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Converting Expressions. Detailed theory (2019)

Converting Expressions

We often hear this unpleasant phrase: “simplify the expression.” Usually we see some kind of monster like this:

“It’s much simpler,” we say, but such an answer usually doesn’t work.

Now I will teach you not to be afraid of any such tasks. Moreover, at the end of the lesson, you yourself will simplify this example to (just!) an ordinary number (yes, to hell with these letters).

But before you start this lesson, you need to be able to handle fractions and factor polynomials. Therefore, first, if you have not done this before, be sure to master the topics “” and “”.

Have you read it? If yes, then you are now ready.

Basic simplification operations

Now let's look at the basic techniques that are used to simplify expressions.

The simplest one is

1. Bringing similar

What are similar? You took this in 7th grade, when letters instead of numbers first appeared in mathematics. Similar are terms (monomials) with the same letter part. For example, in total similar terms- this is i.

Do you remember?

To bring similar means to add several similar terms to each other and get one term.

How can we put the letters together? - you ask.

This is very easy to understand if you imagine that the letters are some kind of objects. For example, a letter is a chair. Then what is the expression equal to? Two chairs plus three chairs, how many will it be? That's right, chairs: .

Now try this expression: .

To avoid confusion, let different letters represent different objects. For example, - is (as usual) a chair, and - is a table. Then:

chairs tables chair tables chairs chairs tables

The numbers by which the letters in such terms are multiplied are called coefficients. For example, in a monomial the coefficient is equal. And in it is equal.

So, the rule for bringing similar ones is:

Examples:

Give similar ones:

Answers:

2. (and similar, since, therefore, these terms have the same letter part).

2. Factorization

This is usually the most important part in simplifying expressions. After you have given similar ones, most often the resulting expression needs to be factorized, that is, presented as a product. This is especially important in fractions: in order to be able to reduce a fraction, the numerator and denominator must be represented as a product.

You went through the methods of factoring expressions in detail in the topic “”, so here you just have to remember what you learned. To do this, decide a few examples(needs to be factorized):

Solutions:

3. Reducing a fraction.

Well, what could be more pleasant than crossing out part of the numerator and denominator and throwing them out of your life?

That's the beauty of downsizing.

It's simple:

If the numerator and denominator contain the same factors, they can be reduced, that is, removed from the fraction.

This rule follows from the basic property of a fraction:

That is, the essence of the reduction operation is that We divide the numerator and denominator of the fraction by the same number (or by the same expression).

To reduce a fraction you need:

1) numerator and denominator factorize

2) if the numerator and denominator contain common factors, they can be crossed out.

The principle, I think, is clear?

I would like to draw your attention to one typical mistake when abbreviating. Although this topic is simple, many people do everything wrong, not understanding that reduce- this means divide numerator and denominator are the same number.

No abbreviations if the numerator or denominator is a sum.

For example: we need to simplify.

Some people do this: which is absolutely wrong.

Another example: reduce.

The “smartest” will do this: .

Tell me what's wrong here? It would seem: - this is a multiplier, which means it can be reduced.

But no: - this is a factor of only one term in the numerator, but the numerator itself as a whole is not factorized.

Here's another example: .

This expression is factorized, which means you can reduce it, that is, divide the numerator and denominator by, and then by:

You can immediately divide it into:

To avoid such mistakes, remember easy way how to determine whether an expression is factorized:

The arithmetic operation that is performed last when calculating the value of an expression is the “master” operation. That is, if you substitute some (any) numbers instead of letters and try to calculate the value of the expression, then if the last action is multiplication, then we have a product (the expression is factorized). If the last action is addition or subtraction, this means that the expression is not factorized (and therefore cannot be reduced).

To consolidate, solve a few yourself examples:

Answers:

1. I hope you didn’t immediately rush to cut and? It was still not enough to “reduce” units like this:

The first step should be factorization:

4. Adding and subtracting fractions. Reducing fractions to a common denominator.

Addition and subtraction ordinary fractions- the operation is well known: we look for a common denominator, multiply each fraction by the missing factor and add/subtract the numerators. Let's remember:

Answers:

1. The denominators and are relatively prime, that is, they do not have common factors. Therefore, the LCM of these numbers is equal to their product. This will be the common denominator:

2. Here the common denominator is:

3. First thing here mixed fractions we turn them into incorrect ones, and then follow the usual pattern:

It's a completely different matter if the fractions contain letters, for example:

Let's start with something simple:

a) Denominators do not contain letters

Everything here is the same as with ordinary numerical fractions: find the common denominator, multiply each fraction by the missing factor and add/subtract the numerators:

Now in the numerator you can give similar ones, if any, and factor them:

Try it yourself:

b) Denominators contain letters

Let's remember the principle of finding a common denominator without letters:

· first of all, we determine the common factors;

· then we write out all the common factors one at a time;

· and multiply them by all other non-common factors.

To determine the common factors of the denominators, we first factor them into prime factors:

Let us emphasize the common factors:

Now let’s write out the common factors one at a time and add to them all the non-common (not underlined) factors:

This is the common denominator.

Let's get back to the letters. The denominators are given in exactly the same way:

· factor the denominators;

· determine common (identical) factors;

· write out all common factors once;

· multiply them by all other non-common factors.

So, in order:

1) factor the denominators:

2) determine common (identical) factors:

3) write out all the common factors once and multiply them by all other (non-underlined) factors:

So there's a common denominator here. The first fraction must be multiplied by, the second - by:

By the way, there is one trick:

For example: .

We see the same factors in the denominators, only all with different indicators. The common denominator will be:

to a degree

to a degree

to a degree

to a degree.

Let's complicate the task:

How to make fractions have the same denominator?

Let's remember the basic property of a fraction:

Nowhere does it say that the same number can be subtracted (or added) from the numerator and denominator of a fraction. Because it's not true!

See for yourself: take any fraction, for example, and add some number to the numerator and denominator, for example, . What did you learn?

So, another unshakable rule:

When you reduce fractions to common denominator, use only the multiplication operation!

But what do you need to multiply by to get?

So multiply by. And multiply by:

We will call expressions that cannot be factorized “elementary factors.” For example, - this is an elementary factor. - Same. But no: it can be factorized.

What about the expression? Is it elementary?

No, because it can be factorized:

(you already read about factorization in the topic “”).

So, the elementary factors into which you expand the expression with letters are an analogue prime factors, into which you decompose the numbers. And we will deal with them in the same way.

We see that both denominators have a multiplier. It will go to the common denominator to the degree (remember why?).

The factor is elementary, and they do not have a common factor, which means that the first fraction will simply have to be multiplied by it:

Another example:

Solution:

Before you multiply these denominators in a panic, you need to think about how to factor them? They both represent:

Great! Then:

Another example:

Solution:

As usual, let's factorize the denominators. In the first denominator we simply put it out of brackets; in the second - the difference of squares:

It would seem that there are no common factors. But if you look closely, they are similar... And it’s true:

So let's write:

That is, it turned out like this: inside the bracket we swapped the terms, and at the same time the sign in front of the fraction changed to the opposite. Take note, you will have to do this often.

Now let's bring it to a common denominator:

Got it? Let's check it now.

Tasks for independent decision:

Answers:

Here we need to remember one more thing - the difference of cubes:

Please note that the denominator of the second fraction does not contain the formula “square of the sum”! The square of the sum would look like this: .

A is the so-called incomplete square of the sum: the second term in it is the product of the first and last, and not their double product. The partial square of the sum is one of the factors in the expansion of the difference of cubes:

What to do if there are already three fractions?

Yes, the same thing! First of all, let's make sure that maximum quantity the factors in the denominators were the same:

Please note: if you change the signs inside one bracket, the sign in front of the fraction changes to the opposite. When we change the signs in the second bracket, the sign in front of the fraction changes again to the opposite. As a result, it (the sign in front of the fraction) has not changed.

We write out the entire first denominator into the common denominator, and then add to it all the factors that have not yet been written, from the second, and then from the third (and so on, if there are more fractions). That is, it turns out like this:

Hmm... It’s clear what to do with fractions. But what about the two?

It's simple: you know how to add fractions, right? So, we need to make two become a fraction! Let's remember: a fraction is a division operation (the numerator is divided by the denominator, in case you forgot). And there is nothing easier than dividing a number by. In this case, the number itself will not change, but will turn into a fraction:

Just what you need!

5. Multiplication and division of fractions.

Well, the hardest part is over now. And ahead of us is the simplest, but at the same time the most important:

Procedure

What is the procedure for calculating a numerical expression? Remember by calculating the meaning of this expression:

Did you count?

It should work.

So, let me remind you.

The first step is to calculate the degree.

The second is multiplication and division. If there are several multiplications and divisions at the same time, they can be done in any order.

And finally, we perform addition and subtraction. Again, in any order.

But: the expression in brackets is evaluated out of turn!

If several brackets are multiplied or divided by each other, we first calculate the expression in each of the brackets, and then multiply or divide them.

What if there are more brackets inside the brackets? Well, let's think: some expression is written inside the brackets. When calculating an expression, what should you do first? That's right, calculate the brackets. Well, we figured it out: first we calculate the inner brackets, then everything else.

So, the procedure for the expression above is as follows (the current action is highlighted in red, that is, the action that I am performing right now):

Okay, it's all simple.

But this is not the same as an expression with letters?

No, it's the same! Only instead of arithmetic operations you need to do algebraic, that is, the actions described in the previous section: bringing similar, adding fractions, reducing fractions, and so on. The only difference will be the action of factoring polynomials (we often use this when working with fractions). Most often, to factorize, you need to use I or simply put the common factor out of brackets.

Usually our goal is to represent the expression as a product or quotient.

For example:

Let's simplify the expression.

1) First, we simplify the expression in brackets. There we have a difference of fractions, and our goal is to present it as a product or quotient. So, we bring the fractions to a common denominator and add:

It is impossible to simplify this expression any further; all the factors here are elementary (do you still remember what this means?).

2) We get:

Multiplying fractions: what could be simpler.

3) Now you can shorten:

Well, that's all. Nothing complicated, right?

Another example:

Simplify the expression.

First, try to solve it yourself, and only then look at the solution.

First of all, let's determine the order of actions. First, let's add the fractions in parentheses, so instead of two fractions we get one. Then we will do division of fractions. Well, let's add the result with the last fraction. I will number the steps schematically:

Now I’ll show you the process, tinting the current action in red:

Finally, I will give you two useful tips:

1. If there are similar ones, they must be brought immediately. At whatever point similar ones arise in our country, it is advisable to bring them up immediately.

2. The same applies to reducing fractions: as soon as the opportunity to reduce appears, it must be taken advantage of. The exception is for fractions that you add or subtract: if they now have the same denominators, then the reduction should be left for later.

Here are some tasks for you to solve on your own:

And what was promised at the very beginning:

Solutions (brief):

If you have coped with at least the first three examples, then you have mastered the topic.

Now on to learning!

CONVERTING EXPRESSIONS. SUMMARY AND BASIC FORMULAS

Basic simplification operations:

• Bringing similar: to add (reduce) similar terms, you need to add their coefficients and assign the letter part.
• Factorization: rendering common multiplier beyond brackets, application, etc.
• Reducing a fraction: The numerator and denominator of a fraction can be multiplied or divided by the same non-zero number, which does not change the value of the fraction.
  1) numerator and denominator factorize
  2) if the numerator and denominator have common factors, they can be crossed out.

  IMPORTANT: only multipliers can be reduced!

• Adding and subtracting fractions:
  ;
• Multiplying and dividing fractions:
  ;

Ministry of Education of the Republic of Belarus

Educational institution

"Gomel state university them. F. Skorina"

Faculty of Mathematics

Department of MPM

Identity transformations expressions and methods of teaching students how to perform them

Executor:

Student Starodubova A.Yu.

Scientific supervisor:

Cand. physics and mathematics Sciences, Associate Professor Lebedeva M.T.

Gomel 2007

Introduction

1 The main types of transformations and stages of their study. Stages of mastering the use of transformations

Conclusion

Literature

Introduction

The simplest transformations of expressions and formulas, based on the properties of arithmetic operations, are carried out in elementary school and 5th and 6th grades. The formation of skills and abilities to perform transformations takes place in an algebra course. This is due both to the sharp increase in the number and variety of transformations being carried out, and to the complication of activities to justify them and clarify the conditions of applicability, to the identification and study of the generalized concepts of identity, identical transformation, equivalent transformation.

1. Main types of transformations and stages of their study. Stages of mastering the use of transformations

1. Beginnings of algebra

An undivided system of transformations is used, represented by rules for performing actions on one or both parts of the formula. The goal is to achieve fluency in completing tasks for solving simple equations, simplifying formulas that define functions, and rationally carrying out calculations based on the properties of actions.

Typical examples:

Solve equations:

A) ; b) ; V) .

Identical transformation (a); equivalent and identical (b).

2. Formation of skills in applying specific types of transformations

Conclusions: abbreviated multiplication formulas; transformations associated with exponentiation; transformations associated with various classes of elementary functions.

Organization whole system transformations (synthesis)

The goal is to create a flexible and powerful device suitable for use in solving a variety of educational assignments . The transition to this stage is carried out during the final repetition of the course in the course of understanding the already known material learned in parts, by certain types transformations add transformations of trigonometric expressions to the previously studied types. All these transformations can be called “algebraic”; “analytical” transformations include those that are based on the rules of differentiation and integration and transformation of expressions containing passages to limits. The difference of this type is in the nature of the set that the variables in identities (certain sets of functions) run through.

The identities being studied are divided into two classes:

I – identities of abbreviated multiplication valid in a commutative ring and identities

fair in the field.

II – identities connecting arithmetic operations and basic elementary functions.

2 Features of the organization of the system of tasks when studying identity transformations

The main principle of organizing the system of tasks is to present them from simple to complex.

Exercise cycle– combining in a sequence of exercises several aspects of studying and techniques for arranging the material. When studying identity transformations, a cycle of exercises is associated with the study of one identity, around which other identities that are in a natural connection with it are grouped. The cycle, along with executive ones, includes tasks, requiring recognition of the applicability of the identity in question. The identity under study is used to carry out calculations on various numerical domains. The tasks in each cycle are divided into two groups. TO first These include tasks performed during initial acquaintance with identity. They serve as educational material for several consecutive lessons united by one topic.

Second group exercises connects the identity being studied with various applications. This group does not form a compositional unity - the exercises here are scattered on various topics.

The described cycle structures refer to the stage of developing skills for applying specific transformations.

At the stage of synthesis, the cycles change, groups of tasks are combined in the direction of complication and merging of cycles related to various identities, which helps to increase the role of actions to recognize the applicability of a particular identity.

Example.

Cycle of tasks for identity:

I group of tasks:

a) present in the form of a product:

b) Check the equality:

c) Expand the parentheses in the expression:

.

d) Calculate:

e) Factorize:

f) simplify the expression:

.

Students have just become familiar with the formulation of an identity, its writing in the form of an identity, and its proof.

Task a) is associated with fixing the structure of the identity being studied, with establishing a connection with numerical sets(comparison of sign structures of identity and transformed expression; replacement of a letter with a number in an identity). IN last example it is still necessary to reduce it to the species being studied. In the following examples (e and g) there is a complication caused by the applied role of identity and the complication of the sign structure.

Tasks of type b) are aimed at developing replacement skills on . The role of task c) is similar.

Examples of type d), in which it is necessary to choose one of the directions of transformation, complete the development of this idea.

Group I tasks are focused on mastering the structure of an identity, the operation of substitution in the simplest, fundamentally most important cases, and the idea of ​​​​the reversibility of transformations carried out by an identity. Very important has also enrichment linguistic means showing various aspects identities. The texts of the assignments give an idea of ​​these aspects.

II group of tasks.

g) Using the identity for , factor the polynomial .

h) Eliminate irrationality in the denominator of the fraction.

i) Prove that if - odd number, then divisible by 4.

j) The function is given analytical expression

.

Get rid of the modulus sign by considering two cases: , .

k) Solve the equation .

These tasks are aimed at as much as possible full use and taking into account the specifics of this particular identity, presuppose the formation of skills in using the identity being studied for the difference of squares. The goal is to deepen the understanding of identity by considering its various applications in different situations, combined with the use of material related to other topics in the mathematics course.

or .

Features of task cycles related to identities for elementary functions:

1) they are studied on the basis of functional material;

2) the identities of the first group appear later and are studied using already developed skills for carrying out identity transformations.

The first group of tasks in the cycle should include tasks to establish connections between these new numeric domains with the original domain of rational numbers.

Example.

Calculate:

;

.

The purpose of such tasks is to master the features of records, including symbols of new operations and functions, and to develop mathematical speech skills.

Much of the use of identity transformations associated with elementary functions, falls on the solution of irrational and transcendental equations. Sequence of steps:

a) find the function φ for which given equation f(x)=0 can be represented as:

b) substitute y=φ(x) and solve the equation

c) solve each of the equations φ(x)=y k, where y k is the set of roots of the equation F(y)=0.

When using the described method, step b) is often performed implicitly, without introducing a notation for φ(x). In addition, students often prefer different ways leading to finding the answer, choose the one that leads to the algebraic equation faster and easier.

Example. Solve the equation 4 x -3*2=0.

2)(2 2) x -3*2 x =0 (step a)

(2 x) 2 -3*2 x =0; 2 x (2 x -3)=0; 2 x -3=0. (step b)

Example. Solve the equation:

a) 2 2x -3*2 x +2=0;

b) 2 2x -3*2 x -4=0;

c) 2 2x -3*2 x +1=0.

(Suggest for independent solution.)

Classification of tasks in cycles related to the solution of transcendental equations, including exponential function:

1) equations that reduce to equations of the form a x =y 0 and have a simple, general answer:

2) equations that reduce to equations of the form a x = a k, where k is an integer, or a x = b, where b≤0.

3) equations that reduce to equations of the form a x =y 0 and require explicit analysis of the form in which the number y 0 is explicitly written.

Tasks in which identity transformations are used to construct graphs while simplifying formulas that define functions are of great benefit.

a) Graph the function y=;

b) Solve the equation lgx+lg(x-3)=1

c) on what set is the formula log(x-5)+ log(x+5)= log(x 2 -25) an identity?

The use of identity transformations in calculations. (Journal of Mathematics at School, No. 4, 1983, p. 45)

Task No. 1. The function is given by the formula y=0.3x 2 +4.64x-6. Find the values ​​of the function at x=1.2

y(1,2)=0.3*1.2 2 +4.64*1.2-6=1.2(0.3*1.2+4.64)-6=1.2(0 .36+4.64)-6=1.2*5-6=0.

Task No. 2. Calculate leg length right triangle, if the length of its hypotenuse is 3.6 cm, and the other leg is 2.16 cm.

Task No. 3. What is the area of ​​a rectangular plot having dimensions a) 0.64 m and 6.25 m; b) 99.8m and 2.6m?

a)0.64*6.25=0.8 2 *2.5 2 =(0.8*2.5) 2;

b)99.8*2.6=(100-0.2)2.6=100*2.6-0.2*2.6=260-0.52.

These examples make it possible to identify the practical application of identity transformations. The student should be familiarized with the conditions for the feasibility of the transformation (see diagrams).

image of a polynomial, where any polynomial fits into round contours. (Diagram 1)

-

the condition for the feasibility of transforming the product of a monomial and an expression that allows transformation into a difference of squares is given. (scheme 2)

-

here the shadings mean equal monomials and an expression is given that can be converted into a difference of squares. (Scheme 3)

an expression that allows for a common factor.

Students’ skills in identifying conditions can be developed using following examples:

Which of the following expressions can be transformed by taking the common factor out of brackets:

2)

3) 0.7a 2 +0.2b 2 ;

5) 6,3*0,4+3,4*6,3;

6) 2x 2 +3x 2 +5y 2 ;

7) 0,21+0,22+0,23.

Most calculations in practice do not satisfy the conditions of satisfiability, so students need the skills to reduce them to a form that allows calculation of transformations. In this case, the following tasks are appropriate:

when studying taking the common factor out of brackets:

this expression, if possible, convert into an expression, which is depicted in diagram 4:

4) 2a*a 2 *a 2;

5) 2n 4 +3n 6 +n 9 ;

8) 15ab 2 +5a 2 b;

10) 12,4*-1,24*0,7;

11) 4,9*3,5+1,7*10,5;

12) 10,8 2 -108;

13)

14) 5*2 2 +7*2 3 -11*2 4 ;

15) 2*3 4 -3*2 4 +6;

18) 3,2/0,7-1,8*

When forming the concept of “identical transformation”, it should be remembered that this means not only that the given and resulting expression as a result of the transformation take equal values for any values ​​of the letters included in it, but also that during an identical transformation we move from an expression that defines one method of calculation to an expression that defines another method of calculating the same value.

You can illustrate diagram 5 (the rule for converting the product of a monomial and a polynomial) with examples

0.5a(b+c) or 3.8(0.7+).

Exercises to learn how to take a common factor out of brackets:

Calculate the value of the expression:

a) 4.59*0.25+1.27*0.25+2.3-0.25;

b) a+bc at a=0.96; b=4.8; c=9.8.

c) a(a+c)-c(a+b) with a=1.4; b=2.8; c=5.2.

Let us illustrate with examples the formation of skills in calculations and identity transformations. (Journal of Mathematics at School, No. 5, 1984, p. 30)

1) skills and abilities are acquired faster and retained longer if their formation occurs on a conscious basis (the didactic principle of consciousness).

1) You can formulate a rule for adding fractions with same denominators or first, using specific examples, consider the essence of adding equal shares.

2) When factoring by taking the common factor out of brackets, it is important to see this common factor and then apply the distribution law. When performing the first exercises, it is useful to write each term of the polynomial as a product, one of the factors which is common for all terms:

3a 3 -15a 2 b+5ab 2 = a3a 2 -a15ab+a5b 2 .

It is especially useful to do this when one of the monomials of a polynomial is taken out of brackets:

II. First stage formation of a skill - mastery of a skill (exercises are performed with detailed explanations and records)

(the issue of the sign is resolved first)

Second stage– the stage of automating the skill by eliminating some intermediate operations

III. Strength of skills is achieved by solving examples that are varied in both content and form.

Topic: “Putting the common factor out of brackets.”

1. Write down the missing factor instead of the polynomial:

2. Factorize so that before the brackets there is a monomial with a negative coefficient:

3. Factor so that the polynomial in brackets has integer coefficients:

4. Solve the equation:

IV. Skill development is most effective when some intermediate calculations or transformations are performed orally.

(orally);

V. The skills and abilities being developed must be included in the previously formed system of knowledge, skills and abilities of students.

For example, when teaching how to factor polynomials using abbreviated multiplication formulas, the following exercises are offered:

Factorize:

VI. The need for rational execution of calculations and transformations.

V) simplify the expression:

Rationality lies in opening the parentheses, because

VII. Converting expressions containing exponents.

No. 1011 (Alg.9) Simplify the expression:

No. 1012 (Alg.9) Remove the multiplier from under the root sign:

No. 1013 (Alg.9) Enter a factor under the root sign:

No. 1014 (Alg.9) Simplify the expression:

In all examples, first perform either factorization, or subtraction of the common factor, or “see” corresponding formula abbreviations.

No. 1015 (Alg.9) Reduce the fraction:

Many students experience some difficulty in transforming expressions containing roots, in particular when studying equality:

Therefore, either describe in detail expressions of the form or or go to a degree with a rational exponent.

No. 1018 (Alg.9) Find the value of the expression:

No. 1019 (Alg.9) Simplify the expression:

2.285 (Skanavi) Simplify the expression

and then plot the function y For

No. 2.299 (Skanavi) Check the validity of the equality:

Transformation of expressions containing a degree is a generalization of acquired skills and abilities in the study of identical transformations of polynomials.

No. 2.320 (Skanavi) Simplify the expression:

The Algebra 7 course provides the following definitions.

Def. Two expressions whose corresponding values ​​are equal for the values ​​of the variables are said to be identically equal.

Def. Equality is true for any values ​​of the variables called. identity.

No. 94 (Alg.7) Is the equality:

a)

c)

d)

Description definition: Replacing one expression with another identically equal expression is called an identical transformation or simply a transformation of an expression. Identical transformations of expressions with variables are performed based on the properties of operations on numbers.

No. (Alg.7) Among the expressions

find those that are identically equal.

Topic: “Identical transformations of expressions” (question technique)

The first topic of “Algebra-7” - “Expressions and their transformations” helps to consolidate the computational skills acquired in grades 5-6, systematize and generalize information about transformations of expressions and solutions to equations.

Finding the values ​​of numeric and literal expressions makes it possible to repeat with students the rules of action with rational numbers. The ability to perform arithmetic operations with rational numbers is fundamental to the entire algebra course.

When considering transformations of expressions, formal and operational skills remain at the same level that was achieved in grades 5-6.

However, here students rise to a new level in mastering theory. The concepts of “identically equal expressions”, “identity”, “identical transformations of expressions” are introduced, the content of which will constantly be revealed and deepened when studying transformations of various algebraic expressions. It is emphasized that the basis of identity transformations is the properties of operations on numbers.

When studying the topic “Polynomials”, formal operational skills of identical transformations of algebraic expressions are formed. Abbreviated multiplication formulas contribute to the further process of developing the ability to perform identical transformations of whole expressions; the ability to apply formulas for both abbreviated multiplication and factorization of polynomials is used not only in transforming whole expressions, but also in operations with fractions, roots, powers with a rational exponent .

In the 8th grade, the acquired skills of identity transformations are practiced in actions with algebraic fractions, square root and expressions containing powers with an integer exponent.

In the future, the techniques of identity transformations are reflected in expressions containing a degree with a rational exponent.

Special group identical transformations are trigonometric expressions and logarithmic expressions.

TO mandatory results Tuition for algebra courses in grades 7-9 include:

1) identity transformations of integer expressions

a) opening and enclosing brackets;

b) reduction similar members;

c) addition, subtraction and multiplication of polynomials;

d) factoring polynomials by putting the common factor out of brackets and abbreviated multiplication formulas;

e) factorization of a quadratic trinomial.

“Mathematics at school” (B.U.M.) p.110

2) identity transformations rational expressions: addition, subtraction, multiplication and division of fractions, as well as apply the listed skills when performing simple combined transformations [p. 111]

3) students should be able to perform transformations of simple expressions containing powers and roots. (pp. 111-112)

The main types of problems were considered, the ability to solve which allows the student to receive a positive grade.

One of the most important aspects of the methodology for studying identity transformations is the student’s development of goals for performing identity transformations.

1) - simplification numerical value expressions

2) which of the transformations should be performed: (1) or (2) Analysis of these options is a motivation (preferable (1), since in (2) the scope of definition is narrowed)

3) Solve the equation:

Factoring when solving equations.

4) Calculate:

Let's apply the abbreviated multiplication formula:

(101-1) (101+1)=100102=102000

5) Find the value of the expression:

To find the value, multiply each fraction by its conjugate:

6) Graph the function:

Let's select the whole part: .

Prevention of errors when performing identity transformations can be obtained by varying examples of their implementation. In this case, “small” techniques are practiced, which, as components, are included in a larger transformation process.

For example:

Depending on the directions of the equation, several problems can be considered: multiplication of polynomials from right to left; from left to right - factorization. Left side is a multiple of one of the factors on the right side, etc.

In addition to varying the examples, you can use apologia between identities and numerical equalities.

The next technique is the explanation of identities.

To increase students' interest, we can include finding in various ways problem solving.

Lessons on studying identity transformations will become more interesting if you devote them to searching for a solution to the problem .

For example: 1) reduce the fraction:

3) prove the formula “ complex radical»

Consider:

Let's transform right side equality:

-

the sum of conjugate expressions. They could be multiplied and divided by their conjugate, but such an operation would lead us to a fraction whose denominator is the difference of the radicals.

Note that the first term in the first part of the identity is a number greater than the second, so we can square both parts:

Practical lesson №3.

Topic: Identical transformations of expressions (question technique).

Literature: “Workshop on MPM”, pp. 87-93.

Sign high culture calculations and identity transformations, students have a solid knowledge of the properties and algorithms of operations on exact and approximate quantities and their skillful application; rational techniques calculations and transformations and their verification; ability to justify the use of methods and rules of calculations and transformations, automaticity of skills error-free execution computing operations.

At what grade should students begin working on developing the listed skills?

The line of identical transformations of expressions begins with the use of techniques rational calculation begins with the use of techniques for rationally calculating the values ​​of numerical expressions. (5th grade)

When studying such topics school course mathematics should be given to them special attention!

Students’ conscious implementation of identity transformations is facilitated by understanding the fact that algebraic expressions do not exist on their own, but in unbreakable connection with some numerical set, are generalized records of numerical expressions. Analogies between algebraic and numerical expressions (and their transformations) are logical; their use in teaching helps prevent students from making mistakes.

Identity transformations are not any a separate topic school mathematics course, they are studied throughout the course of algebra and the beginnings of mathematical analysis.

The mathematics program for grades 1-5 is propaedeutic material for studying identical transformations of expressions with a variable.

In the 7th grade algebra course. the definition of identity and identity transformations is introduced.

Def. Two expressions whose corresponding values ​​are equal for any values ​​of the variables are called. identically equal.

ODA. An equality that is true for any values ​​of the variables is called an identity.

The value of identity lies in the fact that it allows a given expression to be replaced by another that is identically equal to it.

Def. Replacing one expression with another identically equal expression is called identical transformation or just transformation expressions.

Identical transformations of expressions with variables are performed based on the properties of operations on numbers.

The basis of identity transformations can be considered equivalent transformations.

ODA. Two sentences, each of which is a logical consequence of the other, are called. equivalent.

ODA. Sentence with variables A is called. consequence of a sentence with variables B, if the domain of truth B is a subset of the domain of truth A.

Another definition of equivalent sentences can be given: two sentences with variables are equivalent if their truth domains coincide.

a) B: x-1=0 over R; A: (x-1) 2 over R => A~B, because areas of truth (solution) coincide (x=1)

b) A: x=2 over R; B: x 2 =4 over R => domain of truth A: x = 2; truth domain B: x=-2, x=2; because the domain of truth of A is contained in B, then: x 2 =4 is a consequence of the proposition x = 2.

The basis of identity transformations is the ability to represent the same number in different forms. For example,

-

This representation will help when studying the topic “basic properties of fractions.”

Skills in performing identity transformations begin to develop when solving examples similar to the following: “Find the numerical value of the expression 2a 3 +3ab+b 2 with a = 0.5, b = 2/3,” which are offered to students in grade 5 and allow for propaedeutics concept of function.

When studying abbreviated multiplication formulas, you should pay attention to their deep understanding and strong assimilation. To do this, you can use the following graphic illustration:

(a+b) 2 =a 2 +2ab+b 2 (a-b) 2 =a 2 -2ab+b 2 a 2 -b 2 =(a-b)(a+b)

Question: How to explain to students the essence of the given formulas based on these drawings?

A common mistake is to confuse the expressions “square of the sum” and “sum of squares.” The teacher's indication that these expressions differ in the order of operation does not seem significant, since students believe that these actions are performed on the same numbers and therefore the result does not change by changing the order of actions.

Assignment: Create oral exercises to develop students’ skills in using the above formulas without errors. How can we explain how these two expressions are similar and how they differ from each other?

The wide variety of identical transformations makes it difficult for students to orient themselves as to the purpose for which they are performed. Fuzzy knowledge of the purpose of carrying out transformations (in each specific case) negatively affects their awareness and serves as a source massive errors students. This suggests that explaining to students the goals of performing various identity transformations is important. integral part methods for studying them.

Examples of motivations for identity transformations:

1. simplification of location numerical value expressions;

2. choosing a transformation of the equation that does not lead to the loss of the root;

3. When performing a transformation, you can mark its calculation area;

4. use of transformations in calculations, for example, 99 2 -1=(99-1)(99+1);

To manage the decision process, it is important for the teacher to have the ability to give an accurate description of the essence of the mistake made by the student. Accurate error characterization is key to the right choice subsequent actions taken by the teacher.

Examples of student errors:

1. performing multiplication: the student received -54abx 6 (7 cells);

2. By raising to a power (3x 2) 3 the student received 3x 6 (7 grades);

3. transforming (m + n) 2 into a polynomial, the student received m 2 + n 2 (7th grade);

4. By reducing the fraction the student received (8 grades);

5. performing subtraction: , student writes down (8th grade)

6. Representing the fraction in the form of fractions, the student received: (8 grades);

7. Removing arithmetic root the student received x-1 (grade 9);

8. solving the equation (9th grade);

9. By transforming the expression, the student receives: (9th grade).

Conclusion

The study of identity transformations is carried out in close connection with numerical sets studied in a particular class.

At first, you should ask the student to explain each step of the transformation, to formulate the rules and laws that apply.

In identical transformations of algebraic expressions, two rules are used: substitution and replacement by equals. Substitution is most often used, because It is based on calculation formulas, i.e. find the value of the expression a*b with a=5 and b=-3. Very often, students neglect parentheses when performing multiplication operations, believing that the multiplication sign is implied. For example, the following entry is possible: 5*-3.

Literature

1. A.I. Azarov, S.A. Barvenov “Functional and graphic methods solving exam problems”, Mn..Aversev, 2004

2. O.N. Piryutko " Common mistakes on centralized testing", Mn..Aversev, 2006

3. A.I. Azarov, S.A. Barvenov “Trap tasks in centralized testing”, Mn..Aversev, 2006

4. A.I. Azarov, S.A. Barvenov “Methods of solution trigonometric problems", Mn..Aversev, 2005

Lesson type: lesson of generalization and systematization of knowledge.

Lesson objectives:

• Improve the ability to apply previously acquired knowledge to prepare for the State Examination in 9th grade.
• Teach the ability to analyze and approach a task creatively.
• To cultivate a culture and efficiency of thinking, cognitive interest to mathematics.
• Help students prepare for the State Examination.

• Systematize theoretical knowledge students.
• Strengthen the practical orientation of this topic in preparation for the State Examination.
• Build mental skills - search rational ways solutions.

Equipment: multimedia projector, worksheet, clock.

Lesson plan: 1. Organizational moment.

1. Updating knowledge.
2. Development of theoretical material.
3. Lesson summary.
4. Homework.

1) Greeting from the teacher.

Cryptography is the science of ways to transform (encrypt) information in order to protect it from illegal users. One of these methods is called “grid”. It is one of the relatively simple ones and is closely related to arithmetic, but one that is not studied in school. A sample of the lattice is in front of you. Someone will figure out how to use it.

- the solution to the message.

“Everything that stops working out stops attracting.”

Francois Larachefoucauld.

2) Messages about the topic of the lesson, lesson objectives, lesson plan.

– slides in the presentation.

II. Updating knowledge.

1) Oral work.

1. Numbers. What numbers do you know?

– natural numbers are numbers 1,2,3,4... which are used when counting

– integers are numbers…-4,-3,-2,-1,0,1, 2… natural numbers, their opposites and the number 0.

– rational numbers are whole and fractional numbers

– irrational – these are infinite decimal non-periodic fractions

– real – these are rational and irrational.

2. Expressions. What expressions do you know?

– numerical are expressions consisting of numbers connected by arithmetic symbols.

– alphabetic – this is an expression containing some variables, numbers and action signs.

– Integers are expressions consisting of numbers and variables using the operations of addition, subtraction, multiplication and division by a number.

– fractional ones are whole expressions using division by an expression with a variable.

3. Transformations. What are the main properties used when performing transformations?

– commutative – for any numbers a and b it is true: a+b=b+a, ab=va

– associative – for any numbers a, b, c, the following is true: (a+b)+c=a+(b+c), (ab)c=a(c)

– distributive – for any numbers a, b, c it is true: a(b+c)=av+ac

4. Do:

– arrange the numbers in ascending order: 0.0157; 0.105; 0.07

– arrange the numbers in descending order: 0.0216; 0.12; 0.016

– one of the points marked on the coordinate line corresponds to the number v68. What point is this?

– what point do the numbers correspond to?

– the numbers a and b are marked on the coordinate line. Which one the following statements is correct?

III. Development of theoretical material.

1. Work in notebooks, at the board.

Each teacher has a worksheet where tasks are written down for work in notebooks during the lesson. In the right column of this sheet there are assignments for work in class, and in the left column there is homework.

Students come out to work at the board.

Task No. 1. In which case is the expression converted to identically equal.

Task No. 3. Factor it out:

a 3 – av – a 2 c + a 2; x 2 y – x 2 -y + x 3.

2x+ y + y 2 – 4x 2; a – 3c +9c 2 -a 2 .

2. Independent work.

On the worksheets you have independent work, below after the text there is a table in which you enter the number under the correct answer. It takes 7 minutes to complete the job.

Test “Numbers and Conversions”

1. Write 0.00019 in standard form.

1)0,019*10 -2 ; 2)0,19*10 -3 ; 3)1,9*10 -4 ; 4)19*10 -5

2. One of the points marked on the coordinate line corresponds to the number

3. About numbers a and b it is known that a>0, b>0, a>4b. Which one the following inequalities wrong?

1) a-2a>-3b; 2) 2a>8b; 3) a/4>b-2; 4) a+3>b+1.

4.Find the value of the expression: (6x – 5y): (3x+y), if x=1.5 and y=0.5.

1) 1,5; 2) 1,3; 3) 1,33; 4) 2,5.

5.Which of the following expressions can be converted into (7 – x)(x – 4)?

1)– (7 – x)(4 – x); 2) (7 – x)(4 – x);

3) – (x – 7)(4 – x); 4) (x – 7)(x-4).

6. Lesson summary.

Today in class you solved tasks selected from collections to prepare for the State Examination. This is a small part of what you need to repeat to pass the exam perfectly.

- The lesson is over. What did you find useful from the lesson?

“An expert is a person who no longer thinks, he knows.” Frank Hubbard.

7. Homework

On the sheets of paper are tasks to complete at home.

Numerical and algebraic expressions. Converting Expressions.

What is an expression in mathematics? Why do we need expression conversions?

The question, as they say, is interesting... The fact is that these concepts are the basis of all mathematics. All mathematics consists of expressions and their transformations. Not very clear? Let me explain.

Let's say you have an evil example in front of you. Very big and very complex. Let's say you're good at math and aren't afraid of anything! Can you give an answer right away?

You will have to decide this example. Consistently, step by step, this example simplify. By certain rules, naturally. Those. do expression conversion. The more successfully you carry out these transformations, the stronger you are in mathematics. If you don't know how to do the right transformations, you won't be able to do them in math. Nothing...

To avoid such an uncomfortable future (or present...), it doesn’t hurt to understand this topic.)

First, let's find out what is an expression in mathematics. What's happened numeric expression and what is algebraic expression.

What is an expression in mathematics?

Expression in mathematics- this is very broad concept. Almost everything we deal with in mathematics is a set of mathematical expressions. Any examples, formulas, fractions, equations, and so on - it all consists of mathematical expressions.

3+2 is a mathematical expression. c 2 - d 2- this is also a mathematical expression. And a healthy fraction, and even one number - that’s all mathematical expressions. For example, the equation is:

5x + 2 = 12

consists of two mathematical expressions connected by an equal sign. One expression is on the left, the other on the right.

IN general view term " mathematical expression"is used, most often, to avoid mooing. They will ask you what an ordinary fraction is, for example? And how to answer?!

First answer: "This is... mmmmmm... such a thing... in which... Can I write a fraction better? Which one do you want?"

Second answer: " Common fraction- this is (cheerfully and joyfully!) mathematical expression , which consists of a numerator and a denominator!"

The second option will be somehow more impressive, right?)

This is the purpose of the phrase " mathematical expression "very good. Both correct and solid. But for practical application need to be well versed in specific types expressions in mathematics .

The specific type is another matter. This It's a completely different matter! Each type of mathematical expression has mine a set of rules and techniques that must be used when making a decision. For working with fractions - one set. For working with trigonometric expressions - the second one. For working with logarithms - the third. And so on. Somewhere these rules coincide, somewhere they differ sharply. But don't be scared by these scary words. We will master logarithms, trigonometry and other mysterious things in the appropriate sections.

Here we will master (or - repeat, depending on who...) two main types of mathematical expressions. Numerical expressions and algebraic expressions.

Numeric expressions.

What's happened numeric expression? This is a very simple concept. The name itself hints that this is an expression with numbers. Yes, that's how it is. A mathematical expression made up of numbers, brackets and arithmetic symbols is called a numerical expression.

7-3 is a numerical expression.

(8+3.2) 5.4 is also a numerical expression.

And this monster:

also a numerical expression, yes...

Regular number, fraction, any example of calculation without X's and other letters - all these are numerical expressions.

Main sign numerical expressions - in it no letters. None. Only numbers and mathematical symbols (if necessary). It's simple, right?

And what can you do with numerical expressions? Numeric expressions can usually be counted. To do this, it happens that you have to open the brackets, change signs, abbreviate, swap terms - i.e. do expression conversions. But more on that below.

Here we will deal with such a funny case when with a numerical expression you don't need to do anything. Well, nothing at all! This pleasant operation - do nothing)- is executed when the expression doesn't make sense.

When does a numerical expression make no sense?

It’s clear that if we see some kind of abracadabra in front of us, like

then we won’t do anything. Because it’s not clear what to do about it. Some kind of nonsense. Maybe count the number of pluses...

But there are outwardly quite decent expressions. For example this:

(2+3) : (16 - 2 8)

However, this expression also doesn't make sense! For the simple reason that in the second brackets - if you count - you get zero. But you can’t divide by zero! This is a forbidden operation in mathematics. Therefore, there is no need to do anything with this expression either. For any task with such an expression, the answer will always be the same: "The expression has no meaning!"

To give such an answer, of course, I had to calculate what would be in brackets. And sometimes there’s a lot of stuff in parentheses... Well, there’s nothing you can do about it.

There are not so many forbidden operations in mathematics. There is only one in this topic. Division by zero. Additional restrictions arising in roots and logarithms are discussed in the corresponding topics.

So, an idea of ​​what it is numeric expression- received. Concept the numeric expression doesn't make sense- realized. Let's move on.

Algebraic expressions.

If letters appear in a numerical expression, this expression becomes... The expression becomes... Yes! It becomes algebraic expression. For example:

5a 2; 3x-2y; 3(z-2); 3.4m/n; x 2 +4x-4; (a+b) 2; ...

Such expressions are also called literal expressions. Or expressions with variables. It's practically the same thing. Expression 5a +c, for example - both literal and algebraic, and an expression with variables.

Concept algebraic expression - wider than numeric. It includes and all numerical expressions. Those. a numerical expression is also an algebraic expression, only without letters. Every herring is a fish, but not every fish is a herring...)

Why alphabetic- It's clear. Well, since there are letters... Phrase expression with variables It’s also not very puzzling. If you understand that numbers are hidden under the letters. All sorts of numbers can be hidden under letters... And 5, and -18, and anything else. That is, a letter can be replace on different numbers. That's why the letters are called variables.

In expression y+5, For example, at - variable quantity. Or they just say " variable", without the word "magnitude". Unlike five, which is a constant value. Or simply - constant.

Term algebraic expression means that to work with this expression you need to use laws and rules algebra. If arithmetic works with specific numbers, then algebra- with all the numbers at once. A simple example for clarification.

In arithmetic we can write that

But if we write such an equality through algebraic expressions:

a + b = b + a

we'll decide right away All questions. For all numbers in one fell swoop. For everything infinite number. Because under the letters A And b implied All numbers. And not only numbers, but even other mathematical expressions. This is how algebra works.

When does an algebraic expression not make sense?

Everything about the numerical expression is clear. You can't divide by zero there. And with letters, is it possible to find out what we are dividing by?!

Let's take for example this expression with variables:

2: (A - 5)

Does it make sense? Who knows? A- any number...

Any, any... But there is one meaning A, for which this expression exactly doesn't make sense! And what is this number? Yes! This is 5! If the variable A replace (they say “substitute”) with the number 5, in brackets you get zero. Which cannot be divided. So it turns out that our expression doesn't make sense, If a = 5. But for other values A does it make sense? Can you substitute other numbers?

Certainly. In such cases they simply say that the expression

2: (A - 5)

makes sense for any values A, except a = 5 .

The whole set of numbers that Can substituting into a given expression is called region acceptable values this expression.

As you can see, there is nothing tricky. Let's look at the expression with variables and figure out: at what value of the variable is the forbidden operation (division by zero) obtained?

And then be sure to look at the task question. What are they asking?

doesn't make sense, our forbidden meaning will be the answer.

If you ask at what value variable expression makes sense(feel the difference!), the answer will be all other numbers except for the forbidden.

Why do we need the meaning of the expression? He is there, he is not... What's the difference?! The point is that this concept becomes very important in high school. Extremely important! This is the basis for such solid concepts as the domain of acceptable values ​​or the domain of a function. Without this, you will not be able to solve serious equations or inequalities at all. Like this.

Converting Expressions. Identity transformations.

We were introduced to numerical and algebraic expressions. We understood what the phrase “the expression has no meaning” means. Now we need to figure out what it is expression conversion. The answer is simple, to the point of disgrace.) This is any action with an expression. That's all. You have been doing these transformations since first grade.

Let's take the cool numerical expression 3+5. How can it be converted? Yes, very simple! Calculate:

This calculation will be the transformation of the expression. You can write the same expression differently:

Here we didn’t count anything at all. Just wrote down the expression in a different form. This will also be a transformation of the expression. You can write it like this:

And this too is a transformation of an expression. You can make as many such transformations as you want.

Any action on expression any writing it in another form is called transforming the expression. And that's all. It's very simple. But there is one thing here very important rule. So important that it can safely be called main rule all mathematics. Breaking this rule inevitably leads to errors. Are we getting into it?)

Let's say we transformed our expression haphazardly, like this:

Conversion? Certainly. We wrote the expression in a different form, what’s wrong here?

It's not like that.) The point is that transformations "at random" are not interested in mathematics at all.) All mathematics is built on transformations in which appearance, but the essence of the expression does not change. Three plus five can be written in any form, but it must be eight.

Transformations, expressions that do not change the essence are called identical.

Exactly identity transformations and allow us, step by step, to transform complex example into a simple expression, keeping the essence of the example. If we make a mistake in the chain of transformations, we make a NOT identical transformation, then we will decide another example. With other answers that are not related to the correct ones.)

This is the main rule for solving any tasks: maintaining the identity of transformations.

Example with numerical expression I brought 3+5 for clarity. In algebraic expressions, identity transformations are given by formulas and rules. Let's say in algebra there is a formula:

a(b+c) = ab + ac

This means that in any example we can instead of the expression a(b+c) feel free to write an expression ab + ac. And vice versa. This identical transformation. Mathematics gives us a choice between these two expressions. And which one to write - from concrete example depends.

Another example. One of the most important and necessary transformations is the basic property of a fraction. You can see more details at the link, but here I’ll just remind you of the rule: If the numerator and denominator of a fraction are multiplied (divided) by the same number, or an expression that is not equal to zero, the fraction will not change. Here is an example of identity transformations using this property:

As you probably guessed, this chain can be continued indefinitely...) Very important property. It is this that allows you to turn all sorts of example monsters into white and fluffy.)

There are many formulas defining identical transformations. But the most important ones are quite a reasonable number. One of the basic transformations is factorization. It is used in all mathematics - from elementary to advanced. Let's start with him. In the next lesson.)

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

Lesson motto: m

Lesson type:

Goals:

Tasks:

Lesson progress

1. Organizational moment.

Who doesn't notice anything

He doesn't study anything

Who doesn't study anything

He's always whining and bored.

2.

(numeric and alphabetic)

3. .Updating knowledge.

1)Rules for opening brackets.

2)1. The rule for multiplying a monomial by a polynomial.

Find and fix the error:

( )

Find and fix the error:

( )

3)

Tasks Answers

4) Factorization.

B) method of grouping;

PHYSICAL MINUTE!!!

a) Reducing a fraction

b) Sum and difference of fractions.

4. Fixing the material.

Exercise.

5. Results. Reflection.

6. Homework.

View document contents
"Repetition: Expressions and Their Transformations"

Topic: “Repetition: expressions and their transformations”

Lesson motto: m You can’t learn mathematics by watching your neighbor do it.

Lesson type: consolidation and generalization of the studied material.

Goals: a) systematize the knowledge of students for the algebra course grades 7-9, generalize their knowledge and skills on this topic, remember and consolidate methods of working with algebraic expressions: rules for opening parentheses, rules for multiplying a monomial by a polynomial and a polynomial by a polynomial, abbreviated multiplication formulas, decomposition polynomial into factors, actions on rational fractions;

b) education of learning motives, positive attitude to knowledge, discipline;

c) development of analytical and synthesizing thinking, skills to apply knowledge in practice, accuracy, precision in performing actions, and independence.

Tasks: remember and apply when solving training exercises the above rules for working with algebraic expressions.

Lesson progress

Organizational moment.

The poet Roman Sef wrote jokingly:

Who doesn't notice anything

He doesn't study anything

Who doesn't study anything

He's always whining and bored.

We won't be bored today. Do you agree? Write down the date in your notebooks, great job and the topic of the lesson “Expressions and their transformations.”

Setting goals and objectives for the lesson.

Look carefully at the topic of the lesson.

What types of expressions do you know? (numeric and alphabetic)

What transformations are you familiar with? (rules for opening parentheses, rules for multiplying a monomial by a polynomial and a polynomial by a polynomial, abbreviated multiplication formulas, factoring a polynomial, operations on rational fractions)

So what is the purpose of our work today? ( remember and consolidate methods of working with algebraic expressions)

Thus, we will systematize and generalize the knowledge and skills on this topic for the 7-9 grade algebra course as a whole.

Repetition educational material .Updating knowledge.

1) Rules for opening brackets.

One type of expression transformation is the expansion of parentheses. It can be convenient to move from an expression with parentheses to identically equal to the expression, which no longer contains these parentheses.

Please formulate a rule for opening brackets preceded by a “+” sign: If there is a “+” sign in front of the brackets, then you can omit the brackets and this “+” sign, preserving the signs of the terms in the brackets.

Now formulate the rule for opening parentheses preceded by a “−” sign: if the parentheses are preceded by a “−” sign, then the parentheses are omitted, and the terms in the brackets change their sign to the opposite.

2) 1. Rule for multiplying a monomial by a polynomial.

Let's remember the rule for multiplying a monomial by a polynomial: To multiply a monomial by a polynomial, you need to multiply this monomial by each term of the polynomial and add the resulting products.

Find and fix the error:

()

2. The rule for multiplying a polynomial by a polynomial.

Please remind us of the rule for multiplying a polynomial by a polynomial: To multiply a polynomial by a polynomial, you need to multiply each term of one polynomial by each term of another polynomial and add the resulting products.

Find and fix the error:

()

3) Abbreviated multiplication formulas.

It's time to remember the abbreviated multiplication formulas. Fill in the blanks in the formulas.

Now let's complete the next task. Connect the tasks and answers with lines.

Tasks Answers

4) 4)

6) 6)

7) 7)

Key: 1-2; 2-4; 3-3; 4-6; 5-7; 6-5; 7-1.

If you did it correctly, then put “+”; if you made a mistake, then put “-” and correct the mistake.

Raise your hand if you did everything right. Where were the mistakes made?

4) Factorization.

Look carefully at the examples written on the board. Answer the question: what do the examples below have in common?

Answer: the answers result in works.

So what is factorization?

Answer: Representing a polynomial as a product of two or more polynomials is called factorization.

Based on these examples, name methods for factoring a polynomial:

A) placing the common factor out of brackets;

B) method of grouping;

C) using abbreviated multiplication formulas;

D) factorization formula quadratic trinomial.

PHYSICAL MINUTE!!!

5) Actions on rational fractions.

And now I propose to play mathematical lotto. We work in pairs. You need to select and combine a rule and an example corresponding to it.

a) Reducing a fraction

b) Sum and difference of fractions.

c) Product and quotient of fractions.

Let's check it as follows. I show an example, and you voice the corresponding rule.

So we repeated theoretical material and move on to the practical part.

Fixing the material.

Exercise. Insert the following monomials or signs in place of the gaps so that the resulting equality is an identity:

Results. Reflection.

As Evgeniy Domansky says: “Those who have managed to reflect on reality receive advantages in moving forward.” Therefore, we will also conduct a reflection.

Let's go back to the beginning of our lesson. Look at the purpose of the lesson. Have we achieved it? We achieved it because...

Homework.

Please open your diaries and write down homework:

B 69, 70 (9) (collection exam tasks)

Exercise. Consider the solution to the example and find errors:

The right decision write on the board: