Municipal educational institution
"Novoukolovsk secondary school"
Krasnensky district of the Belgorod region
Algebra lesson in grade 11
"Application of several transformations leading to a corollary equation"
Prepared and conducted
Mathematic teacher
Kharkovskaya Valentina Grigorievna
Algebra Grade 11
Subject: Application of several transformations leading to an equation - a consequence.
Target: create conditions for fixing the material on the topic: "Application of several transformations leading to an equation - a consequence"; Rto develop independence, to cultivate literacy of speech; to form the computational skills of students; complete tasks corresponding to the level of the exam.
Equipment: textbook, computer, cards
Lesson type: lesson on the complex application of ZUN
During the classes
Organizational moment (Slide 1)
Good afternoon guys! Look at these pictures and choose which one you like the most. I see that you, like me, came to the lesson in a good mood, and I think it will remain the same until the end of the lesson. I want to wish you fruitful work.
Guys, each of you has evaluation sheets on the table in which you will evaluate yourself at each stage of the lesson.
Checking homework. (Slide 2)
Highlight the solutions on the slide and the children rate themselves in
self-check sheet. No errors - "5", if 1 error - "4", 2
errors - "3". If you get a lot of children who have 2
mistakes, then solve this task at the blackboard.
Announcement of the topic of the lesson (Slide 3). lesson goal setting
You can see the topic of our lesson on the slide. What do you think
Are we going to be in class today?
Well, guys, let's remember the material covered .
Let's start with speaking :
Oral work (Slide 4)
What equations are called corollary equations? (if any root of the first equation is the root of the second, then the second equation is called a consequence of the first);
What is called the transition to the consequence equation? (replacement of the equation by another equation, which is its consequence);
What transformations lead to the corollary equation? Give examples. (raising an equation to an even power; potentiating a logarithmic equation; freeing an equation from a denominator; bringing like terms of an equation; applying formulas).
Solve the equations (Slide 5)
(equations are displayed on the screen):
1) = 6; (answer: 36)
2) = 3; (answer: 11)
3) = 4; (answer: 6)
4) = - 2; (answer: no solutions, since the left side of the equation takes only non-negative values)
5) = 9; (answer: -9 and 9)
6) = -2; (answer: no solutions, since the sum of two
non-negative numbers cannot be negative)
Guys, I think you noticed that when doing homework and oral work, we met tasks corresponding to the demo version, specification and USE codifier.
4. Completing tasks
Guys, let's work in notebooks:
№8.26 (a) - at the blackboard
№8.14 (c) - at the blackboard
Fizminutka for the eyes (music)
№8.8 (c)-at the board
№8.9-(e)-at the board
5. Independent work (Slide 6)
Independent work solution (Slide 7)
6. Homework: complete No. 8.14 (g), task USE B5 in options 21,23,25 (Slide 8)
7. The results of the lesson (Slide 9)
8. Reflection (Slide 10)
Questionnaire.
1. I worked at the lesson
2. With my work in the lesson, I
3. The lesson seemed to me
4. For the lesson I
5. My mood
6. The material of the lesson was
7. Do you think you will cope with such tasks in the exam?
8. Homework seems to me
active / passive
happy / not happy
short / long
not tired / tired
got better / got worse
clear / not clear
useful / useless
interesting / boring
yes/no/don't know
easy / difficult
interesting / uninteresting
Resources used:
Nikolsky S.M., Potapov K.M., . Algebra and the beginnings of mathematical analysis, grade 11 M .: Education, 2010
Collection of tasks for preparing for the exam in mathematics
The lesson was conducted by the teacher of mathematics MBOU secondary school No. 6 Tupitsyna O.V.
Topic and lesson number in the topic:“Application of several transformations leading to an equation-consequence”, lesson No. 7, 8 in the topic: “Equation-consequence”
Academic subject:Algebra and the beginnings of mathematical analysis - Grade 11 ( specialized training according to the textbook by S.M. Nikolsky)
Type of lesson: "systematization and generalization of knowledge and skills"
Lesson type: workshop
The role of the teacher: direct cognitive activity students to develop skills to independently apply knowledge in a complex to select the desired method or methods of transformation, leading to an equation - a consequence and application of the method in solving the equation, in new conditions.
Required technical equipment:multimedia equipment, webcam.
The lesson used:
- didactic learning model- creating a problematic situation,
- pedagogical means- sheets indicating training modules, a selection of tasks for solving equations,
- type of student activity- group (groups are formed in the lessons - "discoveries" of new knowledge, lessons No. 1 and 2 from students with different degrees of learning and learning), joint or individual problem solving,
- personality oriented educational technologies : modular training, problem-based learning, search and research methods, collective dialogue, activity method, work with a textbook and various sources,
- health-saving technologies- to relieve stress, physical education is carried out,
- competencies:
- educational and cognitive basic level - students know the concept of an equation - a consequence, the root of an equation and the methods of transformation leading to an equation - a consequence, are able to find the roots of equations and perform their verification on productive level;
- at an advanced level- students can solve equations using well-known methods of transformations, check the roots of equations using the area of \u200b\u200badmissible values of equations; calculate logarithms using exploration-based properties; informational - students independently search, extract and select the information necessary for solving educational problems in sources of various types.
Didactic goal:
creating conditions for:
Formation of ideas about equations - consequences, roots and methods of transformation;
Formation of the experience of meaning-creation based on the logical consequence of the previously studied methods for transforming equations: raising an equation to an even degree, potentiation logarithmic equations, the release of the equation from the denominators, the reduction of similar terms;
Consolidation of skills in determining the choice of the transformation method, further solving the equation and choosing the roots of the equation;
Mastering the skills of setting a problem based on known and learned information, generating requests to find out what is not yet known;
Formation cognitive interests, intellectual and creative abilities of students;
Development logical thinking, creative activity of students, project skills, the ability to express their thoughts;
Formation of a sense of tolerance, mutual assistance when working in a group;
Awakening interest in independent solution of equations;
Tasks:
Organize the repetition and systematization of knowledge about how to transform equations;
- to ensure mastery of methods for solving equations and checking their roots;
- contribute to the development of analytical and critical thinking students; compare and choose optimal methods for solving equations;
- create conditions for the development of research skills, group work skills;
Motivate students to use the studied material to prepare for the exam;
Analyze and evaluate your work and the work of your comrades in the performance of this work.
Planned results:
*personal:
Skills of setting a task based on known and learned information, generating requests to find out what is not yet known;
The ability to choose the sources of information necessary to solve the problem; development of cognitive interests, intellectual and creative abilities of students;
The development of logical thinking, creative activity, the ability to express one's thoughts, the ability to build arguments;
Self-assessment of performance results;
Teamwork skills;
*metasubject:
The ability to highlight the main thing, compare, generalize, draw an analogy, apply inductive methods of reasoning, put forward hypotheses when solving equations,
Ability to interpret and apply the acquired knowledge in preparation for the exam;
*subject:
Knowledge of how to transform equations,
The ability to establish a pattern associated with various types of equations and use it in solving and selecting roots,
Integrating lesson objectives:
- (for the teacher) Formation in students of a holistic view of the ways of transforming equations and methods for solving them;
- (for students) Development of the ability to observe, compare, generalize, analyze mathematical situations associated with types of equations containing various functions. Preparation for the exam.
Stage I of the lesson:
Updating knowledge to increase motivation in the field of application of various methods of transforming equations (input diagnostics)
The stage of updating knowledgecarried out in the form verification work with self check. Developmental tasks are proposed, based on the knowledge acquired in previous lessons, requiring active mental activity from students and necessary to complete the task at this lesson.
Verification work
- Choose equations that require the restriction of unknowns on the set of all real numbers:
a) = X-2; b) 3 \u003d X-2; c) =1;
d) ( = (; e) = ; e) +6 =5;
g) = ; h) = .
(2) Specify the range of valid values of each equation, where there are restrictions.
(3) Choose an example of such an equation, where the transformation may cause the loss of the root (use the materials of the previous lessons on this topic).
Everyone checks the answers independently according to the ready-made ones highlighted on the screen. The most complex tasks are analyzed and addressed Special attention students on examples a, c, g, h, where restrictions exist.
It is concluded that when solving equations, it is necessary to determine the range of values allowed by the equation or to check the roots in order to avoid extraneous values. The previously studied methods of transforming equations leading to an equation - a consequence are repeated. That is, the students are thus motivated to find the right way to solve the equation proposed by them in further work.
II stage of the lesson:
Practical application of their knowledge, skills and abilities in solving equations.
The groups are given sheets with a module compiled on the issues of this topic. The module includes five learning elements, each of which is aimed at performing certain tasks. Students with different degrees of learning and learning independently determine the scope of their activities in the lesson, but since everyone works in groups, there is a continuous process of adjusting knowledge and skills, pulling those who are lagging behind to compulsory, others to advanced and creative levels.
In the middle of the lesson, a mandatory physical minute is held.
No. of educational element | Educational element with assignments | Guide to the development of educational material |
UE-1 | Purpose: To determine and justify the main methods for solving equations based on the properties of functions.
Specify the transformation method for solving the following equations: A) )= -8); b) = c) (=( d) ctg + x 2 -2x = ctg +24; e) = ; f) = sinx. 2) Task: Solve at least two of the proposed equations. Describe what methods were used in the solved equations. | Clause 7.3 p.212 Clause 7.4 p.214 Clause 7.5 p.217 Clause 7.2 p. 210 |
UE-2 | Purpose: To master rational techniques and methods of solving Exercise: Give examples from the above or self-selected (use materials from previous lessons) equations that can be solved using rational methods of solution, what are they? (emphasis on the way to check the roots of the equation) | |
UE-3 | Purpose: Using the acquired knowledge in solving equations high level difficulties Exercise: = (or ( = ( | Clause 7.5 |
UE-4 | Set the level of mastery of the topic: low - solution of no more than 2 equations; Medium - solution of no more than 4 equations; high - solution of no more than 5 equations | |
UE-5 | Output control: Make a table in which to present all the methods of transforming equations you use and for each method write down examples of the equations you solved, starting from lesson 1 of the topic: “Equations - consequences” | Abstracts in notebooks |
III stage of the lesson:
day off diagnostic work, representing the reflection of students, which will show readiness not only for writing control work, but also readiness for the exam in this section.
At the end of the lesson, all students, without exception, evaluate themselves, then comes the teacher's assessment. If there are disagreements between the teacher and the student, the teacher can offer an additional task to the student in order to objectively be able to evaluate it. Homeworkaimed at reviewing the material before the control work.
In the presentation, we will continue to consider equivalent equations, theorems, and dwell in more detail on the stages of solving such equations.
First, let's recall the condition under which one of the equations is a consequence of the other (slide 1). The author cites once again some theorems on equivalent equations that were considered earlier: on the multiplication of parts of an equation by same value h(x); raising the parts of the equation to the same even power; obtaining an equivalent equation from the equation log a f (x) = log a g (x).
On the 5th slide of the presentation, the main stages are highlighted, with the help of which it is convenient to solve equivalent equations:
Find solutions to an equivalent equation;
Analyze solutions;
Check.
Consider example 1. It is necessary to find a consequence of the equation x - 3 = 2. Find the root of the equation x = 5. Write the equivalent equation (x - 3)(x - 6) = 2(x - 6), applying the method of multiplying the parts of the equation by (x - 6). Simplifying the expression to the form x 2 - 11x +30 = 0, we find the roots x 1 = 5, x 2 = 6. each root of the equation x - 3 \u003d 2 is also a solution to the equation x 2 - 11x +30 \u003d 0, then x 2 - 11x +30 \u003d 0 is a consequence equation.
Example 2. Find another consequence of the equation x - 3 = 2. To obtain an equivalent equation, we use the method of raising to an even power. Simplifying the resulting expression, we write x 2 - 6x +5 = 0. Find the roots of the equation x 1 = 5, x 2 = 1. x \u003d 5 (the root of the equation x - 3 \u003d 2) is also a solution of the equation x 2 - 6x +5 \u003d 0, then the equation x 2 - 6x +5 \u003d 0 is also a consequence equation.
Example 3. It is necessary to find a consequence of the equation log 3 (x + 1) + log 3 (x + 3) = 1.
Let us replace 1 = log 3 3 in the equation. Then, applying the statement from Theorem 6, we write the equivalent equation (x + 1)(x +3) = 3. Simplifying the expression, we obtain x 2 + 4x = 0, where the roots are x 1 = 0, x 2 = - 4. So the equation x 2 + 4x = 0 is a consequence for the given equation log 3 (x + 1) + log 3 (x + 3) = 1.
So, we can conclude: if the domain of definition of the equation is expanded, then an equation-consequence is obtained. We single out the standard actions in finding the equation-consequence:
Getting rid of the denominators that contain the variable;
Raising the parts of the equation to the same even power;
Exemption from logarithmic signs.
But it is important to remember: when the domain of definition of the equation is expanded during the solution, it is necessary to check all the roots found - whether they will fall into the ODZ.
Example 4. Solve the equation presented on slide 12. First, find the roots of the equivalent equation x 1 \u003d 5, x 2 \u003d - 2 (first stage). It is imperative to check the roots (second stage). Checking the roots (third stage): x 1 = 5 does not belong to the area of admissible values of the given equation, therefore the equation has one solution only x = - 2.
In example 5, the found root of the equivalent equation is not included in the ODZ of the given equation. In example 6, the value of one of the two found roots is not defined, so this root is not a solution to the original equation.
School lecture
“Equivalent Equations. Corollary equation»
methodological comments. The concepts of equivalent equations, corollary equations, theorems on the equivalence of equations are important issues related to the theory of solving equations.
By 10th grade, students have gained some experience in solving equations. In grades 7-8, linear and quadratic equations, there are no non-equivalent transformations here. Further, in the 8th and 9th grades, rational and simplest irrational equations are solved, it turns out that in connection with the release from the denominator and squaring both parts of the equation, extraneous roots may appear. Thus, there is a need for the introduction of new concepts: the equivalence of equations, equivalent and non-equivalent transformations of an equation, extraneous roots, and verification of roots. Based on the experience accumulated by students in solving the above classes of equations, it is possible to determine a new relation of equivalence of equations and “discover” theorems on the equivalence of equations together with students.
The lesson, the summary of which is presented below, precedes the consideration of topics related to the solution of irrational, exponential, logarithmic and trigonometric equations. The theoretical material of this lesson serves as a support for solving all classes of equations. In this lesson, it is necessary to define the concept of equivalent equations, corollary equations, to consider transformation theorems that lead to such types of equations. The material under consideration, as noted above, is a kind of systematization of students' knowledge about the transformations of equations, it is characterized by a certain complexity, therefore the most acceptable type of lesson is a school lecture. The peculiarity of this lesson is that the educational task (goals) set on it is solved over the course of many subsequent lessons (identifying transformations over equations leading to the acquisition of extraneous roots and the loss of roots).
Each stage of the lesson occupies an important place in its structure.
On the update stage students remember the main theoretical provisions related to the equation: what is an equation, the root of the equation, what does it mean to solve the equation, the range of acceptable values (ODV) of the equation. They find the ODZ of specific equations that will serve as a support for the “discovery” of theorems in the lesson.
Target stage of motivation- to create a problem situation, which consists in finding the correct solution of the proposed equation.
Decision learning task (operational-cognitive stage) in the presented lesson lies in the "discovery" of theorems on the equivalence of equations and their proof. The main attention in the presentation of the material is given to the definition of equivalent equations, equations-consequences, the "search" of theorems on the equivalence of equations.
The notes that the teacher makes during the lesson are presented directly in the abstract. Registration of notes by students in notebooks is given at the end of the lesson summary.
Lesson summary
Subject. Equivalent equations. Equation-consequence.
(Algebra and the Beginnings of Analysis: A Textbook for Grades 10-11 educational institutions/Sh.A. Alimov, Yu.M. Kolyagin, Yu.V. Sidorov and others - M .: Education, 2003).
Lesson goals. AT joint activities with students to identify the equivalence relation on the set of equations, “discover” theorems on the equivalence of equations.
As a result, the student
knows
Definition of equivalent equations,
Definitions of the equation-corollary,
Statements of the main theorems;
can
From the proposed equations, choose equivalent equations and equations-consequences,
Apply definitions of equivalent equations and corollary equations in standard situations;
understands
What transformations lead to equivalent equations or to equations-consequences,
That there are transformations, as a result of which the equation can acquire extraneous roots,
That as a result of some transformations, the loss of roots may occur.
Lesson type. School lecture (2 hours).
Lesson structure.
I. Motivational and orienting part:
Knowledge update,
Motivation, setting a learning task.
II. Operational-cognitive part:
The solution of the educational and research problem (the purpose of the lesson).
III. Reflective-evaluative part:
Summing up the lesson
Issuance of homework.
During the classes
I. Motivational and orienting part.
Today in the lesson we will talk about the equation, but we will not write down the topic yet. Recall the basic concepts associated with the equation. First of all, what is an equation?
(An equation is an analytical record of the problem of finding the values of the arguments for which the values of one function are equal to the values of another function).
What other concepts are related to the equation?
(The root of the equation and what it means to solve the equation. The root of the equation is a number, when substituting into the equation, the correct numerical equality is obtained. Solve the equation - find all its roots or establish that they do not exist).
What is the ODZ equation?
(The set of all numbers for which the functions on the left and right sides of the equation make sense at the same time).
Find the ODZ of the following equations.
5) 
6) 
.
The solution to the equation is written on the blackboard. 
What is the process of solving an equation?
(Performing transformations that bring this equation to an equation of a simpler form, i.e. such an equation, finding the roots of which is not difficult).
True, i.e. there is a sequence of simplifications from equation to equation 
etc. to 
. Let's see what happens to the roots of the equation at each stage of transformations. In the presented solution, two roots of the equation are obtained 
. Check if they are numbers and numbers 
and 
roots of the original equation.
(numbers 
, and are the roots of the original equation, and 
- No).
So, in the process of solving these roots were lost. In general, the transformations performed led to the loss of two roots 
and the acquisition of an extraneous root.
How can you get rid of extraneous roots?
(Make a check).
Is it possible to lose roots? Why?
(No, because solving an equation means finding all its roots).
How to avoid losing roots?
(Probably, when solving the equation, do not perform transformations that lead to the loss of roots).
So, in order for the process of solving an equation to lead to correct results, what is important to know when performing transformations on equations?
(Probably, to know which transformations over the equations preserve the roots, which lead to the loss of roots or the acquisition of extraneous roots. Know what transformations they can be replaced so that there is no loss or acquisition of roots).
That's what we're going to do in this lesson. How would you formulate the goal of the upcoming activity in today's lesson?
(To identify transformations over equations that preserve roots, lead to the loss of roots or the acquisition of extraneous roots. Know what transformations can be replaced so that there is no loss or acquisition of roots).
II . Operational-cognitive part.
Let's go back to the equation written on the blackboard. Let's trace at what stage and as a result of what transformations, two roots were lost and an outsider appeared. (The teacher to the right of each equation puts down the numbers).
Name the equations that have the same set (set) of roots.
(Equations , , 
,
and 
,).
Such equations are called equivalent. Try to formulate a definition of equivalent equations.
(Equations that have the same set of roots are called equivalent).
Let's write down the definition.
Definition 1. Equations
and 
are said to be equivalent if the sets of their roots are the same.
It should be noted that equations without horses are also equivalent.
To denote equivalent equations, you can use the symbol " 
».
The process of solving the equation using the new concept can be reflected as follows:
Thus, the transition from a given equation to an equivalent one does not affect the set of roots of the resulting equation.
And what are the main transformations performed when solving linear equations?
(Opening brackets; transferring terms from one part of the equation to another, changing the sign to the opposite; adding an expression containing an unknown to both parts of the equation).
Have their roots changed?
On the basis of one of these transformations, namely: the transfer of terms from one part of the equation to another, while changing the sign to the opposite, in the 7th grade they formulated a property of equations. Formulate it using a new concept.
(If any term of the equation is transferred from one part of the equation to another with the opposite sign, then an equation equivalent to the given one will be obtained).
What other property of the equation do you know?
(Both sides of the equation can be multiplied by the same non-zero number.)
Applying this property also replaces the original equation with an equivalent one. Let's go back to the equation written on the blackboard. Compare the set of roots of equations and ?
(The root of the equation is the root of the equation).
That is, when passing from one equation to another, the set of roots, although expanded, did not lose the roots. In this case, the equation is called a consequence of the equation. Try to formulate a definition of an equation that is a consequence of this equation.
(If there is no loss of roots when passing from one equation to another, then the second equation is called a consequence of the first equation).
Definition 2. An equation is called a consequence of an equation if each root of the equation is a root of the equation.
- As a result of what transformation did you get the equation from the equation?
(Squaring both sides of the equation).
This means that this transformation can lead to the appearance of extraneous roots, i.e. the original equation is transformed into a consequence equation. Are there any other corollary equations in the presented chain of equation transformations?
(Yes, for example, the equation is a consequence of the equation, and the equation is a consequence of the equation).
What are these equations?
(Equivalent).
Try, using the concept of a consequence equation, to formulate an equivalent definition of equivalent equations.
(Equations are said to be equivalent if each of them is a consequence of the other).
Are there any other corollary equations in the proposed solution of the equation?
(Yes, the equation is a consequence of the equation).
What happens to roots when going from to ?
(Two roots are lost).
What transformation resulted in this?
(Error in applying the identity 
).
Applying the new concept of the equation-corollary, and using the symbol " 
”, the process of solving the equation will look like this:
.
So, the resulting scheme shows us that if equivalent transitions are made, then the sets of roots of the resulting equations do not change. But it is not always possible to apply only equivalent transformations. If the transitions are not equivalent, then two cases are possible: and . In the first case, the equation is a consequence of the equation, the set of roots of the resulting equation includes the set of roots of the given equation, here extraneous roots are acquired, they can be cut off by performing a check. In the second case, an equation was obtained for which this equation is a consequence: , which means that there will be a loss of roots, such transitions should not be performed. Therefore, it is important to ensure that when transforming an equation, each subsequent equation is a consequence of the previous one. What do you need to know so that the transformations are only such? Let's try to install it. Let's write task 1 (it offers equations; their ODZ found at the update stage; the set of roots of each equation is recorded).
Task 1. Are the equations of each group (a, b) equivalent? Name the transformation, as a result of which the first equation of the group is replaced by the second.
a)
b)
Let us turn to the equations of the group a), are these equations equivalent?
(Yes, and they are equivalent).
(We used the identity).
That is, the expression in one part of the equation was replaced by an identically equal expression. Has the ODZ equation changed under this transformation?
Consider the group of equations b). Are these equations equivalent?
(No, the equation is a consequence of the equation).
As a result of what transformation did you get from ?
(We replaced the left side of the equation with an identically equal expression).
What happened to the odz equation?
(ODZ expanded).
As a result of the expansion of the ODZ, we obtained a consequence equation and an extraneous root 
for the equation. This means that the expansion of the ODZ equation can lead to the appearance of extraneous roots. For both cases a) and b) state the statement in general view. (Students formulate, teacher corrects).
(Let in some equation 
, expression 
replaced by the identical expression 
. If such a transformation does not change the ODZ equation, then we pass to the equivalent equation 
. If the ODZ expands, then the equation is a consequence of the equation ).
This statement is a transformation theorem leading to equivalent equations or corollary equations.
Theorem 1., |
|
a) ODZdoes not change | b) ODZ is expanding |
We accept this theorem without proof. Next task. Three equations and their roots are presented.
Task 2. Are the following equations equivalent? Name the transformation, as a result of which the first equation is replaced by the second equation, the third equation.
Which of the following equations are equivalent?
(Only equations and ).
What transformations were performed in order to pass from the equation to the equation , ?
(To both sides of the equation in the first case we added 
, in the second case we added 
).
That is, in each case, some function was added 
. Compare the domain of the function in the equation with the ODZ equation.
(Function 
defined on the ODZ equation ).
What equation was obtained by adding the function to both sides of the equation?
(We get an equivalent equation).
What happened to the ODZ equation compared to the ODZ equation?
(It has narrowed due to the function 
).
What did you get in this case? Will the equation be equivalent to the equation or - the equation-corollary for the equation?
(No, not both).
Having considered two cases of transformation of the equation, which are presented in task 2, try to draw a conclusion.
(If we add to both parts of the equation the function defined on the ODZ of this equation, then we get an equation equivalent to the given one).
Indeed, this statement is a theorem.
Theorem2. , - defined | on the odz equation |
|
But we used a statement similar to the formulated theorem when solving equations. How does it sound?
(The same number can be added to both sides of the equation.)
This property is a particular case of Theorem 2 when 
.
Task 3. Are the following equations equivalent? Name the transformation, as a result of which the first equation is replaced by the second equation, the third equation.
Which of the equations in task 3 are equivalent?
(Equations and ).
As a result of what transformation from the equation are the equations , ?
(Both sides of the equation are multiplied by 
and get the equation. To get the equation, both sides of the equation are multiplied by 
).
What condition must the function satisfy so that by multiplying both sides of the equation by , an equation equivalent to would be obtained?
(The function must be defined on the entire ODZ of the equation).
Have such transformations been performed on equations before?
(Performed, both parts of the equation were multiplied by a number other than zero).
This means that the condition imposed on the function must be supplemented.
(The function must not go to zero for any 
from the ODZ equation).
So, we write in symbolic form a statement that allows us to pass from a given equation to an equivalent one. (The teacher, under the dictation of the students, writes down Theorem 3).
Theorem 3. - defined throughout the ODZ
| |
for any of the ODZLet's prove the theorem. What does it mean that two equations are equivalent?
(It must be shown that all the roots of the first equation are the roots of the second equation and vice versa, i.e. the second equation is a consequence of the first and the first equation is a consequence of the second).
Let us prove that is a consequence of the equation . Let be 
- the root of the equation, what does it mean?
(When substituting in we get the correct numerical equality 
).
At a point, the function is defined and does not vanish. What does this mean?
(Number 
. Therefore, the numerical equality can be multiplied by 
. We get the correct numerical equality ).
What does this equality mean?
( - the root of the equation. This showed that the equation is an equation-consequence for the equation).
Let us prove that is a consequence of the equation . (Students work independently, then after the discussion, the teacher writes the second part of the proof on the board).
Task 4. Are the equations of each group (a, b) equivalent? Name the transformation, as a result of which the first equation of the group is replaced by the second.
a)
b)
Are the equations and ?
(Equivalent).
As a result of what transformation from can be obtained?
(We raise both sides of the equation to a cube).
From the right and left sides of the equation, you can take the function 
. On which set is the function defined? 
?
(On the common part of the sets of function values 
and 
).
Describe the group of equations under the letter b)?
(They are not equivalent, is a consequence, the function was applied to the equation 
and passed to the equation , the function is defined on the common part of the sets of function values 
and 
).
What is the difference between the properties of functions in the group a) and b)?
(In the first case, the function is monotonic, but not in the second).
Let us formulate the following assertion. (The teacher, under the dictation of the students, writes down the theorem).
Theorem 4.
|
|
a) - monotonous | b) - not monotonous |
- is defined on the common part of the sets of function values and 
Let's discuss how this theorem will "work" when solving the following equations.
Example. solve the equation
1) 
; 2) 
.
Which function is applicable to both sides of equation 1)?
(Let's raise both sides of the equation to a cube, i.e., apply the function).
(This function is defined on the common part of the sets of values of functions on the left and right sides of the equation; it is monotonic).
So, by raising both sides of the original equation to a cube, what equation will we get?
(Equivalent to this).
Which function is applicable to both sides of equation 2)?
(Let's raise both sides of the equation to the fourth power, i.e. apply the function 
).
List the properties of this function necessary to apply Theorem 4.
(This function is defined on the common part of the value sets of functions on the left and right sides of the equation; it is not monotonic).
What equation, relative to the original one, will we get by raising this equation to the fourth power?
(Consequence equation).
Will the set of roots of the original equation and the set of roots of the resulting equation differ?
(Extraneous roots may appear. So, a check is necessary).
Solve these equations at home.
III . Reflective-evaluative part.
Today we “discovered” four theorems together. Look at them again and say what equations they say.
(On equivalent equations and the equation-corollary).
Let's write the topic of the lesson. Let's return to the equation that was considered at the beginning of today's conversation. Which of Theorems 1-4 were applied when passing from one equation to another? (Students together with the teacher find out which theorem worked at each step, the teacher marks the number of the theorem on the diagram).
T.2 T.2 T.1 T.4 T.2 T.4
What new did you learn at the lesson today?
(The concepts of equivalent equations, corollary equations, theorems on the equivalence of equations).
What task did we set at the beginning of the lesson?
(Select transformations that do not change the set of roots of the equation, transformations leading to the acquisition and loss of roots).
Have we solved it completely?
We solved the problem in part, we will continue its study in the next lessons when solving new types of equations.
Using the concept of equivalent equations, new for us, reformulate the first part of the task "to select transformations that do not change the set of roots of the equation."
(How to know if going from one equation to another is an equivalent transformation).
What will help answer this question?
(Theorems on the equivalence of equations).
And have any transformations been applied today that lead to the acquisition of extraneous roots?
(Applied, this is the squaring of both parts of the equation; the use of formulas, the left and right parts of which make sense for different values of the letters included in them).
There are other "specific" reasons that lead to both the appearance and loss of the roots of the equation, we talked about some of them. But there are also those that, as a rule, are associated with a certain class of equations, and we will talk about this later.
Let's write homework:
know the definitions of equivalent equations, corollary equations;
know the formulations of theorems 1-4;
carry out, by analogy with the proof of Theorem 3, the proof of Theorems 1 and 2;
4) Nos. 139(4,6), 141(2) - find out if the equations are equivalent; solve equations; .
Notebook entries
Equivalent equations. Equation-consequence.
Definition 1. Equations and are said to be equivalent if the sets of their roots coincide.
Definition 2. An equation is called a consequence of an equation if each root of the equation is a root of the equation. replaced by an identical expression.
Example.solve the equation
Class: 11
Duration: 2 lessons.
The purpose of the lesson:
- (for teacher) the formation of a holistic view of the methods for solving irrational equations among students.
- (for students) Development of the ability to observe, compare, generalize, analyze mathematical situations (slide 2). Preparation for the exam.
First lesson plan(slide 3)
- Knowledge update
- Analysis of the theory: Raising an equation to an even power
- Workshop on solving equations
Plan of the second lesson
- Differentiated independent work by groups "Irrational equations on the exam"
- Summary of lessons
- Homework
Course of lessons
I. Updating knowledge
Target: repeat the concepts necessary for the successful development of the topic of the lesson.
front poll.
What two equations are said to be equivalent?
What transformations of the equation are called equivalent?
- Replace this equation with an equivalent one with an explanation of the applied transformation: (slide 4)
a) x + 2x +1; b) 5 = 5; c) 12x = -3; d) x = 32; e) = -4.
What equation is called the equation-consequence of the original equation?
– Can the consequence equation have a root that is not the root of the original equation? What are these roots called?
– What transformations of the equation lead to the equation-consequences?
What is an arithmetic square root?
Let's dwell today in more detail on the transformation "Raising an equation to an even power".
II. Analysis of the theory: Raising an equation to an even power
Explanation by the teacher with the active participation of students:
Let 2m(mN) – fixed even natural number. Then the consequence of the equationf(x) =g(x) is the equation (f(x)) = (g(x)).
Very often this statement is used in solving irrational equations.
Definition. An equation containing the unknown under the sign of the root is called irrational.
When solving irrational equations, the following methods are used: (slide 5)
Attention! Methods 2 and 3 require mandatory checks.
ODZ does not always help to eliminate extraneous roots.
Conclusion: when solving irrational equations, it is important to go through three stages: technical, solution analysis, verification (slide 6).
III. Workshop on solving equations
Solve the equation:
After discussing how to solve the equation by squaring, solve by passing to an equivalent system.
Conclusion: the solution of the simplest equations with integer roots can be carried out by any familiar method.
b) \u003d x - 2
Solving by raising both parts of the equation to the same power, students get the roots x = 0, x = 3 -, x = 3 +, which are difficult and time-consuming to check by substitution. (Slide 7). Transition to an equivalent system
allows you to quickly get rid of extraneous roots. The condition x ≥ 2 is satisfied only by x.
Answer: 3+
Conclusion: It is better to check irrational roots by passing to an equivalent system.
c) \u003d x - 3
In the process of solving this equation, we obtain two roots: 1 and 4. Both roots satisfy the left side of the equation, but for x \u003d 1, the definition of arithmetic is violated square root. The ODZ equation does not help eliminate extraneous roots. The transition to an equivalent system gives the correct answer.
Conclusion:a good knowledge and understanding of all the conditions for determining the arithmetic square root helps to move on toperforming equivalent transformations.
By squaring both sides of the equation, we get the equation
x + 13 - 8 + 16 \u003d 3 + 2x - x, separating the radical to the right side, we get
26 - x + x \u003d 8. Applying further steps to squaring both parts of the equation, will lead to an equation of the 4th degree. The transition to the ODZ equation gives a good result:
find the ODZ equation:
x = 3.
Check: - 4 = , 0 = 0 is correct.
Conclusion:sometimes it is possible to carry out a solution using the definition of the ODZ equation, but be sure to check.
Solution: ODZ equation: -2 - x ≥ 0 x ≤ -2.
For x ≤ -2,< 0, а ≥ 0.
Therefore, the left side of the equation is negative, and the right side is non-negative; so the original equation has no roots.
Answer: no roots.
Conclusion:having made the correct reasoning on the restriction in the condition of the equation, you can easily find the roots of the equation, or establish that they do not exist.
Using the solution of this equation as an example, show the double squaring of the equation, explain the meaning of the phrase "solitude of radicals" and the need to check the found roots.
h) 
+ = 1.
The solution of these equations is carried out by the method of changing the variable until the return to the original variable. Finish the decision to offer those who will cope with the tasks of the next stage earlier.
test questions
- How to solve the simplest irrational equations?
- What should be remembered when raising an equation to an even power? ( extraneous roots may appear)
- What is the best way to check irrational roots? ( using the ODZ and the conditions for the coincidence of the signs of both parts of the equation)
- Why is it necessary to be able to analyze mathematical situations when solving irrational equations? ( For the correct and quick choice of a method for solving an equation).
IV. Differentiated independent work on groups "Irrational equations on the exam"
The class is divided into groups (2-3 people each) according to the levels of training, each group chooses an option with a task, discusses and solves the selected tasks. When necessary, contact the teacher for advice. After completing all the tasks of their version and checking the answers by the teacher, the group members individually complete the solution of equations g) and h) of the previous stage of the lesson. For options 4 and 5 (after checking the answers and the teacher's decision), additional tasks are written on the board, which are performed individually.
All individual solutions at the end of the lessons are handed over to the teacher for verification.
Option 1
Solve the equations:
a) = 6;
b) = 2;
c) \u003d 2 - x;
d) (x + 1) (5 - x) (+ 2 = 4.Option 5
1. Solve the equation:
a) = ;
b)= 3 - 2x;
2. Solve the system of equations:
Additional tasks:
v. Summary of lessons
What difficulties did you experience in completing the exam tasks? What is needed to overcome these difficulties?
VI. Homework
Repeat the theory of solving irrational equations, read paragraph 8.2 in the textbook (pay attention to example 3).
Solve No. 8.8 (a, c), No. 8.9 (a, c), No. 8.10 (a).
Literature:
- Nikolsky S.M., Potapov M.K., N.N. Reshetnikov N.N., Shevkin A.V. Algebra and beginning of mathematical analysis , textbook for the 11th grade of educational institutions, M .: Education, 2009.
- Mordkovich A.G. On some methodological issues related to the solution of equations. Mathematics at school. -2006. -No. 3.
- M. Shabunin. Equations. Lectures for high school students and entrants. Moscow, "Chistye Prudy", 2005. (library "First of September")
- E.N. Balayan. Workshop on problem solving. Irrational equations, inequalities and systems. Rostov-on-Don, "Phoenix", 2006.
- Mathematics. Preparation for the exam-2011. Edited by F.F. Lysenko, S.Yu. Kulabukhov Legion-M, Rostov-on-Don, 2010.