Lesson plan: quadratic trinomial and its roots. Summary of a lesson in mathematics "Square trinomial and its roots." The path of reflection is the noblest path

The topic “Square trinomial and its roots” is studied in the 9th grade algebra course. Like any other mathematics lesson, a lesson on this topic requires special teaching tools and methods. Visibility is necessary. One of these is this video tutorial, which was designed specifically to make the teacher’s work easier.

This lesson lasts 6:36 minutes. During this time, the author manages to reveal the topic completely. The teacher will only have to select tasks on the topic to reinforce the material.

The lesson begins by showing examples of polynomials with one variable. Then the definition of the root of the polynomial appears on the screen. This definition is supported by an example where it is necessary to find the roots of a polynomial. Having solved the equation, the author obtains the roots of the polynomial.

The following is a remark that quadratic trinomials also include those polynomials of the second degree in which the second, third, or both coefficients, except the leading one, are equal to zero. This information is supported by an example where the free coefficient is zero.

The author then explains how to find the roots of a quadratic trinomial. To do this, you need to solve a quadratic equation. And the author suggests checking this using an example where a quadratic trinomial is given. We need to find its roots. The solution is constructed based on the solution to the quadratic equation obtained from the given quadratic trinomial. The solution is written on the screen in detail, clearly and understandably. In the course of the decision this example the author remembers how to solve a quadratic equation, writes down the formulas, and gets the result. The answer is recorded on the screen.

The author explained finding the roots of a square trinomial based on an example. When students understand the essence, they can move on to more general points, which is what the author does. Therefore, he further summarizes all of the above. In general terms In mathematical language, the author writes down the rule for finding the roots of a square trinomial.

The following is a remark that in some problems it is more convenient to write the quadratic trinomial a little differently. This entry is shown on the screen. That is, it turns out that from a square trinomial one can extract the square of a binomial. It is proposed to consider such a transformation with an example. The solution to this example is shown on the screen. As in the previous example, the solution is constructed in detail with all the necessary explanations. The author then considers a problem that uses the information just given. This is a geometric proof problem. The solution contains an illustration in the form of a drawing. The solution to the problem is described in detail and clearly.

This concludes the lesson. But the teacher can select tasks based on the students’ abilities that will correspond to the given topic.

This video lesson can be used as an explanation of new material in algebra lessons. It's perfect for self-study students for the lesson.

Lesson topic:

The purpose of the lesson:

Systematize students’ knowledge and skills in applying formulas for factoring a quadratic trinomial. Learn to use formulas when reducing fractions;

Promote the development of observation, the ability to analyze, compare and draw conclusions;

Encourage students to self-control and self-analysis educational activities.

Equipment: computer, interactive whiteboard, training cards, assessment sheets, hearts, answer sheets, tests.

Lesson epigraph:

Three paths lead to knowledge:

The path of reflection is the noblest path;

The path of imitation is the easiest path;

The path of experience is the most bitter path.

Confucius.

Lesson plan:

Organizational stage.

Hearts

Score sheets

Epigraph of the lesson

Lesson Plan

Updating basic knowledge:

A) glossary: ​​What terms have you come across in last lesson?

Square trinomial...

Decomposition of a square trinomial into factors... (we will write the formula for the decomposition of a square trinomial on the board).

IN) Oral work:

We write down only the answers on the answer sheets.

1. What is it equal to Square root numbers:

2. Indicate the coefficients of the trinomial

Reduce a fraction: a) (x + 6)(x – 1) b) X 2 + 3x + 2

X 2 – 5x + 6 x + 1

(Let's check the work, give ourselves a grade for the oral work).

Which task did you have difficulty solving?

Students' answer (the last task, it was necessary to factorize)

This leads to the topic of the lesson: Factoring a quadratic trinomial.

Now each of you will set the goal of the lesson.

Student answers.

In the notebooks they wrote down the date, class work, and topic of the lesson.

3. Consolidation stage:

1) Working with the textbook

Find level B. No. 235 (1 and 2) on page 79. Read the task. How will we decide? (Disassemble completely). We do it ourselves. We write in a notebook, observing the rule of recording decisions.

Now we exchanged notebooks and check the correctness of the solution with the solution on the board.

Let's rate our neighbor and write our full name next to it.

2) Physical exercise (voluntary movements to the beat of music).

3) Work in groups. (Divide into groups based on the color of the hearts).

In front of each of you are training cards of multi-level tasks.

Explore. Complete the tasks, following the algorithm for factoring a quadratic trinomial (we do it, starting with the easiest, moving to a more complex level, helping each other)).

Completed and checked with the answers on the board. We rated each group member together.

4) work in groups. No. 237 (1-2). We do it quickly. Right. Beautiful.

The first person to complete writes on the board. What property are we using?

(The main property of a fraction.)

We give ratings together.

And now everyone quickly sat down.

Lesson summary:

The show game “Taxi” will help us summarize the lesson. All students participate.

Rules of the game: You have 2 lives and two clues.

If you make two mistakes, you do not receive a grade for the lesson.

Two tips:

1 hint “Help from a classmate”

Tip 2 “Teacher Help”

Tests in front of you (3 min).

We exchanged sheets of paper. We checked our neighbor's answers.

We will give a rating to our neighbor on the score sheet. Answers on the board.

5.Ratings

Now everyone will give themselves a grade for the lesson based on the evaluation sheet (show the arithmetic average of the grades on the evaluation sheet). And give the sheets to me.

6.D/Z №235 (3-4), 237(4-6)

7. Reflection. Answer the questions. Questions on the board

What did you take from the lesson?

What did you secure?

What is a quadratic function7

What to study for the next lesson.

And now everyone will give themselves a grade for the lesson according to the evaluation sheet (derive the arithmetic average of the grades for the lesson). And give the sheets to me.

Student evaluation sheet ___

Surname____________________

Name _______________

Lesson topic: “Square trinomial. Factoring a quadratic trinomial."

The purpose of the lesson: consolidate students' knowledge of applying the formula for factoring a quadratic trinomial.

Test for 8th grade.

Full name student(s)_____________________

Subject: Square trinomial. Factoring a quadratic trinomial.

The purpose of the lesson: test students' knowledge of applying the formula for factoring a quadratic trinomial.

Underline the correct answer.

I.Theory

A square trinomial is called...

A. ...a monomial of the form ax 2, where x is a variable, a, a coefficient.

IN....a polynomial of the form ax 2 + inx + c, where x is a variable, a, b, c, coefficients, and a≠0

WITH. ...a polynomial of the form ax 2 + inx + c, where x is a variable, a, b, c, coefficients, and a = 0

D... equation that can be factorized

If a quadratic trinomial has roots, then...

A....it is factorized.

IN. ...then it cannot be factorized.

WITH. ... then it has one root.

D. ... then it is a polynomial.

3) If a square trinomial can be factorized, then...

A. ...it has one root.

IN. ...that is a monomial.

WITH. ... then it has roots.

D. ... then it is a polynomial.

II.Practice

Factor the quadratic trinomial x 2 – 4x + 3

A. (x – 3)(x + 1)

IN. (x – 5)(x - 1)

WITH. (x – 3)(x - 1)

D. (x + 3)(x + 1)

Which numbers are the roots of a square trinomial

x 2 + 2x – 3

A. x 1 = 1; x 2 = 4

IN. x 1 = 2; x 2 = -3

WITH. x 1 = -1; x 2 = 3

D. x 1 = 1; x 2 = -3

3) Reduce the fraction: X 2 + x - 42

A. x – 6 IN. x - 6 WITH. x + 7 D. x + 7

2 Lesson Objectives: Generalization of properties quadratic function Establishing connections with the most difficult questions of theory (solving inequalities, equations containing a module, parameter) Show examples of using the studied material in solving tasks Test knowledge and skills using a test

“The path to truth is difficult, and therefore daring courage is needed in pure thinking no less than in mountain climbers.” Stage 1 plan. History of quadratic equations. Stage 1. History of quadratic equations. Stage 2. Reproduction of repeated material. Stage 2. Reproduction of repeated material. Stage 3. Systematization and generalization of previously studied. Stage 3. Systematization and generalization of previously studied. Stage 4. Deepening and expanding knowledge. Stage 4. Deepening and expanding knowledge. 3

History of Quadratic Equations The general method for solving quadratic equations was discovered by Indian mathematicians. So, in the 12th century AD. Indian mathematician Bhaskara for general equation ax 2 +bx+c=0 found a solution in the form: X= Moreover negative roots he didn't take it into account.

Stage 2. Reproduction of the material covered 1. Factor the square trinomial: 2x 2 -x-1, we get: a) 2(x-0.5)(x+1); b) (x+0.5)(x-1); c) (2x+1)(x-1); d) (x-0.5)(x+1); e) (2x+1)(2x-2). 2. Let us denote by x 1 and x 2 the larger and smaller roots of the equation 108x 2 -21x+1=0, respectively. Then x 1 - x 2 is equal to: e) 1/12; g) 5/12; h) 1/36; i) 36; j) The graph of the function y=-x 2 -4 is located in the coordinate quarters: o) 1 and 2; n) 2; p) 3 and 4; c) 1 and The vertex of the parabola y=-x 2 -4x+1 is the point with coordinates: k) (2;-5); l) (-4;1); n) (-2;5). 5. Solve the inequality: -x 2 +7x-120 о) (-;3] U р) (-;-4] U [-3;+) 8 CORRECT

Stage 3. Systematization and generalization of previously studied. 1. Find the coordinates of the points of intersection of the parabola y=5x 2 +10x+7 with the coordinate axes and the coordinates of the vertex of the parabola. 3. Find highest value expressions 3-(5+x) 2 4. Create a quadratic equation whose roots are twice as large as the roots of the equation x 2 +x+2=0 2. Calculate the value of the expression x 2 -36x+63 at x=37.

Answers: The Ox axis does not intersect; Oy axis at point (0;7). Vertex coordinates (-1;2) The required equation cannot be compiled, since the original one has no roots.