Lesson plan square trinomial and its roots. Summary of the lesson in mathematics "Square trinomial and its roots." Solving equations and systems
Lesson topic:
The purpose of the lesson:
To systematize the knowledge and skills of students in applying the formulas for factoring a square trinomial into factors. Learn how to use formulas when reducing fractions;
To promote the development of observation, the ability to analyze, compare and draw conclusions;
Encourage students to self-control, self-analysis of their educational activities.
Equipment: computer, interactive whiteboard, training cards, evaluation sheets, hearts, answer sheets, tests.
Epigraph of the lesson:
Three paths lead to knowledge:
The way of reflection is the most noble way;
The way of imitation is the easiest way;
The path of experience is the most bitter path.
Confucius.
Lesson plan:
organizational stage.
hearts
Evaluation sheets
Epigraph of the lesson
Lesson plan
Updating of basic knowledge:
A) glossary: What terms did you meet in the last lesson?
The square trinomial...
Decomposition of a square trinomial into factors ... (we write the formula for the decomposition of a square trinomial on the board).
C) Oral work:
Write down the answers on the answer sheets.
1. What is the square root of the number:
2. Specify the coefficients of the trinomial
| Square trinomial | ||||
| 3y 2 - 5y + 1 | ||||
Reduce the fraction: a) (x + 6)(x - 1) b) X 2 + 3x + 2
X 2 - 5x + 6 x + 1
(Let's check the work, put ourselves a grade for the oral work).
What task did you have difficulty with?
Students' answer (the last task, it was necessary to decompose into factors)
From this follows the topic of the lesson: Decomposition of a square trinomial into factors.
Now each of you will set the goal of the lesson.
Student responses.
In notebooks they wrote down the number, class work, the topic of the lesson.
3. Fixing stage:
1) Work with the textbook
Find on page 79 level V. No. 235 (1 and 2). Let's read the task. How will we decide? (Disassemble completely). We do it ourselves. We write in a notebook, observing the rule of writing decisions.
Now we exchanged notebooks, we check the correctness of the solution with the solution on the board.
| Square trinomial | Discriminant | The roots of a square trinomial | Factorization of a square trinomial |
|
| 6x 2 - 5x + 1 | x = ½, x = 1/3 | 6x 2 - 5x +1 \u003d 6 (x-1/2) (x-1/3) |
||
| x = - 1/5, x = 1 | 5x 2 + 4x + 1-5 (x + 1/5) (x - 1) |
Let's rate the neighbor, write your full name next to it.
2) Fizminutka (voluntary movements to the beat of the music).
3) Work in groups. (The color of the hearts will be divided into groups).
In front of each of you are training cards of multi-level tasks.
Explore. Complete the tasks following the factorization algorithm of the square trinomial (we do it, start from the easiest, move to a more difficult level, help each other)).
Completed, checked with answers on the board. Put the assessment together, to each member of the group.
4) work in groups. No. 237 (1-2). We do it quickly. Correctly. Handsomely.
The first one to complete writes down at the blackboard. What property are we using.
(The main property of a fraction.)
Estimates are made together.
And now everyone quickly sat down in their places.
Lesson Summary:
The show-game "Taxi" will help us to summarize the lesson. All students participate.
Game rules: You have 2 lives and 2 clues.
If you make two mistakes, you don't get a mark for the lesson.
Two hints:
1 hint "Help from a classmate"
2 clue "Teacher Help"
Tests in front of you (3 min).
They exchanged sheets. We checked the neighbor's answers.
We will put an assessment on the neighbor on the evaluation sheet. Answers on the board.
5.Ratings
Now everyone will give himself a grade for the lesson on the grade sheet (display the arithmetic average of grades on the grade sheet). And pass 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 have you fixed?
What is a quadratic function7
What to study for the next lesson.
And now everyone will give himself a mark for the lesson according to the evaluation sheet (display the arithmetic mean of the marks for the lesson). And pass the sheets to me.
Student score sheet ___
Surname____________________
Name _______________
Lesson topic: "Square trinomial. Factorization of a square trinomial”.
The purpose of the lesson: consolidate students' knowledge of applying the formula for factoring a square trinomial into factors.
| Exercise | grade | F.i. student who graded |
|
| oral work | |||
| Group work by training cards | For activity | ||
| For correctness | |||
| For activity | |||
| For correctness | |||
| Final mark for the lesson | |||
Test for grade 8.
F.i. student(s) _____________________
Topic: Square trinomial. Factorization of a square trinomial.
The purpose of the lesson: to test students' knowledge on the application of the formula for factoring a square trinomial into factors.
Underline the correct answer.
I.Theory
It is called a square trinomial ....
BUT. ... a monomial of the form ax 2, where x is a variable, a is a coefficient.
AT.... a polynomial of the form ax 2 + inx + c, where x is a variable, a, b, c, coefficients, and a≠0
FROM. ... a polynomial of the form ax 2 + inx + c, where x is a variable, a, b, c, coefficients, and a \u003d 0
D. ... an equation that is factored
If a square trinomial has roots, then...
BUT.…it factorizes.
AT. …then it cannot be factorized.
FROM. ... then it has one root.
D. … then it is a polynomial.
3) If the square trinomial is factorized, then ...
BUT. ... it has one root.
AT. …that is a monomial.
FROM. … then it has roots.
D. ... then it is a polynomial.
II.Practice
Factor the square trinomial x 2 – 4x + 3
BUT. (x - 3)(x + 1)
AT. (x - 5) (x - 1)
FROM. (x - 3) (x - 1)
D. (x + 3)(x + 1)
Which of the numbers are the roots of a square trinomial
x 2 + 2x - 3
BUT. x 1 = 1; x 2 = 4
AT. x 1 = 2; x 2 = -3
FROM. x 1 \u003d -1; x 2 = 3
D. x 1 = 1; x 2 = -3
3) Reduce the fraction: X 2 + x - 42
BUT. x - 6 AT. x - 6 FROM. x + 7 D. x + 7
ALGEBRA
All lessons for grade 8
Lesson #63
Topic. The final lesson on the topic “Square trinomial.
Solving equations that reduce to quadratic equations and their use for solving text problems "
Purpose: to repeat, systematize and generalize the knowledge and skills of students regarding the possibility and methods of applying the solution of a quadratic equation for decomposing a square trinomial into linear factors, solving biquadratic and fractional-rational equations, as well as text problems of physical and geometric meaning.
Type of lesson: systematization and generalization of knowledge and skills.
Visibility and equipment: reference notes.
During the classes
I. Organizational stage
II. Checking homework
To save time, only the exercises on the application of the algorithm learned in the previous lesson are subject to thorough verification.
III. Formulation of revenge and lesson tasks, motivation of students' learning activities
The main didactic goal and tasks for the lesson quite logically follow from the place of the lesson in the topic - since the lesson is the last, final, the question of repetition, generalization and systematization of knowledge and skills acquired by students in the course of studying the topic is important. This formulation of the goal creates the appropriate motivation for the activities of students.
IV. Repetition and systematization of knowledge
@ Depending on the level of preparation of students, the teacher can organize their work in different ways: either as independent work with theoretical material (for example, repeat the content of the main concepts of the topic after a textbook or abstract of theoretical material, or draw up a diagram reflecting the logical connection between the main concepts of the topic, etc.), or traditionally conduct a survey (in the form of an interactive exercise) with the main questions of the topic.
Doing oral exercises
1. What polynomial is called a square trinomial? Give examples.
2. Name the coefficients of the square trinomial.
3. What is called the root of a square trinomial?
4. How many roots does a square trinomial have if its discriminants are:
a) greater than zero; b) equal to zero; c) less than zero?
5. Give examples of equations that reduce to square ones.
6. What is the plan for solving the equation:
a) x4 - 3x2 + 2 = 0; b) (x - 3)2 + 2(x - 3) + 1 = 0; in) .
7. What is the plan for solving the problem of compiling an equation?
V. Repetition and systematization of skills
@ Usually this stage of the lesson is carried out in the form of group work, the purpose of which is for students to formulate and test a generalized scheme of actions that they must follow in solving typical problems, similar to which they will be put under control.
For example, typical tasks of the topic “Square trinomial. Solving equations that reduce to quadratic equations and their use for solving text problems "tasks:
find the roots of a square trinomial and factorize the square trinomial according to the formula;
Reduce a rational fraction, the numerator and (or) denominator of which contains square trinomials, having previously factored them according to the formula;
· to solve biquadratic (fractional-rational, equations of higher degree), which is reduced to a quadratic one according to a certain algorithm;
Compose and solve in accordance with the conditions of the text problem, the equation is reduced to a quadratic one.
After compiling a list of the main types of tasks, the teacher combines students into working groups (by the number of types of tasks) and the tasks of each of the groups are formulated as “Compose an algorithm for solving the problem ...” (each of the groups receives an individual task). Each of the groups is given a certain time to compile the algorithm, during which the group members must compile the algorithm, write it down in the form of successive steps, and prepare a presentation of their work. At the end there is a presentation of the work performed by each of the groups. After the presentation - a mandatory test of the algorithms: moreover, it is desirable that the groups exchange algorithms and check their application not on one, but on several tasks. After the test - mandatory correction and summing up.
VI. Lesson summary
The result of the lesson of generalization and systematization of students' knowledge and skills is, firstly, the generalized schemes of actions compiled by the students themselves in solving typical problems, and secondly, the implementation by students of the necessary part of conscious mental activity - reflection - reflection by each student of personal perception of success, and most importantly - Issues that still need to be worked on.
VII. Homework
1. Study the algorithms compiled in the lesson.
2. Using the compiled algorithms, complete homework assignments.
Home test
1. The perimeter of a rectangle is 20 cm. Find its sides if its area is 24 cm2.
2. The path from point A to point B, which is 20 km, the tourist must overcome in a certain time. However, he was delayed with an exit for 1 hour, so he was forced to increase the speed by 1 km / h in order to eliminate the delay. With what initial speed should the tourist move?
3. Solve the equation:
a) 9x4 - 37x2 + 4 = 0;
b) (x2 - 2x)2 - 3(x2 - 2x) - 4 = 0;
c) (x - 4) (x - 3) (x - 2) (x - 1) = 24;
G) 
; e)* x2 - 7|x| + 6 = 0.
4. Through one pipe, you can fill the pool 9 hours faster than emptying this pool through the second pipe. If both pipes are turned on at the same time, then the pool will be filled in 40 hours. How many hours can the first pipe fill and the second pipe empty the pool?
2 Lesson objectives: Generalization of the properties of a quadratic function Establishing a connection with the most difficult issues of theory (solving inequalities, equations containing a module, a parameter) Show examples of using the studied material in the course of solving tasks Check knowledge and skills using a test
“The path to truth is difficult, and therefore, in pure thinking, daring courage is needed no less than climbers.” Plan 1 stage. History of quadratic equations. Stage 1. History of quadratic equations. Stage 2. Playing repeatable material. Stage 2. Playing repeatable material. Stage 3. Systematization and generalization of the previously studied. Stage 3. Systematization and generalization of the previously studied. Stage 4. Deepening and expanding knowledge. Stage 4. Deepening and expanding knowledge. 3
History of quadratic equations A general method for solving quadratic equations was discovered by Indian mathematicians. So, in the 12th century AD. Indian mathematician Bhaskara for the general equation ax 2 +bx+c=0 found a solution in the form: X= Moreover, he did not take into account negative roots.
Stage 2. Reproduction of the material covered 1. Factoring 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. Denote by x 1 and x 2, respectively, the larger and smaller roots of the equation 108x 2 -21x + 1 \u003d 0. Then x 1 -x 2 is: e) 1/12; g) 5/12; h) 1/36; i) 36; j) The graph of the function y \u003d -x 2 -4 is located in the coordinate quarters: o) 1 and 2; n) 2; p) 3 and 4; c) 1 and The top of the parabola y \u003d -x 2 -4x + 1 is a point with coordinates: k) (2; -5); k) (-4; 1); m) (-2; 5). 5. Solve the inequality: -x 2 + 7x-120 o) (-; 3] U p) (-; -4] U [-3; +) 8 TRUE
Stage 3. Systematization and generalization of the previously studied. 1. Find the coordinates of the points of intersection of the parabola y=5x2 +10x+7 with the coordinate axes and the coordinates of the parabola vertex. 3. Find the largest value of the expression 3-(5+x) 2 4. Compose a quadratic equation whose roots are twice 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 cross; the y-axis at the point (0;7). Vertex coordinates (-1; 2) The required equation cannot be made, since the original one has no roots.
The topic "Square trinomial and its roots" is studied in the 9th grade algebra course. like any other math lesson, a lesson on this topic requires special tools and teaching methods. Visibility is needed. This includes this video lesson, which is designed specifically to facilitate the work of the teacher.
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 in order to consolidate the material.
The lesson begins by showing examples of polynomials in 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.
This is followed by the remark that square trinomials also include such polynomials of the second degree, in which the second, third, or both coefficients, except for the highest one, are equal to zero. This information is supported by an example where the free factor is zero.
The author then explains how to find the roots of a square trinomial. To do this, you need to solve a quadratic equation. And the author suggests checking this with an example where a square trinomial is given. We need to find its roots. The solution is built on the basis of the solution of 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 solving this example, the author remembers how a quadratic equation is solved, writes down the formulas, and gets the result. The answer is written on the screen.
The author explained finding the roots of a square trinomial based on an example. When students understand the essence, then you can move on to more general points, which 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 remark follows that in some problems it is more convenient to write the square trinomial in a slightly different way. This entry is shown on the screen. That is, it turns out that the square of the binomial can be distinguished from the square trinomial. 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 built in detail with all the necessary explanations. Then the author considers the problem, where the information just given is used. This is a geometric proof problem. The solution contains an illustration in the form of a drawing. The solution to the problem is detailed and clear.
This concludes the lesson. But the teacher can choose, according to the abilities of the students, tasks that will correspond to this topic.
This video lesson can be used as an explanation of new material in algebra lessons. It is perfect for self-preparation of students for the lesson.
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