The lesson "Square threesthals and its roots. Summary of the lesson in mathematics "Square Treshlen and its roots" Theme: "Square Treshlen and its roots"
ALGEBRA
All lessons for 8 classes
Lesson number 63.
Subject. The final lesson on the topic "Square threechel.
Solving equations reduced to square equations and their use for solving text tasks "
Purpose: repeat, systematize and summarize the knowledge and skill of students regarding the possibility and methods of applying the solution of a square equation for decomposition of a square three decisions on linear multipliers, solutions of bіkvadrichi and fractional rational equations, as well as textual problems of physical and geometric meaning.
Type of lesson: systematization and synthesis of knowledge and skills.
Visuality and equipment: reference abstracts.
During the classes
I. Organizational stage
II. Checking homework
In order to save time, only exercises on the use of an algorithm studied in the previous lesson are subject to careful check.
III. The wording of the second and tasks of the lesson, the motivation of educational activities of students
The main didactic goal and the task of the lesson is quite logical to flow out of the lesson in the subject - since the lesson is the last, final, then the question of repeating, generalization and systematization of knowledge and skills acquired by students during the study of the topic is important. This formulation of the target creates appropriate motivation of students.
IV. Repetition and systematization of knowledge
@ Depending on the level of students' training, their work teacher can be organized in different ways: either as an independent work with theoretical material (for example, a textbook or abstract of theoretical material to repeat the content of the main concepts of the topic or to make 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 subjects of the topic.
Performing oral exercises
1. What polynomial is called square trichene? Give examples.
2. Name the coefficients of the square three decar.
3. What is called the root of square three-shoes?
4. How many roots has a square triple, if its discriminants:
a) more zero; b) equal to zero; c) less zero?
5. Give examples of equations that are tailored to square.
6. What a plan to solve the equation:
a) x4 - 3x2 + 2 \u003d 0; b) (x - 3) 2 + 2 (x - 3) + 1 \u003d 0; in) .
7. What plan is the solution to the problem of drawing up the equation?
V. Repetition and systematization of skills
@ Typically, this stage of the lesson is carried out in the form of group work, the purpose of which is that students themselves formulate themselves and experienced a generalized scheme of actions that they must follow in solving typical tasks like which will be carried out to control.
For example, typical tasks of the topic "Square three-melen. Solving equations that are reduced to square equations and their use for solving text tasks "Tasks:
· Find the roots of square three declections and decompose the square threefold on the factors by the formula;
· Reduce the rational fraction, numerator and (or) the denominator of which contains square trichenes, decomposing them pre-for multipliers by the formula;
· Solve bіkvadtrat (fractional rational, higher degree equation), which comes down to a square according to a specific algorithm;
· Compile and decide in accordance with the conditions of the text problem, the equation comes down to square.
After drawing up the list of basic types of tasks, the teacher unites students in working groups (by the number of types of tasks) and the task of each group is formulated as "draw up an algorithm for solving the problem ..." (each group receives an individual task). The compilation of the algorithm for each of the groups is given a certain time for which the participants of the group must compile an algorithm, write it in the form of consecutive steps, prepare a presentation of their work. At the end, a presentation of the work performed by each group occurs. After presentation - a mandatory test of algorithms: it is desirable that the groups will be exchanged algorithms and checked their use on one, but on several tasks. After testing - mandatory correction and summing up.
Vi. The results of the lesson
The result of the lesson of generalization and systematization of knowledge and skills of students is, first, compiled by the students themselves generalized schemes of actions in solving typical tasks, and secondly - the implementation of the necessary part of conscious mental activity - reflection - reflections by each student of personal perception of success, and most importantly - Problems over which still work.
VII. Homework
1. Examine the algorithms compiled in the lesson.
2. Using the compiled algorithms, perform the tasks of home testing.
Home testing
1. The perimeter of the rectangle is 20 cm. Find it the parties if its area is 24 cm2.
2. The path from the point A to the point B, which is 20 km, the tourist must overcome for a certain time. However, it was detained with a 1 hour exit, so he was forced to increase the speed of 1 km / h to eliminate lateness. How starting speed should the tourist move?
3. Decide the equation:
a) 9x4 - 37x2 + 4 \u003d 0;
b) (x2 - 2x) 2 - 3 (x2 - 2x) - 4 \u003d 0;
c) (x - 4) (x - 3) (x - 2) (x - 1) \u003d 24;
d) 
; e) * x2 - 7 | x | + 6 \u003d 0.
4. Through one pipe, you can fill the pool for 9 hours faster than through the second empty this pool. If you have both pipes at the same time, the pool will be filled in 40 hours. How many hours is the first pipe can be filled, and the second is to empty the pool?
Development of a lesson on a single-level cycle technology on the topic:
"Square three-half and its roots" in grade 9 in the textbook of authors Makarychev Yu.N., Mindyuk N.G. et al. (Development author - E.A.Beshmel)
Theme lesson : "Square three-half and its roots."
The purpose of the lesson : To acquaint students with the concept of square three-shredded and its roots, improve their skills and skills in solving tasks for the selection of the square of the square of the square three declections.
The lesson includes four main stages:
- Knowledge control
- Explanation of the new material
- Reproductive consolidation.
- Training consolidation.
- Reflection.
Stage 1. Knowledge control.
The teacher conducts a mathematical dictation "under the copy" by the material of the previous cycle. For dictation, the cards of two colors are used: blue - for 1 option, red -2 \u200b\u200boption.
Tasks.
- From these analytical models of functions, select only quadratic.
Embodiment 1. U \u003d ah + 4, y \u003d 45-4x, y \u003d x² + 4x-5, y \u003d x³ + x²-1.
Embodiment 2. U \u003d 8x-B, y \u003d 13 + 2x, y \u003d -x² + 4x, y \u003d -x³ + 4x²-1.
- Picture schematic quadratic functions. It is possible to unambiguously determine the position of the quadratic function on the coordinate plane. Answer Try to argue.
- Decide square equations.
Option 1. A) x² + 11x-12 \u003d 0
B) x² + 11x \u003d 0
Option 2. A) x² -9x + 20 \u003d 0
B) x 20 x \u003d 0
4. Not solving equations, find out whether it has roots.
Option 1. A) x² + x + 12 \u003d 0
Option 2. A) x² + x - 12 \u003d 0
The teacher's answers received checks the first two pairs. The incorrect answers received are discussed by the whole class.
Answers.
2 stage . Let's make a cluster. What associations do you have when considering square three-melan?
Compilation of cluster.
? ?
Square threechlen
Possible answers:
- square three-stakes are used to consider the rank. functions;
- you can find zeros sq. Functions
- the value of the discriminant estimate the number of roots.
- Describe real processes, etc.
Explanation of a new material.
Paragraph 2. Clause 3 p. 19-22.
Expressions are considered, and the definition of square three-shredded and the root of the polynomial (during the discussion of the previously reviewed expressions)
- The determination of the root of the polynomial is formulated.
- The definition of square three declared is formulated.
- Examples of the solution of three decisions are distinguished:
- Find the roots of square three-shoes.
3x² + 4x-5 \u003d 0
- We highlight the square of the bouncer of the square three decar.
3x²-36x + 140 \u003d 0.
- A diagram of the estimated basis of action is drawn up.
The algorithm for the release of twisted of the square three decletes.
1. Place the numerical value of the older square coefficientthree.
A ≠ 1 A \u003d 1
2. Perform identical and 2. Convert an expression,
Equivalent transformations using formula
(To make a common factor behind the brackets; Square amount and difference.
convert expression in brackets
Completing it to the sum of the sum of the sum
Or difference)
Remember!
A² + 2av + c² \u003d (a + c) ² a²-2Av + c² \u003d (A-B) ²
3 stages . The solution of typical tasks from the textbook (No. 60 A, B; 61 A, 64 A, B) are made at the board and comment.
4 stages . Independent work at 2Variant (No. 60a, b; 65 A, b). Students are checked with sample solutions on the board.
Homework: P.3 (the theory to learn, № 56, 61g, 64 g)
Reflection . The teacher gives the task: to evaluate your progress at every stage of the lesson using the drawing and pass the teacher. (The task is performed on separate sheets, the sample is issued).
Sample: ignorance
1 stage lesson
2 stage lesson
3 stage lesson
4 stage lesson
Using, the order of the elements in the figure, determine at what stage of the lesson your ignorance prevailed. Highlight this stage in red.
Designer of the lesson of mathematics: micromoduli.
p \\ P. | Sections of the lesson | The main functional blocks micromoduli |
Beginning of the lesson | Mathematical dictation |
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Oral work. Actualization of reference knowledge. Setting the goals of the lesson | Compilation of cluster |
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Explanation of the new material | Problem dialogue (discussion of cluster compilation results) |
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Fastening, training | Interconnection |
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Development of skills and skills | Commented solution tasks |
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Systematic repetition | Indicative answer |
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Control | Work with operational verification |
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Homework | Discussing homework |
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End of lesson (reflection) | Survey-result |
Project Educational Situation
common data | ||
Full Name | Foreignless Elena Aleksandrovna |
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Academic subject | Mathematics |
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Educational theme (When choosing the topic, make a link to the number page "Fundamental core ...") | Square threesthal and its roots |
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Pupil age (class) | Grade 9. |
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Planned results of studying the educational topic (When describing / specifying planned results, you can use the wording of the abilities of human qualities of the 21st century) | ||
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MetaPermet |
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Subject |
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Educational situations, the activities of students in which will lead to the achievement of planned results | ||
(Below write a brief annotation of an academic situation) | (specify the planned results of studying the topic for the proposed learning situation) |
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6.1. Beginning of the lesson: Situation 1. Teacher: Today we will continue to get acquainted with square three-stale. And so that our work is productive, let's remember everything that we need today. Each row is envelopes with tasks. Tasks for repetition of the material passed. | Personal : Productive work in a pair; Communicative skills. Metapled : Creativity and curiosity; The ability to analyze I. solve the delivered problem Subjects: Square Three Pile |
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6.2. Situation 2. On the basis of the results received and voiced by the student, the teacher and students are cluster. During this work, students remember all the information about the square threestyle. Next, the teacher formulates the concept of square three declections and its roots. Situation 3. Students, together with the teacher, the scheme of the algorithm for the release of the square of the square of the square. Three. | Personal: productive work in the team; Communicative skills; The focus on self-development. Subjects: an idea of \u200b\u200ba square threestyle and its roots; Knowledge of the algorithm for finding the roots of square. three-shredded and the selection of a square of bicker from square three declared; The ability to apply theoretical knowledge in practice. |
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6.3. The teacher offers students to fulfill tasks from the textbook using the composed scheme. | Personal: Communicative skills; The focus on self-development. MetaPered: Creativity and curiosity; The ability to analyze I. solve the delivered problem; Critical and systemic thinking Subject: knowledge of the algorithm; Ability to apply theoretical knowledge in practice |
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Development of one of the training situations | ||
Name | Drawing up a diagram of an algorithm for the selection of a square of binary from square. bounce |
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Planned learning results | Formation of creativity and curiosity students; Ability to analyze I. solve the delivered problem. Development of critical and systemic thinking. The formation of the ability to analyze the results obtained and compile the schemes. |
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Brief description of the situation | The teacher emphasizes the attention of students on the properties of the senior coefficient of kV. Three stages resembles the need to know the formulas of abbreviated multiplication. Students analyze the responses received and make up the schemes. |
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Tasks for students whose execution will lead to the achievement of planned results (take advantage of help Designer tasks. File "Task Designer»Located in the campus portfolio) |
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Actions of the teacher to create the conditions for achieving scheduled results (use the verbs of action: make, record, use, organize, plan, draw up, offer, prepare, spend, distribute, ask, develop, ensure, create the opportunity, etc.. For example: prepare a scheme for ..., offer students ...., Use a camera for ...etc.) | 1. Prepare cards with tasks. 2. Create an opportunity for students to freely add, discussing the task with a member of your group. |
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Criteria for estimating the task "Bring the descriptions of your (compiled earlier) algorithm in the form of a block diagram" |
Algorithm does not contain blocks | The algorithm contains one of the required blocks. | The algorithm contains all the required blocks. |
Block diagram elements are not connected arrows | Some block diagram elements are connected by arrows. | All schema elements are consistently connected by arrows. |
Dana Description of what kind of transformations with square threestyle | Dana Description of the execution of transformations with square threestyle, without taking into account the sequence | It is given a description of the implementation of transformations with square three-stroke, taking into account all steps. |
The block diagram is performed inactively and does not have a vertical location. | The block diagram is performed inactively, but has a vertical location. | The block diagram is accurate and has a vertical location. |
Personal and metapredal objectives / planned results are thoroughly thought out and prescribed in curricula related to the study of school subjects. When studying educational topics, they can be specified and achieved in part or in a specific context. In other words, the achievement of personal and metap looms cannot be fully and adequately appreciated when mastering only part of the curriculum.
When specifying personal and meta-delicent results, it is possible to use the following wording:are aimed at ..., contribute ..., allow ..., etc. Also within the framework of one educational topic for different learning situations, these planned results may be repeated.
Sections: Mathematics
The purpose of the lesson. To summarize the knowledge of students on the use of three decisions and solving various tasks.
During the classes.
1. Orgmoment
2. Square triple.
but). Continue or add a statement:
- To find the roots of the square three decar AX 2 + ..., it is necessary to solve the equation of the form ...
- The discriminant of the square equation is on the formula d \u003d ...
1 O) Square three decrease is called a polynomial of the form ... where x is variable, ... - Some numbers, and a ...
2) and the roots of the square equation are by the formula x \u003d ...
3) The root of the square three decletes is called the value of the variable in which the values \u200b\u200bof this three-stale ...
4) If x 1 and x 2 are known - the roots of the square three decar, it can be decomposed on the factors by the formula ...
b). C / p with test elements.
Answer: Yes, no, I do not know.
- D.<0. Уравнение имеет 2 корня.
- The number 2 is the root of the equation x 2 + 3x-10 \u003d 0.
- Are there any values \u200b\u200bof T, in which the square three-half 4T 2 -11t + 16 takes a value equal to 10?
Answer: a) not ex.; b) yes; x 1 \u003d 3/4, x 2 \u003d 2; c) yes; T 1 \u003d -2, T 2 \u003d -3 / 4.
- D\u003e 0. The equation has 2 roots.
- The number 3 is the root of the square equation x 2 --x-12 \u003d 0.
- There are such values \u200b\u200bof x, in which 2x 2 -7x-54 and x 2 -8x-24 and x 2 -8x-24 take equal values.
Answers to tasks are written on the back of the board.
c) Spread on the multipliers of square three stages:
- x 2 -6x-7;
- 3x 2 + 11x-4;
- x 2 + 7x-8;
- 3x 2 -4x-4.
d) Reduce the fraction:
e) highlight the square of the bounced:
- x 2 -2x-3;
- x 2 + 6x + 7.
3. Quadratic function, its schedule and properties.
- What is the function called quadratic? What is the name of the schedule of the function?
- How is the chart of a quadratic function if a<0.
- Parabola branches are directed up. What is the number a?
- In one coordinate system, you can map schematically
5 A) whether the graphics belong to Y \u003d 20X 2 B (0.5; 5), Y \u003d -50x 2 A (-0.2; -2).
5) Parabola y \u003d 2x 2 was moved down to 4 units. And right on 3 units., And the branches were sent down. Write the equation of the enchantable.
6) C / p with test elements.
a) write down the coordinates of the vertices:
b) build a function schedule
y \u003d -x 2 -8x-14; y \u003d x 2 -6x + 8;
4. Inequalities with one variable.
1) Decide inequality:
I. -5A 2 + 6A + 8<0
II. 4x 2 + x-3≥0
2) Decide the interval method:
- 2x 2 -18x\u003e 0
- x 2 -0.25≤0.
- x (2x + 9) (7-x)<0
3) Find the field definition areas
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Is it true inequality?
with x (-1; 2/5)
at x [-3; 1/2]
5. Solution of equations and systems.
1) with what value and the equation AX 2 + 4X + 4 \u003d 0 does not have roots?
2) Decide equation:
a) 2x 4 -19x 2 + 12 \u003d 0; b) 
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3) depictures schematically graphics, find out how many roots have an equation
4) Decide the system of equations the most rational way.
The theme "Square three-shredd and its roots" is studied in the course of class 9 algebras. Like any other lesson of mathematics, the lesson on this topic requires containment and training methods. Need clarity. This can be attributed to this video tutorial, which is designed specifically in order to facilitate the work of the teacher.
This lesson lasts 6:36 minutes. During this time, the author time to reveal the topic completely. The teacher will only need to choose tasks on the topic to secure the material.
The lesson begins with the demonstration of 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 the polynomial. Deciding the equation, the author gets the roots of the polynomial.
Further, a comment follows that such polynomials of the second degree belong to the square three of them, in which the second, third or both coefficients, besides the older, are zero. This information is supported by an example where the free coefficient is zero.
Then the author explains how to find the roots of square three decletes. To do this, it is necessary to solve the square equation. And check this author offers an example where a square three-step is given. It is necessary to find his roots. The solution is based on the solution of a square equation obtained from a given square three declest. The solution is written on the screen in detail, clearly and understandable. In the course of the solution of this example, the author recalls how the square equation is solved, writes formulas, and receives the result. The answer is recorded.
Finding the roots of the square three declared author explained on the basis of the example. When students understand the essence, you can go to more general moments that the author does. Therefore, he further summarizes all of the above. General words on the mathematical language, the author records the rule of finding the roots of square three decletes.
Next, the comment follows that in some tasks it is more convenient to record a bit different way. This entry is given on the screen. That is, it turns out that the square of the bounce can be distinguished from square three decar. Such a transformation is invited to consider on the example. Solution This example is shown on the screen. As in the past example, the solution is built in detail with all the necessary explanations. The author then considers the task where only information issued is used. This is a geometric task for evidence. The solution is present in the form of drawing. The problem solving is written in detail and understandable.
This lesson completes. But the teacher may pick up for the abilities of students who will comply with this topic.
This video tutorial can be used as an explanation of the new material in the algebra lessons. It is perfect for independent training of students to the lesson.
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