source;

o Clarifying existing knowledge of oral sources.

Forms:

o Seminar;

o Lecture.

Means:

o Entries on the board;

o Speaking;

o Handout.

Defining the form of comments

Task 1: Are the records of 2.4 2.40;

4.05 to 4.050?

What is rounded up?

(10 min.) The proposed version of the frontal discussion of children briefly recorded on the board.

Must state the following (offered student or teacher would say): Recording 2.4 means that the figures are only valid integers and decimals (ie, the true value of the number may be, for example, 2.43 or 2.38). Record 2.40 means that true and hundredths (the true number may be 2.403 or 2.398, but 2.421 and 2.382).

Task 2: Read the rules and examples. Answer the questions. Clear whether the rule, if not, then ask questions. Come up with your own example.

(15 min.) Independent work. frontal discussion. Pupils given text:

True name figures when they submitted the result is an error of not more than ? LSB.

Example 1: If x = 20,04 and this value has three winning numbers, then we can assume that 19,95 <x <20,05.

Example 2: x = 4,323 have two correct numbers. What is the range of variation x?

Example 3: x = 4,3230 has 5 correct digits. What is the range of variation x?

(5 minutes).

Task 3: Perform.

1) The approximate value of x is equal to 3.6647. If the absolute error is 0.0007, then what numbers are the numbers correct?

2) The approximate value of x is equal to 0.029560. If the absolute error is 0.00003, what are the numbers of the faithful? Independent work. frontal discussion. The task for the detention.

(8 min).

Task 4: Listen to the rule and do the job.

Assignment: What numbers are significant in numbers?

0.09862;

652;

87,200;

0.064504. Front explanation.

Frontal discussion. Teacher explains: Meaningful called all the faithful of the figures, except for leading zeros before the number.

Summing up. (2 min.)

Topic 5: "The error of the sum and difference. The accumulation of rounding errors in the preliminary. "

(40 min)

Objectives:

o Formation of understanding the possibility of accumulation of error.

o Select the conditions under which the accumulation of errors.

Methods:

o The method of heuristic knowledge.

o Forms:

o Seminar.

Means:

o Entries on the board;

o Handout.

Defining the form of comments

Task 1: Find the sum of

25.3 + 0.442 + 2.741

a) no rounding terms;

b) the sum of rounded to tenths;

c) rounded to the tenth of each term. Independent work of students. (5 minutes). Students can work either in pairs or individually.

(Reference to the outstanding student cards).

Task 2: Find the sum of

52.861 + 0.2563 + 8.1 + + 57.35 + 0.0087

a) no rounding terms;

b) the sum of rounded to tenths;

c) rounded to the tenth of each term. Independent work of students. (5 minutes). Students can work either in pairs or individually.

Task 3:

a) Compare the results in reference 1;

b) compare the results in the job 2;

c) what is common in the results?

d) how can we explain the results?

d) what the results are accurate? Independent work of students. (5 minutes).

Frontal discussion. (12 min.) The work is divided into two stages:

In the first phase students are considering the results, discuss in pairs. In the second stage students discuss together because of what will get different answers.

In discussing the need to note that the results differ by 0.1. In conclusion, must be formulated that different responses were obtained due to the accumulation of error.

Task 4: Make an example that would be the accumulation of errors in subtraction. How did you come up with an example? Compare the example of the other examples, can provide something in common? Independent work (3 min).

Frontal discussion (8 min.) Students must present their examples tell us how they were designed.

Summing up. (2 min.)

Topic 6: "The relative error, limiting the absolute and relative errors."

(40 min)

Objectives:

o Presentation of the relative error;

o The idea of limiting absolute and relative errors;

Methods:

o The method of heuristic knowledge;

o Clarifying existing knowledge from a written source.

Forms:

o Seminar;

o Self-study.

Means:

o Entries on the board;

o Handout.

Defining the form of comments

Task 1: Read the definition and do the job. Independent work,

Frontal discussion (students offer their ways, choose the most successful type of writing). (10 min.) Handout 1:

Definition 1: The relative error of the approximate number is the ratio of the absolute error of the approximate number to the number itself.

Zadanie1: Make a formula for finding the relative error.

Note: The relative error is sometimes written as a percentage. To do this, multiply the result by 100%.

Task 2: Read the definition and example, to answer the questions provided. Then discuss the answers together. Independent work,

Frontal discussion.

(10 min.) Handout 2:

Opredelenie2: Number, clearly exceeds the relative error or equal to it, called the limiting relative error.

Opredelenie3: Number, clearly exceeds the absolute error or equal to it, called the limiting absolute error.

Legend:

D - "delta" - limiting absolute error;

d - "small delta" - maximum relative error.

Example 1: The seller weighs a watermelon on a beam balance. In a set of weights the least - 50g. Weighing gave 6300g. Seller found the exact weight of the watermelon?

6300 - the approximate number. Absolute error does not exceed 50g.

Why not?

The relative error does not exceed 50/6300 ? 0,008. Why?

In this example, the number of which can be taken as limiting the absolute and relative error?

Task 3: Come up with an example similar proposal has, and explain on what grounds a similar example. Independent work,

Frontal discussion. (10 min.) Pupils come up with examples, and then offer them to others for a general discussion.

Task 4: Measure the length and width of the notebook sheet. What is the limiting relative error of each measurement? Independent work. (5 minutes).

Task 5: Compare the measurements resulting from students. Frontal discussion. (5 minutes). Pupils should get different results - an illustration of a possible error.

Theme 7: "The error of the product."

(40 min)

Objectives:

o The concept of error works.

Methods:

o The method of heuristic knowledge.

Forms:

o Seminar;

o Self-study.

Means:

o Entries on the board;

o Handout.

Defining the form of comments

Task 1: Find the area of the desk. What is the confusion? Independent work. (5 minutes). Students work in pairs.

Question: How can you find the area? Found to measurement error? Front poll. (5 minutes). The result should provide two ways of finding errors:

Method 1: Find the error in each measurement (- limiting relative error factors).

Method 2: First, find the area, and then the error.

Task 3: Find the area under the most adverse circumstances (two areas). Compare the three areas together. Independent work. (10 min.) S - area found by multiplying the measurements;

S <- square, if the true value is greater than to receive.