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'll 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 a 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. s 2 - d 2- this is also a mathematical expression. Both a healthy fraction and even one number are 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 humming. 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 use you need to have a good understanding of specific types of 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 afraid of 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. That is 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 math icons(if it's 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 - To 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- got. 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 - broader 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 value. 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 stroke. For everything infinite. 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. We 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 of a variable the expression has the meaning(feel the difference!), the answer will be all other numbers except for what is 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 transformation of expressions. 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. Everything is 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.
I gave an example with the numerical expression 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 depends on the specific example.
Another example. One of the most important and necessary transformations is the basic property of a fraction. You can look at the link for more details, 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...) A 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.
The numbers and expressions that make up the original expression can be replaced by identically equal expressions. Such a transformation of the original expression leads to an expression that is identically equal to it.
For example, in the expression 3+x, the number 3 can be replaced by the sum 1+2, which will result in the expression (1+2)+x, which is identically equal to the original expression. Another example: in the expression 1+a 5, the power a 5 can be replaced by an identically equal product, for example, of the form a·a 4. This will give us the expression 1+a·a 4 .
This transformation is undoubtedly artificial, and is usually a preparation for some further transformations. For example, in the sum 4 x 3 +2 x 2, taking into account the properties of the degree, the term 4 x 3 can be represented as a product 2 x 2 2 x. After this transformation, the original expression will take the form 2 x 2 2 x+2 x 2. Obviously, the terms in the resulting sum have a common factor of 2 x 2, so we can perform the following transformation - bracketing. After it we come to the expression: 2 x 2 (2 x+1) .
Adding and subtracting the same number
Another artificial transformation of an expression is the addition and simultaneous subtraction of the same number or expression. This transformation is identical because it is essentially equivalent to adding zero, and adding zero does not change the value.
Let's look at an example. Let's take the expression x 2 +2·x. If you add one to it and subtract one, this will allow you to perform another identical transformation in the future - square the binomial: x 2 +2 x=x 2 +2 x+1−1=(x+1) 2 −1.
Bibliography.
- Algebra: textbook for 7th grade general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 17th ed. - M.: Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.
- Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
- Mordkovich A. G. Algebra. 7th grade. At 2 p.m. Part 1. Textbook for students educational institutions/ A. G. Mordkovich. - 17th ed., add. - M.: Mnemosyne, 2013. - 175 p.: ill. ISBN 978-5-346-02432-3.
Ministry of Education of the Republic of Belarus
Educational institution
"Gomel State University them. F. Skorina"
Faculty of Mathematics
Department of MPM
Identical transformations of expressions and methods of teaching students how to perform them
Executor:
Student Starodubova A.Yu.
Scientific adviser:
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 primary 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 of an integral system of transformations (synthesis)
The goal is to create a flexible and powerful apparatus suitable for use in solving a variety of educational tasks. 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; for certain types of transformations, transformations of trigonometric expressions are added 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 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 the last example, we still have to reduce it to the form 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. The enrichment of linguistic means showing various aspects of identity is also very important. 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 is an odd number, then it is divisible by 4.
j) The function is given by an 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 numerical areas and the original area 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.
A significant part 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 the 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 the length of one leg of a 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 practical use 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 the 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:
convert this expression, if possible, into an expression that 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 the resulting expression as a result of the transformation take on equal values for any values of the letters included in it, but also that during the identical transformation we move from the expression that defines one way of calculating to an expression defining another way of calculating the same value.
Scheme 5 (the rule for converting the product of a monomial and a polynomial) can be illustrated 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.(zh. 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 like denominators or preliminarily 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 of which is common to 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 skill formation – mastery of a skill (exercises are performed with detailed explanations and notes)
(the issue of the sign is resolved first)
Second phase– 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 part of 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” the corresponding reduction formula.
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 operation with rational numbers. Ability to perform arithmetic operations with rational numbers are the basis for 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 on operations 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.
Mandatory learning outcomes for an algebra course in grades 7-9 include:
1) identity transformations of integer expressions
a) opening and enclosing brackets;
b) bringing 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) decomposition quadratic trinomial by multipliers.
“Mathematics at school” (B.U.M.) p.110
2) identical transformations of 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 degrees 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 of the numerical value of the expression
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. The 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.
Next appointment– 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 of the “complex radical”
Consider:
Let's transform the right side of the equality:
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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.
A sign of a high culture of calculations and identity transformations among students is a strong knowledge of the properties and algorithms of operations on exact and approximate quantities and their skillful application; rational methods of calculations and transformations and their verification; the ability to justify the use of methods and rules of calculations and transformations, automatic skills of error-free execution of computational operations.
At what grade should students begin working on developing the listed skills?
The line of identical transformations of expressions begins with the application of rational calculation techniques. It begins with the application of rational calculation techniques for the values of numerical expressions. (5th grade)
When studying such topics in a school mathematics course, you need to pay attention to them. Special attention!
Students' conscious implementation of identity transformations is facilitated by the understanding of the fact that algebraic expressions do not exist on their own, but in inextricable connection with a certain numerical set, they 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 simply 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,
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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) has a negative impact on their awareness and serves as a source of massive errors among students. This suggests that explaining to students the goals of performing various identical transformations is an important part of the methodology for studying them.
Examples of motivations for identity transformations:
1. simplification of finding the numerical value of an expression;
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 Calculation using formulas is based on it, 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 graphical methods for solving examination problems”, Mn..Aversev, 2004
2. O.N. Piryutko “Typical errors in 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 for solving trigonometric problems”, Mn..Aversev, 2005
