Approximate calculations in mathematics curriculum

Размещено на http://allbest.ru/

Table of contents

Introduction

Chapter 1. Approximate calculations in mathematics curriculum

1. Mathematical problems which lead to the need for development of the apparatus of approximate calculations

2. Topic "Approximate Calculations" in school mathematics

3. Concepts associated with the approximate calculations

4. Analysis of school textbooks

5. Approximate calculations in school mathematics and their possible place

Chapter 2. Elective course "Approximate calculation" for 7-8 classes

1. Elective courses as a form of further education students

2. The goals, objectives, structure, optional course

3. Description of the course content

4. Testing course analysis of the results

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

1. Creativity and mathematical creativity

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

Conclusion

Literature

Apps



Introduction

In the school program of the most fully represented the concept of numbers and functions. Approximate calculations are affected much less offered to study, mainly, only two algorithms: round and finding errors. Need to study the approximate calculations emphasized Bradis VM [8]: "... the lack of school education in a special section on approximate calculations, is a serious defect of these programs are very adversely affect the mathematical culture of young people graduating from high school." Schoolchildren theme is represented as an auxiliary, not important, not of interest to the study. There is a problem: approximate calculation - an independent and very interesting branch of mathematics - is not presented to students. Extended our students about the field of mathematics, to show that the approximate calculations are a separate area, enrich the research experience of students possible under additional educational forms, for example, in the form of an optional course. Therefore the aim of the thesis is the development of an optional course and detection training and research tasks on the material approximate calculations.

To achieve the goal the following tasks:

- Identify areas of approximate calculations in mathematics curriculum. For this analysis has been done of methodological literature, analysis of the scientific literature on these issues, analysis of school textbooks.

- Selection of the material to an optional course.

- The allocation of research tasks that bring students to the concepts associated with the approximate calculations.

- Explore the possibility of introducing the material in the form of teaching and research tasks.

Thesis consists of three chapters, introduction, conclusion, bibliography of 29 titles and five annexes.

In the first chapter, we define the place occupied by approximate calculations in the school curriculum in mathematics as a science. As a result of this analysis of scientific and methodological literature, were found the directions in which no approximate calculations do virtually impossible:

1) to find numerical solutions of applications (for example, the study of the phenomena of nature);

2) the approximate location of irrational numbers, finding solutions of algebraic and transcendental equations;

3) approximate formulas;

4) The approximation of a function.

An analysis of the educational literature, it was found that the direction of the school curriculum are not represented. There is a disparity represent approximate calculations in the school curriculum with the role they play in both theoretical and applied mathematics. At school, children are taught the two algorithms (rounding and location of error), the substance is not considered approximate calculations. We found an association of theoretical mathematics from the school program. Approximate solution of equations, in particular, square, can lead students to the concept of approximate calculations, to open for them a new field of knowledge.

The second chapter is devoted to the elaboration of an optional course. The objectives of the elective course include:

1. Improved knowledge of students about mathematics.

2. Creating the conditions for students independent academic research.

In order to develop an elective course was analyzed material of scientific tasks selected suitable for pupils to choose a relevant age. Elective course is aimed at school children 7 - 8 classes. Choosing age explains the features of the school curriculum.

Designed elective course consists of two units.

In the first section examines the basic concepts highlighted in the analysis of textbooks. The basic concepts are introduced through the introduction of the concepts of logic, approximate calculations, developed Kovaleva SA. [15] In the second block are offered training and research objectives:

- "Error of the sum and difference. The accumulation of rounding errors in the preliminary. " Pupils are several examples of 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 (two possible ways). In finding the value of the first method, you must first round up to the nearest tenth the terms, and then add or subtract. In finding the value of the second method first add or take away, and then rounded to the nearest tenth. The result is a different answer. The question arises as to why this happened. After analyzing each round, students must come to the conclusion that there was accumulation of error.

- "Error of the product." In the task to take measurements to find the error of each measurement, and then the error of the work. Next you need to find the error of not finding work errors of each measurement. As a result, you have to come to the formula for finding the error works.

- "The approximate solution of equations." Proposed to solve a quadratic equation by different methods: selection, successive approximation, halving the interval. In the problem we formulate the problem. Which method: selection or successive approximations, the most effective? Which method: selection, successive approximation, bisection, the most effective? Is any equation can be solved by successive approximations? Which equations method works?

Elective course was tested in high school № 3 of Krasnoyarsk, in the 7th grade, for three months.

The developers of the course there was a hypothesis that the themes of creative works are research problems of approximate calculations. The third chapter is devoted to the creative task. Here is the experience of writing a creative work on "The Study of the rate of convergence of different methods for solving quadratic equations." The work of several methods for the approximate location of the roots of a quadratic equation was selected the most effective. Then it was discovered that the method does not work for all equations, after it was found the conditions under which the method works. The work was performed as part of the "School for Young Scientists" at the Gymnasium № 1 "Universal" is protected in the school conference. The paper noted the literacy PROGRESS study.

Thus, a number of problems associated with the approximate calculations can be entered in the optional courses and to offer as those creative works, which will enhance the representation of students and open a new area for research.



Chapter 1. Approximate calculations in mathematics curriculum

1. Mathematical problems which lead to the need for development of the apparatus of approximate calculations

To understand the role of approximate calculations in school mathematics look at their role in science. We emphasize the importance and widespread use of approximate calculations.

In the following analysis [6, 13, 16, 17, 19, 29] literature we have identified a number of areas, which are associated with the need to approximate calculations.

1) to find numerical solutions of applications (for example, the study of natural phenomena), [6];

2) the approximate location of irrational numbers, finding solutions of algebraic and transcendental equations [13, 16, 19, 29];

3) approximate formulas [17];

4) The approximation of a function, [13, 17].

Let us examine each direction more.

1. Finding a numerical solution of applied problems

In finding the numerical solution of applied problems (eg, the study of natural phenomena, to obtain their mathematical description, ie, the mathematical model of the phenomenon and its study) can not do without the approximate calculations. Analysis of complicated models requires special, numerical methods for solving problems. Necessary to know how a particular method is accurate, but it needs to address the approximate calculations.

Finding solutions of algebraic and transcendental equations

Approximate location values of irrational numbers, finding solutions of algebraic and transcendental equations - it is the task of the theory of numbers. In solving this problem it is necessary to assess the accuracy of the approximation, resulting in the development apparatus associated with the error.

Ability to operate with approximate numbers allows for an approximate solution of equations (algebraic and transcendental). The manual [27, p. 78] called algebraic complex or real number x0, satisfying the equation of the form, where the numbers a0, a1, ..., an are integers, not all zero, and n - natural. Any real or complex number that is not algebraic is called transcendental.

In finding the solutions of algebraic and transcendental equations to solve two common problems:

1) to get a method that gives the opportunity to improve the approximation;

2) to obtain an approximate solution with a predetermined level of accuracy.

In [13] This technique for finding approximate roots of algebraic and transcendental equations.

Finding the roots of an algebraic equation

For the approximate determination of the roots of an algebraic equation to the rule of Descartes determine the number of positive and negative roots, then separate them. Separating the root, we have the opportunity, as its approximate value take any number of the selected segment.

Department of real roots of the equation F (x) = 0 is very convenient to carry graphic. The values of the real roots of the equation F (x) = 0 are the abscissae of the points of intersection of the graph of y = f (x) with the axis Ox. To specify the segments to be only one root of the equation, does not require great precision.

To improve the approximation of the roots of an algebraic equation using four methods:

Method 1. Newton's Method (method of tangents).

The root must be separated, ie, to determine the interval [a, b], which is the only real root. In the first approximation of the root should take the value of the end of the segment on which the sign of the same sign as its second derivative.

For example, we find the root of the equation x2 - x - 1 = 0. Real root lies in the interval [2, 3].

2.Sposob way linear interpolation (the way the chords).

To calculate the (n + 1) - th approximation used by root

Note that xi and xn - values between which the desired root. In the first approximation of the root can take the value of any of the endpoints, which are separated by the root.

Method 3. Used to determine the approximate value of the maximum and minimum absolute value of the root of an algebraic equation.

If you have an equation, the simple, the largest in absolute value root can be found approximately from the equation. The approximate value is smaller in absolute value of the root can be found from the equation.

For example,.

1) - 1 = 0 = 1, the approximate larger in absolute value root.

2) - 1 = 0, = - 1 approximate the smaller the absolute value of the root.

Method 4. In the equation we select the last three members and solve the quadratic equation.

The roots are real, and then solve the equation for the first approximation to take root =.

The left side of the equation is divided by. Dividing by the scheme Horner. Division spend as long as there will be the binomial of the form, which is not divisible by. . From the equation we find the second approximation of the root = -. Left side of the equation is divided by Horner scheme and the remainder is in the form, etc.

Usually, this process leads to a number of values that approach required radical. After we stopped for some approximation of the root and took it to the desired value of the root, divide the left-hand side of the equation to. Get a polynomial of degree one less than the left-hand side of the equation. Equate this polynomial is zero, and with the resultant new equation we proceed as described above.

When solving algebraic equations using the methods of successive approximation (iterative method) [13] and the bisection of the interval [16, 29].

Approximation method

In order to use the method of successive approximations, the equation must be converted to a form where

y (x) = x, j (x) = f (x).

Substituting the values in the series, we find - th approximation to the root of the equation.

Note that if the sequence x0, x1, x2, ..., xn, ... converges, ie have a limit, this limit will be the root of the equation.

For example, we solve the equation x2 - x - 1 = 0

x = 1 + 1 / x j (x) = 1 + 1 / x.

x0 = 2, the first approximation of the root;

x1 = 1.5 second approximation to the root;

x2 = 1, the third approximation of the root, and so on

Bisection of the interval

Represent the equation F (x) = 0 in the form y (x) = j (x);

1. We construct graphs with = y (x) and y = j (x);

2. The value of x will be the point of intersection of the graphs are roots.

3. We choose a closed interval [a, b], containing the point of intersection.

4. Interval [a, b] is divided by the two-point z1 = (a + b) / 2;

5. If F (z1) = 0 then z1 - the desired root. If F (z1) № 0, then the two intervals [a, z1] and [z1, b] we choose one for which the value of the function f (x) at the ends have different signs, and is denoted by [a1, b1] . Now if we take the point z2 = (a1 + b1) / 2, then again, or F (z2) = 0 or F (z2) № 0, etc.

For example,

x2 - x - 1 = 0.

x = 1 + 1 / x.

The point of intersection of the graphs located on the interval [2, 3].

Interval [1, 2] contains the point of intersection graphs.

1) z1 = (1 +2) / 2 = 1.5;

Received two segments: [1, 1.5] and [1.5, 2].

For the interval [1.5, 2], the values of have different signs. In fact,

12 - 1 - 1 = -1;

1.52 - 1.5 - 1 = -0.25;

22 - 2 - 1 = 1;

2) z2 = (1.5 + 2) / 2 = 1.75;

Received two segments: [1.5, 1.75] and [1.75, 2].

For the interval [1.5, 1.75] function values have different signs. In fact,

1.52 - 1.5 - 1 = -0.25;

1.752 - 1.75 - 1 = 0.3125;

22 - 2 - 1 = 1.

Thus, the root is in the interval [1.5, 1.75]. Continuing the process can find the root of a certain specified degree of accuracy.

Finding roots of transcendental equations

In the solution of transcendental equations to the equation F (x) = 0 in the form j (x) = y (x). Following are two ways to use the approximate solutions of equations:

1) The graphical solution.

Plotting the curves y = j (x) and y = y (x); abscissas of intersection are the desired roots of the equation. Then use the method to find the roots of algebraic equations.

2) iterative method.

Let x = j (x) y (x) = j (x).

a) graphically or by trial are the first approximation of the root

x = x0, x0 = first approximation of the root.

b) to the right side of the equation x = ? (x) x0 and then substitute x1 = j (x).

x1 - the second approximation of the root.

c) substituted in the right-hand side of the equation x = j (x) instead of the value of x1

x, x2 = j (x1), x2 - third approximation of the root.

g) such approximations are obtained as follows:

x1 = j (x0);

x2 = j (x1);

x3 = j (x2);

x4 = j (x3), etc.

It is important to note that the transcendental number can be represented by means of a series. For example in the encyclopedia [29], the sum of 1 - 1/3 + 1/5 - 1/7 + 1/9 - ... is Х / 4, the sum of 1/12 +1 / 22 + 1/32 + ? 2 + ... is Х 2/6. These amounts are allowed to approximate the number of Х any, given in advance, the degree of accuracy (if you take a lot of the series). Accuracy will be determined by using the concepts of true and significant figures.

Approximate formulas

There is another one which is closely associated with the approximate calculations - the approximate formulas. The encyclopedia [17, s.489] approximate formula is defined as "the formula

f (x) » f * (x),

obtained from the formulas of the form

f (x) = f * (x) + e (x),



where e (x ) is treated as an error and discarded after evaluation. " Approximate formulas allow for calculations with approximate numbers to quickly find the approximate answer. We present some of the most commonly used approximation formulas, and note under what restrictions on | x | k formula will give the exact decimal places.

Annex 1 of this thesis work are graphs of functions that let you see how close to each other the exact and approximate roots of equations.

The textbook Bashmakova MI [7] presented formulas for computing the approximate values of the function

f (x) - y0 » f / (x0) D x; y » y0 + dy; » at y0 + f / (x0) (x - x0).

Applying the above formula we can construct some approximate formulas.

- Dana the power function y = xn. Fix a point x0 and apply the formula:

(x0 + D x) n » h0n + nx0n-1 D x.

In the BES [17, p. 487] approximation of functions defined as "finding for the function f of g of a certain class, in some sense close to f, it gives a rough idea." The problem of approximating a function - it is the task of replacing some functions other functions. This problem always arises in mathematics and its applications, because there are theoretical and practical requirements for the solution.

Theoretical:

approximation of functions is one of the most powerful means of investigating the properties of the functions themselves. There is a section of complex analysis - approximation of functions of a complex variable - student questions approximate representation of complex function through special classes of analytic functions. In the BES [17, p. 489] noted that the approximation theory is closely related to other sections of the complex analysis (the theory of conformal mapping, integral representations). Many theorems, formulated in terms of approximation theory, are, in fact, profound results on the properties of analytic functions and analytic nature.

Application:

there is a need to replace complicated functions easier, such a problem arises, for example, when it is necessary to calculate the value of the function.

- Want to replace this function approximates the function belonging to a given family of functions, defined by the physical conditions of the problem.

- The variation of the function is only known with a certain error, on the basis of this information we can determine the function of only approximately, is the origin of the so-called empirical formulas directly related to the processing of observations.

The encyclopedia [19, p. 415] describes the steps into which the actual solution to every problem of approximating functions.

1) The choice of approach, that is, the choice of a family of functions with which to give an approximation of a given function. Note that the classic way of approximating functions are algebraic polynomials of fixed degree n, rational functions, where the polynomials of degree n and m, the trigonometric polynomials of a given order n. In general, as a means of approximation is usually chosen polynomials of the form, where a given function.

2) Select a method of measuring deviations from the given function to the approximating function, ie, selection of the method to judge when approximating function is close to the set. Way to measure the deviation is defined by specifying the measures deviations from a given approximating function, that is a number that characterizes this deviation. Identify the following measures evasion:

- If it is important to approach a function on the whole interval [a, b] uniformly differed little from the given function:

- If it is important to approach a function of the average differed little from that set, and it is permissible to have existed a very short segments, in which the deviation reaches a significant value:

- If the values are not our function, and want to know the approximate value of the integral of the function:

3) The choice of approach, ie, the choice of the rule that the family of approximating functions given one approximating function. Note that there are the following methods:

- Interpolation;

- Best methods of approximation;

- Sums of Fourier;

- The partial sums of the series.

4) The actual construction of the approximating function. (The difficulty of constructing approximating function depends on the approximation method).

5) Evaluation of error arising from the replacement of a given function approximating function. (Algebraic polynomial on any finite interval [a, b] and the system of trigonometric functions with respect to all continuous periodic functions have the property that the approximation error can be made arbitrarily small by choosing the number of parameters that affect the family of approximating functions, large enough).

Thus, it is important to development of the apparatus of approximate calculations for applied mathematics and theoretical problems. The work was allocated to four areas, which do not dispense with approximate calculations. Of these areas for students available approximation of the function, because it uses a lot of new concepts. However, an approximate solution of the equations is quite accessible for students, this theoretical material related to the school curriculum.

2. Topic "Approximate Calculations" in school mathematics

In this section list the concepts of the theory of approximate calculations, students are familiar with from 1 to 11.

Approximation. In the references you can find several formulations.

1) So, in encyclopedias [17, s.487] and [19, s.316] a broader concept - approximation of - replacing some other mathematical objects, in some sense similar to the original. Approximation allows us to study the numerical characteristics and quality properties of the object, reducing the problem to the study of simpler or more convenient objects (eg, the characteristics of which are easily calculated or properties are already known).

2) The encyclopedia [8, p.20] also considered an approximation to the lack and excess.

3) The encyclopedia [19, s.249] approach is considered as a replacement of, and differs little from him many a * - his approach.

The available language will be understood as a substitute for the approach of some other mathematical objects, in some sense similar to the original.

If the approximate value of less precise, it is the approximate value of the shortage, if more - then the excess. The term "approach" will be used in the sense of the approximate value of the quantity.

Rounding. Rounds will be understood as an approximate representation of the number in decimal (or other such binary) number system with a finite number of digits. Such a definition is presented in the encyclopedia [19, s.238]. It also said the approach of rounding, but there is no clear definition. In the methodological literature, the definition of "rounding" is not offered, the term is explained in terms of rounding. In the literature there are three types of rules:

1) formal rounding algorithm [8, 11, 12];

2) the rules of rounding whole numbers and decimals, [22];

3) the rule of even numbers [19, 8, 11, 12].

Annex 2 to this work the formulation of rules.

Different formulations of the rules mean the same thing. The textbooks used mainly formal rounding algorithm.

Error. In reference literature deals with various errors. To determine the error is important to know about the sources of its occurrence. In reference [6, p.17] identified the following causes of errors in problem solving:

1) a mathematical description of the problem is unclear, particularly given the source data inaccurate description;

2) used for the solution method is often not accurate: get the exact solution of a mathematical problem arises requiring unlimited or unacceptably large number of arithmetic operations, so instead of exact solutions of the problem has to resort to approximate;

3) arithmetic operations rounding.

4) A typology of errors in accordance with the causes, ie, there are three types of error.

3. Concepts associated with the approximate calculations

approximate calculation mathematics school

Types of error corresponding to these reasons:

1) A fatal error - this error, which is a consequence of the inaccuracy of numerical data included in the mathematical description of the problem;

2) the accuracy of the mathematical model - is the uncertainty that is a result of the mismatch of the mathematical description of the problem of reality;

3) the accuracy of the method;

4) computational error.

We introduce the formal definition.

Let

I - seek out the exact value of the parameter,

I * - this value, consistent with the mathematical description,

I * h - the solution obtained in the implementation of the numerical method assuming no rounding,

I * h * - approach to the problem, obtained in real calculations.

Then

r 1 = I * - I unrecoverable error

r 2 = I * h - I * error in the method,

r 3 = I * h * - I * h computational error

r 0 = I * h * - I complete error.

The total error satisfies 0 = r--r 1 + r 2 + r 3.

In many cases, the term accuracy of a species does not understand the difference between the above approximations, and some measure of affinity between them. For example:

r 0 = | I * h * - I |

r 1 = | I * - I |

r 2 = | I * h - I * |

r 3 = | I * h * - I * h |

The following groups of errors:

1) Measurement error and the approximation error.

Some sources [25, p.142] for the measurement error of understanding the difference x - a, where x - the true value of the measured quantity, and - the measurement result. Under the approximation error of understanding the difference between the number of x and approximate values. For example, the approximate values of P.

2) Errors in absolute, relative and marginal.

Thus, in [15, p.13] states that the absolute error - the absolute difference | x - a |, where a - This number, which is regarded as an approximate value of a certain value, the exact value of which is equal to x.

Under the relative error we mean the ratio of the absolute error of the approximate number to the number itself.

In reference [11, p. 95] The concept of limiting error.

A number that clearly exceeds the absolute error (or at worst equal s), called the limiting absolute error. Number, obviously higher than the relative error (or at worst equal to it), is called the limiting relative error.

3) Errors resulting from arithmetic operations on numbers.

Note the error works, sum and difference quotient.

In reference [11, p.98 - 100] states that the maximum absolute error is the sum of the amount of the limit of the absolute errors of the individual terms. With a significant number of terms is usually a mutual compensation of errors, so the true amount of error only in exceptional cases coincides with the utmost accuracy, or close to it.

Limiting absolute error of the difference is the sum of the absolute errors limit the minuend and the subtrahend.

Limiting the amount of relative error is between the minimum and maximum of the relative errors of the terms. If all the terms have the same (or nearly the same) the limiting relative error, the sum is the same (or nearly the same) maximum relative error. Ie the accuracy of the amount not less than the accuracy of terms. With a significant amount of the same number of terms, as a rule, much more precise terms.

The difference between the approximate numbers may be less accurate than the minuend and subtrahend. "The loss of accuracy" is particularly large in the case where the minuend and subtrahend differ little from each other.

Here in [11, p.100] on the error works written, limiting relative error is approximately equal to the amount of work limiting relative error factors. The rule for the two factors can be written as: d--»--d 1 + d 2. The exact expression is d:--d--=--d 1 + d 2 + d 1 d 2, ie, the marginal product of the relative error is always more than the sum of the limiting relative error factors, it is more than the sum of the relative errors on the product of the factors. This excess is usually so small that it does not have to be considered.

Accuracy in private [11, p.106 - 107] is in two ways:

1) The maximum relative error is approximately equal to the sum of the private marginal relative errors of the dividend and the divisor.

2) Let the dividend and divisor are each in k digits. Then the absolute error in the worst case, the private close to 1.05 units (k - 1) - the first sign (this value is never reached).

The boundaries of the absolute and relative errors. In [15, p.13-14] gave the following definition:

The boundary of the absolute error - the number D (a) such that | x - a | Ј--D (a).

The highest and lowest limits exact value.

Upper limit x: (x HS): g = a + D a.

The lower limit of x: (x NG): p = a - D a.

If you find value with the required accuracy, in finding the error associated with arithmetic operations on numbers are important concepts of true and significant figures. In [16, p.24] shows the following definition of correct digits: true name figures when they submitted the result has an error of no more than ? LSB. In reference [11, p.93] meaningful name all the faithful of the figures, except leading zeros before the number. Faithful and significant figures represent different. Here is an example. So, if x = 20.024, and this value has three digits of the faithful, we can assume that 19.95 <x <20.05.

Most of these concepts encountered in the school curriculum.

4. Analysis of school textbooks

To determine the role of the topic "Approximate computing" in the curriculum was reviewed by three textbooks for grades 5 and 8, and view tutorials for other classes to be used approximate calculations. Application has been found in a textbook for 11th grade.

Textbook for fifth grade [18]

Theme "Rounding".

Uses the concepts of rounding up units and the approximate value in excess.

The new material is introduced for the problem: How many cans of paint need to buy in order to paint the floor in the apartment of 148 m2, if we know that a 10 m2 floor needed 1 can of paint? With the problems the author would like to stress the need to rounding. But the task is chosen to fail, as a practical matter it is possible rounding only to a larger number, regardless of the rules of rounding.

Presented are two ways to introduce the concept of rounding and rounding numbers to one.

The input method is a particular example, the concept of rounding too. "The replacement of the approximate value of 14.8 15 called rounding that number to a few" (rounding of other numbers says nothing.) Give two examples of rounding and allocated a special case.

Special case - it is 14.5 equidistant from 14 and 15. Accepted approximate value in excess, is 15. Earlier about the approximate values ??with the excess does not say anything. Used a concept that has not been introduced. In addition, in explaining operated numbers 14 and 15, there is no reference that can be rounded up to another number. It concludes with a rounding rule and examples of its application. The examples also introduced the sign "approximately equal."

The analysis found that:

- About the lack of approximation says nothing;

- About the approach to the excess referred to in passing;

- Only rounded decimals, rounding of integers does not say anything;

- The word "close", but does not say when rounded number should be as close as possible to the original number;

- Not different rounding and rounding only in a big way.

Textbook for grade 5 [9]

Topic: "The approximate values of numbers. Rounding "

Uses the concepts of approximate values disadvantage with an approximate value of excess and rounding numbers to integers.

The author offers two illustrated examples. In the first example offered two solutions, from their comparison shows the necessity of rounding. Example matched successfully, corresponds with the children. Example 1: A mass of pumpkin more than 3 kg, but less than 4 kg. If we denote the weight of the pumpkin (in kilograms) by the letter x, 3 <x <4. The second example confirms the first. Example 2: The length of the segment AB lies between 6 cm and 7 cm If the length of the interval x, 6 <x <7. With the help of examples Vilenkin N. introduces the concept of an approximate value with excess and a lack of an approximate value.

Next, a general definition. The word "closer": "If the length of the segment closer to 6 cm, then to 7, it is approximately equal to 6." We consider several possible cases of the first example. Shows the possibility of rounding off numbers to a different one and the same number. Rounding rule is formulated using the word "closer".

Noting that the number can be rounded, not only to whole, but also to other places. Formulated a rule that must be applied by rounding up some of the discharge.

In conclusion, the author gives two examples:

- By rounding to the nearest tenth;

- By rounding integer.

The analysis found that:

- Showing rounding integers and decimals;

- Gives an idea of the problem and the possibility of rounding rounding both to lack or in excess.

- There are two rules: to be rounded to the nearest whole number and to round up the fractional part.

Textbook for grade 5 [22]

Offers two themes: "Rounding integers" and "Rounding decimals."

"Rounding integers"

The concepts of rounding and estimation are approximately equal.

Initially, the author offers a solution to the problem. It reflects the need for rounding, but for students in the fifth grade tricky. (Not many children faced with a census of the population). Objective: On the day of the census number of residents equal to 57,328 persons. But the number of people in the city is constantly changing (arrival, departure, birth, death). So, that number is soon to be incorrect. Therefore we can say that in the city of about 57,000 lives.

In the example of the concepts of rounding numbers to thousands. Emphasizes the possibility of rounding up to tens, hundreds, etc.

It is noted that the rounded number should be as close to the original. From this follows the rule of rounding.

The following two examples. Sign "approximately equal" is introduced after. The good thing is that the author explains how the sign is pronounced.

After application of the rounding of integers shows where and how student can actually apply the ability to round.

"Rounding decimals"

No new concepts not used. The material is based on rounding whole numbers.

The example of rounding:

1) is defined between any numbers entered number to be rounded;

2) is determined to which of them is closer to round number, therefore, it is the result of rounding.

Emphasizes the possibility of rounding up any discharge. And rounding rule is formulated. Attention is paid to the students record 32.0. Described that 0 can not be cast as the number rounded to the nearest tenth, not to units (note that there is a difference).

The general analysis of textbooks for grade 5

In the fifth grade topic "Approximate calculation" is introduced in two ways: one paragraph that includes rounding all numbers and separately rounding whole numbers and decimals.

In the textbooks, the theme is different, "Rounding", "approximate values of numbers. Rounding "," Rounding of natural numbers. Rounding decimal numbers. In various books contain different information about rounding, but can be distinguished from all common:

- Rounding the number to a few;

- The approximate value in abundance;

- The approximate value of a disability;

- Approximately equal numbers;

- Estimation.

Textbook for class 8 [4].

Topic: "The approximate calculations"

Topic is presented in four sections: "The approximate values of variables. Approximation error "," Error estimate, "" Rounding "," relative error. "

The four sections are about the same structure. So, first the necessity of introducing the concept, then, is the problem with the use of the term (it comes bundled with the solution described in detail), then the result is generalized in (built form that can be used to solve other problems), is the problem, showing how to apply the formula and provides exercises to perfect.

In this tutorial include the following concepts:

- Approximate value of various sizes;

- Absolute error;

- Estimate of the absolute error;

- The approximate value of a disability;

- The approximate value in abundance;

- Accuracy of measurement;

- Rounding;

- Relative error;

Let us examine each section details:

§ 1. "Approximate values of. The error of approximation "

We introduce the concept - the approximate value of the different variables. Further examples are available (they include exact and approximate values of the variables).

But the student does not say that for example. The textbook is written: "Let's see some examples of" further enumerated. The recording may confuse the student, as many people think that these are examples of values with approximate values.

Following the examples of answers, where the values calculated exactly, and where approximately.

Next is the problem and its solution. Its example introduces the concept of the absolute error of approximation. Following the formula for calculating the absolute error (generalization problem, replacement numbers letters).

Here is a problem and its solution that requires finding the absolute error using the formula.

Exercises to include testing of the following tasks: in the examples indicate which numbers are the exact value, and which approximate, finding the absolute error of approximation.

But there are exercises that do not match the theoretical part: you need to add some approximate values. But it was not mentioned that the approximate values may be several - inconsistency.

§ 2. "Assessment of errors"

Students are introduced to when we can assess the absolute error and that it needs. There three new concepts:

- Estimate of the absolute error;

- Closer to the excess;

- Approach to disability;

Offered problem and its solution (see solutions of how to estimate the absolute error). Provides a method of recording of an equal number of x, up to a h (but first this record is entered in the particular example). Of approximations to the lack and excess of said very little. The text of these terms are not allocated, not represented in the form of a definition, only one example.

A large section is devoted to a discussion of the accuracy of measuring devices. After about the use of the approximate values by replacing common fractions and decimals is an example. Exercise is repeated examples of a theoretical part (mostly changed only digits).

§ 3. "Rounding"

Say where rounding is used and an example. Attention is paid to write (x ? a). After the problem with the proposed solution. The answer turned out 3.125 (say, in practice, such a result is rounded to the nearest tenth. - Is not entirely true, because you can round up to whole, but it is not mentioned.)

On example is a rounding rule. The result suggests two possibilities: round and round with an excess of a disadvantage. Further rounding rule in general and a few examples.

§ 4. "Relative error"

The need of the relative error is illustrated by two examples. The concept of relative error is introduced in the form of the definition, then written to a formula. Also given the task to use the formula.

The textbook for the eighth grade. [20].

Topic: "The approximate value of the real numbers."

Theme includes the following concepts:

- The approximate solution;

- Approximate value of the shortage;

- The approximate value for the number of excess;

- Approximate value of up to ...;

- Rounding;

- Absolute error;

- Error of approximation.

By the example of finding the points of intersection of the graphs shows the approximate solution of the equation. Here we show the record (x ? a). The rationale for introducing the concept of an approximate value of a real number:

- To find a solution graphically;

- A real number - is infinite decimal, but use of the record in practice uncomfortable.

On the example introduces the concept of a lack and excess of a given accuracy. Describes a possible approach with different accuracy (up to 0.0001, 0.01, etc.). Understand examples of finding approximate values of lack and excess of a given accuracy. Introduce the concept of rounding numbers as generalizing the approach of lack and excess. The concepts of the approximation error (absolute error) is introduced as a definition.

Importantly, the author raises the question: which approach is better? The shortage or excess (draws attention to avoid further confusion).

Further rounding rule and examples of the application of the rules.

The author acknowledges the important part: there is no approximation to within h (stressed that the accuracy can be any).

Exercises to perfect reflect the theoretical material:

- To find the approximate values of the shortage and excess of a given accuracy;

- Calculate a given accuracy;

- To evaluate the error of the approximate equality.

The textbook for the eighth grade. [1].

Topic: "Approximate computation."

Topic is presented in two sections: "Record the approximate values" and "Actions approximate numbers."

You can select only one concept: true figures. In this tutorial, the concept of absolute and relative errors are not introduced, it is assumed that they are known, the author handles them in explaining the record of the approximate values.

§ 1. "Record the approximate values"

Mainly shows a record with a given accuracy. Next, we introduce the definition of the correct figures and examples to them. Next are examples of finding and assessment of absolute and relative error.

§ 2. "The actions of the approximate numbers"

Examples are given to rounding when addition, subtraction, multiplication and division.

Thus, the material in this tutorial does not correspond to the material proposed by other authors. A prospective study of absolute and relative error in the seventh grade. The content is complicated.

The general analysis of textbooks for grade 8

All books subject title includes the phrase "Approximate computation."

But the content in the three different books:

Textbook [1] is not fully in line with other books. In the book [4] studied the approximation: the absolute and the relative, the estimate of the absolute error and rounding numbers. In the book [20] to study the proposed approximate values for lack and excess, and absolute rounding error.

Of these tutorials can be identified main content:

- The approximate value of the shortage and excess;

- Rounding;

- Absolute error;

- Relative error.

General characteristics of the textbooks for 5, 8 classes

Generally in grades 5 and 8, the theme of "The approximate calculation" includes concepts:

- Round;

- The approximate value of the shortage;

- The approximate value of the excess;

- Absolute error;

- Estimate of the absolute error;

- Relative error.

But there is a drawback. The author of each tutorial includes concepts that wants. As a result, and in the fifth and eight grades introduced approximation of lack and excess. There is no separation into classes.

Analyzing the content of school textbooks in the textbook for grade 11 [7] found the job under which the use of knowledge on approximate calculations.

- To find the approximate value of using the graphs of functions;

- To find the values of the MC lg, log, trigonometric functions and record up to h;

- Calculate the approximate value of the formulas;

- Approximate formulas;

5. Approximate calculations in school mathematics and their possible place

Topic "Approximate calculation" in the school curriculum introduced in classes V and VIII, and the material is not related to each other. What you type in the class V, re-entered in the VIII, but on other grounds. Consider only some of the tasks that lead to approximate calculations, not all authors. "Approximate calculations" are reduced to round and finding the absolute and relative errors. From the above it can be concluded that the concept of a holistic approach is not exactly represented in school mathematics, and thus a certain place (except for some items), approximate calculations are not.

Thus, we can identify duplicate content themes:

- Overall, the mandatory proposed instill authors of school textbooks;

- Extra, which is not entered in a special way, but may be useful in the study of other subjects and help in solving the required tasks.

As noted above, in the curriculum theme is introduced in the fifth and eighth grades. And it is given an average of two hours. After analyzing the textbooks, it was found that for compulsory study proposed two algorithms: round and finding errors.

Thus, in the fifth and eighth grades study the same concepts, and students is not known what role the subject plays in mathematics.

At the same time, in school mathematics approximate calculations are present. There are issues that can not do without the concept of accuracy of the approximation. We list these topics:

- Irrational numbers;

- Infinite decimals;

- Calculate the root n - degree;

- Logarithms;

- Quadratic equations;

- Approximate formulas;

- Plotting functions;

- Limit.

In almost all of these topics require knowledge of the range of variation. Often without the help of the MC student can not find the meaning of the root n - degree, but this can be done by applying knowledge of the theme "The approximate calculation".

With the approximate formulas, students are facing in the eighth and eleventh grades. In eighth grade, the guide [25, p.151 - 153], the following approximate formula:

For small values of a and b holds approximate formula

(1 + a) (1 + b) » 1 + a + b, if a--=--b, we obtain (1 + a)--2--»--1--+--2--a.

It follows that if a | b | is small compared to | a |, then (a + b) 2 » a2 + 2ab.

The textbook for grade 11 [7] contains a number of laboratory work, under which students are faced with the approximate calculations. In laboratory studies found the following tasks:

- Draw a rough graph speed, change the slope of the tangent;

- Calculate the approximate slope of the tangent;

- Calculate with given accuracy;

- Find the approximate value of the root of the approximate formula;

- Find the approximate value of the root of the method of successive approximations;

- Study the impact of the error on the calculation of t computation error at

- Calculate the approximate value of the natural logarithm of a number using the formula (ax) = (ln a) ax;

- Calculate the approximate value of the integral by Riemann sums;

- Decide the approximate differential equations by Euler's method. (From a construction point of polygon with a given step. Value of the function is given by equation of the line, the slope is found from the differential equation).

- Calculate the approximate roots of the equation:

A) the method of bisection;

C) the method of tangents;

C) the method of chords.

Comparing the curriculum material on "Approximate computation" with the problems leading to the idea of ??approximate calculations, it is clear that the proposed algorithms (rounding error accumulation) does not reflect the trend. Hidden from the students possible research tasks. In fact, the problem of the approximation of functions requires a large amount of additional knowledge and is not available for students. However, we have found material linking the curriculum with the theory. Approximate solution of equations, in particular, quadratic equations, can lead students to the concept of approximate calculations, to open for them a new field of knowledge. There was a hypothesis that the problem of approximate solution of quadratic equations can be research.



Chapter 2. Elective course "Approximate calculation" for 7-8 classes

1. Elective courses as a form of further education students

Elective courses in mathematics in high school, is a form of extracurricular activities to deepen and broaden the student's knowledge of mathematics, develop an interest in the subject, to inculcate a taste for independent acquisition of knowledge, to join the research.

In [5] the following characteristics of an optional course: "The material thematically elective rigidly connected to the lesson. The lesson is given application context. The content of the lesson easier elective, does not require retention of complexly structured mathematical objects. Worked through the art of methods, techniques and knowledge acquired in the classroom, are solved non-trivial and entertaining tasks, some Olympiad problems. "

There are other, non-traditional, a look at the content of an optional course. It is in the manual [26], in which the collected experience of teachers Saratov Pedagogical Institute. Let us dwell on this understanding elective course details.

Ivanova NN in the manual [26] points out the issues that need attention in a training:

1. The importance of emotional teaching (vivid story, viewing filmstrips, students experiment with models, game situations);

2. creation of problem situations and their resolution (to develop a system of questions to guide student thinking to find solution to the problem, to attach students to self-exploration, discovery, the ability to creatively interpret studied);

3. system search problems. Search task - the task is not presented which students do not know in advance how to resolve it. Students solve all the problems on their own or with a little help from the teacher.

4. inclusion of historical material. It is appropriate for the following reasons:

- Striking fact biography of the scientist can cause the students desire deeply acquainted with his life and work;

- It is useful to show the character of the problem, the difficulties that arose for scientists, attempts to overcome the difficulties and why it failed. The above steps set though very simplified, but the pattern of movement in science.