Common problem-solving methods in junior high school mathematics
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 unary quadratic equation ax2+bx+c=0(a, B, C belong to R, A? 0) Discrimination of roots, △=b2-4ac is not only used to judge the properties of roots, but also widely used as a problem-solving method in algebraic deformation, solving equations (groups), solving inequalities, studying functions and even geometric and trigonometric operations.
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.
Different problem solving methods
Multiple choice question:
When doing multiple-choice questions, you can use various methods to solve problems, such as direct method, special value method, exclusion method, verification method, graphic method, hypothesis method and hands-on operation method (such as folding and folding, measurement and measurement, etc.). ). The choice with "or" in multiple-choice questions must be vigilant to see if you want to choose.
Fill in the blanks:
Pay attention to special circumstances such as multiple solutions to one question.
Consider various simple solutions to the problem. This is especially true for multiple-choice questions and fill-in-the-blank questions (direct method is the last consideration), especially for multiple-choice questions, some of which can be solved by exclusion method, special value method and drawing method, without having to calculate every question.
Answer the question:
1. Pay attention to the standard answer, and write the standard for the process and conclusion. Carefully examine the questions, take your time, the first is easy, the second is difficult, and you can't ignore any conditions in the questions.
2. The calculation questions must be careful, and the final answer should be the simplest and absolutely correct.
3. Simplify the problem before evaluating it, and simplify it to the simplest. When substituting for evaluation, pay attention to: denominator is not zero; Give due consideration to technologies such as total replacement.
4. Solve the right triangle problem. Pay attention to the practice of explaining auxiliary lines and problem-solving steps. Pay attention to right angles and special angles. When you take an approximate value, you must follow the requirements of the topic.
5. Practical application questions, long questions, read more questions, find the correct relationship, and list equations, inequalities (groups) or functional relationships according to the meaning of the questions. Finally, we must test the solution of the equation.
6. Proof problem: auxiliary lines should be written for tangent proof, and dotted lines should be used for auxiliary lines; When encountering the proportional formula and product formula of line segments, it is necessary to prove that the triangles in which the line segments are located are similar, and at the same time pay attention to the equivalent replacement of line segments (pay attention to the multiple relationship of line segments).
7. Scheme design topic: To see the design requirements of the topic clearly, consider the simplest scheme that meets the requirements, not the complicated and beautiful scheme.
8. If the last question in the finale is really at a loss, you can give it up. Why don't you spend your time testing other topics? For the existing problems, you should pay attention to several situations that cannot be missed. For the problem of moving point, we should pay attention to defining the motion process by drawing more sketches, and also consider several possible situations.
When solving all kinds of big problems, we must reflect in our minds. The problems are similar in peacetime, and we should reflect the feeling of deja vu, not deja vu.
A method of solving problems is summarized as 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, which 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 tool and mathematical method, it plays an important role in solving algebra, geometry and trigonometric functions. 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.
Univariate quadratic equation aX? +bX+c=0(a、b、c？ r，a？ 0) Discriminant of roots△ = b? -4ac is not only used to judge the nature of roots, but also as a problem-solving method, which is widely used in algebraic deformation, solving equations (groups), solving inequalities, studying functions and even analytic geometry and trigonometric function operations.
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 important methods commonly used 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. Equal product (area or volume) method
The area (volume) formula in plane (solid) geometry and the property theorems related to the calculation of area (volume) derived from the area (volume) formula can be used not only to calculate area (volume), but also to prove (calculate) geometric problems that sometimes get twice the result with half the effort. The method of using area (volume) relationship to prove or calculate geometric problems is called equal (surface or volume) product method, which is a common method in geometry.
The difficulty in proving geometric problems by induction or analysis lies in adding auxiliary lines. The characteristic of equal (surface or volume) product method is to connect the known quantity with the unknown quantity by area (volume) formula, and to achieve the verification result through operation. Therefore, using equal (surface or volume) product method to solve geometric problems, the relationship between geometric elements becomes the relationship between quantities, which only needs calculation, sometimes without 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. Methods to solve objective problems
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.
1. Introduce common methods through examples: (1) Direct deduction method: directly starting from the conditions given by propositions, 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.