Mathematics is one of the compulsory subjects, so we should study it seriously from the first day of junior high school. So, how can we learn math well? Introduce several methods for your reference:

First, pay attention to the lecture in class and review it in time after class.

The acceptance of new knowledge and the cultivation of mathematical ability are mainly carried out in the classroom, so we should pay attention to the learning efficiency in the classroom and seek correct learning methods. In class, you should keep up with the teacher's ideas, actively explore thinking, predict the next steps, and compare your own problem-solving ideas with what the teacher said. In particular, we should do a good job in learning basic knowledge and skills, and review them in time after class, leaving no doubt. First of all, we should recall the knowledge points the teacher said before doing various exercises, and correctly master the reasoning process of various formulas. If we are not clear, we should try our best to recall them instead of turning to the book immediately. In a sense, you should not create a learning way of asking questions if you don't understand. For some problems, because of their unclear thinking, it is difficult to solve them at the moment. Let yourself calm down and analyze the problems carefully and try to solve them by yourself. At every learning stage, we should sort out and summarize, and combine the points, lines and surfaces of knowledge into a knowledge network and bring it into our own knowledge system.

Second, do more questions appropriately and develop good problem-solving habits.

If you want to learn math well, it is inevitable to do more problems, and you should be familiar with the problem-solving ideas of various questions. At the beginning, we should start with the basic problems, take the exercises in the textbook as the standard, lay a good foundation repeatedly, and then find some extracurricular exercises to help broaden our thinking, improve our ability to analyze and solve problems, and master the general rules of solving problems. For some error-prone topics, you can prepare a set of wrong questions, write your own problem-solving ideas and correct problem-solving processes, and compare them to find out your own mistakes so as to correct them in time. We should develop good problem-solving habits at ordinary times. Let your energy be highly concentrated, make your brain excited, think quickly, enter the best state, and use it freely in the exam. Practice has proved that at the critical moment, your problem-solving habit is no different from your usual practice. If you are careless and careless when solving problems, it is often exposed in the big exam, so it is very important to develop good problem-solving habits at ordinary times.

Third, adjust the mentality and treat the exam correctly.

First of all, we should focus on basic knowledge, basic skills and basic methods, because most of the exams are basic topics. For those difficult and comprehensive topics, we should seriously think about them, try our best to sort them out, and then summarize them after finishing the questions. Adjust your mentality, let yourself calm down at any time, think in an orderly way, and overcome impetuous emotions. In particular, we should have confidence in ourselves and always encourage ourselves. No one can crush me except ourselves, and no one can crush my pride.

Be prepared before the exam, practice routine questions, spread your own ideas, and avoid improving the speed of solving problems on the premise of ensuring the correct rate before the exam. For some easy basic questions, you should have a 12 grasp and get full marks; For some difficult questions, you should also try to score, learn to score hard in the exam, and make your level normal or even extraordinary.

It can be seen that if you want to learn mathematics well, you must find a suitable learning method, understand the characteristics of mathematics and let yourself enter the vast world of mathematics.

How to improve the ability of solving mathematical problems

Any knowledge includes knowledge and ability. In mathematics, ability is much more important than specific knowledge. Of course, we can't overemphasize ability and ignore knowledge learning. We should learn some problem-solving skills while learning some knowledge.

What is the ability? The psychological definition is that ability refers to the personality and psychological characteristics that directly affect the efficiency of people's activities and make them complete smoothly. In mathematics, I think ability is the intelligence to solve problems.

First, how to improve their ability to solve problems?

The first is imitation. Solving problems is a skill, just like swimming, skiing and playing the piano. At first, you can only learn by imitation.

The second is practice. If you don't go swimming by yourself, you'll never learn to swim. Therefore, if you want to get the ability to solve problems, you must do more problems.

Thirdly, to improve your problem-solving ability, it is not enough to imitate, you must use your head. For example, it is not enough to understand the solution and proof of theorems in textbooks. You must understand how people came up with that solution and why they solved the problem like that. Is there any other way to solve the problem? I think this is the most important thing. If you really understand other people's problem-solving ideas, then you can innovate on this basis and improve your problem-solving ability.

Second, what kind of ability should we pay attention to in learning mathematics?

1 computing power.

2. Spatial imagination.

3 logical thinking ability.

4 the ability to abstract practical problems into mathematical problems.

5. The ability to combine shapes and numbers and convert them to each other.

6 Ability to observe, experiment, compare, guess and summarize problems.

7. Ability to study and discuss problems and innovation.

Third, what is the key to improve the ability to solve problems in mathematics?

Flexible use of mathematical thinking method is the key to improve problem-solving ability. Our predecessors mathematicians have created many mathematical thinking methods for us. We should know it well, understand it and use it flexibly. For junior high school mathematics, there are mainly the following four kinds of mathematical ideas (the so-called ideas are the theoretical methods that guide our practice, here mainly refer to ideas or methods): 1 transform ideas. 2 equation thought. 3. Combination of form and number. 4 function thought. 5. holistic thinking 6. Classified discussion thinking 7. Statistical thinking. As long as we can deeply understand the above thinking methods and flexibly apply them to specific problem-solving practice, we can greatly improve your problem-solving ability.

Improve your ability of classified discussion

Classified discussion is an important way of thinking in middle school mathematics. Every year, the senior high school entrance examination will involve classified discussion related issues, and many students often miss solutions and have incomplete discussions in the process of answering questions. Near the senior high school entrance examination, this paper analyzes some missing solutions among students, hoping to help students improve their ability of classified discussion.

The concept is unclear, leading to missing solutions.

The concept of the knowledge learned is unclear and the understanding is not deep enough, which leads to incomplete answers.

Example: (a-3) x >; 6. Find the value range of x..

Analysis: According to the nature of inequality, "when both sides of inequality are multiplied or divided by negative numbers that are not zero, the direction of inequality should be changed", but the symbol of (a-3) in this question is uncertain, so the positive and negative problems of (a-3) should be discussed separately.

Example: if y2+(k+2)y+ 16 is completely flat, find k.

Analysis: There are two situations in the completely flat mode: (AB) 2 = A2aAB+B2, and students tend to ignore the solution of k+2=-8.

The mindset leads to the lack of solutions.

In the process of daily problem solving, many students are often influenced by habitual thinking in their usual study, which leads to incomplete problem solving.

For example, the height of isosceles trisection is higher than half of the waist length, and the bottom angle is found.

Analysis: according to the meaning of the question, the waist height may be within the triangle, or it may be outside the triangle because the isosceles trisection can neither be an acute isosceles trisection nor an obtuse isosceles triangle. Influenced by habitual thinking, most students ignore the possibility of extratriangular height.

For example, if the three sides of a right triangle are 3, 4 and C respectively, find the value of C. ..

Analysis: C in this question does not necessarily represent a hypotenuse, but may also be a right-angled side. Some students mistakenly confuse it with C in Pythagorean Theorem, and think that C must be a hypotenuse, which leads to a missing solution.

Example: The radius of circle O is 5cm, and the lengths of two parallel chords are 6cm and 8cm respectively. Find the distance between two chords.

Analysis: the positional relationship between the two strings in a circle may be on the same side of the center of the circle or on both sides of the center of the circle, so you can't think according to your own habits when answering.

Summary of the law of auxiliary line of mathematics in senior high school entrance examination (ingenious formula)

People say that geometry is difficult, and it is difficult in auxiliary lines. Auxiliary line, how to add it? Master theorems and concepts.

We must study hard and find out the rules by experience. There is an angular bisector in the picture, which can be perpendicular to both sides.

You can also look at the picture in half, and there will be a relationship after symmetry. Angle bisector parallel lines, isosceles triangles add up.

Angle bisector plus vertical line, try three lines. Perpendicular bisector is a line segment that usually connects the two ends of a straight line.

It needs to be proved that the line segment is double-half, and extension and shortening can be tested. The two midpoints of a triangle are connected to form a midline.

A triangle has a midline and the midline extends. A parallelogram appears and the center of symmetry bisects the point.

Make a high line in the trapezoid and try to translate a waist. It is common to move diagonal lines in parallel and form triangles.

The card is almost the same, parallel to the line segment, adding lines, which is a habit. In the proportional conversion of equal product formula, it is very important to find the line segment.

Direct proof is more difficult, and equivalent substitution is less troublesome. Make a high line above the hypotenuse, which is larger than the middle term.

Calculation of radius and chord length, the distance from the chord center to the intermediate station. If there are all lines on the circle, the radius of the center of the tangent point is connected.

Pythagorean theorem is the most convenient for the calculation of tangent length. To prove that it is tangent, carefully distinguish the radius perpendicular.

Is the diameter, in a semicircle, to connect the chords at right angles. An arc has a midpoint and a center, and the vertical diameter theorem should be remembered completely.

There are two chords on the corner of the circle, and the diameters of the two ends of the chords are connected. Find tangent chord, same arc diagonal, etc.

If you want to draw a circumscribed circle, draw a vertical line in the middle on both sides. Also make a dream circle with inscribed circle and bisector of inner angle.

If you meet an intersecting circle, don't forget to make it into a string. Two circles tangent inside and outside pass through the common tangent of the tangent point.

If you add a connector, the tangent point must be on the connector. Adding a circle to the equilateral angle makes it not so difficult to prove the problem.

Ten Practical Methods of Solving Mathematical Problems

The following are the most commonly used methods to solve problems in junior high school mathematics, and some methods are also required to be mastered in the middle school syllabus. These methods can also give you some help in your present study. Please save it as data. Of course, it is best to learn to understand it all in the future and keep it in your mind.

1, matching method

The so-called formula is to change some items of an analytical formula into the sum of positive integer powers of one or more polynomials by using the method of constant deformation. The method of solving mathematical problems with formulas is called matching method. Among them, the most common method is to make it completely flat. Matching method is an important method of constant deformation in mathematics. It is widely used in factorization, simplifying roots, solving equations, proving equality and inequality, finding extreme values of functions and analytical expressions.

2, factorization method

Factorization is to transform a polynomial into the product of several algebraic expressions. Factorization is the basis of identity deformation. As a powerful mathematical tool and method, it plays an important role in solving algebra, geometry and trigonometry problems. There are many methods of factorization, such as extracting common factors, formulas, grouping decomposition, cross multiplication and so on. Middle school textbooks also introduce the use of decomposition and addition, root decomposition, exchange elements, undetermined coefficients and so on.

3. Alternative methods

Method of substitution is a very important and widely used method to solve problems in mathematics. We usually refer to unknowns or variables as elements. The so-called method of substitution is to replace a part of the original formula with new variables in a complicated mathematical formula, thus simplifying it and making the problem easy to solve.

4. Discriminant method and Vieta theorem.

The root discrimination of unary quadratic equation ax2+bx+c=0(a, B, C belongs to R, a≠0) and△ = B2-4ac is not only used to judge the properties of roots, but also widely used in algebraic deformation, solving equations (groups), solving inequalities, studying functions and even geometric and trigonometric operations as a problem-solving method.

Vieta's theorem not only knows one root of a quadratic equation, but also finds another root. Knowing the sum and product of two numbers, we can find the symmetric function of the root, calculate the sign of the root of quadratic equation, solve the symmetric equation and solve some problems about quadratic curve. , has a very wide range of applications.

5, undetermined coefficient method

When solving mathematical problems, it is first judged that the obtained results have a certain form, which contains some undetermined coefficients, then the equations about undetermined coefficients are listed according to the problem setting conditions, and finally the values of these undetermined coefficients or some relationship between these undetermined coefficients are found. This method is called undetermined coefficient method to solve mathematical problems. It is one of the commonly used methods in middle school mathematics.

6. Construction method

When solving problems, we often use this method to construct auxiliary elements by analyzing conditions and conclusions, which can be a figure, an equation (group), an equation, a function, an equivalent proposition and so on. And establish a bridge connecting conditions and conclusions, so that the problem can be solved. This mathematical method of solving problems is called construction method. Using construction method to solve problems can make algebra, trigonometry, geometry and other mathematical knowledge permeate each other, which is beneficial to solving problems.

7. reduce to absurdity

Reduction to absurdity is an indirect proof method. First, a hypothesis contrary to the conclusion of the proposition is put forward, and then from this hypothesis, through correct reasoning, contradictions are led out, thus denying the opposite hypothesis and affirming the correctness of the original proposition. The reduction to absurdity can be divided into reduction to absurdity (with only one opposite conclusion) and exhaustive reduction to absurdity (with more than one opposite conclusion). The steps to prove a proposition by reduction to absurdity can be roughly divided into: (1) reverse design; (2) return to absurdity; (3) conclusion.

Anti-design is the basis of reduction to absurdity. In order to make correct anti-design, we need to master some commonly used negative expressions, such as: yes/no; Existence/non-existence; Parallel/non-parallel; Vertical/not vertical; Equal to/unequal to; Large (small) inch/small (small) inch; Both/not all; At least one/none; At least n/ at most (n-1); At most one/at least two; Only/at least two.

Reduction to absurdity is the key to reduction to absurdity. There is no fixed model in the process of derivation of contradiction, but it must be based on reverse design, otherwise the derivation will become passive water and trees without roots. Reasoning must be rigorous. There are the following types of contradictions: contradictions with known conditions; Contradicting with known axioms, definitions, theorems and formulas; There are dual contradictions; Contradictions

8. Find the area method

The area formula in plane geometry and the property theorems related to area calculation derived from the area formula can be used not only to calculate the area, but also to prove that plane geometry problems sometimes get twice the result with half the effort. The method of proving or calculating plane geometric problems by using area relation is called area method, which is commonly used in geometry.

The difficulty in proving plane geometry problems by induction or analysis lies in adding auxiliary lines. The characteristic of area method is to connect the known quantity with the unknown quantity by area formula, and achieve the verification result through operation. Therefore, using the area method to solve geometric problems, the relationship between geometric elements becomes the relationship between quantities, and only calculation is needed. Sometimes there may be no auxiliary lines, even if auxiliary lines are needed, it is easy to consider.

9, geometric transformation method

In the study of mathematical problems, the transformation method is often used to transform complex problems into simple problems and solve them. The so-called transformation is a one-to-one mapping between any element of a set and the elements of the same set. The transformation involved in middle school mathematics is mainly elementary transformation. There are some exercises that seem difficult or even impossible to start with. We can use geometric transformation to simplify the complex and turn the difficult into the easy. On the other hand, the transformed point of view can also penetrate into middle school mathematics teaching. It is helpful to understand the essence of graphics by combining the research of graphics under isostatic conditions with the research of motion.

Geometric transformation includes: (1) translation; (2) rotation; (3) symmetry.

10, objective problem solving method

Multiple-choice questions are questions that give conditions and conclusions and require finding the correct answer according to a certain relationship. Multiple-choice questions are ingenious in conception and flexible in form, which can comprehensively examine students' basic knowledge and skills, thus increasing the capacity and knowledge coverage of test papers.

Fill-in-the-blank question is one of the important questions in standardized examination. Like multiple-choice questions, it has the advantages of clear test objectives, wide knowledge coverage, accurate and fast marking, and is conducive to examining students' analytical judgment and calculation ability. The difference is that the fill-in-the-blank question does not give an answer, which can prevent students from guessing the answer.

In order to solve multiple-choice questions and fill-in-the-blank questions quickly and correctly, in addition to accurate calculation and strict reasoning, there are also methods and skills to solve multiple-choice questions and fill-in-the-blank questions. The following examples introduce common methods.

(1) Direct deduction method: Starting directly from the conditions given by the proposition, using concepts, formulas, theorems, etc. Carry out reasoning or operation, draw a conclusion and choose the correct answer. This is the traditional method of solving problems, which is called direct deduction.

(2) Verification method: find out the appropriate verification conditions from the questions, and then find out the correct answer through verification, or substitute alternative answers into the conditions for verification to find out the correct answer. This method is called verification method (also called substitution method). This method is often used when encountering quantitative propositions.

(3) Special element method: substitute appropriate special elements (such as figures or numbers) into the conditions or conclusions of the topic, so as to get the solution. This method is called the special element method.

(4) Exclusion and screening method: for multiple-choice questions with only one correct answer, according to mathematical knowledge or reasoning and calculus, the incorrect conclusion is excluded and the remaining conclusions are screened, so that the solution to make the correct conclusion is called exclusion and screening method.

(5) Graphic method: The method of judging and making a correct choice through the properties and characteristics of the graphics or images that meet the conditions of the topic is called graphic method. Graphic method is one of the common methods to solve multiple-choice questions.

(6) Analysis method: directly through the conditions and conclusions of multiple-choice questions, make detailed analysis, induction and judgment, so as to select the correct result, which is called analysis method.