First, the basic review stage-systematic arrangement, building a mathematical knowledge network
The first round of review, also called "knowledge articles", is roughly the first semester of senior three. At this stage, the teacher will lead the students to review the courses they have learned in senior one and senior two, but this is not just a simple repetition of what they have learned before, but an important process of understanding the old knowledge from a higher angle. Because senior one and senior two, the teacher gives lectures in turn with knowledge points as the main clue. Because the relevant knowledge behind has not been learned, it can't be connected vertically, so what you learn is often fragmentary and scattered knowledge points. In the first round of review, the teacher's main clue is the vertical connection and horizontal connection of knowledge, and the scattered knowledge points are connected in series by chapters, which is systematic and comprehensive. Usually review should pay attention to the basic knowledge and skills such as concepts, theorems and formulas in textbooks; At the same time, it pays more attention to the development and formation of knowledge, the analysis of examples and the process of solving problems. Review should be based on textbooks, lay a solid foundation, focus on textbooks, comprehensively sort out knowledge and methods, and pay attention to the reorganization and generalization of knowledge structure. It is an important link in mathematics review to systematically sort out the mathematics knowledge learned in senior high school, organically link the basic knowledge in the form of concise charts, and build a knowledge network, so that students can have a comprehensive understanding and grasp of the whole senior high school mathematics system, which is convenient for the storage, extraction and application of knowledge, and also helps to cultivate and improve students' thinking quality. The first round focuses on reviewing the "three basics" (basic knowledge, basic skills and basic methods), with the goal of being comprehensive, solid, systematic and flexible. It is easy for students to ignore the mathematical thinking methods contained in reviewing important examples in textbooks. For example, the college entrance examination in Shanghai once appeared "What is the important thinking method of analytic geometry", and the college entrance examination in Jiangsu once appeared "Finding the derivative of a function by definition" and other questions. The exam notes clearly point out that the proportion of easy, medium and difficult questions is controlled at about 3: 5: 2, that is, the middle and low-grade questions account for about 80% of the total score, which determines that the college entrance examination review must grasp the foundation and persevere. Only when the foundation is laid can the concept of doing middle and low-grade questions be clear and handy, and the thinking and calculation can be clear and accurate. So everyone should do the following in the review process:
① Based on textbooks, quickly activate all the knowledge points learned. I suggest you read through the textbooks of Grade One and Grade Two in the summer vacation before Grade Three. )
(2) Pay attention to the changes in the coverage of the knowledge points used in the topic, and consciously think about and study the position of these knowledge points in the textbook and the relationship between them. I noticed that the comprehensiveness of the teacher's topic selection is constantly strengthening.
③ Understand the knowledge structure of textbooks from front to back, and frame and network the whole knowledge system.
Throughout the high school mathematics textbooks, it is composed of a continent, a peninsula and an archipelago. This continent is the shape and number of two-dimensional space, involving sets, mappings and functions, equations and inequalities, sequence and its limit, points and number pairs in rectangular coordinate system, intersection of curves and equations, parametric equations and related parameters, derivatives and their applications; This peninsula refers to solid geometry. Its system comes down in one continuous line with plane geometry, both of which are classical axiomatic systems. After strict reasoning and argumentation, solid geometry problems are generally solved by plane geometry problems. Of course, we should pay special attention to the role of vectors and summarize the basic mode of using vectors to solve solid geometry problems. This archipelago refers to some pearls of discrete mathematics scattered in middle school textbooks, such as permutation and combination, binomial theorem, probability statistics, mathematical induction and so on. The structure of middle school mathematics content can be regarded as a collection of numbers and points. The set of numbers forms four blocks: algebraic expression, function, complex set and permutation and combination. The collection of points forms a graph, which is divided into three parts: plane graph (plane geometry), space graph (solid geometry) and coordinate plane graph (analytic geometry). Specific contents and points are listed below each block, which are connected vertically and horizontally, forming a middle school mathematics knowledge network.
Can refine the knowledge points used in solving problems and tell their sources.
(4) often summarize the most used knowledge points, study the chapters where the key knowledge is located, and understand the position and role of each chapter in the textbook. The main points of each chapter are listed below for your reference.
1. Functions and Inequalities (Topic). Algebra is dominated by functions, and the combination of inequalities and functions is a "hot spot".
The properties of (1) function, such as monotonicity, parity, periodicity (often based on trigonometric function), symmetry, inverse function, etc., can be tested everywhere. Specific functions are often intuitively developed by combining the geometry of images, and sometimes they are properly abstracted. This kind of problem is difficult, and finding common functions that meet the conditions is a good way to solve this kind of problem.
(2) The univariate quadratic function is the most important, and the training of its nature and application should be thorough and extensive. The research on the function value domain (maximum) should focus on the quadratic function or the value domain transformed into quadratic function, and the quadratic function value domain with parameter variables. Methods Stress method, substitution method and basic inequality method were mainly introduced. The distribution and discussion of the roots of the unary quadratic equation, the discussion of the solution of the unary quadratic inequality and the intersection of the quadratic curve are closely related to the unary quadratic function, which should occupy a large proportion in training. Strengthen the review of "three quadratic formulas"
(3) Proof of inequality. The inequality related to function proves that combining mathematical induction is the key. Methods The comparative method and the formula method using basic inequalities were emphasized. Although scaling method is not the focus of college entrance examination, it is necessary to master several simple scaling skills because scaling method will be used more or less in exam questions over the years. Proving inequalities should be good at analyzing the structural characteristics of formulas, looking for the differences between known proofs, and thus finding the connection with related theorems as a breakthrough in solving problems.
(4) In solving inequalities, flexible transformation and classified discussion are emphasized in order to master unary quadratic inequalities and comprehensive problems that can be transformed into unary quadratic inequalities. There are often letters in solving inequalities, which need to be discussed, and methods such as transformation, combination of numbers and shapes, ideas of functions and equations, and general solutions of eight common inequalities need to be mastered.
2. Series (main body). The general term, sum and limit of sequence are investigated with arithmetic and equal proportion as carriers. As for the abstract sequence (given by recursive relation), it is not limited to "induction and proof", but needs to be strengthened. Several common methods of summation of series, such as union, division, split term and dislocation subtraction, must be mastered (pay attention to the discussion of Q).
3. Triangle (non-subject). "Adjustment Opinions" and "Eight formulas of sum, difference, multiplication and difference need not be recited". The difficulty of the examination questions is not diminished. Attention should be paid to the skillful use of basic formulas in training, with emphasis on positive, negative and variant use. There are two main forms of triangle problem: one is to find some properties of complex trigonometric function expression; The second is about the angle in a triangle. Any problem of trigonometric formula transformation can be solved by analyzing the differences in angle, function type and formula substructure characteristics.
4. plural (non-subject, liberal arts is not tested). In recent years, it has shown a cooling trend. The types, methods and difficulty of training can reach the level of teaching materials.
5. Solid geometry (subject).
Emphasize "space" and "three-dimensional", that is, put the positional relationship of lines, lines, faces and faces into a geometric scene. Geometry focuses on pyramids and prisms, while prisms focus on triangular prisms and cubes. The pyramid focuses on one side edge or the side perpendicular to the bottom surface, and the combination of prism and pyramid should also be paid attention to. The positional relationship focuses on judging or proving verticality, highlighting the flexible application of the three vertical theorems and inverse theorems.
Spatial angle focuses on dihedral angle, which strengthens the angle determination method based on three perpendicular lines theorem. Spatial distance focuses on point distance and line distance, and the combination of the two is particularly important. Equal product transformation and equidistant transformation are the most commonly used methods. The calculation of angle and distance is finally transformed into a triangle.
In the calculation of area and volume, most problems involve pyramids (especially triangular pyramids), because the volume of triangular pyramids is flexible and the ideas are wide.
6. Analytic geometry (discipline).
The equations, related properties and mutual positional relationship of straight lines and conic curves are important contents. Objective questions should take care of all aspects and answers should be comprehensive. The positional relationship between straight line and conic curve is the main question in college entrance examination. It is a common problem to highlight the intersection point, midpoint, chord length and trajectory of a straight line and a conic curve, but the range problem with parameters is a difficult one. Highlight the connection with functions and vectors.
Second, the comprehensive review stage-comprehensive deepening, mastering mathematical thinking methods.
The second round of review is usually called "Methods". From the second semester to the end of mid-April. At this stage, teachers will focus on methods and skills, mainly learning mathematical thinking methods. In the review, we should pay attention to improving mathematical ability, and improve logical thinking ability, calculation ability, spatial imagination ability, problem analysis and problem solving ability, mathematical inquiry and innovation ability. Expand new horizons, improve the knowledge structure required by the college entrance examination, optimize the quality of thinking, and fundamentally improve mathematics literacy. These are the directions and goals that mathematics review must break through. Learning mathematics requires solving problems, but solving problems is not the whole of mathematics. Mathematical thinking method is the soul of mathematics. Solving problems without mastering mathematical thinking methods is foolhardy, and learning mathematics without solving problems is "entering Baoshan empty-handed" and unable to grasp the true meaning of mathematics. Teachers' review no longer focuses on the order of knowledge structure, but aims at improving students' ability to solve and analyze problems. They put forward, analyzed and solved problems by methods such as "method, undetermined coefficient method, method of substitution, elimination method, combination of numbers and shapes, and classified discussion". The second round of review is generally a special intensive training, with the goal of improving students' ability to answer questions in the college entrance examination. At this stage, students should not indulge in the rehearsal of test papers, but should, under the guidance of teachers, take typical examples as carriers, take the flexible use of mathematical thinking methods as clues, and stress problem-solving strategies to consolidate, improve, synthesize and improve themselves on the basis of the first round of review. In the important stage, we should strengthen the cultivation of thinking quality and comprehensive ability, mainly focusing on knowledge reorganization, and establish a complete structure of knowledge and ability, including the method ability, thinking ability and expression ability of the subject. But this must be based on the ability to remember knowledge, understand the source of knowledge and its mathematical thinking method, grasp the vertical and horizontal connection of knowledge, and cultivate the ability to explore and study problems. In the second round of review, we should cultivate the consciousness of mathematics application, learn to connect theory with the scenes and problems of materials, and find the combination point with the main knowledge according to the materials given in the topic. We should learn to form systems and methods, that is, problem-solving ideas, including the extraction of effective information, the methods and skills needed to solve problems, the analysis and judgment of factual materials, and the evaluation and reflection of conclusions. "Hard work" without paying attention to methods is tantamount to brute force. It is necessary to do problems on the basis of clarifying the basic concepts and knowledge structure. Sometimes doing problems can deepen the understanding of basic knowledge. It is inefficient and sometimes meaningless to solve problems without paying attention to summing up the law of solving problems and mathematical thinking methods. Students should do the following:
① Take the initiative to split, process and reorganize related knowledge. Find out that a certain knowledge point will appear in a series of problems, and a certain method can solve a class of problems.
(2) When analyzing the topic, from the original emphasis on knowledge points, gradually explore ideas and methods to solve the problem.
From now on, solving problems must be standardized. As the saying goes, "if you are not afraid of scoring difficult questions, you are afraid of deducting points for each question." Therefore, everyone must write the problem-solving process clearly and structurally.
(4) Appropriate selection of simulated test papers and previous college entrance examination questions, and gradually clarify the scope and focus of the college entrance examination.
Third, strengthen the review stage-strengthen training and improve the actual combat ability of the exam.
The third round of review, about a month's time, is also called "strategy". The teacher mainly talks about "the solution of multiple-choice questions, the solution of fill-in-the-blank questions, the solution of applied questions, the solution of inquiry propositions, the solution of comprehensive questions and the solution of innovative questions", teaching students some special methods and skills to improve the speed of solving problems and the strategy of taking exams. The third round is generally simulated reinforcement, which aims to adjust students' intelligence, emotion, will and other factors, so that students can gradually become familiar with the requirements of mathematics college entrance examination for students. At this stage, students should strengthen their reflection after solving problems, and be willing to spend some time studying the exam outline, exam instructions, previous college entrance examination questions and simulation questions all over the country, master the information and proposition dynamics of college entrance examination, improve the accuracy and practice speed, and sublimate to perfection in practice. Pay attention to the following points when practicing: solve problems in a standardized way. As the saying goes, "If you are not afraid of scoring difficult problems, you are afraid of deducting points for each problem", so you must write the problem-solving process clearly and structurally. What matters is the quality of solving problems, not the quantity. We should refine our own problems selectively. Not satisfied with knowing how to do it, but paying more attention to reflection after solving problems and understanding the essence of problem-solving strategies and thinking methods, especially some college entrance examination questions, new questions and slightly difficult questions. This kind of reflection is more important. Thinking more and understanding more can often yield the essence. Students should do the following:
(1) When solving problems, we will choose the most time-saving and trouble-saving method from various methods, and strive to think about problems from multiple angles and angles, and gradually adapt to the requirements of the college entrance examination for "simple thinking".
Pay attention to the speed of solving problems, examine questions slowly, think comprehensively, write accurately and answer questions quickly. Sometimes it's just a symbol error, which will make you feel the taste of "a little miss, a thousand miles away". If it is at a critical time, it will make you regret it for life. The return of the American space shuttle Columbia to the ground was fatal because a heat insulation tile failed. These learning qualities will benefit you for life in your future work.
Develop the habit of analyzing the proposer's intention in the process of solving problems, and think about how the proposer organically combines the knowledge points examined, which thinking methods are compounded, what the proposer wants to test me, and what I should know.
Fourth, preparing for the exam-psychological adjustment to adapt to the college entrance examination
Finally, it is the sprint stage, which is called "preparing for the exam". At this stage, the teacher will give you the initiative to review. The emphases, difficulties, methods and ideas of previous study are all based on the teacher's will. However, now you should study the exam instructions directly and actively, study the college entrance examination questions in recent years, master the college entrance examination information and proposition dynamics, and do:
(1) Search your own knowledge system, seize the weak points, and do special training and surprise measures (ask the teacher to help you recite); Lock in the top priority and master the most important knowledge to the point of perfection.
② Grasp the error-prone points in thinking and pay attention to typical questions.
Browse the exercises and papers you have done before, recall the course of learning relevant knowledge, and do a good job of "re-"error correction.
(4) extensive reading, extensive study, memory, make yourself well informed, pay attention to those new backgrounds, new methods, and representative knowledge issues.
⑤ Don't do difficult questions, off-topic questions and strange questions, keep emotional stability and prepare for the exam with confidence. Pre-test guidance mainly includes four aspects: the basic knowledge of making mistakes frequently; Common problem-solving methods; Test problem-solving skills; Psychological guidance for exams.
The secret of improving your grades
In a sense, the math college entrance examination is about "difficult" and "fast". To get good grades, "accuracy" and "speed" are the guarantees. Every student should systematically sort out his knowledge according to his actual level and situation, find out his weaknesses and dig up the root causes. If you have problems with your knowledge understanding, you should read the textbook repeatedly, understand the context of the concept word by word, and deeply understand the problem-solving ideas, problem-solving methods and connotation extension of the textbook examples and exercises. If you have problems with your study attitude and study habits, you should look for those non-intellectual factors that interfere with you, find out the main contradictions and secondary contradictions, and rule them out one by one. If there is a problem with the method of solving problems, students must be concise and understand the ways and methods of solving problems, so as to have the effect of drawing inferences from others. Generally speaking, we should first adjust our mentality during the exam, and we should not let the difficulty, weight and familiarity of the test questions affect our emotions. We should try our best to make sure that the questions we can do will not be deducted and the questions we can't do will be scored as much as possible. Then read the questions carefully, calculate the questions carefully, and standardize the answers. Secondly, it should be completed within the specified time, paying attention to speed and accuracy. Usually, problems should be clear, clear and accurate, that is, pay attention to clear thinking, rigorous thinking, orderly narration and accurate results. Of course, the strategy of taking the exam varies from person to person. For example, students with good foundation can control the fill-in-the-blank questions and multiple-choice questions in about 45 minutes, while students with poor foundation may need 65,438+0 hours or more, mainly depending on how to handle them to achieve the best results. After each exam, students should sum up themselves carefully, and teachers should make comments as much as possible. The teacher's comments should best include four aspects: ① What knowledge points were examined in this question? 2 how to review the questions? How to open the way to solve the problem? ③ What methods and skills are mainly used in this question? Where are the key steps? ④ What are the typical mistakes in students' answers? What are the intellectual, logical, psychological or strategic reasons? Teachers themselves have to consider a problem, that is, how to adjust review strategies, master test-taking skills, improve psychological quality, and make review more targeted and targeted. Therefore, from the first round of review, we should attach great importance to the cultivation of problem-solving norms and operational ability. We should also pay attention to cultivate self-confidence, keep a calm mind, grasp the overall situation, take the exam calmly from easy to difficult, pay attention to the examination, calculate carefully, avoid unnecessary mistakes and give full play to its due level.
Several problems that should be paid attention to in mathematics review
Pay attention to the training of knowledge crossing. The intersection of knowledge, that is, the vertical and horizontal organic connection between knowledge, not only embodies the ability and conception of the college entrance examination in mathematics, but also is the "hot spot" of the college entrance examination proposition, which is precisely the "weakness" of students' usual study.
Pay attention to the cultivation of thinking process. The expression of mathematical thinking process is the concentrated expression of mathematical thinking method and the link between teachers and students. In review, teachers should let students discuss and communicate with each other and improve together.
Strengthen the mutual translation of mathematical languages. In the review of senior three, teachers should strengthen the guidance and training of students' mathematical language translation, so that students can understand the meaning of the questions and translate them, so as to answer the questions correctly.
Strengthen the examination of applied problems. Mathematics application based on real life, modern science and technology and social hot issues is one of the hot topics in the college entrance examination. The topic is often not very difficult. The key is to examine the ability to understand topic information and solve mathematical problems. This is the general direction that the college entrance examination will adhere to in the future, but it will not form a rigid model that is hard to find. It is not appropriate to collect a large number of application questions during review, nor to conduct special training at random. Instead, we should pay attention to the training of mutual translation ability between language forms and symbol forms of questions, which should run through the whole review process.
Aim for hot spots. The combination of middle school teaching content and advanced mathematics. Example: the concept of compound function and its monotonicity, translation, expansion, symmetry transformation of image, and the maximum value of closed interval of quadratic function; Using quadratic function to study the distribution of roots of equations, the summation of sequences and so on. These are the foundations for further study of advanced mathematics.
Grab a key. No one can take the place of studying and studying hard. Therefore, whether students' internal factors can be mobilized will directly affect the review effect. When reviewing, we must pay attention to the following questions: (1) Cultivate students' sense of participation. (2) Teaching students in accordance with their aptitude. (1) must start from the learning situation. ② Arouse students' enthusiasm and make them full of confidence and interest in learning. (3) Control poor students, grasp basic training, grasp speed, grasp accuracy, and prevent losing points. ④ Control the difficulty. (3) To fully expose the thinking process is not to replace students' thinking with teachers' thinking, but to let students constantly master the basic ideas and methods of mathematics under the guidance of teachers. (4) Improve efficiency and give timely feedback.
There are several principles for doing problems: easy first, difficult later, simple first, and complicated later. There is no need to stick to the order of doing problems. Mature first, do those questions whose structure and content are familiar, and then do those questions whose type, content and even language are unfamiliar. For the former, we should not fall into the trap in a hurry because of impulse, and pay more attention to the difference when we encounter deja vu. For the latter, there is no need to panic. Comfort yourself in time if you have any questions. It may be more difficult for others. The third is to do high-scoring questions first and then low-scoring questions when the difficulty is roughly the same. Don't focus on high-scoring questions, so as not to cause the embarrassing situation of "high is not enough and low is not enough". Adhere to the basic principles of "easy first, difficult later, ripe later, same later, small later, high later, low later".
Keep the best review mentality. Mentality is even more important than learning methods. Learning mentality is the psychological state of students when they study. Mathematical activities are not only "mathematical cognitive activities", but also sensory activities with emotion and mentality. Successful math activities are often accompanied by the best mentality. So how to form the best mentality of reviewing mathematics? We should constantly create a sense of relaxation, pleasure, rigor and accomplishment for ourselves in the process of reviewing mathematics. Psychological research shows that when people are relaxed, neurons in the cerebral cortex can form an excitation center, which makes the channels for nerve cells to transmit information unimpeded and makes their thinking quick and agile. Pleasure is the psychological expression of positive emotions, has the tendency of active learning, and is the catalyst of the best mentality in mathematics learning. With the pleasure of learning, learning will be full of interest and initiative, and the operation of thinking mechanism will also be accelerated. The sense of rigor refers to the emotion of pursuing scientific work style, which can make people well-founded and meticulous. Psychology tells us that rigorous style will be transferred to mathematics learning activities, and mathematics learning activities can form rigorous style. Therefore, in the process of solving problems, we must have clear thinking, clear cause and effect, accurate and standardized, and there must be no omissions and ambiguities, that is, "we must get full marks for what we can do." The sense of success is the "internal motivation" of learning and a great spiritual force to promote creative thinking. Therefore, we should have a unique sense of success, happiness, self-appreciation and intoxication about our achievements. Only in this way can we maintain a positive and enterprising attitude. Therefore, the best learning mentality is mainly composed of four elements: relaxation, pleasure, preciseness and success, which are interrelated and promote each other. Relaxation is the engine of success in mathematics activities, pleasure is the catalyst of success, rigor is the monitor of success, and success is both the key and the ultimate goal.
Review the materials carefully. Don't review more than two sets of materials, and always pay attention to its systematicness during use. Don't be greedy. If you have too much information, you will not only be trapped in the ocean of problems, but also your knowledge system will not be continued because you care about one thing and can't see another.
Some students ignore mistakes in homework and exams and simply attribute them to carelessness. This is a very serious misconception, and all our mistakes have their inevitability. We must get to the bottom of it, find out the real reason, correct it in time, and remember this lesson.
Never drill into difficult problems, digressions or strange questions. "College entrance examination is based on ability", and the ability here refers to: thinking ability, observation and analysis ability of real life, creative imagination, hands-on ability of exploratory experiments, ability of understanding and applying practical problems, ability of analyzing and solving problems, ability of processing and applying information, ability of understanding new materials, new situations and new problems, and its focus is on the process of understanding the formation and laws of conceptual views, which are often hidden in the simplest and most simple. Not the ability to get out of a dead end.
Don't believe in guessing questions. Treat the guesses and information of teachers and all walks of life reasonably, and don't be superstitious Because they are not gods, we can only rely on our own strength and wisdom to go to the examination room. Therefore, we should do a good job in preparing for the exam in a down-to-earth manner.