Functional thinking refers to analyzing, transforming and solving problems with the concepts and properties of functions. Equation thinking starts with the quantitative relationship of the problem, transforms the conditions in the problem into mathematical models (equations, inequalities or mixed groups of equations and inequalities) with mathematical language, and then solves the problem by solving equations (groups) or inequalities (groups). Sometimes, functions and equations are transformed and related to each other to realize solutions.
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 is equality, there is equality; Where there is a formula, there is an equation; The evaluation problem is realized by solving equations ... and so on; Inequality is also closely related to equations. When applying the idea of equations, it is very important to consider the characteristics of equations, solve equations and study equations.
Function describes the relationship between quantities in nature. Function thought 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 function is to use the properties of function to construct a function to solve problems. The commonly used properties are monotonicity, parity, periodicity, maximum and minimum of f(x). What we are required to master skillfully are 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 idea to mine the implicit conditions in the problem and construct the properties of resolution function and ingenious function. Only by in-depth, full and comprehensive observation, analysis and judgment of the given problem can there be a trade-off relationship and a function prototype be constructed. Besides,
Function knowledge involves many knowledge points and has certain requirements in concept, application and understanding, so it is the focus of college entrance examination. Several common problems when we use the function idea are: when we encounter variables, we construct the function relationship to solve the problem; 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.
A combination of numbers and shapes
"Numbers are invisible, not very intuitive, and there are countless shapes, making it difficult to be nuanced." Using the combination of numbers and shapes can simplify the problems to be studied, and combine algebra with geometry, for example, solving geometric problems by algebraic methods and solving algebraic problems by geometric methods. This method is most commonly used in analytic geometry, such as finding the root sign ((A- 1)2+(B- 1)2)+ root sign (A- 1)2)+ root sign ((A- 1)).
Classified discussion thinking
When a problem may lead to different results because of different situations of a quantity, it is necessary to discuss various situations of this quantity in different categories, such as solving inequality |a- 1| >4. It is necessary to discuss the value of a.
Equal thinking
When a problem may be related to an equation, we can solve it by constructing the equation and studying its properties. For example, when we prove Cauchy inequality, we can transform Cauchy inequality into a discriminant of quadratic equation.
Holistic thinking
Starting from the overall nature of the problem, we should highlight the analysis and transformation of the overall structure of the problem, find out the overall structural characteristics of the problem, and be good at treating some formulas or figures as a whole with the "overall" vision, grasping the relationship between them, and making a purposeful and conscious overall treatment. The holistic thinking method is widely used in algebraic simplification and evaluation, solving equations (groups), geometric solution and so on. , the whole substitution superposition.
Change idea
It is to transform unknown, unfamiliar and complex problems into known, familiar and simple problems through deduction and induction. Mathematical theories such as trigonometric function, geometric transformation, factorization, analytic geometry, calculus, and even rulers and rulers of ancient mathematics are permeated with the idea of transformation. Common transformation methods include: general special transformation, equivalent transformation, complex and simple transformation, number-shape transformation, structural transformation, association transformation, analogy transformation and so on.
Implicit conditional thinking
A condition that is not explicitly expressed but can be inferred from an existing explicit expression, or is not explicitly expressed, but the condition is a routine or truth.
Analogical thinking
Comparing two (or two) different mathematical objects, if they are found to have similarities or similarities in some aspects, it is inferred that they may also have similarities or similarities in other aspects.
Modeling thinking
In order to describe an actual phenomenon more scientifically, logically, objectively and repeatedly, people use a generally strict language to describe various phenomena. This language is mathematics. What is described in mathematical language is called a mathematical model. Sometimes we need to do some experiments, but these experiments often use abstract mathematical models as substitutes for actual objects and carry out corresponding experiments. The experiment itself is also a theoretical substitute for the actual operation.
Change ideas
The idea of transformation is to turn the unknown into the known, simplify the complex and turn the difficult into the easy. For example, fractional equations become integral equations, algebraic problems become geometric problems, and quadrilateral problems become triangular problems. The methods to realize this transformation are: undetermined coefficient method, collocation method, whole replacement method, and the transformation idea of turning dynamic into static and abstract into concrete.
Inductive reasoning thinking
Some objects of a certain kind of things have certain characteristics, and all objects of this kind of things have the inference of these characteristics, or the inference that generalizes general conclusions from individual facts is called inductive reasoning (induction for short). In short, inductive reasoning is from part to whole, from individual to general reasoning.
In addition, there are mathematical ideas such as probability and statistics. For example, probability statistics refers to solving some practical problems through probability statistics, such as the winning rate of lottery tickets and the comprehensive analysis of an exam. In addition, some area problems can be solved by probability method.