1 solving skills of the finale of junior high school mathematics
Function synthesis problem
Based on the given rectangular coordinate system and geometric figure, the analytic expression of the function is obtained first, and then the figure is studied to obtain the coordinates of points or some properties of the figure.
The main methods for finding the known resolution function are the undetermined coefficient method, including the key to finding the point coordinates, and the basic methods for finding the point coordinates are the geometric method (graphical method) and the algebraic method (analytical method).
Geometric synthesis problem
Given the geometric figure, the calculation is made according to the known conditions, and the corresponding line segment, area and other changes are often generated by the moving point or the moving shape, and the analytical formula of the corresponding (unknown) function is obtained, and the value range of the independent variable of the function is obtained. Finally, the function relationship is explored and studied. Generally, there are: under what conditions is the figure an isosceles triangle, a right triangle, a quadrilateral a parallelogram, a diamond, a trapezoid, etc. , or to explore what conditions two triangles meet, such as congruence and similarity, or to explore the relationship between the number and position of line segments, or to find the value of X when the area meets a certain relationship, or to find the value of independent variables when a straight line (circle) is tangent to a 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), which is a comprehensive problem of geometry and algebra. In junior high school, the main methods to find equivalence relations are Pythagorean theorem, triangle similarity and equal area. The value range of function independent variables is mainly to find the special position (extreme 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. It is the choice trapezoid of the finale.
2 junior high school math application problem solving skills
Examine the problem carefully
Many students are often eager to find the available conditions after seeing the application questions, so they often focus on some data and ignore the text description, especially in the case of tight examination time. Many students often eat application questions alive when reading them, and they are eager to answer them without examining the meaning of the questions, which leads to mistakes. Therefore, in order to do a good job in application problems, we must first carefully examine the problems and clarify the meaning expressed by the problems, so as to carry out the next problem-solving activities.
Inductive problem
After reading the topic, the first thing students should do is to summarize the topic and make clear what type the topic belongs to, so that the actual problem can be transformed into a mathematical model according to different types. In junior high school, we will come into contact with many types of application problems, including travel problems, engineering problems, production problems, marketing and strategy problems, growth rate problems, geometry problems and so on. After reading and classifying the problems, we can find the necessary conditions in the problems purposefully according to different types of problems. For example, when solving the distance problem, we should find out the quantity and relationship of distance, speed and time in the topic, and when solving the marketing and strategy problems, we should make clear the conditions such as unit price, quantity and total price. In short, only through scientific induction can we use previous knowledge to solve problems on this basis.
Location fault
The so-called problem discovery is to find out what we need to find in this application problem, and then use reverse thinking to infer what conditions are needed to solve these problems, so that we can return to the problem based on this information and try to find these conditions to prepare for solving the problem.
Organize data information
In order to improve students' ability of analysis and induction, many application questions will deliberately set some fog for students and give some conditions or data unrelated to the topic. Therefore, in order to solve the problem, we should try our best to sort out the required data under the given conditions, and then sort out and analyze these conditions or data according to the requirements of the topic.
3 Math problem-solving skills in senior high school entrance examination
Positive thinking is the most common way.
In other words, after reviewing the topic, do a little verification from front to back according to the requirements of the topic. This is the basic method to prove the problem, which is widely used in problems with medium difficulty and simple difficulty. Reverse thinking is the opposite of positive thinking. Starting with verification, what conditions are needed to achieve this result, step by step reverse analysis. Reverse thinking is very helpful to solve the problem of not knowing where to start after reading the requirements. From the conclusion, sometimes the problem is easier.
For example, to prove that two sides are equal in length, it is only necessary to prove that the triangles they exist are equal by combining graphics; What kind of angle conditions are needed to prove that these two triangles are congruent? In order to find out the relationship between angles, where do we need to draw an auxiliary line ... In this way, we actually have all the conditions we need. This method of solving problems should give students more exercise in solving problems.
Positive and negative combination
This is the key problem-solving idea in difficult topics. For some cases where it is difficult to get a complete idea from the conclusion and you don't know where to start, we should choose the method of combining positive and negative. In junior high school mathematics, basically all the known conditions given by the topic are useful, so we must not let go of every condition and do more extension.
For example, given the midpoint of one side of a triangle, we should consider whether to make the center line, given the trapezoid, whether to make it high, whether to translate the waist or diagonal, whether to make up some kind of figure, and so on.
4 junior high school mathematics proof problem-solving skills
Examine the problem carefully and determine its meaning.
Examining questions is the first step in doing them. This process is like the working principle of a translation machine, which needs to transform the pure text language into a mathematical model that we understand. First, read the questions carefully, mark the key words, and distinguish the known from the verified. For example, the requirement in the title is rewritten as "If the bisectors of the two base angles are made into isosceles triangles, it can be concluded that the two bisectors are equal in length". If you have a picture, you'd better combine it with graphics. If there is no picture in the topic, ask the students to make reasonable graphics and establish a graphic model according to the meaning of the topic. Don't just imagine, you must start painting. Third, write "known" and "verified" by understanding mathematical languages and symbols. "Known" is the condition of the proposition, and "verification" is the conclusion of the proposition. We must pay attention to the known and verified expressions are mathematical languages and symbols.
It should be noted that in the examination of the topic, in addition to marking the key points of the topic, we must also learn the appropriate extension. In the process of examining questions, some basic theorems, basic graphics, special graphics and problems learned in class are combined to improve the accuracy and speed of solving problems in the later stage. This also puts forward higher requirements for students to build a knowledge system.
No weight or leakage, check carefully.
After the analysis process is completed, it is the highlight of the answer, and the whole proof process is expounded with mathematical language and symbols. The writing process requires rigor and meticulousness, and it can't be made out of nothing, nor can it be nonsense and messy. It should be well-founded, well-founded Choose one from several problem-solving ideas and express it completely according to the problem-solving ideas.
Another reason for the high rate of wrong questions among middle school students is that they have not developed the good habit of checking. The rigor of mathematics is vividly reflected in the proof questions. Every step should be reasonable, and axioms, theorems or inferences that can prove the conclusion should be written, which cannot be fabricated out of thin air or inferred at will. In the process of proof, every step should be carefully checked, and there should be no omissions, few conditions, and no mistakes caused by carelessness such as writing answers and misreading requirements. Only by careful investigation can we ensure that it is well founded and not lose a point.
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