Application of various methods of factoring a polynomial. Factorization

LESSON PLAN

Lesson type : lesson learning new material based on problem-based learning

9 Purpose of the lesson

create conditions for practicing the skills and abilities of factoring a polynomial using various methods.

10. Tasks:

Educational

repeat the algorithms of operations: taking the common factor out of the bracket, grouping method, abbreviated multiplication formulas.

build skills:

– apply knowledge on the topic "factorization of a polynomial in various ways";

– perform tasks according to the chosen method of action;

– choose the most rational way to rationalize calculations, transform polynomials.

Educational

to promote the development of cognitive abilities, attention, memory, thinking of students through the use of various exercises;

develop skills of independent work and group work; keep students interested in mathematics

educators

keep students interested in mathematics

11.Shaped UUD

Personal: awareness of the purpose of the activity (expected result), awareness or choice of the method of activity (How will I do it? How will I get the result?), analysis and evaluation of the result; assessment of their capabilities;

Regulatory: take into account the rule in planning and controlling the way of solving, planning, evaluating the results of work;

Cognitive: choosing the most effective ways to solve problems, structuring knowledge;converting information from one form to another.

Communicative: planningeducational cooperation with the teacher and peers, compliance with the rules of speech behavior, the ability to express andsubstantiate their point of view, take into account different opinions and strive to coordinate various positions in cooperation.

12 .Methods:

by sources of knowledge: verbal, visual;

regarding the nature of cognitive activity: reproductive, partially exploratory.

13. Forms of student work: frontal, individual, group.

14. Necessary Technical equipment: computer, projector, interactive whiteboard, handouts (self-control sheet, task cards), electronic presentation made in the programpowerpoint

15.Planned results :

Personal fostering a sense of self- and mutual respect; development of cooperation when working in groups;

Metasubject speech development; development of students' independence; development of attentiveness when looking for errors.

subject development of skills to work with information, mastery of solutions

During the classes:

1. Greeting students. Checking the readiness of the class for the lesson by the teacher; organization of attention; evaluation sheet tutorialAnnex 1 , refinement of evaluation criteria.

Checking homework and updating knowledge

1. 3a + 6b= 3(a + 2b)

2. 100 - 20s + s 2 = (10 + s) 2

3. with 2 - 81 \u003d (s - 9) (s + 9)

4. 6x 3 – 5x 4 = x 4 (6x - 5)

5. ay - 3y - 4a + 12 \u003d y (a - 3) - 4 (a - 3)

6. 0.09x 2 - 0.25y 2 \u003d (0.03x - 0.05y) (0.03x + 0.05y)

7. c (x - 3) -d(x - 3) \u003d (x - 3) (s -d)

8. 14x 2 - 7x \u003d 7x (7x - 1)

9. -1600 + a 12 = (40 + a 6 ) (40 - a 6 )

10.9x 2 – 24xy + 16y 2 = (3x - 4y) 2

11.8s 3 – 2s 2 + 4s - 1 =

2s 2 (4s - 1) + (4s - 1) = (4s - 1)2s 2

12. b 4 + with 2 – 2 b 2 c = (b – c) 2

(Assignments for homework are taken from the textbook, include factorization in different ways. In order to complete this work, students need to remember previously studied material)

The answers recorded on the slide contain errors, students learn to see ways, and also, noticing errors, remember ways to act,

Students in groups, after checking their homework, give points for the work done.

2 RelayAnnex 2 (team members take turns completing the task, while the arrow connects the example and the way it is decomposed)

3a-12b = 3(a – 4 b)

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

9a 2 – 16b 2 = ( 3a - 4 b)(3a + 4b)

16a 2 - 8ab+b 2 = (4а – b) 2

7a 2 b-14ab 2 + 7ab = 7ab(a - 2b + 1)

a 2 + ab- a - ac- bc + c = (a + b - 1) (a - c)

25a 2 + 70ab + 49b 2 = ( 5a + 7 b) 2

5x 2 - 45y 2 \u003d 5 (x - 3y) (x + 3y)

Does not factorize

Grouping method

With the help of the slide, the work done is checked, and attention is drawn to the fact that the last example must be combined with two decomposition methods (bracketing the common factor and the abbreviated multiplication formula)

Students evaluate the work done, enter the results into the assessment sheets, and also formulate the topic of the lesson.

3. Completing tasks (students are invited to complete the task. Discussing the solution in a group, the guys come to the conclusion that several ways are required to factorize these polynomials. The team that first offers the correct decomposition has the right to write down their solution on the board, the rest write it down in a notebook .. The team has established work to help students who find it difficult to cope with the task)

1) 2a 2 - 2b 2

5) 5m 2 + 5n 2 – 10mn

9) 84 - 42y - 7xy + 14x

13) x 2 y+14xy 2 + 49y 3

2) 3a 2 + 6ab + 3b 2

6) cx 2 – cy 2

10) -7b 2 – 14bc – 7c 2

14) 3ab 2 – 27a

3) x 3 – 4x

7) -3x 2 + 12x - 12

11) 3x 2 - 3

15) -8a 3 b+56a 2 b 2 – 98ab 3

4) 3ab + 15b - 3a - 15

8) x 4 – x 2

12) c 4 - 81

16) 0 , 09t 4 – t 6

4. Final stage -

Factoring a polynomial

Taking the common factor out of brackets

Grouping method

Abbreviated Multiplication Formula

Summary of the lesson. Students answer the questions:What task did we set? Were we able to solve our problem? How? What were the results? How can a polynomial be factored? For what tasks can this knowledge be applied? What did you do well in class? What else needs to be worked on?

During the lesson, students assessed themselves, at the end of the lesson they are asked to add up the points received and rate them in accordance with the proposed scale.

Final word of the teacher: Today in the lesson we learned to determine what methods need to be applied in order to factorize polynomials. To consolidate the work done

Homework: §19, #708, #710

Additional task:

Solve the x equation 3 + 4x 2 = 9x + 36

• Formation of skills to apply various methods for factorization.
• Contribute to the education of a culture of speech, accuracy of recording, independence.
• Formation of skills of partial search activity: to be aware of the problem, to analyze, to draw conclusions.

Equipment: textbook, blackboard, notebook, task cards.

Lesson type: Lesson of application of ZUN.

Teaching method: problematic, partially exploratory.

Form of organization of educational activities: group, frontal, individual, work in pairs.

Duration: 1 lesson (45 min)

Lesson plan:

1. Organization of the beginning of the lesson. (1 minute)
2. Checking homework. (2 minutes)
3. Actualization. (5 minutes)
4. Learning new material. (10 min)
5. Consolidation of new material. (15 minutes)
6. Control and self-examination of knowledge. (8 min)
7. Summarizing. (2 minutes)
8. Homework. (2 minutes)

During the classes

I. Organizational moment

Hello guys.

The topic of the lesson is “Application of various methods for factorization”. Today we will form the skills of using various methods of factorization and once again we will be convinced of the usefulness of the ability to factor a polynomial into factors.

I wish you to work actively in the lesson. (Write the topic in a notebook).

II. Checking homework

Before the start of the lesson, students hand over notebooks with completed homework for verification. Issues that caused difficulties are discussed.

III. Updating of basic knowledge.

Before we start solving problems, we will check how ready we are for this. Let's remember what we know about the topic of the lesson.

3.1. Front poll:

a) What does it mean to factor a polynomial?
b) What basic methods of factoring a polynomial do you know?
c) Any polynomial can be factorized? For instance?
d) In what tasks is it sometimes useful to use factorization?

3.2. Draw lines to connect the polynomials with their corresponding factorization methods.

3.3. Find the wrong statement:

a) a 2 + b 2 - 2ab \u003d (a - b) 2

b) m 2 + 2mn - n 2 \u003d (m - n) 2

c) –2pt + p 2 + t 2 = (p - t) 2

d) 25 - 16 s 2 = (5 - 4s) (5 - 4s) (errors b, d)

3.4. Present as a product: a) 64x 2 - 1; b) (d - 3) 2 - 36;

3.5. Solve the Equation x 2 - 16 = 0 (4; -4)

3.5. Find the value of an expression 34 2 – 24 2 (580)

IV. Studying the material

To factorize polynomials, we used parentheses, grouping, and abbreviated multiplication formulas.

What do you think, are there situations in which it is possible to factorize a polynomial by applying successively several methods?

The following task will help us find the answer to this question:

Factor the polynomial and indicate which methods were used in this case. ( Work in pairs with the subsequent solution at the blackboard)

Example 1. 9x 3 - 36x used 2 methods:

Example 2. a 2 + 2ab + b 2 - c 2 used 2 methods:

• grouping;
• use of abbreviated multiplication formulas.

Example 3. y 3 - 3y 2 + 6y - 18 used 3 methods:

• grouping;
• use of abbreviated multiplication formulas;
• taking the common factor out of brackets.

Example 4. x 3 + 3x 2 + 2x used 3 ways:

• taking the common factor out of brackets;
• preliminary transformation;
• grouping.

We conclude: sometimes it is possible to factorize a polynomial by applying successively several methods. In order to successfully solve such examples, today let's develop a plan for consistently applying them:

1. Take the common factor out of the bracket (if any).
2. Try to factorize the polynomial using the abbreviated multiplication formulas.
3. Try to apply the grouping method (if the previous methods did not lead to the goal).

V. Exercises to consolidate the stated topic

5.1. The combination of various methods of factoring allows you to easily and gracefully perform arithmetic calculations, solve equations of the form ax 2 + bx + c \u003d 0 (a ≠ 0) (such equations are called quadratic, we will study them in grade 8).

* Solve the equation: a) x 2 - 17x + 72 = 0, b) x 2 + 10x + 21 = 0

Hint: Some term of the polynomial is decomposed into the necessary terms or supplemented by adding some term to it. In the latter case, so that the polynomial does not change, the same term is subtracted from it.

(Two students solve equations on their own in a notebook. Answer: a) 8; 9; b) - 1; - 5).

Complete the exercise from the textbook No. 1016 (c), 1017 (c), p. 186

(Two students decide on the board, the rest according to the options in the notebook).

5.2. Solve equations ( Pupils work in pairs, followed by self-examination)

No. 949, p.177 a) x 3 - x = 0 b) 9x - x 3 = 0 c) x 3 + x 2 = 0 d) 5x 4 - 2x 2 = 0

** (Individual tasks for more prepared students)

Card 1	Card 2	Card 3
Solve the equation and write the sum of the roots x 2 + 3x + 6 + 2x = 0		Solve the equation and write the sum of the roots

x(x+3) +2(3+x) =0 the sum is -5

The sum of the roots of this equation:

The sum of the roots of the equation:.

VI. Control and self-examination of knowledge.

The topic under consideration is an integral part of the GIA in mathematics. To control and self-test knowledge on this topic, you are invited to complete test tasks from the GIA training tasks. Circle your answer on the test questions.

Individual work on cards: (Students perform GIA test tasks, + self test)

Which of these expressions are identically equal to 4x-10y 2(2x-5y) -2(5y-2x) -10y-4x -10y+4x? a) 1; 3; b) all; c) 1;2;4; oppression	Which of these expressions are identically equal - 3 (-2a + y) -3(-y+2a) 6a-3y 3(2a-y) 3u-6a? and all; b) 2; y) 2;3; c)1;4	Which of these expressions are identically equal to -6a + 12p -6(a-2p) 12r-6a 6(-a+2p) -6(-p+a) ? a) 1; at all; c) 2;4; d)1;3
3a 3 -3a 2 -5a + 5. a) (a-1) (3a 2 +5); b) (a + 1) (3a 2 -5); c) (a-1) (5-3a 2); e) (a-1) (3a 2 +5).	Express as a product of polynomials 13ah-26x-5av + 10v. e) (a-2) (13x-5c); b) (a + 2) (3x-5c); c) (3a-6)(4x-c); d) (a-2) (5c-3x).	Express as a product of polynomials bу-6b-5у 2 +30у. a) (6-y) (b-5y); b) (y -6) (b + 5y); c) (y-6)(b-5y); d) (y -6) (5y - b).
Follow the steps: (5a-c) 2 . a) 25a 2 + 10ac + s 2; b) 25a 2 + 10ac-c 2; p) 25a 2 -10ac + c 2; d) 25a 2 -5ac + s 2.	Do the following: (5x + 2y) 2 . a) 25x 2 + 20xy + 4y 2; success

Teacher: Let's check the answers. Read the words you have. These are exactly the words that accompany seventh graders in preparation for the GIA in grade 9.

VII. Summing up the lesson

The teacher conducts a frontal review of the main stages of the lesson, evaluates the work of students and orients students in homework.

VIII. Homework: 38, No. 950 (p. 177), No. 1016 (g), 1017 (g), p. 186.

** Find the value of the expression (x+3)2 -2 (x+3) (x-3) +(x-3)2 at x=100.

The value of this expression does not depend on the choice of x.

The lesson is over. Thank you for the lesson and remember that knowledge that is not replenished daily decreases every day.

Used Books:

1. Textbook "Algebra Grade 7". Yu.N. Makarychev, N.G. Mindyuk and others. Ed. S.A. Telyakovsky. – M.; Enlightenment, 2009.
2. Collection of test tasks for thematic and final control. Algebra 7. I.L. Guseva and others - M.; Intellect Center, 2009.
3. State final certification (according to the new form): Grade 9. Thematic training tasks. Algebra / FIPI author-compiler: V.L. Kuznetsova. – M.: Eksmo, 2010.

The factorization of polynomials is an identical transformation, as a result of which a polynomial is transformed into a product of several factors - polynomials or monomials.

There are several ways to factorize polynomials.

Method 1. Bracketing the common factor.

This transformation is based on the distributive law of multiplication: ac + bc = c(a + b). The essence of the transformation is to single out the common factor in the two components under consideration and “put it out” of the brackets.

Let us factorize the polynomial 28x 3 - 35x 4.

Solution.

1. We find a common divisor for elements 28x3 and 35x4. For 28 and 35 it will be 7; for x 3 and x 4 - x 3. In other words, our common factor is 7x3.

2. We represent each of the elements as a product of factors, one of which
7x 3: 28x 3 - 35x 4 \u003d 7x 3 ∙ 4 - 7x 3 ∙ 5x.

3. Bracketing the common factor
7x 3: 28x 3 - 35x 4 \u003d 7x 3 ∙ 4 - 7x 3 ∙ 5x \u003d 7x 3 (4 - 5x).

Method 2. Using abbreviated multiplication formulas. The "mastery" of mastering this method is to notice in the expression one of the formulas for abbreviated multiplication.

Let us factorize the polynomial x 6 - 1.

Solution.

1. We can apply the difference of squares formula to this expression. To do this, we represent x 6 as (x 3) 2, and 1 as 1 2, i.e. 1. The expression will take the form:
(x 3) 2 - 1 \u003d (x 3 + 1) ∙ (x 3 - 1).

2. To the resulting expression, we can apply the formula for the sum and difference of cubes:
(x 3 + 1) ∙ (x 3 - 1) \u003d (x + 1) ∙ (x 2 - x + 1) ∙ (x - 1) ∙ (x 2 + x + 1).

So,
x 6 - 1 = (x 3) 2 - 1 = (x 3 + 1) ∙ (x 3 - 1) = (x + 1) ∙ (x 2 - x + 1) ∙ (x - 1) ∙ (x 2 + x + 1).

Method 3. Grouping. The grouping method consists in combining the components of a polynomial in such a way that it is easy to perform operations on them (addition, subtraction, taking out a common factor).

We factorize the polynomial x 3 - 3x 2 + 5x - 15.

Solution.

1. Group the components in this way: the 1st with the 2nd, and the 3rd with the 4th
(x 3 - 3x 2) + (5x - 15).

2. In the resulting expression, we take the common factors out of brackets: x 2 in the first case and 5 in the second.
(x 3 - 3x 2) + (5x - 15) \u003d x 2 (x - 3) + 5 (x - 3).

3. We take out the common factor x - 3 and get:
x 2 (x - 3) + 5 (x - 3) \u003d (x - 3) (x 2 + 5).

So,
x 3 - 3x 2 + 5x - 15 \u003d (x 3 - 3x 2) + (5x - 15) \u003d x 2 (x - 3) + 5 (x - 3) \u003d (x - 3) ∙ (x 2 + 5 ).

Let's fix the material.

Factor the polynomial a 2 - 7ab + 12b 2 .

Solution.

1. We represent the monomial 7ab as the sum 3ab + 4ab. The expression will take the form:
a 2 - (3ab + 4ab) + 12b 2 .

Let's open the brackets and get:
a 2 - 3ab - 4ab + 12b 2 .

2. Group the components of the polynomial in this way: the 1st with the 2nd and the 3rd with the 4th. We get:
(a 2 - 3ab) - (4ab - 12b 2).

3. Let's take out the common factors:
(a 2 - 3ab) - (4ab - 12b 2) \u003d a (a - 3b) - 4b (a - 3b).

4. Let's take out the common factor (a - 3b):
a(a – 3b) – 4b(a – 3b) = (a – 3b) ∙ (a – 4b).

So,
a 2 - 7ab + 12b 2 =
= a 2 - (3ab + 4ab) + 12b 2 =
= a 2 - 3ab - 4ab + 12b 2 =
= (a 2 - 3ab) - (4ab - 12b 2) =
= a(a - 3b) - 4b(a - 3b) =
= (а – 3 b) ∙ (а – 4b).

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Sections: Mathematics

Lesson type:

• according to the method of conducting - a practical lesson;
• for the didactic purpose - a lesson in the application of knowledge and skills.

Target: form the ability to factorize a polynomial.

Tasks:

• Didactic: systematize, expand and deepen the knowledge, skills of students, apply various methods of factoring a polynomial into factors. To form the ability to apply the decomposition of a polynomial into factors by a combination of various techniques. To implement knowledge and skills on the topic: “Decomposition of a polynomial into factors” to complete tasks at a basic level and tasks of increased complexity.
• Educational: to develop mental activity through solving problems of various types, to learn to find and analyze the most rational ways of solving, to contribute to the formation of the ability to generalize the studied facts, to clearly and clearly express one's thoughts.
• Educational: develop skills of independent and team work, self-control skills.

Working methods:

• verbal;
• visual;
• practical.

Lesson equipment: interactive whiteboard or overhead scope, tables with abbreviated multiplication formulas, instructions, handout for group work.

Lesson structure:

1. Organizing time. 1 minute
2. Formulating the topic, goals and objectives of the lesson-practice. 2 minutes
3. Checking homework. 4 minutes
4. Updating the basic knowledge and skills of students. 12 minutes
5. Fizkultminutka. 2 minutes
6. Instructions for completing the tasks of the workshop. 2 minutes
7. Performing tasks in groups. 15 minutes
8. Checking and discussing the performance of tasks. Work analysis. 3 minutes
9. Setting homework. 1 minute
10. Reserve assignments. 3 minutes

During the classes

1. Organizational moment

The teacher checks the readiness of the classroom and students for the lesson.

2. Formulation of the topic, goals and objectives of the lesson-practice

• Message about the final lesson on the topic.
• Motivation of educational activity of students.
• Formulating the goal and setting the objectives of the lesson (together with students).

3. Checking homework

On the board are examples of solving homework exercises No. 943 (a, c); No. 945 (c, d). The samples were made by the students of the class. (This group of students was identified in the previous lesson, they formalized their decision at recess). The students prepare to “defend” the solutions.

Teacher:

Checks for homework in student notebooks.

Invites the students of the class to answer the question: “What difficulties did the assignment cause?”.

Offers to compare their solution with the solution on the board.

Invites the students at the blackboard to answer the questions that the students had in the field when checking on the samples.

He comments on the answers of students, supplements the answers, explains (if necessary).

Summarizes homework.

Students:

Present homework to the teacher.

Change notebooks (in pairs) and check with each other.

Answer the teacher's questions.

Check your solution with samples.

They act as opponents, make additions, corrections, write down a different method if the solution method in the notebook differs from the method on the board.

Ask for the necessary explanations to the students, to the teacher.

Find ways to check the results.

Participate in the assessment of the quality of the tasks at the blackboard.

4. Updating the basic knowledge and skills of students

1. Oral work

Teacher:

Answer the questions:

1. What does it mean to factor a polynomial?
2. How many decomposition methods do you know?
3. What are their names?
4. What is the most common?

2. Polynomials are written on the board:

1. 14x 3 - 14x 5

2. 16x 2 - (2 + x) 2

3. 9 - x 2 - 2xy - y 2

4.x3 - 3x - 2

Teacher invites students to factorize polynomials No. 1-3:

• Option I - by taking out a common factor;
• Option II - using abbreviated multiplication formulas;
• III variant - by way of grouping.

One student is offered to factorize the polynomial No. 4 (an individual task of increased difficulty, the task is performed on the A 4 format). Then a sample solution for tasks No. 1-3 (done by the teacher), a sample solution for task No. 4 (done by the student) appears on the board.

3. Warm up

The teacher gives instructions to factorize and choose the letter associated with the correct answer. By adding the letters you will get the name of the greatest mathematician of the 17th century, who made a huge contribution to the development of the theory of solving equations. (Descartes)

5. Physical education The students read the statements. If the statement is true, then the students should raise their hands up, and if it is not true, then sit down at the desk. (Annex 2)

6. Instruction on how to complete the tasks of the workshop.

On an interactive whiteboard or a separate poster, a table with instructions.

When decomposing a polynomial into factors, the following order must be observed:

1. put the common factor out of brackets (if any);

2. apply abbreviated multiplication formulas (if possible);

3. apply the grouping method;

4. check the result obtained by multiplication.

Teacher:

Offers instruction to students (emphasizes step 4).

Offers the implementation of workshop assignments in groups.

Distributes worksheets into groups, sheets with carbon paper for completing assignments in notebooks and their subsequent verification.

Determines the time for work in groups, for work in notebooks.

students:

They read the instructions.

Teachers listen carefully.

They are seated in groups (4-5 people each).

Prepare for practical work.

7. Performing tasks in groups

Worksheets with tasks for groups. (Annex 3)

Teacher:

Manages independent work in groups.

Evaluates the ability of students to work independently, the ability to work in a group, the quality of the design of the worksheet.

students:

Perform tasks on sheets of carbon paper enclosed in a workbook.

Discuss rational solutions.

Prepare a worksheet for the group.

Prepare to defend your work.

8. Checking and discussing the assignment

Answers on the whiteboard.

Teacher:

Collects copies of decisions.

Manages the work of students reporting on worksheets.

Offers to conduct a self-assessment of their work, compare answers in notebooks, worksheets and samples on the board.

Recalls the criteria for grading for work, for participation in its implementation.

Provides clarification on emerging decision or self-assessment issues.

Summarizes the first results of practical work and reflection.

Summarizes (together with students) the lesson.

Says that the final results will be summed up after checking copies of the work done by students.

students:

Give copies to the teacher.

Worksheets are attached to the board.

Reporting on the performance of work.

Perform self-assessment and self-assessment of work performance.

9. Setting homework

Homework is written on the board: No. 1016 (a, b); 1017 (c, d); No. 1021 (d, e, f)*

Teacher:

Offers to write down the obligatory part of the assignment at home.

Gives a comment on its implementation.

Invites more prepared students to write down No. 1021 (d, e, f) *.

Tells you to prepare for the next review review lesson

LESSON PLAN algebra lesson in 7th grade

Teacher Prilepova O.A.

Lesson Objectives:

Show the application of various methods for factoring a polynomial

Repeat the methods of factorization and consolidate their knowledge during the exercises

To develop the skills and abilities of students in the application of abbreviated multiplication formulas.

Develop students' logical thinking and interest in the subject.

Tasks:

in the direction personal development:

Development of interest in mathematical creativity and mathematical abilities;

Development of initiative, activity in solving mathematical problems;

Cultivating the ability to make independent decisions.

in the meta-subject direction :

Formation of general ways of intellectual activity, characteristic of mathematics and being the basis of cognitive culture;

Use of ICT technology;

in the subject area:

Mastering the mathematical knowledge and skills necessary to continue education;

Formation in students the ability to look for ways to factorize a polynomial and find them for a polynomial that is factorized.

Equipment:handouts, route sheets with evaluation criteria,multimedia projector, presentation.

Lesson type:repetition, generalization and systematization of the material covered

Forms of work:work in pairs and groups, individual, collective,independent, frontal work.

During the classes:

Stages	Plan		UUD
Org moment.	Breakdown into groups and couples: Students choose a mate according to the following criterion: I communicate with this classmate the least. Psychological mood: Choose an emoticon of your choice (the mood at the beginning of the lesson) and under it look at the grade that you would like to receive today in the lesson (SLIDE). - Put yourself in the notebook in the margins of the grade that you would like to receive today in the lesson. You will mark your results in the table (SLIDE). Route sheet. Exercise total Grade Evaluation criteria: 1. I solved everything correctly, without errors - 5 2. When solving, I made from 1 to 2 mistakes - 4 3. Made 3 to 4 mistakes while solving - 3 4. Made more than 4 mistakes when solving - 2		New approaches to teaching (dialogue)
Actualization.	Collective work. - Today at the lesson you will be able to demonstrate your knowledge, participate in mutual control and self-control of your activities Match (SLIDE): On the next slide, pay attention to the expressions, what do you notice? (SLIDE) 15x3y2 + 5x2y Taking the common multiplier out of brackets p 2 + pq - 3 p -3 q Grouping method 16m2 - 4n2 Abbreviated multiplication formula How can these actions be united in one word? (Methods of expansion of polynomials) Statement by students of the topic and purpose of the lesson as their own learning task (SLIDE). Based on this, let's formulate the topic of our lesson and set goals. Questions for students: Name the topic of the lesson; Formulate the purpose of the lesson; Everyone has cards with the name of the formulas. (Work in pairs). Give formulas to all formulas
Application of knowledge	Work in pairs. Checking the slide 1. Choose the correct answer (SLIDE). Cards: Exercise Answer (x+10)2= x2+100-20x x2+100+20x x2+100+10x (5y-7)2= 25y2+49-70y 25u2-49-70u 25y2+49+70 x2-16y2= (x-4y)(x+4y) (x-16y)(x+16y) (x+4y)(4y-x) (2a+c)(2a-c)= 4a2-v2 4а2+в2 2a2-b2 a3-8v3 a2+16-64v6 (a-8c)(a+8c) (a-2c) (a2 + 2av + 4c2)	2. Find errors (SLIDE): Cards No. Checking the slide 1 pair: o ( b- y)2 = b2 - 4 by+y2 o 49- c2=(49-c)(49+s) 2 pair: o (r- 10) 2=r2- 20r+10 o (2a+1)2=4a2+2a+1 3 pair: o (3y+1)2=9y+6y+1 o ( b- a) 2 =b²- 4ba+a2 4 pair: o x²- 25= ( x-25)( 25+x) o (7- a) 2 \u003d 7- 14a + a²	Education in accordance with age characteristics
	3. Each pair is given tasks and a limited time to solve it (SLIDE) We check on the answer cards 1. Follow the steps: a) (a + 3c) 2; b) x 2 - 12 x + 36; c) 4v2-y2. 2. Factorize: a) ; b) ; in 2 x - a 2 y - 2 a 2 x + y 3. Find the value of the expression: (7 p + 4)2 -7 p (7 p - 2) at p = 5.		Management and leadership
	4. Group work. Look, make no mistake (SLIDE). Cards. Let's check the slide. (а+…)²=…+2…с+с² (... + y)² \u003d x² + 2x ... + ... (... + 2x)² \u003d y² + 4xy + 4x² (…+2 m)²=9+…+4 m² (n + 2v)²= n ²+…+4v²		Teaching critical thinking. Management and leadership
	5. Group work (consultation on the solution, discussion of tasks and their solutions) Each member of the group is given tasks of level A, B, C. Each member of the group chooses a feasible task for himself. Cards. (Slide) Checking with answer cards Level A 1. Factor it out: a) c 2 - a 2 ; b) 5x2-45; c) 5a2 + 10av + 5v2; d) ax2-4ax + 4a 2. Do the following: a) (x - 3) (x + 3); b) (x - 3)2; c) x (x - 4). Level B 1. Simplify: a) (3a + p) (3a-p) + p2; b) (a + 11) 2 - 20a; c) (a-4) (a + 4) -2a (3-a). 2. Calculate: a) 962 - 862; b) 1262 - 742. Level C 1. Solve the equation: (7 x - 8) (7x + 8) - (25x - 4)2 + 36(1 - 4x)2 =44 1. Solve the equation: (12 x - 4) (12 x + 4) - (12 x - 1)2 - (4 x - 5) = 16. 1.		Teaching the talented and gifted
Lesson summary	- Let's sum up, we will derive estimates according to the results of the table. Compare your scores with your estimated score. Choose the emoticon that matches your rating (SLIDE). c) the teacher evaluates the work of the class (activity, level of knowledge, skills, self-organization, diligence) Independent work in the form of a test with a RESERVE check		Assessment for Learning and Assessment for Learning
Homework	Continue teaching abbreviated multiplication formulas.
Reflection	Guys, please listen to the parable: (SLIDE) A sage was walking, and three people were meeting him, carrying carts with Stones for the construction of the Temple. The sage stopped and asked each Question. The first asked: - What did you do all day? And he replied with a smirk that he had been carrying cursed stones all day. The second asked: “And what did you do all day? ” And he replied: “I did my job conscientiously.” And the third smiled at him, his face lit up with joy and pleasure, and answered “A I took part in the construction of the Temple.” What is your Temple? (Knowledge) Guys! Who has worked since the first person? (show emoticons) (Score 3 or 2) (SLIDE) Who worked in good faith? (Score 4) And who took part in the construction of the Temple of Knowledge? (Score 5)		Critical Thinking Training