Title: Mathematics. Intensive preparation course for the Unified State Exam.
The book is addressed primarily to graduates taking the Unified State Exam, traditional final or entrance exams, as well as teachers and students, starting from the 8th grade, and heads of electives.
The manual contains effective (not always standard) methods for solving the most “problematic” equations and inequalities of algebra; it contains 20 training versions of the Unified State Exam, some of the problems for which are taken from the entrance exams at Moscow State University and MIPT. Solutions to many tasks are given, including completely two options, and answers to all tasks are given.
The author of the book has been teaching mathematics at MIPT for several decades, has extensive experience working with high school students, teaches at the correspondence school of physics and mathematics at MIPT, has worked at school No. 463 in Moscow for more than ten years, and is a Soros teacher.
Some of the tasks included in the manual go beyond the scope of the school mathematics curriculum and exceed the level of complexity of the Unified State Examination, the real level of complexity of which can be judged from the demonstration versions of the Unified State Exam 2006-2007 available in the manual.
More than three years have passed since the release of the first edition. The author continues to actively work in promoting effective and quick solution methods based on the equivalence of transformations. During this time, changes have also occurred in the methodological literature - equivalent transformations are becoming more common. In particular, familiarization with the materials of KIM shows that authors are increasingly using equivalent transformations, which are not studied in most schools, and formalizing solutions with their help (sometimes using an icon, sometimes writing this fact down in words). During this time, the author's second book was published, in which a large number of complex problems of this book were solved.
The structure of the exam has also changed somewhat - that’s why we changed the structure of the options in accordance with the Unified State Exam 2007: the number of tasks in series A has been reduced, the number of tasks in series B has been increased, and a task on stereometry has been added to series C. Along with the change in the structure, the last tasks in series B have become more difficult and more difficult series C assignments. Our options do not coincide with the Unified State Exam options for any year. This was done intentionally - after all, you cannot seriously prepare for next year’s exam, even having solved all the options of the previous one.
CONTENT
Preface to the new edition 3
From the author 5
SECTION ONE
EFFECTIVE METHODS FOR SOLVING BASIC TYPES OF ALGEBRA AND ANALYSIS PROBLEMS
The concept of equivalence of equations and inequalities 13
Power with rational exponent 17
Equation of the form xn = a 17
Rational equations and inequalities 23
Quadratic equations. Equations that reduce to quadratic 23
Equations of the form Pn(x) = 0, 3 and reducible to them 26
Quadratic inequalities. Interval method for rational functions 28
Irrational equations 33
Equations of the form y/f(x) = d(x) 35
Equations of the form y/f(x) = Jg(x) 38
Irrational inequalities 39
Inequalities of the form ax + b d 40
Inequalities of the form yax2 + bx + c -f 43
Inequalities of the form 0 (or ^0) 46
Inequalities of the form Jf(x) > (<) g(x) 47
Inequality of the form Jf(x) Jg(x) 50
More complex inequalities 52
Equations containing module 53
Equations of the form f(x) = g(x) 53
Equations of the form f(x) = g(x) 56
Equation of the form ax -f b = ex -f d 57
Inequalities containing modulus 58
The simplest inequalities 58
Inequalities of the form \f(x)\< g(x) 60
Inequalities of the form \f(x)\ > g(x) 61
Inequalities of the form \f(x)\< g(x) 64
Exponential and logarithmic equations 66
Introduction. Basic properties of logarithm 66
Logarithm and potentiation 70
Exponential Equations 71
Logarithmic Equations 73
Interval method for logarithmic and exponential inequalities 75
Exponential inequalities 76
Logarithmic inequalities 78
An inequality of the form logaF(x) > 0 (< 0) 78
Inequality of the form logaf(x) > logag(x) 81
More complex inequalities 82
Complex exponent and logarithm with variable base 85
Logarithms with variable base 90
Exponential inequalities with variable base 93
Inequalities for logarithms with variable base 96
Trigonometry 106
Inverse trigonometric functions 109
Basic types of trigonometric equations and methods for solving them 118
Elements of mathematical analysis 131
Function Domain 131
Set of function values 133
Basic concepts of mathematical analysis in a high school course 136
Derivative of a function and its calculation 139
Geometric meaning of the derivative of a function at point 141
Increasing and decreasing function on an interval, Extrema of functions 142
Indefinite integral. Area of a curved trapezoid 150
SECTION TWO
SAMPLES OF USE TRAINING OPTIONS
Structure of the Unified State Exam (Unified State Exam) in mathematics 152
Demo version 2007 166
Demo version 2006 171
Option 1 175
Option 2 180
Option 3 184
Option 4 188
Option 5 192
Option 6 197
Option 7 200
Option 8 204
Option 9 208
Option 10 212
Option 11 216
Option 12 220
Option 13 224
Option 14 228
Option 15 232
Option 16 235
Option 17 239
Option 18 243
SECTION THREE
SOLUTIONS
Solution to option 1 247
Solution to option 12 260
Solving selected problems from other options 271
SECTION FOUR
ANSWERS 294
Download the e-book for free in a convenient format, watch and read:
Download the book Mathematics. Intensive preparation course for the Unified State Exam. Kolesnikova S.I. 2008 - fileskachat.com, fast and free download.
Despite the fact that many are skeptical about education as such, I consider it an extremely useful and necessary thing for a successful person. We are not talking about the actual presence of a diploma or certificate, but rather about a systematic presentation of the chosen direction. Education allows you to see the connections or, as I call it, get a bird's eye view of the work. This is the topic of a separate article and if you don’t want to miss it, subscribe to updates.
Why am I talking about education? The fact is that admission to a good university and the quality of obtaining theoretical knowledge directly depend on Unified State Exam scores. Personally, I think this is a good practice, since thanks to it everyone will be able to go where they want (with due effort, of course). However, getting high scores is not so easy. You need to know how to prepare for the Unified State Exam, what topics to study, and so on.
Why do I think that I am sufficiently qualified in this matter? Of course, I do not have a pedagogical education (maybe I will in the future), but I have good practical results. Yes, according to social science I received the coveted 100 points, and in the Russian language 87, which is also a good result. At the same time, I did not sit at textbooks all day long, but lived a completely ordinary life, working and walking on the street.
When I started preparing for the Unified State Exam in social studies, I had practically nothing in my head. Initially, I did not plan to take this subject, because I wanted to study to become a linguist-translator, but later I changed my choice. This really disappointed my social studies teacher because she didn't like me and didn't think I could prepare properly. I had to prove her wrong.
So, I decided to act according to the following plan:
- Analyze the Unified State Examination of previous years (they can be easily found on the Internet);
- Find all the necessary information about taking the exam (how long, etc.);
- View the areas of knowledge that need to be involved (economics, political science, etc.);
- Find as many tasks as possible (on the Internet, books with tests, etc.);
- Make a preparation plan;
- Bring it to life.
In principle, there is nothing complicated about this. There were still about 7 months left before the deadline, so I understood that I would have time to cope with my goals.
It is worth noting that I am telling you how to prepare for the Unified State Exam on your own. Of course, you can contact tutors. Especially if you need to get high scores in difficult (at least for me) subjects like physics, chemistry, biology, etc. I have not taken exams on them and do not know the specifics of the preparation, so I am telling you the general principles that I myself adhered to and which helped me achieve results.
Understand what is required of you
To achieve your goal you need to know where to go. Of course, it is best to set yourself a specific, measurable and realistic goal, but this can have a demotivating effect, so you just need to understand what is needed from you. There are several stages to be distinguished here.
- stage. A brief overview of the contents of the examination part. That is, you must understand how many different types of tasks there will be, what skills you need to develop for this, and what you have to do in general. For example, in Russian you need to complete a test part, a short answer part and write an essay. There is no test part in mathematics (at least for now), and literature requires a lot of writing. You need to understand the specifics of your exam.
- stage. View specific assignments and identify areas of training. For example, let's take the same Russian language. In task A1 it is necessary to indicate the stress correctly. That is, you write down for yourself (in a notebook or a separate notebook) what exactly is required in this task. As a result, you should end up with a list in which the first column will contain the exercise number, and the second column will contain what needs to be done. If it is convenient for you, you can create a separate table in Word or Excel. This is very important if you want to understand how to prepare for the Unified State Exam from scratch on your own.
- stage. Identify the gaps and indicate what you need to learn. For example, you realize that you absolutely do not know the correct stress in words. In the third paragraph, you set yourself the task: “find all the words found in this exercise and learn their stress,” and then “expand the base of words with complex stress,” just to be sure. Thanks to this, you will understand exactly what you should do and where to move.
As a result, you should end up with a table something like this:
Don't skip these steps. Learning everything in a row is far from the best option. You should also avoid knowledge tests that are not related to the exam. For example, in the same social studies you can learn a LOT, since it is a complex of a large number of sciences. It is clear that the student has many gaps in these disciplines. However, to understand how to prepare for the Unified State Exam, you need to study only what will be useful in the exam.
Understand that mental resources are limited. You won't be able to learn a lot of material in a short period of time. But you will have to prepare not for one subject, but for several. Therefore, you need to narrow the range of preparation for passing the exam as much as possible. I know that teachers will punish me for such lines, but narrow specialization will allow me to get much more in the future than knowing everything, but a little bit at a time. Therefore, it is better to decide in advance on your future profession and move in this direction.
Prepare a folder with theoretical material
Now you know what to do and where to go. It's time to start preparing for the Unified State Exam. It must begin with the collection of theoretical materials. There are a huge number of them: textbooks, articles, information from the Internet, analysis of practical tasks, etc. You must find as much information as possible, read it carefully, analyze it and make a short summary.
For example, In task A1 in the Russian language you need to choose the correct stress in the word. There are certain rules, like “if a word contains the letter е, then the emphasis always falls on it.” Read them, write down the main ones or those you didn’t know. Then make a list of words that cause you difficulty. Now sign the sheet and paste it into the folder.
As a result, you will have your own reference guide for any task. Why is it needed? Firstly, while you are composing it, a lot of useful knowledge will accumulate in your head. If you just studied the theory, you wouldn't be able to remember as much information. Secondly, by working through the options for assignments, you will be able to discover theoretical material and understand what answer to write. This way you can solve more complex tasks without looking at the answers. Thirdly, you can prove to teachers that you really work, and they don’t have to interfere with you. Of course this is a joke.
It is best to do it in print rather than on a computer. This is much more convenient, and you can easily find the material you need. Moreover, there are a lot of distractions on the computer, and it is better to solve exam options in an atmosphere where nothing distracts you.
Now you will need to follow the following algorithm:
- Solve each task separately several times. For example, 50 times A1, then 50 times A2;
- After this, proceed to the group decision. For example, 30 times tasks from A1 to A7;
- Then solve in whole blocks. For example, solve the entire part A 10 times;
- The penultimate step is to solve the exam options completely. For example, solve 5 ready-made options;
- At the end, conduct the finished exam, observing all time intervals.
- You will be able to carefully go through each component without skipping parts that seem easy;
- You will be able to automate the solution of individual tasks or parts, it will be more clear to you which tasks you are not succeeding in and what exactly can be improved;
- You will be able to get reliable results and track dynamics.
Keep track of the number of errors you get. Especially if you are wondering how to prepare for the Unified State Exam in history or mathematics. You can make a chart in Excel, if you know how, that will show the decreasing error rate - this is motivating.
Also, after each solution, you will need to find shortcomings and carefully read the theory again. By receiving information in this way, you will be able to better understand the issue and you will not have to memorize anything. Moreover, all this can be turned into a game, with the help of which preparation will become an exciting activity.
When and how much to exercise
It’s difficult to give specific recommendations here, since a lot depends on your specific goals, workload, study shift, and much more. We can highlight only a number of basic recommendations that really work and are used in training. By the way, I have excellent material about that on my blog, you can follow the link and read it.
It’s better to practice a little bit, but constantly. For example, instead of studying with a tutor for 2 hours once a week, it is better to spend 15 minutes every day. You say that the total will be only 1 hour 45 minutes, but the effect will be many times better.
I also recommend studying in the morning, when your head is not yet full of information, and before bed, so that the knowledge quickly goes into long-term memory. Of course, no one forbids you to exercise during the day, and you even need to do it. Just before going to bed, briefly review the material you have studied and convince yourself that you have already memorized it.
If you are familiar with the Pomodoro technique, then it is better to use it. The idea is that you study something intensely for 25 minutes, and then rest for 5 minutes. This approach allows you to learn much more while spending less time and effort.
- Watch the video analysis. This is very helpful in developing the correct algorithm for solving problems. Especially in the exact sciences like physics and mathematics. It describes in sufficient detail step by step what needs to be done and in what order to solve a problem or example. You can easily find similar video analysis on the Internet;
- Write down everything that seems unclear to you. However, be sure to read as much information as possible before doing so. I recommend creating a separate page in your notebook or folder. Then approach your teacher or tutor with these questions and ask them to explain them in detail;
- Subscribe to VKontakte public pages. At one time I was even one of the active participants: I posted tasks and solutions to them. Explore different options, engage in discussions, and help explain the theory to others. This will definitely help and count for you;
- Read more self-development materials. Thanks to this, you will be able to study more effectively, and generally improve the quality of your life. By the way, I have new useful material published on my blog every week, so you can subscribe to updates so as not to miss something really important.
Now you know how to prepare for the Unified State Exam. If you have any questions, feel free to ask them in the comments. Bye!
We know the reasons for the low
Unified State Exam points:
Lots of spaces from
school curriculum
Making tasks more difficult
Unified State Exam last year
Tension and excitement
before the exam
Multiple gaps in the general education school curriculum arise due to:
- Missed and unlearned topics that are subsequently left unattended by the school teacher;
- Lack of an individual approach: huge classes of 30-35 people simply physically do not allow this to be done;
- A frivolous attitude towards additional classes on the part of both the student and the teacher, who at the end of the school day are tired and simply “sit out” the time specially allocated for “Additional Additional Education”;
Increased complexity of Unified State Exam tasks last year
Every year, assignments in subjects become more complex, and the criteria for assessing graduate work become more stringent. School teachers either simply do not take this into account, or pay attention to it too late, “a month before the exam.” As a result, the most complex standard tasks in mathematics, physics, chemistry, biology, and the Russian language baffle the examinee.
Stress and anxiety before an exam
The Unified State Exam marks the transition of a schoolchild to another status - an adult, an applicant, a student. It is natural that during such a period of life a teenager experiences stress, anxiety, excitement, and depression. In addition, the examinee is “under the magnifying glass” of attention from the outside - another school, a search at the entrance, strangers, video surveillance, various inspectors, etc. It is very difficult for children to overcome this on their own before the Unified State Exam.
Lessons with a tutor as a way to prepare
A very popular opinion is that the key to success on the Unified State Exam is preparation from a tutor. However, practice proves that among tutors there are often amateurs - non-professionals who have a vague idea of the form and content of the upcoming exam; they also often do not have competent methods of preparation for the subject, so the results of their work are questionable or non-existent.
Little time is given during the exam itself
The written work of the Unified State Exam has recently become more difficult. In addition, during the exam you will need to fill out three forms - a registration form and two answer forms - and you need to fill them out in block letters, which is unusual for many. On the exam, you need to answer questions in monosyllables and write detailed answers: solutions, essays, essays. Even a graduate who does well in the subject may not have enough time for this.
Preparation for the Unified State Exam in courses at different universities
One of the popular types of preparation for the Unified State Exam is preparatory courses at universities. Despite the fact that the university is a serious organization interested in good results, their preparation still leaves much to be desired, because:
- A university teacher usually has a vague idea of the latest trends in the content of examination work, does not keep track of the latest changes in the school curriculum, has little idea of the Federal State Educational Standard (educational standard) of the new generation, etc.;
- The university does not conduct a thorough competitive selection of teachers, and the courses are taught by teachers who have a shortage of teaching hours, or by undergraduates or graduate students;
- Groups at preparatory courses are large in size and have different levels of students, which greatly reduces the effectiveness of such work;
- The form of university training, “out of habit,” is reduced to lectures and seminars, which is absolutely not suitable for preparing a schoolchild;
- University preparatory courses are attractive for their price, but for a university, preparing for an exam is not the main activity, therefore, even the most respected universities are most often unable to organize such classes at a high level and achieve real improvement in students’ knowledge.
Lack of interest in studies, poor attention
After 11 years at school, many students experience boredom, absent-mindedness, lack of motivation, lethargy in the perception of educational materials, etc. Achieving interest and motivation is not an easy task. It should be decided not only by parents, but also by teachers.
Preliminary work before the start of training
How and why will you achieve high results?
If you are currently studying in the 10th or 11th grade, then you probably have a question: why can you get more than 80 Unified State Exam points in 32 lessons on courses, if you were not able to do this at school in 3 years (for more than 150 hours of studying the subject) ?
The thing is that the result in the subject largely depends on the student’s desire to study and do homework. Despite the fact that everyone knows about this, no one works on the student’s desire to learn, in particular on the motivation to do homework.
We conduct preliminary activities for schoolchildren in grades 8-11 to set a serious mood and convey a positive attitude towards learning.
Definition of entry level in exam format
It is impossible to move somewhere without determining the starting position. The upcoming amount of work depends on the outcome of the entrance testing.
Setting goals for the school year. We do not prepare for the Unified State Exam, we prepare for the Unified State Exam for a certain score. This means that the plan and intensity of your preparation depends on your goals. The load if the goal is 70 points is very different from the load if the goal is 90 points.
Personal system for achieving results, taking into account the characteristics of the student.
As a result of a personal conversation, you receive a roadmap with your new schedule for the whole week with class times, homework, rest, and sports. You will learn how to manage time correctly.
Working on time management.
The biggest mistake students make is that they incorrectly distribute their working time, rest, and viewing social media. networks or YouTube. Of course, we will not demand that we give up familiar things, for example, social media. networks. We will simply tell you how to use them correctly so as not to get tired, so that you have energy and time to study, how not to be lazy and not put everything off for later.
Work on concentration.
There are several very simple ways to study or work much more effectively, which we will share with students. For example, one of them is the “Pomodoro method,” the essence of which is that the student is engaged in one task and is not distracted by other things under any circumstances for 25 minutes. It's not difficult, but it brings great results.
Meet the team of super-teachers at the Hodograph center
What do you need to know about our team?
Ours is small! 32 teachers in the central branch on Tverskaya and another 20 teachers in other branches. But we are 100% confident in everyone.
What teaching skills does each teacher at the center have?
We base our pedagogical professionalism on the ability to explain a subject to schoolchildren from zero to 95 points. Of course, this does not mean that schoolchildren will be able to achieve +95 points in a year. This means that each stage requires a different pedagogical approach. The basis for 100% completion of a topic is the following principle: we will not go further until each student in the group learns to solve all variations of a specific task from the exam. Only after we have fully mastered the topic do we move on.
What experience do the teachers have?
To begin with, experience is nothing compared to student results. You can teach classes poorly for 10 years, or you can learn to teach in a year so that all students pass with 80+ points. We work with a group of teachers who have proven their professionalism by teaching at least 50 students to success. The experience of our teachers ranges from 3 to 25 years. That's why our team is small.
How do we select them?
We pay great attention to the selection of teachers. Each candidate undergoes a 4-stage selection process:
— Interview to identify personal qualities and special charisma.
— Testing teaching skills.
- ability to explain material.
— Testing knowledge of the school curriculum in the subject.
— Testing your knowledge of the structure and features of the Unified State Exam/Unified State Examination.
In 6 years, we have received more than 2,000 resumes, conducted more than 400 interviews, and in the end we have only 52 permanent teachers.
How are teachers trained?
We have developed our own teaching system, which includes our 6 years of experience as a training center, as well as the experience of the best world pedagogical literature, adapted to the realities of our country and to the characteristics of the age of graduates. In classes, teachers work on key aspects:
— Motivation
- Knowledge
— Problem solving skills
— Discipline
— Working with text
— Ability to build logical chains
— Attention and concentration
- Ability to fill out forms
Do we monitor teachers during the school year?
There is no limit to perfection! Therefore, unlike most centers, we constantly check the work of each teacher. An employee from the quality department periodically appears at classes. Based on the results of this check, the lesson is adjusted. We evaluate more than 50 aspects of a teacher's work.
Are our teachers strict?
Yes. Without rigor, students may not always complete their homework. It is the fully completed training that is the main lever for super results. Despite this, all teachers are friends and communicate with students. Therefore, for many, Hodograph becomes something more than just Unified State Exam courses.
No lectures and maximum emphasis on results in points!
Our center has 8 key rules that allow you to make lessons much more effective, even in individual lessons.
Distribution of students into groups according to knowledge level
Dividing into groups according to the level of initial training makes classes as effective as individual lessons with a tutor. Without such a distribution, group classes may be of low effectiveness. In order to achieve high scores, we usually divide students into 3 categories:
Entry level groups (<35 баллов ЕГЭ)
In such groups, topics from grades 5-9 are first studied to form a knowledge base. Only after this does targeted preparation for the Unified State Exam and the Unified State Exam begin. Goal – 65+ points
Intermediate level groups (35-65 Unified State Examination points)
The main task of the teacher is to identify the student’s weak points. After this, you need to repeat the theory of previous classes. Only after this can you begin a detailed analysis of parts 1 and 2 of the Unified State Exam or OGE tasks. Goal – 85+ points
Strong level groups (65+ Unified State Exam points)
In groups of this level, students know the material of the school curriculum well, so most of the time is spent on analyzing complex problems and part 2. The goal is 95+ points
Purpose of the course
What is the most important thing when preparing students for courses? There are many different answers to this question:
— Classes should be interesting
— All topics must be completed
— Students should like the lessons
— The material should be presented simply and clearly
All these answers, strictly speaking, are incorrect, since only one thing is important in the end - the result in the next exam. We can achieve the highest possible results using one of the points listed above or all at once - it doesn’t matter. The main thing is that we choose the shortest path. Exam scores are most important to us. Therefore, the program, communication format, methodological support - everything is aimed exclusively at results. This is the only way to make a breakthrough of 40-50 points.
Purpose of the lesson
But it is impossible to prepare for such an exam in one go. We go over a certain topic in class, but the result of the lesson is expressed only in the ability to solve specific tasks from the exam. That is, each lesson you will learn to solve certain problems from the Unified State Exam. The purpose of the lesson is written in the upper right corner of the board. This allows all students to know what they will learn at the end of the lesson.
When working according to this scheme, progress will be noticeable after each lesson!
Work on the topic until all students learn to solve the task without errors.
Probably the biggest mistake of many tutors and courses is to follow the program and not pay special attention to the performance of the group or a specific student. But this scheme is a losing one, because if a student can theoretically lose points in a topic, then he will lose them. There is no task to go through the entire program! The goal is to get the maximum score. An example in favor of this reasoning is the school. All topics are covered at school, but can students of even a specialized class solve any problem from the Unified State Exam? But they can’t only because of gaps in knowledge and inability to solve problems of a certain format. We will completely close these gaps.
The principle of high expectations for entry-level students.
The most difficult students have always been those who, for some reason, are unable to progress quickly in the subject. The idea of working with such students is to achieve COMPLETELY correct answers, including to complex questions that require detailed reasoning. In any case, we always achieve a 100% correct answer without any concessions or discounts for a weak level of mastery of the subject.
No lectures! Format of constant communication, questions or reasoning.
If a teacher is giving a lecture, it is very easy to get distracted by your own problems and affairs. Our principle of work involves communication with students. If the topic is new, then it is:
- Or a series of simple questions to each student on their knowledge of the school curriculum.
- Or please repeat after the teacher the material you have just covered.
For the topic covered, a question and answer is used in random order for all students in the group. The random order makes it impossible to guess who will be next, so the whole group is focused and ready to answer questions.
Changes the pace of the lesson every 15 minutes
It is very difficult for schoolchildren to concentrate on one task for a long time. Therefore, the lesson format (lesson tempo) changes every 15 minutes. Example lesson:
— Greeting, discussion of general topics (5 minutes before the lesson)
— Discussion of the previous remote control
— Questions about remote sensing
— A test at the beginning of the lesson in the EST (Unified Testing System) format to test knowledge on previous topics
— Discussion of the test
— Discussion of the purpose of the lesson, USE tasks on the current topic
— Walkthrough of a new topic in the format of questions and discussions
— Completing assignments on a new topic in writing
— Completing tasks orally
— The creative part of the lesson is at the discretion of the teacher (10 minutes)
— Pinning a new topic
— Test to check the purpose of the lesson
— New homework
The course program takes into account all changes in the 2019 Unified State Exam and is approved by the Department of Education.
Annual update of the course program
We are adjusting the program in accordance with the FIPI project in August and official changes in the demo version in the Unified State Exam and Unified State Exam for 2019 in September.
The course program is approved by the Department of Education
A practice block after each completed subtopic in exam format
A topic is considered passed not when students “know” it or can tell it, but when they solve all the tasks from the exam on this topic. Only after ALL students have successfully solved this block of problems from the Unified State Exam, the group moves on to a new topic. This is how all the exam problems are solved during the course.
Extensive homework
The strongest lever that allows you to achieve super results is homework completed 100%. In class, all the necessary theory is covered and numbers from the exam are solved. Material for fastening is given to the house. Average volume of d.z. – 30-40 tasks from the first part of the Unified State Exam. Completing the task takes 2-3 hours. When the student completes all 40 problems without errors, the material is considered completed. If the student does not do this, then we conclude that errors are possible in the exam. The task of the teachers of the Hodograph center is to achieve complete fulfillment of the d.z. In this case, the breakthrough will be quick and guaranteed.
Cyclic repetition of material throughout the course
No one can accurately calculate how many points students lose because they simply do not remember how to complete a particular task. The system of cyclic repetitions is that every fourth lesson a test is conducted to repeat the material covered a long time ago. In April and May we spend time reviewing each lesson. In this way, we gradually increase the speed of completing work and train the ability not to make mistakes in tasks.
Working on the formal aspects of the exam
We teach not only the subject, but also how to pass this subject: what tricks there are in questions, how to read assignments, how to check answers, how to fill out forms, etc.
We are very grateful that you read our site very carefully, so we want to give you a gift:
— How not to lose points on the Unified State Exam
— How not to lose points on the OGE
Just download the documents.
Support 24/7. Full control of students’ progress in Unified State Examination scores after each lesson.
Constant support from the teacher and group
Each group has a common dialogue. The teacher is always in touch and ready to help solve even the most difficult problems.
