First, make a reasonable review plan
The first round, review the basic knowledge system.
1。 According to four modules: number and algebra, space and geometry, statistics and probability, practice and comprehensive application; According to the curriculum standards, we should reorganize knowledge points, that is, memory, understanding and application.
2。 Through the explanation of typical examples and exercises, students can master the learning methods and change conditions, conclusions, figures, formulas and expressions by analogy.
3。 Regular testing and timely feedback. Practice should be targeted, typical and hierarchical, and the amount of practice should not be blindly increased.
The second round, special review.
The topic review is divided into fill-in-the-blank questions, multiple-choice questions, regular questions, inquiry questions, reading material questions and open questions according to the types of the senior high school entrance examination. When reviewing these topics, according to the characteristics of the examination papers of senior high school entrance examination over the years, some novel and representative questions are carefully selected for special training, and some information is collected from the following aspects for special training of the characteristics of senior high school entrance examination: ① practical application questions; (2) Highlighting the chart information problem of scientific and technological development and information resource transformation; ③ Reading comprehension questions reflecting self-study ability test; ④ Graphic change questions and open questions examine students' adaptability; ⑤ Induce conjectures and operational inquiry questions to examine students' thinking ability and innovative consciousness; ⑥ Comprehensive examination questions of geometry algebra, etc. When reviewing these topics, teachers should guide students to start from all aspects, and classify, analyze and study the test questions according to the above-mentioned topics in recent years, so as to truly grasp the direction and law of their propositions, and then formulate test-taking countermeasures.
The third round, comprehensive training (simulation exercise).
The key point is to check and fill the gaps and improve students' comprehensive problem-solving ability. Train students' problem-solving strategies through comments, strengthen problem-solving guidance, and improve students' ability to take exams.
Second, teach students to master review strategies and improve the review effect.
1。 Teach students to think. Let students form the good habit of independent thinking, and don't rely too much on classmates and teachers.
2。 Select, refine, reflect and improve: select the best products and stress the effect. If you think and realize, you will find, improve and innovate.
3。 Create a memo: prepare a notebook for yourself to record some typical problems, difficult problems, problems that are easy to make mistakes and forget, and problems that cannot be solved at the moment, so as to solve them in daily study.
4。 Pay attention to experience and summarize the mathematical methods and ideas in the topic. Mathematics examination questions in senior high school entrance examination pay special attention to the examination of mathematical thinking methods. The basic methods commonly used in junior middle school mathematics are: collocation method, method of substitution method, undetermined coefficient method, observation method and so on. Mathematical thoughts include: function thought, combination of numbers and shapes, classified discussion thought, reduction thought and so on.
5。 Teachers should guide students' learning methods from reviewing lectures, doing questions (test questions), correcting test papers and summarizing. , so that students can do what they can in every aspect of learning, make rational use of time and give full play to learning efficiency. Make students learn the law, enhance their self-confidence and cultivate their interest, and get twice the result with half the effort.
Breakthrough 1: I can't do it, find similarity, have similarity, and use similarity.
The finale involves many knowledge points and it is difficult to transform knowledge. Students often don't know how to start, and they should always look for similar triangles according to the meaning of the question.
Cut-in point 2: the graph or basic graph needed to construct the theorem.
In the process of solving problems, it is sometimes necessary to add auxiliary lines. In the senior high school entrance examination in Beijing, only one simple proof question does not need to be added with auxiliary lines, and the rest involve adding auxiliary lines. The requirements for students to add lines in the senior high school entrance examination are quite high, but almost all of them follow the principle of constructing graphs required by theorems or construct some common basic graphs.
Breakthrough point 3: stick to the invariants and be good at using the methods or conclusions adopted in the previous questions.
When the movement of the figure changes, the position, size and direction of the figure may change, but in this process, there are often some two line segments, or some two angles, or some two triangles corresponding to the position or quantitative relationship does not change.
Point 4: Find the information of multiple solutions in the topic.
There are more than one situation in which a graph may meet the conditions in motion, that is, two solutions or multiple solutions. How to avoid missing answers is also a headache for candidates. The information of multiple solutions can actually be found in the topic, which requires us to dig deep into the topic, in fact, it is repeated and serious examination.
In short, there are many entry points for the final exam of senior high school mathematics. You don't have to look for so much in the exam. You often only need to find one or two. The key is to dare to do it after finding it.
There are four sections in the senior high school entrance examination, which are relatively easy to score. To this end, Bian Xiao introduced the following problem-solving skills to candidates.
● Connecting with practical problems
The general steps to solve practical problems can be divided into the following steps.
Examine the questions. Read the topic carefully, find out the meaning of the topic and straighten out the relationship. When reading questions, we should pay attention to language, refine and process, and grasp key words and expressions.
Modeling. Select basic variables, abstract written language into mathematical language, and establish mathematical model according to relevant definitions, axioms and mathematical knowledge.
Not modeled. According to mathematical knowledge and methods, the mathematical model is solved and the results of mathematical problems are obtained.
Test (regression). Regression of mathematical results to practical problems, through analysis, judgment and verification, the results of practical problems are obtained. When returning, we should use practical conditions to test and select and find out the correct result.
Mathematical models commonly used in junior high schools mainly solve application problems by classifying the established models into equations (groups); Column inequality (group) to solve application problems; Establish analytical expressions, images and charts of functions to solve application problems, and use statistics (mean, median, mode and variance) and one table and five charts (statistical table, fan chart, line chart, bar chart, frequency histogram and frequency histogram) to solve application problems; Establish a right triangle to solve the application problem of acute triangle ratio; Establish geometric model, triangular model and rectangular coordinate system model (in fact, linear programming) to solve application problems, covering most middle school mathematical model problems.
● Geometry demonstration questions
The difficulty of geometric essay questions is controlled in the senior high school entrance examination, but as an important aspect of examining the thinking ability of candidates, geometric essay questions still occupy a considerable proportion in the senior high school entrance examination. Taking the key knowledge of geometry as the carrier, it is still one of the key points of the senior high school entrance examination proposition to require candidates to design a reasoning process with a certain level and length according to the meaning of the question, so as to test candidates' logical thinking ability, basic graphic analysis ability and mathematical language expression ability. Geometric demonstration questions highlight the mastery of basic geometric figures, mathematical logical thinking ability and mathematical expression ability. The geometric figures appearing in the test questions are all common basic figures in students' usual study. Filling auxiliary lines also reflects the general requirements. Geometric proof is set up layer by layer, based on the investigation of the mastery of conventional ideas. Focus on the logic and accuracy of students' problem-solving methods and geometric language expressions.
All the questions focus on the examination of basic knowledge, basic skills and basic thinking methods. If students don't have a solid foundation in mathematics, it is difficult to get good grades by guessing questions and surprise attacks. Therefore, all candidates must learn the basic concepts and their nature, basic skills and basic thinking methods, truly understand and master them, and form a reasonable network structure. Pay attention to the conventional thinking of solving geometric problems and the addition of conventional auxiliary lines. Pay attention to basic reasoning, writing, drawing and other skills, explore laws, and accumulate generality and generality in geometry learning. Pay attention to the accuracy and standardization of geometric language expression. In addition, geometric calculation should pay equal attention to geometric demonstration. Because the geometric argument problem is a thinking training problem, it depends on students' long-term thinking training, not on rote memorization and temporary assault. Candidates are advised to do one or two geometric essay questions every day, and choose those questions that can't be read at first for training. They can think independently, discuss with each other, and ask the teacher if they have difficulties. But we must reflect on ourselves and sum up the general rules of geometric argumentative questions: remember geometric theorems, remember basic figures, master the law of adding lines, and express accurately and concisely. As long as a certain number of basic graphics and basic methods are stored in the brain, they can be activated in the exam and solved easily.
● Function synthesis questions
Function describes the dependence of quantity in nature, and reflects the relationship and law between one thing and another. The thinking method of function is to extract the mathematical characteristics of the problem, put forward the mathematical object with the viewpoint of contact change, abstract its mathematical characteristics, establish the function relationship, and study and solve the problem by using the properties of the function.
The thinking method of function mainly includes the following aspects: using the related properties of function to solve some problems of function; From the point of view of movement change, the quantitative relationship in specific problems is analyzed and studied, the functional relationship is established, and the problem is solved by using the knowledge of functions; After proper mathematical transformation and construction, a non-functional problem is transformed into a functional form, and the problem is dealt with by using the properties of functions.
In the past two years' senior high school entrance examination, the function synthesis questions accounted for a certain proportion, especially in the last 50 points. In 2006, there were two function comprehensive questions in the comprehensive questions of the senior high school entrance examination, accounting for 24 points.
So in what form does the function synthesis question of the senior high school entrance examination appear?
First, given the rectangular coordinate system and geometric figure, we can find the analytical formula of the (known) function (that is, we know the type of the function before solving it), and then we can study the figure, find the coordinates of points or study some properties of the figure. The functions known in junior high school are ① linear function (including proportional function) and constant function, and their corresponding images are straight lines; ② Inverse proportional function, whose corresponding image is hyperbola; ③ Quadratic function, whose corresponding image is parabola. The main method to find the analytic expression of known functions is the undetermined coefficient method, and the key is to find the coordinates of points, while the basic methods to find the coordinates of points are geometric method (graphic method) and algebraic method (analytical method). This kind of questions are basically 24 questions, with a full score of 12, and are basically presented in 2-3 small questions.
● Geometry synthesis questions
This kind of questions often have the characteristics of low starting point but comprehensive requirements in the senior high school entrance examination in the past two years. Comprehensive examination questions often combine number and shape, algebraic calculation and geometric proof, similar triangles's judgment and nature, drawing analysis and solving equations, Pythagorean theorem and function, circle and triangle. At the same time, we will examine the most important mathematical thoughts of students in junior high school mathematics: the idea of combining numbers with shapes, the idea of classified discussion, and the change of geometric movement.
Firstly, the geometric figure is given and calculated according to the known conditions, and then the moving point (or moving line segment) moves, resulting in changes in line segment and area. Thus, we can find the analytical formula of the corresponding (unknown) function (that is, we don't know what the form of the resolution function is before we find it) and the definition domain of the function. Finally, we explore and study according to the sought function relationship. Generally, there are: under what conditions is the figure an isosceles triangle, a right triangle, a quadrangle a diamond, a trapezoid, etc. , or explore what conditions two triangles meet to be similar, or explore the positional relationship between line segments, or explore the relationship between areas to find the value of x, and find the value of the independent variable when the straight line (circle) is tangent to the circle. The key to finding the unknown resolution function is to list the equivalent relationship between independent variables and dependent variables (that is, to list the equations containing X and Y) and write it in the form of Y = F (X). Generally, there are direct method (directly listing the equations containing X and Y) and compound method (listing the equations containing X, Y and the third variable, and then finding the functional relationship between the third variable and X, substituting and eliminating the third variable to get the form of Y = F (X)), and of course, parameter method, which has exceeded the requirements of junior high school mathematics teaching. In junior high school, the methods of finding equivalence relations are mainly using Pythagorean theorem, parallel line cutting proportional line segments, triangle similarity and equal area. Finding the domain is mainly to find the special position (limit position) of the graph and solve it according to the analytical formula. The final exploration problem is ever-changing, but it is essential to analyze and study the graph. Find the value of x by geometric and algebraic methods. The geometry synthesis problem basically appears in the 25th question as the finale, with a full score of 14, which is generally presented in three small questions.