Lesson Objectives:
- educational:
- generalize and consolidate the skills and abilities of solving linear inequalities with one variable and their systems; control the acquired knowledge;
- Educational:
- develop methods of mental activity, attention;
- to form the need to acquire knowledge;
- develop communicative and informational competence of students;
- Educational:
- foster a culture of teamwork;
- development of independence.
Lesson location: after studying the topic "Solution of linear inequalities with one variable and their systems."
Lesson type: lesson of generalization of the studied material.
Equipment: blackboard, textbook, notebooks, cards for self-study, computer, multimedia projector, screen, presentation ( Attachment 1 )
Lesson structure.
1. Organizational moment - 1 min.
2. Actualization of basic knowledge - 10 min.
a) oral work on theory;
b) test.
3. Work in pairs - 5 min.
4. Work at the blackboard and in notebooks - 8 min.
5. Physical education - 1 min.
6. Work with DER - 7 min.
7. Independent work (according to options) - 10 min.
8. Ratings. Homework - 1 min.
9. The result of the lesson. Reflection - 2 min.
DURING THE CLASSES
I. Organizational moment(Attachment 1 , slide 1)
We have finished studying the topic “Linear inequalities with one variable and their systems” and today we have a general lesson. What do you think is the purpose of our lesson? ( Attachment 1
, slide 2)
You have correctly identified the purpose of the lesson and we can begin to implement our plan. ( Attachment 1
, slide 3)
Jan Amos Kamensky said: "Consider that day or that hour unfortunate in which you did not learn anything, did not add anything to your education." ( Attachment 1
, slide 4)
And I hope that today's lesson, and the day will not be miserable and lost for you, because. Each of you will take with you something new, unknown, and informative.
II. Updating of basic knowledge
VII. Independent work on options(Attachment 1 , slide 11)
| I option | II option |
| 1) Solve the inequality: A) 4 + 12 X > 7 + 13X List of used resources:
|
The topic of the lesson is "Solving inequalities and their systems" (mathematics grade 9)
Lesson type: lesson of systematization and generalization of knowledge and skills
Lesson technology: critical thinking development technology, differentiated learning, ICT technologies
The purpose of the lesson: repeat and systematize knowledge about the properties of inequalities and methods for solving them, create conditions for the formation of skills to apply this knowledge in solving standard and creative problems.
Tasks.
Educational:
to promote the development of students' skills to summarize the knowledge gained, to analyze, synthesize, compare, draw the necessary conclusions
organize the activities of students to apply the acquired knowledge in practice
to promote the development of skills to apply the acquired knowledge in non-standard conditions
Developing:
continue the formation of logical thinking, attention and memory;
improve the skills of analysis, systematization, generalization;
creating conditions that ensure the formation of self-control skills in students;
promote the acquisition of the necessary skills for independent learning activities.
Educational:
to cultivate discipline and composure, responsibility, independence, a critical attitude towards oneself, attentiveness.
Planned educational outcomes.
Personal: responsible attitude to learning and communicative competence in communication and cooperation with peers in the process of educational activities.
Cognitive: the ability to define concepts, create generalizations, independently choose the grounds and criteria for classification, build logical reasoning, draw conclusions;
Regulatory: the ability to identify potential difficulties in solving an educational and cognitive task and find means to eliminate them, to evaluate their achievements
Communicative: the ability to express judgments using mathematical terms and concepts, formulate questions and answers in the course of the assignment, share knowledge between group members to make effective joint decisions.
Basic terms, concepts: linear inequality, quadratic inequality, system of inequalities.
Equipment
Projector, teacher's laptop, several netbooks for students;
Presentation;
Cards with basic knowledge and skills on the topic of the lesson (Appendix 1);
Cards with independent work (Appendix 2).
Lesson Plan
| During the classes |
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| Technological stages. Target. | Teacher activity | Student activities | |
| Introductory-motivational component |
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| 1.Organizational Purpose: psychological preparation for communication. | Hello. Good to see you all. Sit down. Check if everything is ready for the lesson. If it's all right, then look at me. | Hello. Check accessories. Getting ready for work. | Personal. Responsible attitude to teaching is formed. |
| 2.Updating knowledge (2 min) Purpose: to identify individual gaps in knowledge on the topic | The topic of our lesson is "Solving inequalities with one variable and their systems." (slide 1) Here is a list of basic knowledge and skills on the topic. Assess your knowledge and skills. Arrange the appropriate icons. (slide 2) | Assess their own knowledge and skills. (Attachment 1) | Regulatory Self-assessment of your knowledge and skills |
| 3.Motivation (2 minutes) Purpose: to provide activities to determine the objectives of the lesson . | In the work of the OGE in mathematics, several questions of both the first and second parts determine the ability to solve inequalities. What do we need to repeat in the lesson in order to successfully cope with these tasks? | Discuss, call questions for repetition. | Cognitive. Identify and formulate a cognitive goal. |
| Reflection stage (content component) |
|||
| 4.Self-assessment and choice of trajectory (1-2 min) | Depending on how you assessed your knowledge and skills on the topic, choose the form of work in the lesson. You can work with the whole class with me. You can work individually on netbooks, using my advice, or in pairs, helping each other. | Determined with an individual learning path. Swap if necessary. | Regulatory identify potential difficulties in solving educational and cognitive tasks and find means to eliminate them |
| 5-7 Work in pairs or individually (25 min) | The teacher advises students working independently. | Students who know the topic well work individually or in pairs with a presentation (slides 4-10) Perform tasks (slides 6.9). | cognitive the ability to define concepts, create generalizations, build a logical chain Regulatory the ability to determine actions in accordance with the educational and cognitive task Communicative the ability to organize educational cooperation and joint activities, work with a source of information Personal responsible attitude to learning, readiness and ability for self-development and self-education |
| 5. Solution of linear inequalities. (10 min) | What properties of inequalities do we use to solve them? Can you distinguish between linear, quadratic inequalities and their systems? (slide 5) How to solve a linear inequality? Execute the solution. (slide 6) The teacher follows the decision at the blackboard. Check if the solution is correct. | They name the properties of inequalities, after answering or in case of difficulty, the teacher opens slide 4. Name the distinguishing features of inequalities. Using the properties of inequalities. One student solves inequality No. 1 at the blackboard. The rest are in notebooks, following the decision of the respondent. Inequalities No. 2 and 3 are performed independently. Check with the prepared answer. | cognitive Communicative |
| 6. Solution of quadratic inequalities. (10 min) | How to solve inequality? What is this inequality? What methods are used to solve quadratic inequalities? Recall the parabola method (slide 7) The teacher recalls the steps for solving an inequality. The interval method is used to solve inequalities of the second and higher degrees. (slide 8) To solve quadratic inequalities, you can choose a method that is convenient for you. Solve inequalities. (slide 9). The teacher monitors the progress of the solution, recalls ways to solve incomplete quadratic equations. The teacher advises individually working students. | Answer: We solve the square inequality using the parabola method or the interval method. The students follow the decision on the presentation. At the blackboard, students take turns solving inequalities No. 1 and 2. Check with the answer. (to solve nerve-va No. 2, you need to remember the way to solve incomplete quadratic equations). Inequality No. 3 is solved independently, checked with the answer. | cognitive the ability to define concepts, create generalizations, build reasoning from general patterns to particular solutions Communicative the ability to present in oral and written form a detailed plan of one's own activities; |
| 7. Solving systems of inequalities (4-5 min) | Recall the steps involved in solving a system of inequalities. Solve the system (Slide 10) | Name the stages of the solution The student decides at the blackboard, checks with the solution on the slide. | |
| Reflective-evaluative stage |
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| 8. Control and verification of knowledge (10 min) Purpose: to identify the quality of assimilation of the material. | Let's test your knowledge on the topic. Solve tasks on your own. The teacher checks the result according to the prepared answers. | Perform independent work on options (Appendix 2) After completing the work, the student reports this to the teacher. The student determines his grade according to the criteria (slide 11). Upon successful completion of the work, he can proceed to an additional task (slide 11) | Cognitive. Build logical chains of reasoning. |
| 9. Reflection (2 min) Purpose: an adequate self-assessment of one's capabilities and abilities, advantages and limitations is formed | Is there an improvement in results? If you still have questions, refer to the textbook at home (p. 120) | They evaluate their own knowledge and skills on the same piece of paper (Appendix 1). Compare with self-esteem at the beginning of the lesson, draw conclusions. | Regulatory Self-assessment of your achievements |
| 10. Homework (2 min) Purpose: consolidation of the studied material. | Determine homework based on the results of independent work (slide 13) | Determine and record an individual task | Cognitive. Build logical chains of reasoning. Produce analysis and transformation of information. |
List of used literature: Algebra. Textbook for grade 9. / Yu.N.Makrychev, N.G.Mindyuk, K.I.Neshkov, S.B.Suvorova. - M.: Enlightenment, 2014
Municipal budgetary educational institution
"Secondary school No. 26
with in-depth study of individual subjects "
city of Nizhnekamsk, Republic of Tatarstan
Summary of the lesson in mathematics
in 8th grade
Solving inequalities with one variable
and their systems
prepared
mathematic teacher
first qualification category
Kungurova Gulnaz Rafaelovna
Nizhnekamsk 2014
Lesson outline
Teacher: Kungurova G.R.
Subject: mathematics
Topic: "Solution of linear inequalities with one variable and their systems."
Grade: 8B
Date: 04/10/2014
Lesson type: lesson of generalization and systematization of the studied material.
The purpose of the lesson: consolidation of practical skills and skills in solving inequalities with one variable and their systems, inequalities containing a variable under the module sign.
Lesson objectives:
Tutorials:
generalization and systematization of students' knowledge about how to solve inequalities with one variable;
extension of the type of inequalities: double inequalities, inequalities containing a variable under the module sign, systems of inequalities;
establishment of interdisciplinary connection between mathematics, Russian language, chemistry.
Developing:
activation of attention, mental activity, development of mathematical speech, cognitive interest among students;
mastering the methods and criteria of self-assessment and self-control.
Educational:
education of independence, accuracy, ability to work in a team
The main methods used in the lesson: communicative, explanatory-illustrative, reproductive, method of programmed control.
Equipment:
a computer
computer presentation
monoblocks (performing an individual online test)
handouts (multi-level individual tasks);
self-control sheets;
Lesson plan:
1. Organizational moment.
4. Independent work
5. Reflection
6. The results of the lesson.
During the classes:
1. Organizational moment.
(The teacher tells the students the goals and objectives of the lesson.).
Today we face a very important task. We must sum up this topic. Again, it will be necessary to work out theoretical questions very carefully, to do calculations, to consider the practical application of this topic in our daily life. And we must never forget about how we reason, analyze, build logical chains. Our speech must always be literate and correct.
Each of you has a self-control sheet on your desk. Throughout the lesson, do not forget to mark with a "+" sign your contribution to this lesson.
The teacher assigns homework, commenting on it:
№1026(a,b), No. 1019(c,d); additionally - No. 1046 (a)
2. Actualization of knowledge, skills, skills
1) Before we begin to perform practical tasks, let's turn to the theory.
The teacher announces the beginning of the definition, and the students must complete the wording
a) An inequality with one variable is an inequality of the form ax>b, ax<в;
b) Solving an inequality means finding all its solutions or proving that there are no solutions;
c) The solution of an inequality with one variable is the value of the variable that turns it into a true inequality;
d) Inequalities are called equivalent if they have the same set of solutions. If they have no solutions, then they are also called equivalent
2) On the board, inequalities with one variable, arranged in one column. And next to it, in another column, their solutions are inscribed in the form of numerical intervals. The task of students is to establish a correspondence between inequalities and corresponding gaps.
Establish a correspondence between inequalities and numerical intervals:
1. 3x > 6 a) (-∞ ; - 0.2]
2. -5x ≥ 1 b) (- ∞ ; 15)
3. 4x > 3 c) (2; + ∞)
4. 0.2x< 3 г) (0,75; + ∞)
3) Practical work in a notebook with self-examination.
On the blackboard, students write a linear inequality with one variable. After completing which one of the students voices his decision and corrects the mistakes made)
Solve the inequality:
4 (2x - 1) - 3 (x + 6) > x;
8x - 4 - 3x - 18 > x;
8x - 3x - x\u003e 4 + 18;
4x > 22;
x > 5.5.
Answer. (5.5 ; +∞)
3. Practical application of inequalities in everyday life (chemical experience)
Inequalities in our daily lives can be good helpers. And besides, of course, there is an inextricable link between school subjects. Mathematics goes shoulder to shoulder not only with the Russian language, but also with chemistry.
(On each desk there is a reference scale for pH, ranging from 0 to 12)
If the value is 0 ≤ pH< 7, то среда кислая;
if pH = 7, then the medium is neutral;
if indicator is 7< pH ≤ 12, то среда щелочная
The teacher pours 3 colorless solutions into different test tubes. From the chemistry course, students are asked to remember the types of solution medium (acidic, neutral, alkaline). Further, empirically, involving students, the environment of each of the three solutions is determined. To do this, a universal indicator is lowered into each solution. The following happens: each indicator is painted in the corresponding color. And according to the color scheme, thanks to the reference scale, students set the environment for each of the proposed solutions.
Conclusion:
1 indicator turns red, value 0 ≤ pH< 7, значит среда первого раствора кислая, т.е. имеем кислоту в 1пробирке
2 the indicator turned green, pH = 7, which means the medium of the second solution is neutral, i.e. we had water in test tube 2
3 indicator turned blue, indicator 7< pH ≤ 12 , значит среда третьего раствора щелочная, значит в 3 пробирке была щелочь
Knowing the limits of the pH indicator, you can determine the level of acidity of the soil, soap, and many cosmetics.
Continued updating of knowledge, skills and abilities.
1) Once again, the teacher begins to formulate definitions, and students must complete them
Continue definitions:
a) Solving a system of linear inequalities means finding all its solutions or proving that there are none
b) The solution of a system of inequalities with one variable is the value of the variable for which each of the inequalities is true
c) To solve a system of inequalities with one variable, you need to find a solution to each inequality, and find the intersection of these intervals
The teacher again reminds the students that the ability to solve linear inequalities with one variable and their systems is the basis, the basis for more complex inequalities to be studied in older grades. The foundation of knowledge is being laid, the strength of which is to be confirmed at the OGE in mathematics after grade 9.
Students write in notebooks to solve systems of linear inequalities with one variable. (2 students complete these tasks on the board, explain their solution, voice the properties of inequalities used in solving systems).
№1012(e). Solve System of Linear Inequalities
0.3 x+1< 0,4х-2;
1.5x-3 > 1.3x-1. Answer. (30; +∞).
№1028(g). Solve a double inequality and indicate all the integers that are its solution
1 < (4-2х)/3 < 2 . Ответ. Целое число: 0
2) Solving inequalities containing a variable under the module sign.
Practice shows that inequalities containing a variable under the module sign cause anxiety and self-doubt in students. And often students simply do not take up such inequalities. And the reason for this is a poorly laid foundation. The teacher sets up the students so that they work on themselves in a timely manner, learn consistently all the steps for the successful fulfillment of these inequalities.
There is oral work. (Front survey)
Solving inequalities containing a variable under the module sign:
1. The module of the number x is the distance from the origin to the point with coordinate x.
| 35 | = 35,
| - 17 | = 17,
| 0 | = 0
2. Solve inequalities:
a) | x |< 3 . Ответ. (-3 ; 3)
b) | x | > 2 . Answer. (-∞; -2) U (2; +∞)
The progress of solving these inequalities is displayed on the screen in detail and the algorithm for solving inequalities containing a variable under the module sign is spoken out.
4. Independent work
In order to control the degree of assimilation of this topic, 4 students take places at the monoblocks and undergo thematic online testing. Testing time 15 minutes. After completion, a self-test is carried out both in points and in percentage terms.
The rest of the students at their desks perform independently independent work.
Independent work (run time 13min)
Option 1
Option 2
1. Solve the inequalities:
a) 6+x< 3 - 2х;
b) 0.8(x-3) - 3.2 ≤ 0.3(2 - x).
3(x+1) - (x-2)< х,
2 > 5x - (2x-1) .
-6 < 5х - 1 < 5
four*. (Additionally)
Solve the inequality:
| 2- 2x | ≤ 1
1. Solve the inequalities:
a) 4+x< 1 - 2х;
b) 0.2 (3x - 4) - 1.6 ≥ 0.3 (4-3x).
2. Solve the system of inequalities:
2(x+3) - (x - 8)< 4,
6x > 3(x+1) -1.
3. Solve the double inequality:
-1 < 3х - 1 < 2
four*. (Additionally)
Solve the inequality:
| 6x-1 | ≤ 1
After completing independent work, students hand in notebooks for verification. Students who worked on monoblocks also hand over notebooks to the teacher for verification.
5. Reflection
The teacher reminds the students about the self-control sheets, on which they had to evaluate their work with the “+” sign throughout the lesson, at its various stages.
But the students will have to make the main assessment of their activity only now, after voicing one ancient parable.
Parable.
A wise man was walking, and 3 people were walking towards him. Under the hot sun, they carried carts with stones to build the temple.
The sage stopped them and asked:
- What did you do all day?
- Carried cursed stones, - answered the first.
“I did my job conscientiously,” replied the second.
- And I took part in the construction of the temple, - proudly answered the third.
In the self-control sheets, in paragraph No. 3, students must enter a phrase that would correspond to their actions in this lesson.
Self-control sheet __________________________________________
№ P / P
Lesson stages
Evaluation of educational activities
Oral work in the lesson
Practical part:
Solving inequalities with one variable;
solution of systems of inequalities;
solution of double inequalities;
solution of inequalities with module sign
Reflection
In paragraphs 1 and 2, mark the correct answers in the lesson with a “+” sign;
in paragraph 3, evaluate your work in the lesson according to the instructions
6. The results of the lesson.
The teacher, summing up the lesson, notes successful moments and problems on which additional work is to be done.
Students are invited to evaluate their work according to self-control sheets, and students receive one more mark based on the results of independent work.
At the end of the lesson, the teacher draws the students' attention to the words of the French scientist Blaise Pascal: "The greatness of a person is in his ability to think."
Bibliography:
1 . Algebra. 8th grade. Yu.N.Makarychev, N.G. Mindyuk, K.E. Neshkov, I.E. Feoktistov.-M.:
Mnemosyne, 2012
2. Algebra.8 class. Didactic materials. Guidelines / I.E. Feoktistov.
2nd edition., Ster.-M.: Mnemosyne, 2011
3. Control and measuring materials. Algebra: Grade 8 / Compiled by L.I. Martyshova.-
M.: VAKO, 2010
Internet resources:
Today in the lesson we will generalize our knowledge in solving systems of inequalities and study the solution of a set of systems of inequalities.
Definition one.
It is said that several inequalities with one variable form a system of inequalities if the task is to find all common solutions of the given inequalities.
The value of the variable, at which each of the inequalities of the system turns into a true numerical inequality, is called a particular solution of the system of inequalities.
The set of all particular solutions to a system of inequalities is a general solution to a system of inequalities (more often they simply say a solution to a system of inequalities).
To solve a system of inequalities means to find all its particular solutions, or to prove that this system has no solutions.
Remember! The solution of a system of inequalities is the intersection of the solutions of the inequalities included in the system.
The inequalities included in the system are combined with a curly bracket.
Algorithm for solving a system of inequalities with one variable:
The first is to solve each inequality separately.
The second is to find the intersection of the found solutions.
This intersection is the set of solutions to the system of inequalities
Exercise 1
Solve the system of inequalities seven x minus forty two less than or equal to zero and two x minus seven greater than zero.
The solution to the first inequality - x is less than or equal to six, the second inequality - x is greater than seven second. We mark these gaps on the coordinate line. The solution of the first inequality is marked with hatching from below, the solution of the second inequality is marked with hatching from above. The solution to the system of inequalities will be the intersection of the solutions of the inequalities, that is, the interval on which both hatchings coincide. As a result, we get a half-interval from seven second to six, including six.
Task 2
Solve the system of inequalities: x squared plus x minus six is greater than zero and x squared plus x plus six is greater than zero.
Solution
Let's solve the first inequality - x squared plus x minus six is greater than zero.
Consider the function y equals x squared plus x minus six. Zeros of the function: the first x is equal to minus three, the second x is equal to two. Schematically depicting a parabola, we find that the solution to the first inequality is the union of open numerical rays from minus infinity to minus three and from two to plus infinity.
Let's solve the second inequality of the system x square plus x plus six greater than zero.
Consider the function y equals x squared plus x plus six. The discriminant is minus twenty-three less than zero, which means that the function has no zeros. The parabola has no common points with the x-axis. Depicting a parabola schematically, we find that the solution of the inequality is the set of all numbers.
Let us depict on the coordinate line the solutions of the inequalities of the system.
It can be seen from the figure that the solution of the system is the union of open numerical rays from minus infinity to minus three and from two to plus infinity.
Answer: the union of open numerical rays from minus infinity to minus three and from two to plus infinity.
Remember! If in a system of several inequalities one is a consequence of another (or others), then the inequality-consequence can be discarded.
Consider an example of solving an inequality by a system.
Task 3
Solve the inequality logarithm of the expression x square minus thirteen x plus forty two base two greater than or equal to one.
Solution
The ODZ inequality is given by x squared minus thirteen x plus forty two greater than zero. We represent the number one as the logarithm of two base two and get the inequality - the logarithm of the expression x square minus thirteen x plus forty two base two is greater than or equal to the logarithm of two base two.
We see that the base of the logarithm is equal to two more than one, then we come to the equivalent inequality x square minus thirteen x plus forty two is greater than or equal to two. Therefore, the solution of this logarithmic inequality is reduced to the solution of a system of two square inequalities.
Moreover, it is easy to see that if the second inequality is satisfied, then the more the first inequality is satisfied. Therefore, the first inequality is a consequence of the second, and it can be discarded. We transform the second inequality and write it in the form: x square minus thirteen x plus forty more than zero. Its solution is the union of two numerical rays from minus infinity to five and from eight to plus infinity.
Answer: the union of two numerical rays from minus infinity to five and from eight to plus infinity.
open number beams
Definition two.
It is said that several inequalities with one variable form a set of inequalities if the task is to find all such values of the variable, each of which is a solution to at least one of the given inequalities.
Each such value of a variable is called a particular solution of the set of inequalities.
The set of all particular solutions of the set of inequalities is general solution of a set of inequalities.
Remember! The solution of a set of inequalities is the union of solutions of inequalities included in the set.
The inequalities included in the set are united by a square bracket.
Algorithm for solving a set of inequalities:
The first is to solve each inequality separately.
The second is to find the union of the found solutions.
This union is the solution to the set of inequalities.
Task 4
zero point two tenths multiplied by the difference of two x and three is less than x minus two;
five x minus seven is greater than x minus six.
Solution
Let's transform each of the inequalities. We get an equivalent set
x is greater than seven thirds;
x is greater than one fourth.
For the first inequality, the set of solutions is the interval from seven thirds to plus infinity, and for the second, the interval from one fourth to plus infinity.
Draw on the coordinate line a set of numbers that satisfy the inequalities x is greater than seven thirds and x is greater than one fourth.
We find that the union of these sets, i.e. the solution to this set of inequalities is an open numerical ray from one fourth to plus infinity.
Answer: an open numerical beam from one-fourth to plus infinity.
Task 5
Solve a set of inequalities:
two x minus one is less than three and three x minus two is greater than or equal to ten.
Solution
Let's transform each of the inequalities. We get an equivalent set of inequalities: x is greater than two and x is greater than or equal to four.
Draw on the coordinate line the set of numbers that satisfy these inequalities.
We find that the union of these sets, i.e. the solution to this set of inequalities is an open numerical ray from two to plus infinity.
Answer: an open number beam from two to plus infinity.
