First, the matching method
Matching method is a technique of directional deformation of mathematical formula (matching into a "complete square"), and the relationship between known and unknown is found through the formula, thus simplifying the complex. When making a formula, we need to make an appropriate prediction and use the skills of "splitting" and "adding", "matching" and "gathering" reasonably to complete the formula. Sometimes called "matching method".
The most common formula is identical deformation, so that the mathematical formula appears completely square. It is mainly suitable for the discussion and solution of known or unknown quadratic equations, quadratic inequalities, quadratic functions and quadratic algebra, or the translation transformation of quadratic curves without xy terms.
The most basic formula used in the matching method is the binomial complete square formula (A+B) 2 = A2+2AB+B2. Using this formula flexibly, we can get various basic formula forms, such as:
Second, alternative methods.
When solving a mathematical problem, we regard a formula as a whole and replace it with a variable, thus simplifying the problem. This is called substitution. The essence of substitution is transformation, the key is to construct elements and set elements, and the theoretical basis is equivalent substitution. The purpose is to change the research object, move the problem to the knowledge background of the new object, standardize non-standard problems, simplify complex problems and become easy to deal with.
Substitution method is also called auxiliary element method and variable substitution method. By introducing new variables, scattered conditions can be linked, implicit conditions can be revealed, or conditions can be linked with conclusions. Or turn it into a familiar form to simplify complicated calculation and derivation.
It can transform high order into low order, fraction into algebraic expression, irrational expression into rational expression, transcendental expression into algebraic expression, and is widely used in the study of equations, inequalities, functions, sequences, triangles and other issues.
The substitution methods include: local substitution, triangle substitution, mean substitution and so on. Local substitution, also known as global substitution, means that an algebraic expression appears many times in the known or unknown, and it is replaced by a letter to simplify the problem. Of course, sometimes it is discovered through deformation.
Triangular substitution is used to remove the root sign, or when it is easy to find the triangular form, it is mainly to use a certain point in the known algebra for substitution.
When using substitution method, we should follow the principle of facilitating operation and standardization. Pay attention to the selection of the new variable range after substitution, and make sure that the new variable range corresponds to the value range of the original variable, which cannot be reduced or expanded.
Third, the undetermined coefficient method
Determine the functional relationship between variables, set some unknown coefficients, and then determine these unknown coefficients according to given conditions, which is called undetermined coefficient method. Its theoretical basis is polynomial identity, that is, the necessary and sufficient conditions for using polynomial f(x)=g(x) are: for any value of a, there exists f (a) = g (a); Or the coefficients of similar terms of two polynomials are correspondingly equal.
The key to solving the problem by undetermined coefficient method is to list the equality or equation correctly according to what is known. Using the undetermined coefficient method is to transform a mathematical problem with a certain form into a set of equations by introducing some undetermined coefficients. Judging whether a problem is solved by undetermined coefficient method mainly depends on whether the solved mathematical problem has a certain mathematical expression, and if so, it can be solved by undetermined coefficient method. For example, factorization, fractional decomposition, summation of series, finding functions, finding complex numbers, finding curve equations in analytic geometry, etc. These problems have clear mathematical expressions and can be solved by undetermined coefficient method.
Using the undetermined coefficient method, the basic steps to solve the problem are:
The first step is to determine the analytical formula of the undetermined coefficient problem;
Secondly, according to the condition of identity, a group of equations with undetermined coefficients are listed.
The third step is to solve the equations or eliminate the undetermined coefficients, so that the problem is solved.
How to list a set of equations with undetermined coefficients is mainly analyzed from the following aspects:
(1) Use the column equation with equal corresponding coefficients;
(2) Substituting the concept of identity into the normal equation by numerical value;
(3) Use the self-defined attribute sequence equation;
(4) Using geometric conditions to set equations.
For example, when solving the conic equation, we can use the undetermined coefficient method to solve the equation: first, set the form of the equation, which contains undetermined coefficients; Then the geometric conditions are transformed into equations or equations with unknown coefficients; Finally, the unknown coefficients are obtained by solving the obtained equations or equations, and the equations of conic curves are obtained by substituting the obtained coefficients into well-defined equation forms.
Fourth, the definition method
The so-called definition method is to solve problems directly with mathematical definitions. Theorems, formulas, properties and laws in mathematics are all derived from definitions and axioms. Definition is a logical method to reveal the connotation of a concept, which is clarified by pointing out the essential attributes of things reflected by the concept.
Definition is the inevitable result of thousands of practices, which scientifically reflects and reveals the essential characteristics of things in the objective world. Simply put, definition is a high abstraction of basic concepts by mathematical entities. Solving problems by definition is the most direct method. Let's go back to definition in this class.
Fifth, mathematical induction.
Induction is a way of thinking, which leads to general principles with special examples. Inductive reasoning can be divided into complete inductive reasoning and incomplete inductive reasoning. Incomplete inductive reasoning only infers the properties of a class of things according to the same properties of some objects, which is not allowed in mathematical reasoning. Complete inductive reasoning is to draw a conclusion after investigating all the objects of a class of things.
Mathematical induction is a reasoning method used to prove some mathematical propositions related to natural numbers, which is widely used to solve mathematical problems. It is a recursive mathematical proof method. The first step of argument is to prove that the proposition holds when n = 1 (or n), which is the basis of recursion. The second step is to assume that the proposition holds when n = k, and then prove that the proposition also holds when n = k+ 1 This is the theoretical basis of infinite recursion. It judges whether the correctness of a proposition can be generalized from special to general. In fact, it is to make the correctness of the proposition break through the limit and reach infinity. These two steps are closely related and indispensable. After completing these two steps, we can conclude that "the conclusion is correct for any natural number (or n≥n and n∈N)". It can be seen from these two steps that mathematical induction is a complete recursive induction.
When proving a problem by mathematical induction, the key is to deduce the proposition of n = k+ 1. The proof of this step should have a sense of purpose, and pay attention to the analysis and comparison with the final goal of solving problems, so as to determine and standardize the direction of solving problems, gradually narrow the differences, and finally achieve the purpose of solving problems.
The following problems can be proved by mathematical induction: identity, algebraic inequality, trigonometric inequality, sequence problem, geometry problem, divisibility problem and so on.
Six, parameter method
Parametric method refers to the process of solving problems by introducing some new variables (parameters) related to the mathematical objects studied in the subject as a medium, and then analyzing and synthesizing them. The parametric equation of straight line and conic is an example of solving problems by parametric method. The substitution method is also a typical example of introducing parameters.
Dialectical materialism affirms that the connection between things is infinite and the ways of connection are rich and varied. The task of science is to reveal the internal relations between things and discover the changing laws of things. The function of parameters is to describe the changing state of things and reveal the internal relations between changing factors. Parameters reflect the idea of movement and change in modern mathematics, and their views have penetrated into all branches of middle school mathematics. It is common to solve problems by parameter method.
The key to solving problems by parameter method is to introduce parameters properly, communicate the internal relationship between known and unknown, and use the information provided by parameters to answer questions smoothly.
Seven. reductio ad absurdum
Different from the above methods, reduction to absurdity belongs to the category of "indirect proof", which is a proof method of thinking about problems from a negative perspective, that is, affirming the topic and denying the conclusion, thus leading to contradictory reasoning. Hadamard, a French mathematician, summed up the essence of reduction to absurdity: "If we affirm the hypothesis of the theorem and deny its conclusion, it will lead to contradictions". Specifically, reduction to absurdity is to start with the conclusion of negative proposition, take the negation of proposition conclusion as the known condition of reasoning, and make correct logical reasoning, so as to compare it with known conditions, known axioms, theorems, laws or propositions that have been proved to be correct. The reason for the contradiction is that the hypothesis is not established, so the conclusion of the proposition is affirmed and the proposition is proved.
The reduction to absurdity is based on the "law of contradiction" and "law of excluded middle" in the laws of logical thinking. In the same thinking process, two contradictory judgments cannot be true at the same time, at least one of them is false, which is the "law of contradiction" in logical thinking; Two contradictory judgments cannot be false at the same time. Simply saying "one or not one" is the "law of excluded middle" in logical thinking. In the process of proving absurdity, contradictory judgments are obtained. According to the law of contradiction, these contradictory judgments cannot be true at the same time, but one of them is bound to be false, and the known conditions, known axioms, theorems, rules or propositions that have been proved to be correct are all true, so the "negative conclusion" is bound to be false. According to "law of excluded middle", the contradictory and mutually negative judgments of conclusion and negative conclusion cannot be false at the same time, and there must be a truth, so we get that the original conclusion must be true. Therefore, reduction to absurdity is based on the basic laws and theories of logical thinking, and reduction to absurdity is credible.
The problem model of reduction to absurdity can be simply summarized as "negation → reasoning → negation". That is to say, starting from the negative conclusion, through correct reasoning, logical contradictions are led out and new negation is achieved. It can be considered that the basic idea of reduction to absurdity is "negation of negation". The three main steps of proof by reduction to absurdity are: denying the conclusion → deducing the contradiction → establishing the conclusion. The specific steps of implementation are:
The first step, reverse design: make assumptions contrary to the verification conclusion;
The second step is to return to absurdity: under the condition of reverse assumption, the contradiction is deduced through a series of correct reasoning;
The third step, conclusion: it shows that the reverse hypothesis is not established, thus affirming the original proposition.
When applying reduction to absurdity, we must use "counter-hypothesis" for reasoning, otherwise it is not reduction to absurdity. When using reduction to absurdity to prove a problem, if only one aspect of the proposition needs to be proved, then refute this situation, which is also called reduction to absurdity; If the conclusion is multifaceted, then all the negative situations must be refuted one by one in order to infer the original conclusion. This method of proof is also called "exhaustive method".
Reduction to absurdity is often used to solve mathematical problems. Newton once said, "Reduction to absurdity is one of the most skilled weapons for mathematicians". Generally speaking, the problems commonly proved by reduction to absurdity are: the proposition that the conclusion appears in the form of "negative form", "at least" or "at most", "unique" and "infinite"; Or the negative conclusion is more obvious. Specific and simple proposition; Or directly prove the difficult proposition, change its thinking direction, and think negatively from the conclusion, and the problem may be solved very simply.
Chapter two: Mathematical thoughts commonly used in senior high school mathematics.
First, the thinking method of combining numbers and shapes
The basic knowledge of middle school mathematics is divided into three categories: one is the knowledge of pure numbers, such as real numbers, algebraic expressions, equations (groups), inequalities (groups), functions and so on. One kind is about pure form knowledge, such as plane geometry, solid geometry and so on. One is the knowledge about the combination of numbers and shapes, which is mainly embodied in analytic geometry.
The combination of numbers and shapes is a mathematical thinking method, which includes two aspects: "helping numbers with shapes" and "helping shapes with numbers". Its application can be roughly divided into two situations: one is to clarify the relationship between numbers with the help of the vividness and intuition of shapes, that is, to use shapes as a means and numbers as the purpose, such as using the image of functions to intuitively explain the nature of functions; Or to clarify some properties of a shape with the help of the accuracy and rigor of numbers, that is, to use numbers as a means and shape as the purpose, for example, to accurately clarify the geometric properties of curves with equations of curves.
Engels once said: "Mathematics is a science that studies the relationship between quantity and spatial form in the real world." The combination of numbers and shapes is based on the internal relationship between the conditions and conclusions of mathematical problems, which not only analyzes its algebraic significance, but also reveals its geometric intuition, so as to skillfully and harmoniously combine the accurate description of numbers with the intuitive image of spatial forms, make full use of this combination, find out the methods to solve the problems, make the problems difficult and easy, simplify the complicated ones, and thus solve them. "Number" and "shape" are a pair of contradictions, and everything in the universe is the unity of their contradictions. Mr. Hua said: if the number is small, it will be less intuitive. If the number is small, it will be difficult to be nuanced. The combination of numbers and shapes is good in all aspects, and everything is closed.
The essence of the combination of numbers and shapes is to combine abstract mathematical language with intuitive images. The key is the mutual transformation between algebraic problems and figures, which can make algebraic problems geometric and algebraic. When analyzing and solving problems by combining numbers and shapes, we should pay attention to three points: first, we should thoroughly understand the geometric meaning of some concepts and operations and the algebraic characteristics of curves, and analyze the geometric meaning and algebraic meaning of conditions and conclusions in mathematical topics; The second is to set parameters reasonably, use parameters reasonably, establish relationships, and transform numbers from numbers to shapes. The third is to correctly determine the range of parameters.
Some knowledge of mathematics itself can be regarded as the combination of numbers and shapes. For example, the definition of acute trigonometric function is defined by right triangle; Define trigonometric function of any angle with rectangular coordinate system or unit circle.
Second, discuss the way of thinking by classification.
When solving some mathematical problems, sometimes there will be many situations, which need to be classified and solved one by one, and then integrated solutions. This is the classified discussion method. Classified discussion is a logical method, an important mathematical thought and an important problem-solving strategy, which embodies the idea of breaking the whole into parts and the method of sorting out. The mathematical problems of classified discussion ideas are obviously logical, comprehensive and exploratory, and can train people's thinking order and generality, so they occupy an important position in the college entrance examination questions.
The main reasons for classified discussion are as follows:
① Classify and define the mathematical concepts involved in the problem. For example, the definition of |a| can be divided into three situations: a>0, a=0 and a<0. This kind of classified discussion questions can be called conceptual.
② Mathematical theorems, formulas, operational properties, laws, limited scope or conditions involved in the problem, or given by classification. For example, the formula of the sum of the first n terms of geometric series can be divided into two cases: Q = 1 and q≠ 1. This kind of classified discussion questions can be called natural type.
③ When solving problems with parameters, we must discuss them according to the range of parameters. For example, solving the inequality ax> at 2 am>0, a = 0 and a.
In addition, some uncertain quantities, the shape or position of uncertain figures, uncertain conclusions, etc. It is mainly discussed through classification to ensure their integrity and make them deterministic.
When discussing classification, we should follow the following principles: determination of classification objects, unification of standards, no omission and repetition, scientific classification, clear priority and no skipping discussion. The most important one is "no leakage and no weight".
When answering classified discussion questions, our basic methods and steps are as follows: first, we must determine the scope and the whole discussion object; Secondly, determine the classification standard, correct and reasonable classification, that is, the standard is unified, no duplication is missed, and the classification is mutually exclusive (no repetition); Then discuss the classification step by step and get the results in stages; Finally, summarize and draw a comprehensive conclusion.
Third, the thinking method of function and equation
Function thought refers to analyzing, reforming and solving problems with the concept and nature of function. The idea of equation is to start with the quantitative relationship of the problem, transform the conditions in the problem into mathematical models (equations, inequalities or mixed groups of equations and inequalities) with mathematical language, and then solve the problem by solving equations (groups) or inequalities (groups). Sometimes, functions and equations are mutually transformed and interrelated, thus solving problems.
Descartes' equation thought is: practical problem → mathematical problem → algebraic problem → equation problem. The universe is full of equality and inequality. We know that where there are equations, there are equations; Where there is a formula, there is an equation; The evaluation problem is realized by solving equations ... and so on; The inequality problem is also closely related to the fact that the equation is a close relative. There is no essential difference between function and multivariate equation. For example, the function y = f (x) can be regarded as a binary equation f (x)-y = 0 about x and y, so it can be said that the learning of the function is inseparable from the equation. Column equation, solving equation and studying the characteristics of equation are all important considerations when applying the idea of equation.
Function describes the relationship between quantities in nature, and the function idea establishes the mathematical model of function relationship by putting forward the mathematical characteristics of the problem, so as to carry out research. It embodies the dialectical materialism view of "connection and change". Generally speaking, the idea of a function is to construct a function so as to use the properties of the function to solve problems. Commonly used properties are monotonicity, parity, periodicity, maximum and minimum, image transformation and so on. F(x) and f(x) are inverse functions, which requires us to master the specific characteristics of linear function, quadratic function, power function, exponential function, logarithmic function and trigonometric function. In solving problems, it is the key to use the function thought to be good at excavating the implicit conditions in the problem and constructing the properties of resolution function and ingenious function. Only by in-depth, full and comprehensive observation, analysis and judgment of a given problem can we have a trade-off relationship and build a functional prototype. In addition, equation problems, inequality problems and some algebraic problems can also be transformed into functional problems related to them, that is, solving non-functional problems with functional ideas.
Function knowledge involves many knowledge points and a wide range, and has certain requirements in concept, application and understanding, so it is the focus of college entrance examination. The common types of questions we use function thought are: when encountering variables, construct function relations to solve problems; Analyze inequality, equation, minimum value, maximum value and other issues from the perspective of function; In multivariable mathematical problems, select appropriate main variables and reveal their functional relationships; Practical application of problems, translation into mathematical language, establishment of mathematical models and functional relationships, and application of knowledge such as functional properties or inequalities to solve them; Arithmetic, geometric series, general term formula and sum formula of the first n terms can all be regarded as functions of n, and the problem of sequence can also be solved by function method.
Fourth, the equivalent transformation of thinking methods.
Equivalence transformation is an important thinking method to transform the problem of unknown solution into a problem that can be solved within the scope of existing knowledge. Through continuous transformation, unfamiliar, irregular and complex problems are transformed into familiar, standardized and even simple problems. Over the years, the idea of equivalent conversion has been everywhere in the college entrance examination. We should constantly cultivate and train our consciousness of transformation, which will help to strengthen our adaptability in solving mathematical problems and improve our thinking ability and skills.
Transformation includes equivalent transformation and non-equivalent transformation. Equivalence transformation requires that causality in the transformation process is sufficient and necessary to ensure that the result after transformation is still the result of the original problem. The process of non-equivalent transformation is sufficient or necessary, so the conclusion needs to be revised (for example, the unreasonable equivalent rational equation needs root test), which can bring people a bright spot of thinking and find a breakthrough to solve the problem. In application, we must pay attention to the different requirements of equivalence and non-equivalence, and ensure its equivalence and logical correctness when realizing equivalence transformation.
C.A. Yatekaya, a famous mathematician and professor at Moscow University, once said in a speech entitled "What is problem solving" to the participants of the Mathematical Olympiad: "Solving a problem means turning it into a solved problem". The problem-solving process of mathematics is the transformation process from unknown to known, from complex to simple.
The equivalent transformation method is flexible and diverse. There is no unified model for solving mathematical problems by using the thinking method of equivalent transformation. Can be converted between number, shape and shape, number and shape; Equivalent conversion can be carried out at the macro level, such as the translation from ordinary language to mathematical language in the process of analyzing and solving practical problems; It can realize transformation within the symbol system, which is called identity deformation. The elimination method, method of substitution, the combination of numbers and shapes, and the problem of evaluation domain all embody the idea of equivalent transformation. We often carry out equivalent transformation among functions, equations and inequalities. It can be said that the equivalent transformation is to raise the algebraic deformation of identity deformation to keep the truth of the proposition unchanged. Because of its diversity and flexibility, we should reasonably design the ways and methods of transformation and avoid copying the questions mechanically.
When implementing equivalent transformation in mathematical operations, we should follow the principles of familiarity, simplification, intuition and standardization, that is, we should turn the encountered problems into familiar ones to deal with; Or turn more complicated and tedious problems into simpler ones, such as from transcendence to algebra, from unreasonable to rational, from fractions to algebraic expressions and so on. Or more difficult and abstract problems are transformed into more intuitive problems to accurately grasp the problem-solving process, such as the combination of numbers and shapes; Or from non-standard to standard. According to these principles, mathematical operations can save time and effort in the process of transformation, just like pushing the boat with the current, often infiltrating the idea of equivalent transformation, which can improve the level and ability of solving problems.