Polynomials. Factoring a polynomial: methods, examples. Algebra lesson "different ways of factoring" Factoring a quadratic trinomial

LESSON PLAN algebra lesson in 7th grade

Teacher Prilepova O.A.

Lesson objectives:

Show the use of various methods for factoring a polynomial

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

Develop students' skills and abilities in using abbreviated multiplication formulas.

To develop students’ logical thinking and interest in the subject.

Tasks:

in the direction personal development:

Developing interest in mathematical creativity and mathematical abilities;

Development of initiative and activity in solving mathematical problems;

Developing the ability to make independent decisions.

in the meta-subject direction :

Formation common methods intellectual activity, characteristic of mathematics and which is the basis of cognitive culture;

Use of ICT technology;

in the subject area:

Mastery of mathematical knowledge and skills necessary for continuing education;

Developing in students the ability to look for ways to factor a polynomial and find them for a polynomial that can be factorized.

Equipment:handouts, route sheets with assessment 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:

Factoring polynomials is an identity transformation, as a result of which a polynomial is transformed into the product of several factors - polynomials or monomials.

There are several ways to factor polynomials.

Method 1. Taking the common factor out of brackets.

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

Let us factor the polynomial 28x 3 – 35x 4.

Solution.

1. Find the elements 28x 3 and 35x 4 common divisor. For 28 and 35 it will be 7; for x 3 and x 4 – x 3. In other words, our common factor is 7x 3.

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

3. We take the common factor out of brackets
7x 3: 28x 3 – 35x 4 = 7x 3 ∙ 4 – 7x 3 ∙ 5x = 7x 3 (4 – 5x).

Method 2. Using abbreviated multiplication formulas. The “mastery” of using this method is to notice one of the abbreviated multiplication formulas in the expression.

Let us factor the polynomial x 6 – 1.

Solution.

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

2. We can apply the formula for the sum and difference of cubes to the resulting expression:
(x 3 + 1) ∙ (x 3 – 1) = (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 is to combine the components of a polynomial in such a way that it is easy to perform operations on them (addition, subtraction, subtraction of a common factor).

Let's factor the polynomial x 3 – 3x 2 + 5x – 15.

Solution.

1. Let's group the components in this way: 1st with 2nd, and 3rd with 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) = x 2 (x – 3) + 5(x – 3).

3. We take the common factor x – 3 out of brackets and get:
x 2 (x – 3) + 5(x – 3) = (x – 3)(x 2 + 5).

So,
x 3 – 3x 2 + 5x – 15 = (x 3 – 3x 2) + (5x – 15) = x 2 (x – 3) + 5(x – 3) = (x – 3) ∙ (x 2 + 5 ).

Let's secure the material.

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

Solution.

1. Let us 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. Let's group the components of the polynomial in this way: 1st with 2nd and 3rd with 4th. We get:
(a 2 – 3ab) – (4ab – 12b 2).

3. Let’s take the common factors out of brackets:
(a 2 – 3ab) – (4ab – 12b 2) = a(a – 3b) – 4b(a – 3b).

4. Let’s take the common factor (a – 3b) out of brackets:
a(a – 3b) – 4b(a – 3b) = (a – 3 b) ∙ (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) =
= (a – 3 b) ∙ (a – 4b).

website, when copying material in full or in part, a link to the source is required.

Public lesson

mathematics

in the 7th grade

"Using various methods to factor a polynomial."

Prokofieva Natalya Viktorovna,

Mathematic teacher

Lesson Objectives

Educational:

1. repeat abbreviated multiplication formulas
2. formation and primary consolidation of the ability to factor polynomials in various ways.

Educational:

1. development of mindfulness, logical thinking, attention, ability to systematize and apply acquired knowledge, mathematically literate speech.

Educational:

1. developing interest in solving examples;
2. nurturing a sense of mutual assistance, self-control, and mathematical culture.

Lesson type: combined lesson

Equipment: projector, presentation, blackboard, textbook.

Preliminary preparation for the lesson:

1. Students should know the following topics:

1. Squaring the sum and difference of two expressions
2. Factoring using the squared sum and squared difference formulas
3. Multiplying the difference of two expressions by their sum
4. Factoring a difference of squares
5. Factoring the sum and difference of cubes

1. Have the skills to work with abbreviated multiplication formulas.

Lesson Plan

1. Organizational moment (focus students on the lesson)
2. Checking homework (error correction)
3. Oral exercises
4. Learning new material
5. Training exercises
6. Repetition exercises
7. Summing up the lesson
8. Homework message

During the classes

I. Organizational moment.

The lesson will require you to know abbreviated multiplication formulas, be able to apply them, and of course, pay attention.

II. Checking homework.

Homework questions.

Analysis of the solution at the board.

II. Oral exercises.

Mathematics is needed
It's impossible without her
We teach, we teach, friends,
What do we remember in the morning?

Let's do a warm-up.

Factorize (Slide 3)

8a – 16b

17x² + 5x

c(x+y)+5(x+y)

4a² - 25 (Slide 4)

1 - y³

ax + ay + 4x + 4y Slide 5)

III. Independent work.

Each of you has a table on the table. Sign your work at the top right. Fill out the table. Work time is 5 minutes. Let's get started.

We're done.

Please swap jobs with your neighbor.

They put down their pens and picked up their pencils.

We check the work - pay attention to the slide. (Slide 6)

We put a mark - (Slide 7)

7(+) - 5

6-5(+) - 4

4(+) - 3

Place the formulas in the middle of the table. Let's start learning new material.

IV. Learning new material

In notebooks we write down the date, class work and topic of today's lesson.

Teacher.

1. When factoring polynomials, sometimes they use not one, but several methods, applying them sequentially.
2. Examples:

1. 5a² - 20 = 5 (a² - 4) = 5 (a-2)(a+2). (Slide 8)

We use the common factor out of brackets and the difference of squares formula.

1. 18x³ + 12x² + 2x = 2x (9x² + 6x + 1) = 2x (3x + 1)². (Slide 9)

What can you do with the expression? What method will we use to factorize?

Here we use bracketing the common factor and the squared sum formula.

1. ab³ – 3b³ + ab²у – 3b²у = b² (ab – 3b + ay – 3y) = b² ((ab – 3b) + (ay – 3y)) = b² (b(a – 3) + y(a – 3)) = b² (a – 3)(b +y). (Slide 10)

What can you do with the expression? What method will we use to factorize?

Here the common factor was taken out of brackets and the grouping method was applied.

1. Order of factorization: (Slide 11)

1. Not every polynomial can be factorized. For example: x² + 1; 5x² + x + 2, etc. (Slide 12)

V. Training exercises

Before we start, we do a physical training session (Slide 13)

They quickly stood up and smiled.

They stretched higher and higher.

Come on, straighten your shoulders,

Raise, lower.

Turn right, turn left,

They sat down and stood up. They sat down and stood up.

And they ran on the spot.

And some more gymnastics for the eyes:

1. Close your eyes tightly for 3-5 seconds, and then open them for 3-5 seconds. Repeat 6 times.
2. Place your thumb at a distance of 20-25cm from your eyes, look with both eyes at the end of the finger for 3-5c, and then look with both eyes at the pipe. Repeat 10 times.

Well done, have a seat.

Lesson assignment:

No. 934 avd

№935 av

№937

No. 939 avd

No. 1007 avd

VI.Repetition exercises.

№ 933

VII. Summing up the lesson

The teacher asks questions, and students answer them at will.

1. Name known methods factoring a polynomial.

1. Take the common factor out of brackets
2. Factoring a polynomial using abbreviated multiplication formulas.
3. grouping method

1. Order of factorization:

1. Place the common factor out of brackets (if there is one).
2. Try to factor a polynomial using abbreviated multiplication formulas.
3. If the previous methods did not lead to the goal, then try to use the grouping method.

Raise a hand:

1. If your attitude to the lesson is “I didn’t understand anything, and I didn’t succeed at all”
2. If your attitude towards the lesson is “there were difficulties, but I managed”
3. If your attitude towards the lesson is “I succeeded in almost everything”

Factor 4 a² - 25 = 1 - y³ = (2a – 5) (2a + 5) (1 – y) (1+y+y ²) Factoring a polynomial using abbreviated multiplication formulas

Factorize ax+ay+4x+4y= =a(x+y)+4(x+y)= (ax+ay)+(4x+4y)= (x+y) (a+4) Grouping method

(a + b) ² a ² + 2ab + b ² Square of the sum a² - b² (a – b)(a + b) Difference of squares (a – b)² a² - 2ab + b² Square of the difference a³ + b ³ (a + b) (a² - ab + b²) Sum of cubes (a + b) ³ a³ + 3 a²b+3ab² + b³ Cube of sum (a - b) ³ a³ - 3a²b+3ab² - b³ Cube of difference a³ - b³ (a – b) (a² + ab + b²) Difference of cubes

SET THE MARKS 7 (+) = 5 6 or 5 (+) = 4 4 (+) = 3

Example No. 1. 5 a² - 20 = = 5(a² - 4) = = 5(a – 2) (a+2) Taking the common factor out of brackets Formula for the difference of squares

Example No. 2. 18 x³ + 12x ² + 2x = =2x (9x ² +6x+1)= =2x(3x+1) ² Taking the common factor out of brackets Formula for squared sum

Example No. 3. ab³ –3b³+ab²y–3b²y= = b²(ab–3b+ay-3y)= =b²((a b -3 b)+(a y -3 y)= =b²(b(a-3)+y(a -3))= =b²(a-3)(b+y) Place the factor outside the brackets Group the terms in the brackets Place the factors outside the brackets Place the common factor outside the brackets

Order of factorization: Place the common factor out of brackets (if there is one). Try to factor a polynomial using abbreviated multiplication formulas. 3. If the previous methods did not lead to the goal, then try to apply the grouping method.

Not every polynomial can be factorized. For example: x² +1 5x² + x + 2

PHYSICAL MINUTE

Lesson assignment No. 934 avd No. 935 avd No. 937 No. 939 avd No. 1007 avd

Raise your hand: If your attitude to the lesson is “I didn’t understand anything, and I didn’t succeed at all” If your attitude to the lesson “there were difficulties, but I did it” If your attitude to the lesson “I succeeded in almost everything”

Homework: p. 38 No. 936 No. 938 No. 954

The concepts of “polynomial” and “factorization of a polynomial” in algebra are encountered very often, because you need to know them in order to easily carry out calculations with large multi-digit numbers. This article will describe several decomposition methods. All of them are quite simple to use; you just need to choose the right one for each specific case.

The concept of a polynomial

A polynomial is a sum of monomials, that is, expressions containing only the operation of multiplication.

For example, 2 * x * y is a monomial, but 2 * x * y + 25 is a polynomial that consists of 2 monomials: 2 * x * y and 25. Such polynomials are called binomials.

Sometimes, for the convenience of solving examples with multivalued values, an expression needs to be transformed, for example, decomposed into a certain number of factors, that is, numbers or expressions between which the multiplication action is performed. There are a number of ways to factor a polynomial. It is worth considering them, starting with the most primitive, which is used in primary school.

Grouping (record in general form)

The formula for factoring a polynomial using the grouping method in general looks like this:

ac + bd + bc + ad = (ac + bc) + (ad + bd)

It is necessary to group the monomials so that each group has a common factor. In the first bracket this is the factor c, and in the second - d. This must be done in order to then move it out of the bracket, thereby simplifying the calculations.

Decomposition algorithm using a specific example

The simplest example of factoring a polynomial using the grouping method is given below:

10ac + 14bc - 25a - 35b = (10ac - 25a) + (14bc - 35b)

In the first bracket you need to take the terms with the factor a, which will be common, and in the second - with the factor b. Pay attention to the + and - signs in the finished expression. We put in front of the monomial the sign that was in the initial expression. That is, you need to work not with the expression 25a, but with the expression -25. The minus sign seems to be “glued” to the expression behind it and always taken into account when calculating.

In the next step, you need to take the multiplier, which is common, out of brackets. This is exactly what the grouping is for. To put outside the bracket means to write before the bracket (omitting the multiplication sign) all those factors that are exactly repeated in all the terms that are in the bracket. If there are not 2, but 3 or more terms in a bracket, the common factor must be contained in each of them, otherwise it cannot be taken out of the bracket.

In our case, there are only 2 terms in brackets. The overall multiplier is immediately visible. In the first bracket it is a, in the second it is b. Here you need to pay attention to the digital coefficients. In the first bracket, both coefficients (10 and 25) are multiples of 5. This means that not only a, but also 5a can be taken out of the bracket. Before the bracket, write 5a, and then divide each of the terms in brackets by the common factor that was taken out, and also write the quotient in brackets, not forgetting about the signs + and - Do the same with the second bracket, take out 7b, as well as 14 and 35 multiple of 7.

10ac + 14bc - 25a - 35b = (10ac - 25a) + (14bc - 35b) = 5a(2c - 5) + 7b(2c - 5).

We got 2 terms: 5a(2c - 5) and 7b(2c - 5). Each of them contains a common factor (the entire expression in brackets is the same here, which means it is a common factor): 2c - 5. It also needs to be taken out of the bracket, that is, terms 5a and 7b remain in the second bracket:

5a(2c - 5) + 7b(2c - 5) = (2c - 5)*(5a + 7b).

So the full expression is:

10ac + 14bc - 25a - 35b = (10ac - 25a) + (14bc - 35b) = 5a(2c - 5) + 7b(2c - 5) = (2c - 5)*(5a + 7b).

Thus, the polynomial 10ac + 14bc - 25a - 35b is decomposed into 2 factors: (2c - 5) and (5a + 7b). The multiplication sign between them can be omitted when writing

Sometimes there are expressions of this type: 5a 2 + 50a 3, here you can put out of brackets not only a or 5a, but even 5a 2. You should always try to put the largest common factor out of the bracket. In our case, if we divide each term by a common factor, we get:

5a 2 / 5a 2 = 1; 50a 3 / 5a 2 = 10a(when calculating the quotient of several powers with equal bases, the base is preserved and the exponent is subtracted). Thus, the unit remains in the bracket (in no case do you forget to write one if you take one of the terms out of the bracket) and the quotient of division: 10a. It turns out that:

5a 2 + 50a 3 = 5a 2 (1 + 10a)

Square formulas

For ease of calculation, several formulas were derived. These are called abbreviated multiplication formulas and are used quite often. These formulas help factor polynomials containing degrees. This is another effective way to factorize. So here they are:

• a 2 + 2ab + b 2 = (a + b) 2 - a formula called the “square of the sum”, since as a result of decomposition into a square, the sum of numbers enclosed in brackets is taken, that is, the value of this sum is multiplied by itself 2 times, and therefore is a multiplier.
• a 2 + 2ab - b 2 = (a - b) 2 - the formula for the square of the difference, it is similar to the previous one. The result is the difference, enclosed in parentheses, contained in the square power.
• a 2 - b 2 = (a + b)(a - b)- this is a formula for the difference of squares, since initially the polynomial consists of 2 squares of numbers or expressions, between which subtraction is performed. Perhaps, of the three mentioned, it is used most often.

Examples for calculations using square formulas

The calculations for them are quite simple. For example:

1. 25x 2 + 20xy + 4y 2 - use the formula “square of the sum”.
2. 25x 2 is the square of 5x. 20xy is the double product of 2*(5x*2y), and 4y 2 is the square of 2y.
3. Thus, 25x 2 + 20xy + 4y 2 = (5x + 2y) 2 = (5x + 2y)(5x + 2y). This polynomial is decomposed into 2 factors (the factors are the same, so it is written as an expression with a square power).

Actions using the squared difference formula are carried out similarly to these. The remaining formula is difference of squares. Examples of this formula are very easy to define and find among other expressions. For example:

• 25a 2 - 400 = (5a - 20)(5a + 20). Since 25a 2 = (5a) 2, and 400 = 20 2
• 36x 2 - 25y 2 = (6x - 5y) (6x + 5y). Since 36x 2 = (6x) 2, and 25y 2 = (5y 2)
• c 2 - 169b 2 = (c - 13b)(c + 13b). Since 169b 2 = (13b) 2

It is important that each of the terms is a square of some expression. Then this polynomial must be factorized using the difference of squares formula. For this, it is not necessary that the second degree be above the number. There are polynomials that contain large degrees, but still fit these formulas.

a 8 +10a 4 +25 = (a 4) 2 + 2*a 4 *5 + 5 2 = (a 4 +5) 2

In this example, a 8 can be represented as (a 4) 2, that is, the square of a certain expression. 25 is 5 2, and 10a is 4 - this is the double product of the terms 2 * a 4 * 5. That is this expression, despite the presence of degrees with large exponents, can be decomposed into 2 factors in order to subsequently work with them.

Cube formulas

The same formulas exist for factoring polynomials containing cubes. They are a little more complicated than those with squares:

• a 3 + b 3 = (a + b)(a 2 - ab + b 2)- this formula is called the sum of cubes, since in its initial form the polynomial is the sum of two expressions or numbers enclosed in a cube.
• a 3 - b 3 = (a - b)(a 2 + ab + b 2) - a formula identical to the previous one is designated as the difference of cubes.
• a 3 + 3a 2 b + 3ab 2 + b 3 = (a + b) 3 - cube of a sum, as a result of calculations, the sum of numbers or expressions is enclosed in brackets and multiplied by itself 3 times, that is, located in a cube
• a 3 - 3a 2 b + 3ab 2 - b 3 = (a - b) 3 - the formula, compiled by analogy with the previous one, changing only some signs of mathematical operations (plus and minus), is called the “difference cube”.

The last two formulas are practically not used for the purpose of factoring a polynomial, since they are complex, and it is rare enough to find polynomials that fully correspond to exactly this structure so that they can be factored using these formulas. But you still need to know them, since they will be required when operating in the opposite direction - when opening parentheses.

Examples on cube formulas

Let's look at an example: 64a 3 − 8b 3 = (4a) 3 − (2b) 3 = (4a − 2b)((4a) 2 + 4a*2b + (2b) 2) = (4a−2b)(16a 2 + 8ab + 4b 2 ).

Quite simple numbers are taken here, so you can immediately see that 64a 3 is (4a) 3, and 8b 3 is (2b) 3. Thus, this polynomial is expanded according to the formula difference of cubes into 2 factors. Actions using the formula for the sum of cubes are carried out by analogy.

It is important to understand that not all polynomials can be expanded in at least one way. But there are expressions that contain greater powers than a square or a cube, but they can also be expanded into abbreviated multiplication forms. For example: x 12 + 125y 3 =(x 4) 3 +(5y) 3 =(x 4 +5y)*((x 4) 2 − x 4 *5y+(5y) 2)=(x 4 + 5y)( x 8 − 5x 4 y + 25y 2).

This example contains as much as the 12th degree. But even it can be factorized using the sum of cubes formula. To do this, you need to imagine x 12 as (x 4) 3, that is, as a cube of some expression. Now, instead of a, you need to substitute it in the formula. Well, the expression 125y 3 is a cube of 5y. Next, you need to compose the product using the formula and perform calculations.

At first, or in case of doubt, you can always check by inverse multiplication. You just need to open the parentheses in the resulting expression and perform actions with similar terms. This method applies to all of the reduction methods listed: both to working with a common factor and grouping, and to working with formulas of cubes and quadratic powers.

Sections: Mathematics

Lesson type:

• according to the method of delivery - a workshop lesson;
• By didactic purpose– a lesson in applying knowledge and skills.

Target: develop the ability to factor a polynomial.

Tasks:

• Didactic: systematize, expand and deepen students’ knowledge and skills, apply various methods of factoring a polynomial. Develop the ability to apply polynomial factorization through combination various techniques. Implement knowledge and skills on the topic: “Factoring a polynomial” to complete tasks at both the basic level and tasks of increased complexity.
• Developmental: to develop mental activity through solving various types of problems, to learn to find and analyze the most rational methods of solution, to contribute to the formation of the ability to generalize the facts being studied, to express one’s thoughts clearly and clearly.
• Educational: develop skills of independent and team work, self-control skills.

Working methods:

• verbal;
• visual;
• practical.

Lesson equipment: interactive whiteboard or overhead projector, tables with abbreviated multiplication formulas, instructions, handouts for working in groups.

Lesson structure:

1. Organizing time. 1 minute
2. Formulating the topic, purpose and objectives of the practical lesson. 2 minutes
3. Checking homework. 4 minutes
4. Updating the basic knowledge and skills of students. 12 minutes
5. Physical education minute. 2 minutes
6. Instruction on how to complete the tasks of the workshop. 2 minutes
7. Doing tasks in groups. 15 minutes
8. Checking and discussing assignments. Job analysis. 3 minutes
9. Setting homework. 1 minute
10. Reserve jobs. 3 minutes

During the classes

1. Organizational moment

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

2. Formulating the topic, purpose and objectives of the workshop lesson

• Message about the final lesson on the topic.
• Motivation for students' learning activities.
• Formulating the goal and setting objectives for the lesson (together with students).

3. Checking homework

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

Teacher:

Checks the presence of homework in students' notebooks.

Invites class students to answer the question: “What difficulties did completing the task cause?”

Offers to check your solution with the solution on the board.

Invites students at the board to answer questions that students have on the spot when checking using samples.

Comments on student answers, supplements answers, and clarifies (if necessary).

Summarizes homework completion.

Students:

Present homework to the teacher.

They exchange 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 the students and the teacher for the necessary explanations.

Find ways to verify the results obtained.

Participate in assessing the quality of tasks performed at the board.

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. Which 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 – 2хy – y 2

4. x 3 - 3x – 2

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

• Option I – by applying a common factor;
• Option II – using abbreviated multiplication formulas;
• Option III - by grouping method.

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

3. Warm up

The teacher gives instructions to factor and select the letter associated with the correct answer. By adding the letters you 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 lesson Statements are read to students. If the statement is true, then students should raise their hands up, and if it is false, then sit down at their desks. (Appendix 2)

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

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

When factoring a polynomial, the following order must be observed:

1. put the common factor out of brackets (if there is one);

2. apply abbreviated multiplication formulas (if possible);

3. apply the grouping method;

4. check the result obtained by multiplication.

Teacher:

Presents instructions to students (focuses on step 4).

Offers the completion of workshop assignments in groups.

Distributes worksheets to groups, sheets with carbon paper for preparing assignments in notebooks and their subsequent checking.

Sets time for working in groups and working in notebooks.

Students:

Read the instructions.

The teachers listen attentively.

Seated in groups (4-5 people).

Getting ready to do practical work.

7. Doing tasks in groups

Worksheets with tasks for groups. (Appendix 3)

Teacher:

Manages independent work in groups.

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

Students:

Complete tasks on sheets of carbon paper included in the workbook.

Discuss ways to make rational decisions.

Prepare a worksheet from the group.

Prepare to defend completed work.

8. Checking and discussing the completion of the task

Answers on the interactive board.

Teacher:

Collects copies of decisions.

Manages student reporting on worksheets.

Offers self-assessment of your work, comparing answers from notebooks, worksheets and samples on the board.

Reminds me of the criteria for assigning marks for work and 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.

It says that the final results will be summed up after checking copies of the work completed by students.

Students:

Give copies to the teacher.

Worksheets are attached to the board.

Report on the completion of work.

Carry out self-examination and self-assessment of work performance.

9. Setting homework

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

Teacher:

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

Gives a comment on its implementation.

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

Tells you to prepare for the next review lesson