Absolute values and their classification.
Absolute valuesare the results statistical observations. In statistics, unlike mathematics, all absolute values have a dimension (a unit of measurement), and can also be positive and negative.
Unitsabsolute values reflect the properties of units of the statistical population and can be simple , reflecting 1 property (for example, the mass of cargo is measured in tons) or complex , reflecting several interrelated properties (for example, ton-kilometer or kilowatt-hour).
Unitsabsolute values can be 3 kinds :
- natural - are used to calculate quantities with homogeneous properties (for example, pieces, tons, meters, etc.). Their disadvantage is that they do not allow summing dissimilar quantities.
- Conditionally natural- applied to absolute values with homogeneous properties, but exhibiting them in different ways. For example, total weight energy sources (wood, peat, coal, petroleum products, natural gas) is measured in tce — tons of reference fuel, since each of its types has a different calorific value, and 29.3 mJ/kg was taken as the standard. Similarly total school notebooks is measured in us.sh.t. — conditional school notebooks 12 sheets in size. Similarly, canning products are measured in a.c.b. - conditional cans with a capacity of 1/3 liter. Similarly products detergents is reduced to a conditional fat content of 40%.
- Cost units of measurement are expressed in rubles or in another currency, representing a measure of the value of an absolute value. They make it possible to summarize even heterogeneous values, but their disadvantage is that it is necessary to take into account the inflation factor, so statistics always recalculates cost values in comparable prices.
Absolute values can be momentary or interval. Momentary absolute values show the level of the studied phenomenon or process on certain moment time or date (for example, the amount of money in your pocket or the value of fixed assets on the first day of the month). Interval absolute values are the final accumulated result for certain period(interval) of time (for example, salary for a month, quarter or year). Interval absolute values, unlike moment ones, allow subsequent summation.
The absolute statistic is denoted X , and their total number in the statistical aggregate - N.
Number of values with the same value sign is denoted f and is called the frequency (recurrence, occurrence).
Absolute in themselves statistics do not give full view about the phenomenon under study, since they do not show its dynamics, structure, relationship between parts. For these purposes, relative statistical values are used.
Kemerovo
MOU "Average comprehensive school No. 37"
elective course optionally
for students in grades 10-11
Equations, inequalities and systems,
Compiled by:
Kaplunova Zoya Nikolaevna
mathematic teacher
Explanatory note………………………………………..page 2
Educational and thematic plan…………………………………...p. 6
List of keywords……………………………………...page 7
Literature for the teacher………………………………………..page 8
Literature for students………………………………...p.8
Explanatory note.
The main task of teaching mathematics at school is to ensure a strong and conscious mastery by students of the system of mathematical knowledge and skills necessary in Everyday life and labor activity every member modern society, sufficient to study related disciplines and continuing education.
Along with the solution of the main task, a deeper study of mathematics provides for the formation of students' sustainable interest in the subject, the identification and development of their mathematical ability, orientation to professions that are essentially related to mathematics, preparation for studying at universities.
The issue of differentiating the teaching of mathematics remains relevant, allowing, on the one hand, to provide basic mathematical training, and on the other hand, to satisfy the needs of everyone who is interested in the subject.
Program this course"Equations, inequalities and systems containing a sign of absolute value" offers the study of such issues that are included in the course of mathematics of the main school not in in full but necessary for further study.
The concept of absolute value (modulus) is one of the most important characteristics numbers both in the realm and in the complex numbers. This concept is widely used not only in various sections of the school course, but also in courses higher mathematics, physics and technical sciences studied in universities. For example, in the theory of approximate calculations, the concepts of absolute and relative errors approximate number. In mechanics and geometry, the concepts of a vector and its length (vector modulus) are studied. In mathematical analysis, the concept of the absolute value of a number is contained in the definitions of such basic concepts as a limit, a bounded function, etc. Problems associated with absolute values are often found in mathematical olympiads, entrance exams in universities and on the exam.
AT school curriculum the course of mathematics does not provide for the generalization and systematization of knowledge about the modules, their properties, received by students over the entire period of study.
Thus, this course "Equations, inequalities and systems containing a sign of absolute value" is intended to expand basic course algebra and the beginning of analysis and gives students the opportunity to become familiar with the basic techniques and methods for completing tasks associated with modules. Awakens research interest in these issues, develops logical thinking, contributes to the acquisition of experience with a task that is higher than the required level of complexity.
The course "Equations, inequalities and systems containing a sign of absolute value" is intended for specialized training students in grades 10-11 and is designed for 34 hours (1 hour per week).
In the process of teaching this course, it is proposed to use various methods revitalization cognitive activity students, as well various forms organizing them independent work.
During this course, students will learn theoretical material and perform practical tasks. The result of mastering the course program is the presentation creative works at the final lesson
When studying the course, a test control is provided.
Course Objectives:
* generalization and systematization, expansion and deepening of knowledge on the topic " Absolute value»;
*acquisition of practical skills for completing tasks with the module;
*level up mathematical training students.
Course objectives
* equip students with a system of knowledge on the topic "Absolute value"
* to form the skills of applying this knowledge in solving problems of varying complexity;
* prepare students for the exam;
* to form skills of independent work, work in groups;
* to form the skills of working with reference literature;
Requirements for the level of assimilation of educational material
As a result of studying the course program, students will be able to
know and understand:
*definitions, concepts and basic algorithms for solving equations of inequalities and systems with a modulus;
*rules for constructing graphs of functions containing the sign of the absolute value;
Be able to:
*apply definition, properties of absolute value real number to the solution of a real number to the solution of specific problems;
* solve equations, inequalities, systems of equations and inequalities containing a variable under the module sign;
* be able to do small research independently.
1.Introduction 1h.
Goals and objectives of the course. Questions covered in the course and its structure. Acquaintance with literature, themes of creative works.
24 hours)
Determination of the absolute value. Geometric interpretation module concepts. Operations on absolute values. Simplification of expressions containing a variable under the module sign. Application of module properties when solving problems.
3. Graphs of functions containing a sign of absolute value. (8 hours)
Rules and algorithms for plotting function graphs. Definition even function. Geometric transformations of graphs of functions containing the modulus sign. Basic plotting on examples of the simplest functions. Graphs of equations: y=f|x|; y=f(-|x|); y=|f(x)|; y=|f|x||; |y|=f(x),where f(x)≥0; |y|=|f(x)|
4.Equations containing absolute values.(10 hours)
Basic methods for solving equations with a modulus. Disclosure of the module by definition, transition from the original equation to an equivalent system, squaring both parts of the equation, the method of intervals, graphic method, using the properties of the absolute value. Equations of the form: |f(x)|=0; f|x|=o; |f(x)|=g(x); |f(x)|=|g(x)|;
Method of change of variables, when solving equations containing absolute values. Interval method for solving equations containing absolute values. Equations of the form:|f(x)|±|f(x)|±|f(x)|±…±|f(x)|=0; |f(x)|±|)f(x)|±|f(x)|±…±|f(x)|=g(x).
A method of successive disclosure of a module when solving equations containing a "module in a module". Graphic solution equations containing absolute values.
5. Inequalities containing absolute values (10 hours)
Inequalities with one unknown. Basic methods for solving inequalities
with module |f(x)|>a. Inequalities of the form a|f(x)|>g(x); |f(x)|>|g(x)|.
6. Final lesson (1 hour)
Presentation of creative works.
Section III. Educational and thematic plan
Titles of sections and topics | Practice | Conduct form | form of control |
|||
Introduction | Knowledge Auction | Questionnaire, records |
||||
Absolute value of a real number | ||||||
Absolute value of a real number | Lecture, workshop | Reference summary, problem solving |
||||
Simplifying expressions containing a variable under the modulo sign | workshop | Problem solving |
||||
Graphs of Equations That Contain Modulus Sign | ||||||
Rules and algorithms for plotting graphs | Workshop | Memo with the rules and algorithms of constructions |
||||
Definition of an even function. Geometric Plot Transformations | Seminar - workshop | Reference summary, task solution |
||||
Graphs of equations: y=f|x|; y=f(-|x|); y=|f(x)|; y=|f|x||; |y|=f(x),where f(x)≥0; |y|=|f(x)| | Checking the execution of plotting |
|||||
Equations Containing Absolute Values | ||||||
Basic methods for solving equations with a modulus | Abstracts, algorithms |
|||||
Equations of the form: |f(x)|=0; f|x|=o; |f(x)|=g(x); |f(x)|=|g(x)|; | workshop | Checking solved tasks |
||||
The method of intervals in solving equations containing the sign of the modulus. Equations of the form:|f(x)|±|f(x)|±|f(x)|±…±|f(x)|=0; |f(x)|±|)f(x)|±|f(x)|±…±|f(x)|=g(x). | Workshop | Reference abstract, verification of solved tasks |
||||
The method of successive disclosure of the module when solving equations containing the "module in the module" | workshop | Abstract, memo, check assignments |
||||
Graphical solution of equations containing absolute values. | Workshop | Chart test |
||||
Inequalities containing absolute values | ||||||
Inequalities with one unknown. Basic Methods for Solving Inequalities with Modulus | abstract |
|||||
Basic Methods for Solving Inequalities with Modulus | workshop | Abstract, solution check |
||||
Inequalities of the form a|f(x)|>g(x); |f(x)|>|g(x)|. | workshop | |||||
The method of intervals in solving inequalities containing the sign of the modulus. | workshop | Test control |
||||
Final lesson | conference | abstracts |
||||
Section IV. Keyword List.
Algorithm, equation, inequality, module, graph, coordinate axes, parallel transfer, central and axial symmetry, interval method, square trinomial, polynomial, factorization of a polynomial, abbreviated multiplication formulas, symmetric equations, reciprocal equations, absolute value properties, domain of definition, domain allowed values.
Section V. Literature for the teacher.
1. Bashmakov M.I. Equations and inequalities. (Text) / M.I. Bashmakov.-M.: VZMSh
at Moscow State University, 1983.-138s.
2. Vilenkin N. Ya and others. Algebra and mathematical analysis Grade 11. (Text) / N.Ya.
Vilenkin-M.: Enlightenment, 2007.-280s.
3. Gaidukov I.I. Absolute value. (Text)/ Gaidukov I.I. –M.: Enlightenment, 1968.-96 p.
4. Gelfand I. M. et al. Functions and graphs. (Text) / I. M. Gelfand- M .: MTsNMO,
5. Goldich V.A. Zlotin S.E.t. 3000 problems in algebra (Text) / V.A. Goldich S.E.-M.:
Eksmo, 2009.-350s.
6. Kolesnikova S.I. Mathematics. Intensive course preparation for the One
State exam. (Text) / Kolesnikova S.I. - M .: Iris-press 2004.-299s.
7. Nikolskaya I.L. Optional course mathematics. (Text) / I.L. Nikolskaya-
M.: Enlightenment, 1995.-80s.
8.Olekhnik S.N. etc. Equations and inequalities. Non-standard methods solutions.
(Text) / .Olekhnik S.N.-M.: Bustard, 2002.-219p.
Section VI. Literature for students
1. Goldich V.A. Zlotin S.E.t. 3000 problems in algebra (Text) / V.A. Goldich S.E.-M.:
Eksmo, 2009.-350s.
2. Kolesnikova S.I. Mathematics. Intensive course of preparation for the One
Document... forchoice one or another subject(within curriculum, chapter: " electivecourses") in 10 -11 classes...and also in system additional education. For these categories students developed and implemented network training courseson everyone...
Activity H 4 51-1 "Improvement of teaching methods in secondary school based on the creation of subject-oriented modules in at least 18 subjects based on the implementation of information technology development of scientific and educational
Report... students. AT this study presented electivewellon mathematics "Beginnings mathematical analysis and their applications" for10 - 11 specialized classes... dependencies and relationships (functions, equations, inequalities etc.). It is usually determined first...
When solving inequalities containing the unknown under the sign of the absolute value, the same technique is used as when solving equations containing the unknown under the sign of the absolute value, namely: the solution of the original inequality is reduced to solving several inequalities considered on the intervals of constancy of expressions under the signs of the absolute value magnified.
Example: Solve the inequality
x 2 - 2 + x< 0. (*)
Solution: Let's consider the intervals of constant sign of the expression x 2 - 2, which is under the sign of the absolute value.
1) Suppose that
then the inequality (*) takes the form
x 2 + x -2< 0.
The intersection of the set of solutions to this inequality and the inequality x 2 -2 0 is the first set of solutions to the original inequality (Fig. 1): x (-2; -].
- 2) Suppose that x 2 - 2
- 2 - x 2 + x
The intersection of the set of solutions of this inequality and the inequality x 2 - 2< 0 дает второе множество решений исходного неравенства (рис. 2): х(-; -1). Объединяя найденные множества решений, окончательно получаем х(-2; -1)
Answer: x(-2; -1).
Unlike equations, inequalities do not allow direct verification. However, in most cases it is possible to verify the correctness of the obtained results. graphically. Indeed, we write the inequality of the example in the form
x - 2< -х.
We construct functions y 1 =x 2 - 2 and y 2 = -x included in the left and right side considered inequality, and find those values of the argument for which y 1 On fig. 3, the shaded area of the x-axis contains the desired x values. The solution of inequalities containing the sign of the absolute value can sometimes be significantly reduced using the equality x 2 \u003d x 2. Figure 3 Example: Solve the inequality Solution: The original inequality for all x -2 is equivalent to the inequality x - 1 > x + 2. (**) Squaring both sides of the inequality (**), after reducing similar terms, we obtain the inequality 6x< -3, т.е. х < -1/2. Taking into account the set of admissible values of the initial inequality determined by the condition x -2, we finally obtain that the inequality (*) is satisfied for all x(-; -2)(-2; -1/2). Answer: (-; -2)(-2; -1/2). Example: Find the smallest integer x that satisfies the inequality: Solution: Since x +1 0 and, by condition, x +1 0, then this inequality is equivalent to the following: 2x + 5 > x +1. The latter, in turn, is equivalent to the system of inequalities - (2x + 5)< х + 1 < 2х + 5, The smallest integer x that satisfies this system of inequalities is 0. Note that x -1, otherwise the expression on the left side of this inequality does not make sense. Example: Solve the inequality: Answer: [-1; one]. Example: Solve the inequality x2 - 3x + 2+ 2x + 1 5. Decision. x 2 - 3x + 2 is negative at 1< x < 2 и неотрицателен при остальных х, 2х + 1 меняет знак при х = -Ѕ. Следовательно, нам надо рассмотреть четыре случая. Answer: 5 - 41 2 ? X? 2. Example: Solve the inequality. x 3 + x - 3- 5 x 3 - x + 8. Decision. Let's solve this inequality in a non-standard way. x 3 + x - 3 - 5 x 3 - x + 8, x 3 + x - 3 - 5 - x 3 + x - 8 x 3 + x - 3 x 3 - x + 13 x 3 + x - 3 - x 3 + x - 3 x 3 + x - 3 x 3 - x + 13, x 3 + x - 3 - x 3 + x - 13, x 3 + x - 3 - x 3 + x - 3, x 3 + x - 3 x 3 - x + 3 No interval solution. Whether equations and inequalities with two or more modules are considered. Equations and inequalities containing the sign of the absolute value in school course mathematics as separate topic not being studied. For the first time, the concept of a module occurs in the 6th grade, where the definition of the module of a number is given. But in textbooks different authors given in various chapters. In the textbooks G.V. Dorofeev's modulus of a number is given by comparing rational numbers for example: the modulus of the number -6.5 is 6.5, the modulus of the number -4 is 4. Then an explanation of the origin of the module and after that the designation |a| is introduced. In the textbook N.Ya. Vilenkin is given in the study of positive and negative numbers as a separate item "Module". The concept of the modulus of a number is introduced as the distance from the point representing this number to the starting point on the coordinate line. Then the rule for finding the modulus of a number is formulated. It is explained that the modulus of a number cannot be negative, because the modulus of a number is the distance, that the modulus of a positive number and zero is equal to the number itself, and for the opposite - to the opposite number and opposite numbers have equal modules |-a|=|a|. According to the textbook Yu.N. Makarychev, the module is found in additional exercises in Chapter 7 "Graphs", paragraph "Functions and their graphs". For example: define the scope y=10/(|x|-1) And in the 8th grade when solving inequalities with one variable and their systems. In Nikolsky's 8th grade textbook, the function y=|x| and its graph y= x if x≥0 X if x≤0 The course of the nine-year school considers the simplest equations with one variable, containing a variable under the modulus sign. These include equations of the form |ax+b|=c. When solving such equations, it is necessary to distinguish between cases: If with< 0, то
уравнение |ах+в|=с не имеет корней. If c = 0, then the equation |ax+b|=c is equivalent to the equation ax+b=0. If c > 0, then the equation |ax+b|=c is equivalent to ax+b= -c or ax+b=c. In addition to the indicated type of equations containing a variable under the modulus sign, students of the 8th grade also come across equations of the form |ax+b|= ax+b or |ax+b|= -(ax+b). For example, the equations √ x²=x, √x²-4x+4=2-x are reduced to such equations. Since the equality |m|=m is true if and only if m≥0, and the equality |m|=-m is true if and only if m ≤ 0, then the equation |ax+b|= ax+b is equivalent to the inequality ax+b≥0, and the equation |ax+b| = -(ax+b) is equivalent to the inequality ax+b≤0. Of the inequalities containing the variable under the modulus sign, only inequalities of the form |ax+b|>b and |ax+b|<в. As additional tasks, more complex tasks are given, for example, a double inequality to<|ах+в|< m.
Это двойное неравенство можно записать
в виде системы |ах+в| >to |ax+v|<
m
и, решив каждое из неравенств
системы, найти пересечение множеств их
решений с помощью координатной прямой. Ways to solve inequalities: 1. The solution is associated with the concept of the distance between the points of the coordinate line. 2. Based on the definition of the module. 3. Visually - graphic technique. 4. In other cases, it is useful to first establish at what points the expressions under the modulus sign vanish. These points divide the numerical axis into intervals, within which the expressions retain a constant sign (intervals of constant sign). This allows us to get rid of the modulus sign on each of these intervals and reduce the problem to solving several equations - one for each interval. This method is called the interval method. secondary school in the village of Oshtorma Yumya Agreed Approved at a meeting of the UMO at a meeting of the expert math teachers commission Protocol No. 1 dated _________ Protocol No. __________ Head of UMO: Chairman of the expert Gilyazeva M.M. groups: Sadikova A.R. "Absolute value (modulus)" (Professional training course for 10th grade students, 34 hours) Mathematics teacher Vasilyeva V.A. 2008 The concept of an absolute value (modulus) is one of the most important characteristics of a number both in the field of real and in the field of complex numbers. This concept is widely used not only in various sections of the school mathematics course, but also in the courses of mathematics, physics and technical sciences studied at universities. For example, in the theory of approximate calculations, the concepts of absolute and relative errors of an approximate number are used. In mechanics and geometry, the concepts of a vector and its length (vector modulus) are studied. In mathematical analysis, the concept of the absolute value of a number is contained in the definitions of such basic concepts as a limit, a bounded function, etc. Problems related to absolute values are often found at mathematical Olympiads, entrance exams to universities, and the Unified State Examination. The program of the school mathematics course does not provide for the generalization and systematization of knowledge about the modules, their properties, received by students over the entire period of study. This will make the program "". The course is designed for profile training of students in grades 10 of general education schools who are interested in studying mathematics. The course will allow schoolchildren to systematize, expand and strengthen knowledge related to the absolute value, prepare for further study of topics using this concept, learn to solve various problems of varying complexity, and contribute to the development and consolidation of computer skills. The course will help the teacher prepare students in the most qualitative way for mathematical olympiads, passing the exam, exams for admission to universities. The program of the elective course involves familiarity with the theory and practice of the issues under consideration and is designed for 34 hours. In the process of studying this course, it is supposed to use various methods of activating the cognitive activity of schoolchildren, as well as various forms of organizing their independent work. The result of mastering the course program is the presentation by schoolchildren of creative individual and group works at the final lesson. Course Objectives:generalization and systematization, expansion and deepening of knowledge on the topic of absolute value, the acquisition of practical skills for completing tasks with a module, increasing the level of mathematical training of schoolchildren.
Course objectives To equip students with a system of knowledge on the topic of absolute value; To form the skills of applying this knowledge in solving various problems of varying complexity; To form the skills of independent work, work in small groups; To form the skills of working with reference literature, with a computer; To form the skills and abilities of research work; Contribute to the development of algorithmic thinking of students; Contribute to the formation of cognitive interest in mathematics. Requirements for the level of assimilation of educational material
As a result of studying the program of the elective course "Absolute value (module)", students get the opportunity know and understand:
Determining the absolute value of a real number; Rules for constructing graphs of functions containing the sign of the absolute value; Algorithms for solving equations, inequalities, systems of equations and inequalities containing a variable under the module sign. Be able to:
apply the definition, properties of the absolute value of a real number to the solution of specific problems; Thematic planning
Introduction
1
lecture presentation Absolute value of a real number a
4
naya work y=f(-|x|), y=|f(x)|, y= |f |х||, |y| =f(x), where f(x) ≥ 0, | y| = |f(x)| naya work |f(x)| = g(x) and f(x)| = | g(x)|. naya work =g(x)
naya work |f(x)| > ≥ ≤ a, where a R..
|f(x)| > ≥ ≤ g(x), |f(x)| > ≥ ≤ |g(x)|. naya work 1. Introduction(1 hour).
Goals and objectives of the elective course. Questions covered in the course and its structure. Acquaintance with literature, themes of creative works. Requirements for course participants. Auction "What I know about the absolute value." 2.
The absolute value of the real number a (4
h).
Absolute value of a real number a. Modules of opposite numbers. Geometric interpretation of the concept | a|.
Modulus of the sum and modulus of the difference of a finite number of real numbers. The modulus of the modulus difference of two numbers. Product module and quotient module. Operations on absolute values. Simplification of expressions containing a variable under the module sign. Application of module properties in solving Olympiad problems. 3.
Graphs of functions whose analytical expression contains the sign of the absolute value(5 hours).
Rules and algorithms for constructing graphs of functions whose analytical expression contains the modulus sign. Graphs of functions y=f |х|, y=f (-|x|), y=|f(x)|, y= |f |x||, |y| =f(x), where f(x) ≥ 0, | y| = |f(x)|. Graphs of some of the simplest functions, defined explicitly and implicitly, whose analytic expression contains the sign of the modulus. Graphs of functions, the analytical expression of which contains the sign of the absolute value in the Olympiad tasks. 4.
Equations containing absolute values (11 h). Basic methods for solving equations with a modulus. Revealing the module by definition, moving from the original equation to an equivalent system, squaring both sides of the equation, the method of intervals, the graphical method, using the properties of an absolute value. Type equations | f(x)| = a,
f\
x\ = a, where a R; |f(x)| = g(x) and | f(x)| = | g(x)|.
Variable change method for solving equations containing absolute values. Interval method for solving equations containing absolute values. Equations of the form |f 1 (x)| ± |f 2 (x)| ±.. .±|f n (x)| = a, where a e R,
|f 1 (x)| ± |f 2 (x)| ±.. .± |f n (x)| =
g (x).
A method of successive disclosure of a module when solving equations containing a "module in a module". Graphical solution of equations containing absolute values. Using the properties of an absolute value in solving equations. Equations with parameters containing absolute values. Protection of the solved tasks of the exam. 5.
Inequalities containing absolute values
(7 h).
Inequalities with one unknown. Basic methods for solving inequalities with a modulus. Inequalities of the form |f(x)| > ≥ ≤ a, where a R..
Inequalities of the form |f(x)| > ≥ ≤ g(x), |f(x)| > ≥ ≤ |g(x)|.
The method of intervals in solving inequalities containing the sign of the modulus. Inequalities with parameters containing absolute values. 6.
Systems of equations and inequalities containing absolute values(4 hours).
7.
Other issues in the solution of which the concept of absolute value is used(1 hour).
8.
Final lesson(1 hour).
Expected results
To be able to apply the definition, properties of the absolute value of a real number to the solution of specific problems; Solve equations, inequalities, systems of equations and inequalities containing a variable under the modulus sign. Literature for the teacher
Literature for students
2. A.G. Mordkovich. Algebra 9. Deep Learning. Textbook.
Municipal educational institution
elective course
Explanatory note
basic operations and properties of absolute value;
№
Topic name
Number of hours
Form of occupation
Methodological support
The control
2
Absolute value of a real number a. Basic theorems
1
lecture
reference cards
3
Operations on absolute values
1
reference cards
4
Simplification of expressions containing a variable under the module sign.
1
workshop
5
Application of module properties in solving Olympiad problems.
1
workshop
Task cards
independent
Function graphs, analytic expression which contains the sign of the absolute value
5
6
Rules and algorithms for constructing graphs of functions whose analytical expression contains the modulus sign
1
lecture, workshop
reference cards
7-8
Graphs of functions y=f |х|,
2
workshop
Task cards
independent
9
Graphs of some of the simplest functions, given explicitly and implicitly, whose analytical expression contains the sign of the modulus
1
workshop
Individual cards
10
Graphs of functions, the analytical expression of which contains the sign of the absolute value in the Olympiad tasks
1
workshop
pretention
Equations Containing Absolute Values
11
11-13
Basic methods for solving equations with a modulus
3
lecture
reference cards
14
Type equations | f(x)| = a,
f\
x\ = a, where a R;
1
workshop
Task cards
independent
15
Variable change method for solving equations containing absolute values
1
workshop
reference cards
16-17
Interval method for solving equations containing absolute values. Equations of the form |f 1 (x)| ± |f 2 (x)| ±.. .±|f n (x)| = a, where a e R,
=
2
lecture, workshop
Task cards
independent
18
The method of successive disclosure of the module when solving equations containing the "module in the module"
1
lecture, workshop
reference cards
19
Graphical solution of equations containing absolute values.
1
workshop
20
Equations with parameters containing absolute values
1
workshop
21
Protection of the solved tasks of the exam
1
decision protection
Table
decision protection
7
22-23
Inequalities with one unknown. Basic Methods for Solving Inequalities with Modulus
2
lecture
reference cards
24
Basic Methods for Solving Inequalities with Modulus
1
seminar
25
Inequalities of the form
1
workshop
26-27
Inequalities of the form
2
workshop
Task cards
independent
28
Inequalities with parameters containing absolute values
1
workshop
29-32
4
lecture, workshop
33
1
workshop
34
Final lesson
1
Task cards
control cut
Total
34
After completing the course, students should:
solutions. 10 - 11 cells. – M.: Bustard, 1995.
S.I. Kolesnikova "Solution of complex USE tasks» 300 tasks with detailed solution. Publishing house Moscow Iris press 2005.
G.A. Voronina Practical guide for the teacher "Elective courses" Publishing house Moscow Iris press 2006
M.I.Skanavi Collection of problems in mathematics M.: ONIKS, 2006
Electronic textbook "Algebra 7 - 11"
Olehnik S.N. etc. Equations and inequalities. Non-standard methods
1. M.I.Skanavi Collection of problems in mathematics, M.: ONIKS, 2006
