How to prove that equality is not an identity. Identity

Teacher: Afonasova Irina Olegovna

Subject: algebra

Grade: Grade 7

Lesson type: learning new material

Topic: Proof of identities

Lesson Objectives:

1. Repeat the definitions of identity and identically equal expressions, identical transformation of expressions.
2. Formation of the skill of choosing a method for proving identities by the method of identical transformation of expressions.
3. Bring up communicative culture students.

During the classes

1 . Organizational stage of the lesson

Before the lesson begins, the students in the class are divided into six groups. study groups mixed composition.

Teacher : Hello guys, I suggest study room turn for a while intoresearch laboratory, and you and I in scientists-masters of mathematical sciences.

But every self-respecting scientist is constantly solving some very important problem, so we, first of all, have to find out: what problem are we going to work on today?

2. Determining the topic of the lesson

To do this, consider the expressions 2x+y and 2xy. Let's find the values ​​of expressions at x=1 and y=2.

teacher l offers to go to the board student and solve this problem, andformulate a conclusion: for x=1 and y=2, the expressions take equal values (4).

Teacher: However, you can specify such values ​​of the variables x and y, in which the values ​​of these expressions are not equal. For example, x=3, y=4.

Student standing at the blackboard checks it.

Teacher: Consider now the expressions 3(x+y) and 3x+3y. Let's find the values ​​of expressions at x=5 and y=4.

Student, standing at the blackboard: solves a problem, formulates a conclusion.

Teacher: For any values variable values these expressions are equal? If yes, why?

Student answers. (Answer: Yes, by the distributive property of multiplication).

The teacher invites the class to remember the name of such expressions, the name of their equality.

After that Slide 1.

Then the teacher asks: “What is the topic of today’s lesson?”

Teacher : Today we will work on the “Proof of Identities”.

The topic of the lesson is recorded: "Proof of identities" ( slide2)

Teacher : Okay, now let's check ourselves. Equalities will appear on the screen, if this equality is an identity, then I suggest you raise your hand. ( slide 3)

1. - (a - c) \u003d - a + c (yes)
2. a (b + c) \u003d av - ac (no)
3. a - (c + c) \u003d a - c + c(No)
4. (a + c) - c \u003d a - c + c(Yes)
5. - (a + c) \u003d - a - c (yes)

3. Determining the purpose of the lesson

Teacher : Well, now it's time for us to turn from theorists into practical scientists, but for this we need to find out what needs to be used in order toprove identity, and here we cannot do without scientific literature, we will find the answer to this question on page 18 of your textbook. Students find the answer in the textbook:"To prove that some equality is an identity, use identical transformations of expressions". Members of other groups show agreement or disagreement with special signals, which were mentioned above. ( slide 4)

Teacher : Well done, but now the next question arises, what isidentity conversion of expressions?

"The replacement of one expression by another identically equal to it is called the identical transformation of the expression"(the teacher invites one of the participants of any group to answer this question) ( slide 5)

Teacher : So, what is the purpose of the lesson? Students name one of the set goals: to learn to prove identities using identical transformations of expressions.

4. Identification of a method for proving identities by the method of identical transformation of expressions

Teacher: Now we are already “ripe” for practical work and I ask you to turn your attention to card . Assignment: "Prove the identity", each group of scientists received an example that they must solve on their own, if there are difficulties, consulting cards will come to the rescue.

Task cards

Card 1

Card 2

Card 3

Card 4

Card 5

Card 6

Now we need to protect our work. (Presentation of completed works at the blackboard, willing group members speak)

Teacher : Great, and now, dear colleagues, it's time to sum up, what do we need to do to prove that equality is an identity? Estimated student responses: ( slide 6)

1. write out left side equality, convert it and make sure it is right.
   or
2. Write out the right side of the equation, transform it and make sure that it is equal to the left side.
   or
3. Convert both the left and right side of the equality and make sure they are equal to the same expression.

Teacher : What conclusion can be drawn in the case when everything that we just said will not be fulfilled? Suggested student response:Equality will not be identical.

5. Summing up the lesson.

Have we achieved our goal? ….

Teacher : In order for the knowledge gained to be strong, we will continue this work at home:Homework(Slide 7):

No. 691 (a), 692 (a), 715 (a), creative task (optional): * Make 3 equalities that will be an identity (illustrate each method of proof).

Teacher : And now it's time for creativity: In the poem you see, insert the missing words ( Slide 8):

There are all sorts of equalities, brothers,
And everyone knows about it, of course.
Yes - with variables, yes - (numeric),
Complex very, very (simple),
But there is a special class among equalities,
We will tell our story about him now.
(Identity) equality is called.
But we still have to prove it.
To do this, we only need to take
And equality is (convert)
It is not difficult, of course, we will find out
Which part do we have to change?
And maybe we'll have to change both,
By equality, the mind is not difficult (to understand)
Hooray! We have been able to apply our knowledge
Finished equality conversion.
And boldly we already say the answer:
So is this identity, or is it not!

Teacher: Thanks for the lesson!

Preview:

Task cards

Slides captions:

Definition of identity: Identity is an equality that is true for any admissible values ​​of its constituent variables. Definition of identically equal expressions: Two expressions whose corresponding values ​​are equal for any values ​​of the variables are said to be identically equal.

Proof of identities

Examples of identities: - (a - c) \u003d - a + c a (b + c) \u003d ab - ac a - (b + c) \u003d a - c + c (a + c) - c \u003d a - c + c - (a + c) \u003d - a - c

What should be used to prove identity? To prove that some equality is an identity, or, as they say otherwise, to prove an identity, identical transformations of expressions are used.

Identity transformation of an expression The replacement of one expression by another, identically equal to it, is called the identity transformation of an expression.

To prove that equality is an identity, you need to: Write down the left side of the equality, transform it and make sure that it is equal to the right side. or Write out the right side of the equality, transform it and make sure that it is equal to the left. or Convert both sides of the equality in turn and make sure they are equal to the same expression.

Homework: No. 691 (a), No. 692 (a), No. 694, Make 3 equalities that will be an identity. *

There are all sorts of equalities, brothers, And everyone, of course, knows about it. There are - with variables, there are - ... Complex very, very ... . But there is a special class among the equalities, We will tell our story about it now. ... this is called equality, But we still have to prove it. To do this, we just need to take AND equality is ... . It will not be difficult, of course, for us to find out Which part we will have to change, Or maybe we will have to change both, By equality of the form it is not difficult ... Hooray! It was possible to apply our knowledge Finished equality transformation. And boldly we already say the answer: So is this identity, or is it not?

In the process of learning, students should develop the skills of proving identities in the following ways.

If you want to prove that A=B, then you can

1. prove that A - B \u003d O,

2. prove that A/B = 1,

3. convert A to form B,

4. convert B to form A,

5. convert A and B to the same form C.

The properties of arithmetic operations are used as a support on which the proofs of identities are built. Sometimes geometric concepts and methods are involved in the proof. Geometric proofs are not only instructive and illustrative, but also help to strengthen interdisciplinary connections.

Identity proofs can be divided into three types, depending on how they satisfy the requirements of rigor:

a) Not completely rigorous reasoning, requiring the use of the method of mathematical induction to give them full rigor. These proofs are used to derive a rule for actions with polynomials, properties of degrees with natural exponents. For instance,

a k a r = (a a······a) (a a·······a) = a a·······a = a k+p

k times p times k+r times

b) Fully rigorous reasoning based on the basic properties of arithmetic operations and not using other properties of the numerical system. The main area of ​​application of such proofs is the identities of reduced multiplication. Many of the statements expressed by the formulas of abbreviated multiplication allow a visual-geometric illustration.

Example For identity The teacher might suggest the following illustration:

c) Completely rigorous reasoning using the solvability conditions for equations of the form Ψ(x) = a, where Ψ is the elementary function under study. Such proofs are typical for the derivation of properties of a degree with a rational exponent and a logarithmic function. For example, when proving the property of an arithmetic root

(1)

we will rely on the reformulation of the definition of arithmetic square root: for non-negative numbers x and y the equality y =
and

y 2 = x are equivalent, so (1) is equivalent to (
) 2 = (
) 2 (2). Whence it follows, and in = (
) 2 (
) 2 = a c.

The method of proof that was used here is used quite rarely, however, it must be emphasized that the main idea of ​​the proof is to compare two operations (or functions) - direct and inverse to it, which will be used already in high school.

Technological chain of formation of algorithms and techniques

identical transformations of expressions in the main school

Assignment for the lecture

After analyzing school textbooks, compile a table of identical equalities indicating the set on which it is performed.

Example
, М 1 – those х for which f(x) makes sense.

Proof of identities. There are many concepts in mathematics. One of them is identity.

• An identity is an equality that holds for all values ​​of the variables that are included in it.

We already know some of the identities. For example, all abbreviated multiplication formulas are identities.

Prove Identity- this means to establish that for any admissible value of the variables, its left side is equal to the right side.

In algebra, there are several various ways identity proofs.

Ways to prove identities

• left side of the identity. If in the end we get the right side, then the identity is considered proven.
• Perform equivalent transformations the right side of the identity. If in the end we get the left side, then the identity is considered proven.
• Perform equivalent transformations left and right sides of the identity. If we get the same result as a result, then the identity is considered proven.
• Subtract the left side from the right side of the identity.
• Subtract the right side from the left side of the identity. We perform equivalent transformations on the difference. And if in the end we get zero, then the identity is considered proven.

It should also be remembered that the identity is valid only for admissible values ​​of variables.

As you can see, there are many ways. Which way to choose in this particular case depends on the identity you need to prove. As you prove various identities, experience will come in choosing the method of proof.

Let's look at a few simple examples

Example 1

Prove the identity x*(a+b) + a*(b-x) = b*(a+x).

Solution.

Since there is a small expression on the right side, let's try to transform the left side of the equality.

• x*(a+b) + a*(b-x) = x*a+x*b+a*b – a*x.

We present like terms and take the common factor out of the bracket.

• x*a+x*b+a*b – a*x = x*b+a*b = b*(a+x).

We got that the left side after the transformations became the same as the right side. Therefore, this equality is an identity.

Example 2

Prove the identity a^2 + 7*a + 10 = (a+5)*(a+2).

Solution.

V this example you can do in the following way. Let's open the brackets on the right side of the equality.

• (a+5)*(a+2) = (a^2) +5*a +2*a +10= a^2+7*a+10.

We see that after the transformations, the right side of the equality has become the same as the left side of the equality. Therefore, this equality is an identity.

Learning goal:

repeat the definitions of the equation, identities;

learn to distinguish between the concepts of equation and identity;

identify ways to prove identities;

repeat the methods of bringing a monomial to a standard form, adding polynomials, multiplying a monomial by a polynomial when proving identities.

Development goal:

develop competent mathematical speech of students (enrich and complicate vocabulary when using special mathematical terms),

develop thinking: the ability to compare, analyze, draw analogies, predict, draw conclusions (when choosing ways to prove identities);

develop the educational and cognitive competence of students.

educational goal:

develop the ability to work in a group, coordinate their activities with other participants in the educational process;

cultivate tolerance.

Lesson type: complex application of knowledge.

Lesson steps: preparatory, application of knowledge, result.

The border of knowledge - ignorance:

can apply the operations of reducing a monomial to a standard form;

addition of polynomials, multiplication of a polynomial by a polynomial.

Distinguish between the concepts of equation and identity;

carry out the proof of identities;

rationally choose and apply methods of proving identities.

Front work

Verbal

visual

Application of knowledge (ensuring the assimilation of new knowledge and methods of action at the level of application in a changed learning situation)

Based on the transformations of the left and right parts of the given

mathematical equality, identify ways to prove identities;

Identify a rational way from the proposed ones and work out the selection of a rational solution according to a given condition of identities

group work

Independent work

Search

Practical

Outcome (analysis and assessment of the success of achieving the goal)

Summing up the work in the lesson by performing individual work, where it is proposed to choose an identity from the presented equalities and prove it in any of the proposed ways (preferably rational);

Then the students self-evaluate their work in the lesson according to the specified (from the beginning of the lesson) criteria.

Frontal

Verbal

Lesson outline (briefly):

1. Stage (preparatory)

Consider the mathematical notation: (front work)

Grade 7 students, as a rule, believe that this is an equation, and, solving it, get linear equation of the form: 0 x = 0, true for any x.

Then, the teacher shows the work of another class, and the children are faced with a contradiction - in the work of another class, the students prove that this is the same.

Conclusion: attention should be paid to the fact that the same equality can be considered as an identity and as an equation. It depends on the condition for the given work: if it is required to establish at what value of the variable equality takes place, then this- the equation. And if you want to prove that equality takes place for any values ​​of the variables -identity.

2. Stage (application)

Finding ways to prove identities: (group work)

Expression written:

Practical task in groups to identify ways to prove identities:

Follow the rules for working in groups (they are printed on the signs put up by the teacher at the students' workplaces)

On Whatman paper, in joint work, perform some transformations according to a certain technology indicated in the task for the group and prove that the given expression does not depend on the values ​​of the variables, which means that it is an identity;

Give an explanation of the work done and conclude: what this method proofs of identities;

Task 1 group:

Move the right side of the equation to the left side. Prove that this expression does not depend on the value of the variables.

Task 2 group:

Transform the left side of the equation. Prove that it is equal to the right one, which means that this expression does not depend on the value of the variables.

Task 3 group:

Transform the left and right sides of the equation at the same time. Prove that this equality does not depend on the value of the variables.

When considering the work done by the guys to prove the identity, it is convenient to depict the results of the applied methods in the form of diagrams on separate sheets of paper, with a number indicator, so that in the future, these diagrams can be used not only in this, but also in other algebra lessons.

3. Stage (result)

a) Identities for choosing a rational solution: (front work)

5)