GENERALIZING LESSON ON THE TOPIC "FUNCTIONS AND THEIR PROPERTIES".
Lesson Objectives:
Methodical: increasing the active-cognitive activity of students through individual-independent work and the use of test tasks of a developing type.
Tutorial: repeat elementary functions, their basic properties and graphs. Introduce the concept of mutually inverse functions. Systematize students' knowledge on the topic; contribute to the consolidation of skills and abilities in the calculation of logarithms, in the application of their properties in solving tasks of a non-standard type; repeat the construction of graphs of functions using transformations and test skills and abilities when solving exercises on their own.
Educational: education of accuracy, composure, responsibility, ability to make independent decisions.
Developing: develop intellectual abilities, mental operations, speech, memory. Develop a love and interest in mathematics; during the lesson to ensure the development of students' independence of thinking in educational activities.
Lesson type: generalization and systematization.
Equipment: board, computer, projector, screen, educational literature.
Epigraph of the lesson:“Mathematics should be taught later, so that it puts the mind in order.”
(M.V. Lomonosov).
DURING THE CLASSES
Checking homework.
Repetition of exponential and logarithmic functions with base a = 2, plotting their graphs in the same coordinate plane, analysis of their relative position. Consider the interdependence between the main properties of these functions (OOF and FZF). Give the concept of mutually inverse functions.
Consider exponential and logarithmic functions with base a = ½ s
in order to ensure that the interdependence of the listed properties is observed and for
decreasing mutually inverse functions.
Organization of independent work of a test type for the development of mental
systematization operations on the topic "Functions and their properties".
FUNCTION PROPERTIES:
one). y \u003d │x│;
2). Increases over the entire domain of definition;
3). OOF: (- ∞; + ∞) ;
four). y \u003d sin x;
5). Decreases at 0< а < 1 ;
6). y \u003d x ³;
7). ORF: (0; + ∞) ;
eight). General function;
9). y = √ x;
ten). OOF: (0; + ∞) ;
eleven). Decreases over the entire domain of definition;
12). y = kx + v;
13). OZF: (- ∞; + ∞) ;
fourteen). Increases when k > 0;
fifteen). OOF: (- ∞; 0) ; (0; +∞) ;
16). y \u003d cos x;
17). Has no extremum points;
eighteen). ORF: (- ∞; 0) ; (0; +∞) ;
19). Decreases at< 0 ;
twenty). y \u003d x ²;
21). OOF: x ≠ πn;
22). y \u003d k / x;
23). Even;
25). Decreases when k > 0;
26). OOF: [ 0; +∞) ;
27). y \u003d tg x;
28). Increases at< 0;
29). ORF: [ 0; +∞) ;
thirty). odd;
31). y = logx;
32). OOF: x ≠ πn/2;
33). y \u003d ctg x;
34). Increases when a > 1.
During this work, conduct a survey of students on individual tasks:
No. 1. a) Graph the function 
b) Graph the function 
No. 2. a) Calculate:
b) Calculate:
Number 3. a) Simplify the expression 
and find its value at
b) Simplify the expression 
and find its value at 
.
Homework: No. 1. Calculate: a) 
;
in) 
;
G) 
.
No. 2. Find the domain of a function: a) 
;
in) 
; G) 
.
n1.doc
OGO SPO Ryazan Pedagogical College
ESSAY
Topic: “Numeric functions and their properties. Direct and inverse proportional dependencies»
Titova Elena Vladimirovna
Specialty: 050709 "Teaching in elementary grades with additional training in the field of pre-school education"
Course: 1 Group: 2
Department: school
Head: Pristuplyuk Olga Nikolaevna
Ryazan
Introduction……………………………………………………………………3
Theoretical part
Numeric functions
1.2 Ways to set functions………………………………………………….6
1.3 Function Properties …………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………
2. Direct and inverse proportions
2.1 The concept of direct proportionality………………..9
2.2 Properties of a direct proportional relationship…………………………………………….10
2.3 The concept of inverse proportionality and its properties…………………………………………………………………-
Practical part
3.1 Functional propaedeutics in the initial course of mathematics ... .11
3.2 Solving problems for proportionally dependent quantities……18
Conclusion……………………………………………………….......21
List of used literature………………………………..22
Introduction
In mathematics, the idea of a function appeared along with the concept of magnitude. It was closely associated with geometric and mechanical representations. The term function (from Latin - performance) was first introduced by Leibniz in 1694. By function, he understood the abscissas, ordinates and other segments associated with a point describing a certain line.
In the first half of the XVIII century. there was a transition from a visual representation of the concept of a function to an analytical definition. The Swiss mathematician Johann Bernoulli, and then academician Leonhard Euler, believed that the function
it analytic expression, composed of a variable and a constant.
In other words, the function is expressed by different types of formulas: y=ax+b, y==axІ+bx+c, etc.
Today we know that a function can be expressed not only in mathematical language, but also graphically. The pioneer of this method was Descartes. This discovery played a huge role in the further development of mathematics: there was a transition from points to numbers, from lines to equations, from geometry to algebra. Thus, it became possible to find common methods for solving problems.
On the other hand, thanks to the coordinate method, it became possible to represent geometrically different dependencies.
Thus, graphs give a visual representation of the nature of the relationship between quantities; they are often used in various fields of science and technology.
The main trends in the development of modern school education are reflected in the ideas of humanization, humanitarization, activity-based and student-centered approach to the organization of the educational process.
At the heart of teaching mathematics in a general education school, the principle of priority of the developmental function of education comes to the fore.
Therefore, the study of the concept of a numerical function in elementary school is a rather significant component in the formation of mathematical representations of schoolchildren. For a primary school teacher, it is necessary to focus on the study of this concept, since there is a direct relationship between the function and many areas of human activity, which in the future will help the children enter the world of science.
Besides , students, as a rule, formally learn the definition of the concept of function, do not have a holistic view of functional dependence, i.e. cannot apply their knowledge to solving mathematical and practical problems; associate a function exclusively with an analytic expression in which the variable at expressed in terms of a variable X; cannot interpret representations of a function on different models; find it difficult when plotting function graphs according to its properties, etc.
The reasons for these difficulties are associated not only and not so much with the method of studying functional material in the course of algebra, but with the unpreparedness of students' thinking for the perception and assimilation of the concept of "function".
This means that before the introduction of the concept of “function”, it is necessary to work on the formation of functional thinking skills, so that “at the moment when the general idea of functional dependence should enter the consciousness of students, this consciousness was sufficiently prepared for the objective and effective, and not just for the formal perception of a new concept and related ideas and skills” (A.Ya. Khinchin)
1. Numeric functions
1.1 Development of the concept of functional dependence in mathematics
Let us analyze the course of development of pedagogical ideas in the field of teaching the most important component of mathematics - functional dependence.
The functional line of the school course in mathematics is one of the leading courses in algebra, algebra and the beginnings of analysis. The main feature of the educational material of this line is that it can be used to establish a variety of connections in teaching mathematics.
Over the course of several centuries, the concept of a function has changed and improved. The need to study functional dependence in the school course of mathematics has been in the focus of attention of the pedagogical press since the second half of the 19th century. Much attention was paid to this issue in their works by such well-known methodologists as M. V. Ostrogradsky, V. N. Shklarevich, S. I. Shokhor-Trotsky, V. E. Serdobinsky, V. P. Sheremetevsky.
The development of the idea of functional dependence proceeded in several stages:
First stage- the stage of introducing the concept of a function (mainly through an analytical expression) into the school mathematics course.
Second phase the introduction of the concept of a function into the course of high school algebra is characterized mainly by the transition to a graphical representation of functional dependence and the expansion of the range of studied functions.
Third stage The development of the Russian school began in the 20s. twentieth century. An analysis of the methodological literature of the Soviet period showed that the introduction of the concept of a function into a school mathematics course was accompanied by heated discussions, and allowed us to identify four main problems around which there were differences in the opinions of methodologists, namely:
1) the purpose and significance of studying the concept of function by students;
2) approaches to defining a function;
3) the issue of functional propaedeutics;
4) the place and volume of functional material in the course of school mathematics.
Fourth stage due to the transfer of the economy of the RSFSR to a planned basis
In 1934, the school received the first stable textbook by A.P. Kiselev "Algebra", revised under the editorship of A.P. Barsukov in two parts.
Sections "Functions and their graphs", "Quadratic function" were included in its second part. In addition, in the section "Generalization of the concept of degree" the exponential function and its graph were considered, and in the section "Logarithms" - the logarithmic function and its graph.
It was in it that the function was defined through the concept of a variable: "That variable, the numerical values of which change depending on the numerical values of another, is called the dependent variable, or a function of another variable." However, it does not reflect the idea of correspondence and there is no mention of an analytic expression, which allows us to conclude that this definition has a significant drawback.
I. Ya. Khinchin paid much attention to this problem in his works.
The scientist regarded the formation of an idea of a function as a manifestation of formalism in teaching. He believed that in high school the concept of function should be studied on the basis of the concept of correspondence.
This period is characterized by lack of time to study functions, ill-conceived exercise systems, students' misunderstanding of the true essence of the concept of function, low level of functional and graphic skills of school graduates.
Thus, the need arose again to reform the teaching of mathematics in secondary schools. The restructuring of all school mathematics on the basis of the set-theoretic approach marked the fifth stage in the development of the idea of functional dependence. The idea of a set-theoretic approach was undertaken by a group of French scientists who came together under the pseudonym of Nicolas Bourbaki. In the city of Roymond (France, 1959), an international conference was held at which the overthrow of all conventional courses was proclaimed. The focus was on the structures and unifications of all school mathematics based on set theory.
An important role in the development of the ideas of the reform was played by the articles of V. L. Goncharov, in which the author pointed out the importance of early and long-term functional propaedeutics, suggested using exercises that consisted in performing a number of pre-specified numerical substitutions in the same given literal expression.
The stabilization of programs and textbooks created the ground for the emergence of positive changes in the quality of students' functional knowledge. In the late sixties and early seventies, along with negative reviews, the press began to appear in which there was a certain improvement in the knowledge of school graduates about functions and schedules. However, the general level of mathematical development of students as a whole remained insufficient. The school curriculum of mathematics continued to devote too much time to formal training and did not pay enough attention to developing students' ability to learn independently.
1.2 Ways to set functions
So, numeric function is a correspondence between the numerical set R of real numbers, in which each number from the set X corresponds to a single number from the set R.
Accordingly, X represents the domain of the function (OOF).
The function itself is denoted by lowercase Latin letters (f, d, e, k).
If the function f is defined on the set X, then the real number y corresponding to the number x from the set X is denoted as f(x) (y=f(x)).
The variable x is called argument. The set of numbers of the form f(x) for all x is called function rangef.
Most often, functions are specified by various types of formulas: y=2x+3, y=xІ, y=3xі, y=?3xІ, where x is a real number, y is the single number corresponding to it.
However, using one formula, you can specify lots of functions, the difference of which is determined only by the domain of definition:
Y= 2x-3, where x belongs to the set of real numbers and y=2x-3,
X - belonging to the set of natural numbers.
Often, when specifying a function using a formula, the OOF is not indicated (OOF is the domain of the expression f (x)).
It is also quite convenient to represent numerical functions visually, i.e. using the coordinate plane. 
1.3 Function properties.
Like many others, numeric functions have properties:
Increasing, decreasing, monotonicity, domain of definition and scope of a function, boundedness and unboundedness, evenness and oddness, periodicity.
Scope and scope of a function.
In elementary mathematics, functions are studied only on the set of real numbers R. This means that the argument of a function can take only those real values for which the function is defined, i.e. it also only accepts real values. The set X of all admissible real values of the argument x for which the function y = f(x) is defined is called the domain of the function. The set Y of all real y values that a function takes is called the function's range. Now we can give a more precise definition of a function: the rule (law) of correspondence between sets X and Y, according to which for each element from set X, one and only one element from set Y can be found, is called a function. 
A function is considered to be given if: the scope of the function X is given; the range of values of the function Y is given; the rule (law) of correspondence is known, and such that for each value of the argument only one value of the function can be found. This requirement of uniqueness of the function is mandatory.
Limited and unlimited functions. A function is called bounded if there exists a positive number M such that | f(x) | M for all x values. If there is no such number, then the function is unbounded.
Even and odd functions. If for any x from the domain of the function the following holds: f (- x) = f (x), then the function is called even; if it takes place: f (- x) = - f (x), then the function is called odd. The graph of an even function is symmetrical about the Y axis (Fig. 5), and the graph of an odd function is symmetrical about the origin (Fig. 6). 
Periodic function. A function f (x) is periodic if there exists a non-zero number T such that for any x from the domain of the function, f (x + T) = f (x). This smallest number is called the period of the function. All trigonometric functions are periodic.
But the most important property to learn function in primary classes is monotone.
Monotonic function. If for any two values of the argument x1 and x2 the condition x2 > x1 implies f (x2) > f (x1), then the function | f(x) | is called increasing; if for any x1 and x2 the condition x2 > x1 implies f (x2)
2. Direct and inverse proportional dependencies.
2.1 The concept of direct proportionality.
In elementary school, the function manifests itself in the form of direct and inverse proportional dependencies.
Direct proportionality is, first of all, function, which can be given using the formula y=kx, where k is a non-zero real number. The name of the function y = kx is associated with the variables x and y contained in this formula. If a attitude two quantities is equal to some number other than zero, then they are called directly proportional.
K is the coefficient of proportionality.
In general, the function y=kx is a mathematical model of many real situations considered in the initial course of mathematics.
For example, let's say that there are 2 kg of flour in one package, and x such packages were bought, then the entire mass of the purchased flour is y. This can be written as a formula like this: y=2x where 2=k.
2.2 Properties of a direct proportional relationship.
Direct proportionality has a number of properties:
The domain of the function y=kx is the set of real numbers R;
A graph of direct proportionality is a straight line passing through the origin;
For k>0, the function y=kx increases over the entire domain of definition (for k
If the function f is a direct proportionality, then (x1,y1),(x2,y2) are pairs of corresponding variables x and y, where x is not equal to zero, then x1/x2=y1/y2.
xseveral times the corresponding positive value of y increases (decreases) by the same amount.
2.3 The concept of inverse proportionality.
Inverse proportionality- this is function, which can be given using the formula y=k/x, where k is a non-zero real number. The name of the function y = k/x is associated with the variables x and y, the product of which is equal to some real number that is not equal to zero.
Inverse Proportional Properties:
The domain of definition and the scope of the function y=k/x is the set of real numbers R;
The graph of direct proportionality is a hyperbole;
For k 0, respectively, decreases over the entire domain of definition, branches - down)
If the function f is inversely proportional, then (x1,y1),(x2,y2) are pairs of corresponding variables x and y, where x is not equal to zero, then x1/x2=y2/y1.
If the values of the variablesxandyare positive real numbers, then
with increasing (decreasing) variablexseveral times the corresponding value of y decreases (increases) by the same amount.
Practical part
3.1 Functional propaedeutics in the initial course of mathematics
The concept of functional dependence is one of the leading in mathematical science, therefore, the formation of this concept in students is an important task in the purposeful activity of the teacher to develop mathematical thinking and creative activity of children. The development of functional thinking presupposes, first of all, the development of the ability to discover new connections, to master general learning techniques and skills.
In the initial course of mathematics, a significant role should be given to functional propaedeutics, which provides for the preparation of students for the study of systematic courses in algebra and geometry, and also educates them in the dialectical nature of thinking, understanding the causal relationships between the phenomena of the surrounding reality. In this regard, we will designate the main directions of propaedeutic work at the initial stage of teaching the subject according to the program of L.G. Peterson:
The concept of sets, the correspondence of elements of two sets and functions. Dependence of the results of arithmetic operations on the change of components.
Tabular, verbal, analytical, graphical ways of setting a function.
Linear dependency.
Coordinate system, first and second coordinate, ordered pair.
Solving the simplest combinatorial problems: compiling and counting the number of possible permutations, subsets of elements of a finite set..
Using a systematic enumeration of natural values of one and two variables in solving plot problems.
Filling tables with arithmetic calculations, data from the conditions of applied problems. Selection of data from the table by condition.
Dependence between proportional values; applied study of their graphs.
The content of the initial course of mathematics allows students to form an idea about one of the most important ideas of mathematics - idea of conformity.When performing assignments for finding the values of expressions, filling out tables, students establish that each pair of numbers corresponds to no more than one number obtained as a result. However, to understand this, the contents of the tables must be analyzed.
Make up all possible examples of adding two single-digit numbers with the answer 12.
When completing this task, students establish a relationship between two sets of terms values. The established correspondence is a function, since each value of the first term corresponds to a single value of the second term at a constant sum.
There are 10 apples in a vase. How many apples will be left if 2 apples are taken? 3 apples? 5 apples? Record your solution in the table. What does the result depend on? How many units does it change? Why?
This problem actually presents the function at = 10 - X, where the variable X takes values 2, 3, 5. As a result of completing this task, students should conclude: the larger the subtrahend, the smaller the value of the difference.
The idea of functional correspondence is also present in exercises of the form:
Connect the mathematical expressions and the corresponding numerical values with an arrow:
15 + 6 27 35
Introduction letter symbols allows you to acquaint students with the most important concepts of modern mathematics - a variable, an equation, an inequality, which contributes to the development of functional thinking, since the idea of functional dependence is closely related to them. When working with a variable, students realize that the letters included in the expression can take on different numerical values, and the literal expression itself is a generalized notation of numerical expressions.
Of great propaedeutic importance is the experience of students communicating with exercises on establishing patterns in numerical sequences and their continuation:
1, 2, 3, 4… (at = X + 1)
1, 3, 5, 7… (at= 2 X + 1)
concept quantities, along with the concept of number, is the main concept of the initial course of mathematics. The material of this section is the richest source for the implementation of indirect functional propaedeutics. Firstly, it is the dependence (inversely proportional) between the chosen unit of quantity (measure) and its numerical value (measure) - the larger the measure, the smaller the number obtained as a result of measuring the value with this measure. Therefore, it is important that when working with each quantity, students gain experience in measuring quantities with different measures in order to consciously choose first a convenient, and then a single measure.
Secondly, when studying the quantities characterizing the processes of movement, work, purchase and sale, ideas are formed about the relationship between speed, time and distance, price, quantity and cost in the process of solving text problems of the following types - to bring to unity (finding the fourth proportional) , finding the unknown by two differences, proportional division.
Of particular difficulty for students is the understanding of the relationship between these quantities, since the concept of "proportional dependence" is not the subject of special study and assimilation. In the program of L.G. Peterson methodically solves this problem by using the following techniques:
- Solving problems with missing data ("open" condition):
Vasya is 540 m from home to school, and Pasha is 480 m. Who lives closer? Who will get there faster?
Sasha bought notebooks for 30 rubles and pencils for 45 rubles. What items did he spend the most money on? What items did he buy more?
When analyzing the texts of these tasks, students find that they lack data and that the answers to the questions depend on the price and speed.
- Fixing the conditions of tasks not only in a table (as suggested in the classical technique), but also in the form of a diagram. This allows you to "visualize" the dependencies considered in the problem. So, if moving objects cover the same distance of 12 km in different times (2 hours, 3 hours, 4 hours, 6 hours), then using the scheme, the inverse relationship is clearly interpreted - the more parts (time), the smaller each part (speed).
- Changing one of the task data and comparing the results of solving problems.
48 kg of apples were brought to the school canteen. How many boxes could be brought if there were equal numbers of apples in all boxes?
Students complete the condition of the problem and fix the relationship between quantities using various means of structuring theoretical knowledge - in a table, diagram and verbally.
Here it is useful to pay attention to the multiple ratio of the quantities under consideration - how many times one of the quantities is greater, the other is the same number of times greater (less) with a constant third.
In elementary school, students are implicitly introduced to tabular, analytical, verbal, graphical ways of setting functions.
So, for example, the relationship between speed, time and distance can be expressed as:
A) verbally: “to find the distance, you need to multiply the speed by the time”;
B) analytically: s= v t;
C) table: v = 5 km / h
d) graphically (using a coordinate beam or angle).
A graphical way of specifying the dependency between v , t, s allows you to form an idea of speed as a change in the location of a moving object per unit of time (along with the generally accepted one - as a distance traveled per unit of time) And a comparison of the motion graphs of two bodies (moving independently of each other) clarifies the idea of \u200b\u200bspeed as a quantity characterizing the speed of movement.
Compound numeric expressions(with and without parentheses), calculating their values according to the rules of the order of actions allows students to realize that the result depends on the order of actions.
Arrange the brackets so that you get the correct equalities.
20 + 30: 5=10, 20 + 30: 5 = 26
In the course of L.G. Peterson, students are implicitly introduced to linear dependence, as a special case of a function. This function can be defined by a formula of the form at= kh + b, where X- independent variable, k and b- numbers. Its domain of definition is the set of all real numbers.
After traveling 350 kilometers, the train began to move for t hours at a speed of 60 km/h. How many kilometers did the train travel in total?(350 + 60 t)
Performing tasks with named numbers, students are aware of the dependence the numerical value of quantities from the use of different units of measurement.
The same segment was measured first in centimeters, then in decimeters. In the first case, we got a number 135 more than in the second. What is the length of the segment in centimeters? (Dependence at= 10 X)
In the process of studying the initial course of mathematics, students form the concept of a natural series of numbers, a segment of a natural series, assimilate the properties of a natural series of numbers - infinity, orderliness, etc., form the idea of the possibility of an unlimited increase in a natural number or a decrease in its share.
In the course of mathematics in grades 3-4, considerable attention is paid to teaching students how to use formulas, their independent conclusion. Here it is important to teach students to present the same information in different forms - graphically and analytically, giving students the right to choose the form in accordance with their cognitive styles.
Of considerable interest to students are tasks related to the analysis of tables of variable values, the "discovery" of dependencies between them and writing in the form of a formula.
When analyzing the numbers presented in the table, students easily notice that the numbers in the first row increase by one, the numbers in the second row increase by four. The task of the teacher is to pay attention to the relationship of the values of the variables a and b. In order to strengthen the applied orientation of mathematical education, it is necessary to “revive” this situation, transfer it to the plot status.
To form students' ability to derive formulas, you need to teach them to write down various statements in mathematical language (in the form of equalities):
A pen is three times the price of a pencil R = to + 3);
Number a when divided by 5 gives a remainder of 2 ( a= 5 b + 2);
The length of the rectangle is 12 cm more than the width ( a = b + 12).
A prerequisite is the discussion of possible options for the values of these quantities with filling in the appropriate tables.
A special place in the course of L.G. Peterson take on assignments related to mathematical research:
Imagine the number 16 as a product of two factors in different ways. For each method, find the sum of the factors. In which case did you get the smallest amount? Do the same with the numbers 36 and 48. What is the guess?
When performing such tasks (to study the relationship between the number of corners of a polygon and the total value of the degree measures of angles, between the value of the perimeter of figures of different shapes with the same area, etc.), students improve their skills in working with a table, since it is convenient to fix the solution in the table. In addition, the tabular method of fixing the solution is used in solving non-standard mathematical problems by the method of ordered enumeration or rational selection.
There are 13 children in the class. Boys have as many teeth as girls have fingers and toes. How many boys and how many girls are in the class? (Each boy has exactly 32 teeth.)
Teaching mathematics according to the program of L.G. Peterson provides students with the assimilation of the relationship between the results and components of arithmetic operations, an idea is formed about The "speed" of changing the result of arithmetic operations depending on the change in the components:
Number Composition Exercises;
Private calculation methods (36 + 19 = 35 + 20; 36 - 19 = 37 - 20; 12 5 = 12 10: 2);
Evaluation of the sum, difference, product, quotient.
When performing such tasks, it is important to present information multisensory.
How will the sum change if one term is increased by 10, and the second is decreased by 5?
How will the area of a rectangle (or the product of two numbers) change if one of the sides (one of the numbers) is increased by 3?
A significant part of students perform similar tasks by substituting specific numerical values. Methodically literate in this situation will graphically and analytically interpret the condition.
(a+ 3) · b = a· b+ 3 ·b
The concept of function in high school is associated with coordinate system. In the course of L.G. Peterson contains material for propaedeutic work in this direction:
Numerical segment, numerical ray, coordinate ray;
Pythagorean table, coordinates on the plane (coordinate angle);
Movement charts;
Pie, column and line charts that visually represent the relationship between discrete values.
So, the study of arithmetic operations, increasing and decreasing the number by several units or several times, the relationship between the components and the results of arithmetic operations, solving problems for finding the fourth proportional, for the connection between speed, time and distance; price, quantity and value; the mass of an individual item, their number and total mass; labor productivity, time and work; etc., on the one hand, underlie the formation of the concept of function, and on the other hand, they are studied on the basis of functional concepts. It should be noted that graphic modeling has a rather large propaedeutic value: graphic interpretation of the problem statement, drawing, drawing, and more. Information presented in graphical form is easier to understand, capacious and rather conditional, designed to carry information only about the essential features of the object, to form students' graphic skills.
In addition, the result of the propaedeutics of functional dependence should be a high mental activity of younger students, the development of intellectual, general subject and specific mathematical skills and abilities. All this creates a solid foundation not only for solving the methodological problems of elementary mathematics - the formation of computational skills, the ability to solve text problems, etc., but also for the implementation of developing opportunities for mathematical content and, no less important, for the successful study of functions in high school.
3.2 Solving problems for proportionally dependent quantities
To solve a problem means through a logically correct sequence of actions.
and operations with explicitly or indirectly available in the problem numbers, quantities,
relations to fulfill the requirement of the task (to answer its question).
The main ones in mathematics are arithmetic and
algebraic ways of solving problems. At arithmetic way
the answer to the question of the problem is found as a result of performing arithmetic
actions on numbers.
Different arithmetic methods for solving the same problem are different
relationships between data, data and unknowns, data and what is sought,
underlying the choice of arithmetic operations, or a sequence
use of these relations when choosing actions.
Solving a text problem in an arithmetic way is a complex activity,
decisive. However, it can be divided into several stages:
1. Perception and analysis of the content of the task.
2. Search and drawing up a plan for solving the problem.
3. Implementation of the solution plan. Formulation of the conclusion on the fulfillment of the requirement
task (answer to the question of the task).
4. Verification of the solution and elimination of errors, if any.
Problems for proportional division are introduced in different ways: you can offer
to solve a ready-made problem, or you can first compose it by transforming the problem
to find the fourth proportional. In both cases, the success of the solution
problems for proportional division will be determined by a solid ability to solve
problem of finding the fourth proportional, therefore, as
training, it is necessary to provide for the solution of problems of the appropriate type for finding
fourth proportional. That is why the second one is preferable.
named options for introducing problems for proportional division.
Moving on to solving ready-made problems from the textbook, as well as problems compiled
teacher, including various groups of quantities, you first need to establish what
quantities referred to in the task, then write the task briefly in the table,
having previously divided the question of the problem into two questions, if it contains the word
each. The decision, as a rule, the students perform on their own, analysis
conducted only with individual students. Instead of a short note, you can do
picture. For example, if the problem talks about pieces of matter, coils of wire and
etc., then they can be depicted as segments by writing the corresponding numerical
the values of these quantities. Note that it is not necessary to perform a short summary every time.
record or drawing, if the student, after reading the problem, knows how to solve it, then
let him decide, and those who find it difficult will use a short note or drawing
To solve the task. Gradually, the tasks should become more difficult by introducing
additional data (for example: “In the first piece there were 16 m of matter, and in the second
2 times less.”) or by asking a question (for example: “How many meters
was there more matter in the first piece than in the second?).
When familiarizing yourself with the solution of the problem of disproportionate division, you can go
in another way: first solve ready-made problems, and later perform
transformation of the problem of finding the fourth proportional to the problem of
proportional division and, after solving them, compare both the tasks themselves and
their decisions.
Generalization of the ability to solve problems of the considered type is helped by exercises
creative nature. Let's name some of them.
Before solving it, it is useful to ask which of the questions of the problem will be answered in the answer.
greater number and why, and after the decision to check whether I correspond to this species
the resulting numbers, which will be one of the ways to check the solution. Can be further
find out whether the same numbers could be obtained in the answer and under what conditions.
Useful exercises for the preparation of problems by students with their subsequent solution,
as well as task transformation exercises. It is, first of all, the compilation
tasks similar to those solved. So, after solving the problem with the quantities: price,
quantity and cost - suggest compiling and solving a similar problem with
the same quantities or with others, such as speed, time and distance.
This is the compilation of tasks according to their solution, written as separate
actions, and in the form of an expression, this is the compilation and solution of problems according to their
brief schematic notation
1 way:
X \u003d 15 * 30 / 8 \u003d 56 rubles 25 kopecks
2 way: the amount of cloth increased by 15/8 times, which means that money will be paid 15/8 times more
X \u003d 30 * 15/8 \u003d 56 rubles 25 kopecks
2. A certain gentleman called a carpenter and ordered the yard to be built. He gave him 20 workers and asked how many days they would build a yard for him. The carpenter replied: in 30 days. And the master needs to build in 5 days, and for this he asked the carpenter: how many people do you need to have, so that you can build a yard with them in 5 days; and the carpenter, bewildered, asks you, arithmetician: how many people does he need to hire to build a yard in 5 days?
An unfinished brief condition is written on the board:
I option: proportion
II option: without proportions
I. 
II. X \u003d 20 * 6 \u003d 120 workers
3. They took 560 soldiers of food for 7 months, and they were ordered to be in the service for 10 months, and they wanted to take people away from themselves so that there would be enough food for 10 months. The question is, how many people should be reduced?
Old task.
Solve this problem without proportion:
(The number of months increases by a factor, which means the number of soldiers decreases by a factor.
560 - 392 = 168 (soldiers must be reduced)
In ancient times, for solving many types of problems, there were special rules for solving them. Problems familiar to us for direct and inverse proportionality, in which it is necessary to find the fourth by three values of two quantities, were called problems for the "triple rule".
If for three values, five values were given, and it was required to find the sixth, then the rule was called "five". Similarly, for the four quantities there was a "rule of septenary". Tasks for the application of these rules were also called tasks for the “complex triple rule”.
4.
Three hens laid 3 eggs in 3 days. How many eggs will 12 hens lay in 12 days?
| Hens | days | eggs |
| 3 | 3 | 3 |
| 12 | 12 | X |
Need to find out:
How many times has the number of chickens increased? (4 times)
How did the number of eggs change if the number of days did not change? (increased 4 times)
How many times has the number of days increased? (4 times)
How did the number of eggs change? (increased 4 times)
X \u003d 3 * 4 * 4 \u003d 48 (eggs)
5
. If a scribe can write 15 sheets in 8 days, how many scribes will it take to write 405 sheets in 9 days?
(the number of scribes increases from the increase in sheets in times and decreases
From the increase in days of work 
(scribes)).
Consider a more complex problem with four quantities.
6.
For lighting 18 rooms, 120 tons of kerosene were spent in 48 days, and 4 lamps burned in each room. How many days will 125 pounds of kerosene last if 20 rooms are illuminated and 3 lamps are lit in each room?
The number of days of using kerosene increases from an increase in the amount of kerosene in
times and from reducing the lamps by half.
The number of days of using kerosene decreases with the increase in rooms in 20 times.
X = 48 * * : = 60 (days)
Finally has X = 60. This means that 125 pounds of kerosene is enough for 60 days.
Conclusion
The methodological system for studying functional dependence in elementary school, developed in the context of modular education, is an integrity made up of the relationship of the main components (target, content, organizational, technological, diagnostic) and principles (modularity, conscious perspective, openness, focus of education on the development of the student's personality). , versatility of methodological consulting).
The modular approach is a means of improving the process of studying functional dependence among primary school students, which allows: students - to master the system of functional knowledge and methods of action, practical (operational) skills; the teacher - to develop their mathematical thinking on the basis of functional material, to cultivate independence in learning.
The methodological support of the process of studying functions in elementary school is built on the basis of modular programs, which are the basis for highlighting fundamental patterns that are essential for understanding the topic, successfully and fully mastering the content of the educational material, and acquiring strong knowledge and skills by students.
Bibliography.
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This is a correspondence in which each element x from the set D, according to some rule, is associated with a certain number y, depending on x. Notation: y = f(x) x y Independent variable or argument dependent variable or function value D(f) E(f) Domain of the function Domain of the function Numeric function with domain D
Evenness of a function A function y=f(x) is called even if for any value x from the domain of definition the equality f(-x)=f(x) is true. The function y=f(x) is called odd if for any value x from the domain of definition the equality f(-x)=-f(x) is true.
Monotonicity of the function (Increase and decrease of the function) The function y \u003d f (x) is called increasing on the set X є D (f), if for any points x 1 and x 2 of the set X such that x 1 f (x 2) f (x 2)">
How to build a graph of a periodic function If the function y \u003d f (x) has a period T, then to plot the function graph, you must first plot a branch (wave, part) of the graph at any interval of length T, and then move this branch along the x axis to the right and left by T, 2T, 3T, etc.
Boundedness of a Function A function y=f(x) is called bounded from below on the set X є D(f) if all values of this function on the set X are greater than a certain number. (that is, if there is a number m such that for any value x є X the following inequality is true: f (x) > m. The function y \u003d f (x) is called bounded from above on the set X є D (f) if all values this function on the set X is less than a certain number (i.e. if there is a number M such that for any value x є X the following inequality is true: f(x) m. The function y=f(x) is called bounded from above on the set X є D(f) if all values of this function on the set X are less than some number (i.e., if there is a number M such that for any value x є X the following inequality holds: f(x)
The largest and smallest value of the function The number m is called the smallest value of the function y \u003d f (x) on the set X є D (f), if: 1) there is a point x o є X such that f (х o) \u003d m; 2) For any value x є X, the inequality f(x)f(x o) is satisfied. The number M is called the greatest value of the function y=f(x) on the set X є D(f) if: that f(x o)=M; 2) For any value x є X, the inequality f (x) f (x o)
Convexity of a function A function is convex upward on the interval X with Dif) if, by connecting any two points of its graph with the abscissas from X by a segment, we find that the corresponding part of the graph lies above the drawn segment. It is considered that a function is convex down on the interval X with D(f) if, by connecting any two points of its graph with the abscissas from X by a segment, we find that the corresponding part of the graph lies below the drawn segment
Continuity of the function The continuity of the function on the interval X means that the graph of the function on this interval does not have break points (i.e., it is a solid line). Comment. In fact, one can speak of the continuity of a function only when it is proved that the function is continuous. But the corresponding definition is complex and beyond our strength for the time being (we will give it later, in § 26). The same can be said about the concept of convexity. Therefore, when discussing these two properties of functions, for the time being we will continue to rely on visual-intuitive representations.
Extremum points and function extremum. The maximum and minimum points of a function are called the extremum points of the function. Definition. The point x 0 is called the minimum point of the function f if for all x from some neighborhood x 0 the inequality f(x) f(x 0) is satisfied. Definition. The point x 0 is called the maximum point of the function f if for all x from some neighborhood x 0 the inequality f(x) f(x 0) is satisfied.
Scheme for studying the function 1 - Domain of definition 2 - even (odd) 3 - smallest positive period 4 - intervals of increase and decrease 5 - points of extrema and extrema of the function 6 - boundedness of the function 7 - continuity of the function 8 - the largest and smallest value of the function 9 - Range of values 10 - convexity of the function
Numeric function such a correspondence between a number set is called X and many R real numbers, in which each number from the set X matches a single number from a set R. Lots of X called function scope
. Functions are denoted by letters f, g, h etc. If f is a function defined on the set X, then the real number y, corresponding to the number X their multitude X, often denoted f(x) and write
y = f(x). variable X is called an argument. The set of numbers of the form f(x) called function range
A function is defined using a formula. For example , y = 2X - 2. If, when defining a function using a formula, its domain of definition is not indicated, then it is assumed that the scope of the function is the domain of the expression f(x).
1. The function is called monotonous on some interval A, if it increases or decreases on this interval
2. The function is called increasing on some interval A, if for any numbers in their set A the following condition is satisfied: .
The graph of an increasing function has a feature: when moving along the abscissa axis from left to right along the interval BUT the ordinates of the graph points increase (Fig. 4).
3. The function is called waning at some interval BUT, if for any numbers their sets BUT condition is fulfilled: .
The graph of a decreasing function has a feature: when moving along the abscissa axis from left to right along the interval BUT the ordinates of the graph points decrease (Fig. 4).
4. The function is called even
on some set X, if the condition is met: 
.
The graph of an even function is symmetrical about the y-axis (Fig. 2).
5. The function is called odd
on some set X, if the condition is met: 
.
The graph of an odd function is symmetrical with respect to the origin (Fig. 2).
6. If function y = f(x)
f(x) f(x), then we say that the function y = f(x) accepts smallest value
at =f(x) at X= x(Fig. 2, the function takes the smallest value at the point with coordinates (0;0)).
7. If function y = f(x) is defined on the set X and there exists such that for any the inequality f(x) f(x), then we say that the function y = f(x) accepts highest value at =f(x) at X= x(Fig. 4, the function does not have the largest and smallest values) .
If for this function y = f(x) all the listed properties are studied, then they say that study functions.
Limits.
The number A is called the limit of f-ii as x tends to ∞ if for any E>0, there exists δ (E)>0 such that for all x the inequality |x|>δ satisfies the inequality |F(x)-A| The number A is called the limit of the function when X tends to X 0 if for any E>0, there exists δ (E)>0 such that for all X≠X 0 the inequality |X-X 0 |<δ выполняется неравенство |F(x)-A| ONE-SIDED LIMITS. When determining the limit, that X tends to X0 in an arbitrary way, that is, from any side. When X tends to X0, so that it is less than X0 all the time, then the limit is called the limit at point X0 on the left. Or left-hand limit. The right-hand limit is defined similarly. Sections:
Maths Class:
9
Lesson type: Lesson of generalization and systematization of knowledge. Equipment: The purpose of the lesson is stated. Checking homework 1. Frontal survey. 2. Blitz survey: mark on the board the correct answer in the test (Appendix 1, pp. 2-3). Doing exercises. 1. Solve No. 358 (a). Solve graphically the equation: . 2. Cards (four weak students decide in a notebook or on the board): 1) Find the value of the expression: a) ; b) 2) Find the domain of definition of functions: a) ; b) y = . 3. Solve No. 358 (a). Solve graphically the equation: .
One student solves on the board, the rest in a notebook. If necessary, the teacher helps the student. A rectangular coordinate system was built on the interactive whiteboard using the AutoGraph program. The student draws the corresponding graphs with a marker, finds a solution, writes down the answer. Then the task is checked: the formula is entered using the keyboard, and the graph must match the one already drawn in the same coordinate system. The abscissa of the intersection of the graphs is the root of the equation. Solution: Answer: 8 Solve #360(a). Plot and read the graph of the function: Students complete the task on their own. The construction of the graph is checked using the “AutoGraph” program, the properties are written on the board by one student (domain, range, parity, monotonicity, continuity, zeros and constant sign, the largest and smallest values of the function). Solution: Properties: 1) D( f) = (-); E( f) = , increases by )
During the classes
1. Organizational moment
I stage of the lesson
II stage of the lesson
III stage of the lesson
.
