1. Function and Equation Thought
The idea of function and equation is the most basic idea in middle school mathematics. The idea of function is to analyze and study the quantitative relationship in mathematics from the point of view of movement change, establish the functional relationship or construct the function, and then analyze and solve the related problems by using the image and nature of the function. The idea of equation is to analyze the equivalence relation in mathematics, construct an equation or an equation, and analyze and solve problems by solving or using the properties of the equation.
2. Combination of numbers and shapes
Numbers and shapes can be changed under certain conditions. For example, some algebraic problems and trigonometric problems often have geometric backgrounds, and we can solve related algebraic trigonometric problems with the help of geometric features; Some geometric problems can often be solved by algebraic methods through quantitative structural characteristics. Therefore, the idea of combining numbers and shapes plays an important role in solving problems.
Type of problem solving
① "Variable cell number": it means that with the help of a given figure, after careful observation and research, it can reveal the quantitative relationship contained in the figure and reflect the inherent properties of the geometric figure.
(2) "From number to shape": according to the conditions of the topic, draw the corresponding figure correctly, so that the figure can fully reflect its corresponding quantitative relationship and prompt the essential characteristics of numbers and formulas.
③ "number-shape transformation": observing the shape of the figure, analyzing the structure of the number and formula, causing association, transforming each other in time, abstracting into intuition, and prompting the implied quantitative relationship.
Step 3 discuss ideas by category
The idea of classified discussion is very important because it is logical, because it covers a wide range of knowledge points, and because it can cultivate students' ability to analyze and solve problems. The fourth reason is that it is often necessary to discuss various possibilities in practical problems.
The key to solve the problem of classified discussion is to break the whole into parts and reduce the difficulty of local discussion.
Common types
1 type: discussion caused by mathematical concepts, such as the positional relationship between real number, rational number, absolute value, point (line, circle) and circle;
The second category: discussions caused by mathematical operations, such as whether both sides of inequality are multiplied by a positive number or a negative number;
The third category: the discussion caused by the restrictive conditions of properties, theorems and formulas, such as the discussion caused by the application of the root formula of the quadratic equation of one variable;
The fourth category: the discussion caused by the uncertainty of graphic position, such as the discussion caused by related problems in right, acute and obtuse triangles.
The fifth category: classification discussion caused by the influence of some letter coefficients on the equation, such as the influence of the number of letters in quadratic function on the image, the influence of quadratic term coefficient on the image opening direction, the influence of linear term coefficient on the vertex coordinates, and the influence of constant term on the intercept.
The idea of classified discussion is a way of thinking to classify mathematical objects and seek answers. Its function is to overcome the one-sidedness of thinking and consider problems comprehensively. Classification principle: classification is neither heavy nor leakage.
4. Change and change ideas
Transformation and transformation is one of the most basic mathematical ideas in middle school mathematics and the core of all mathematical thinking methods. The combination of number and shape embodies the transformation of number and shape; The idea of function and equation embodies the mutual transformation between function, equation and inequality; The idea of classified discussion embodies the mutual transformation between the part and the whole, so the above three ideas are also the concrete manifestations of transformation and transformation.
Transformation includes equivalent transformation and non-equivalent transformation, and equivalent transformation requires the necessary cause and effect in the process of transformation; There is only one case of unequal conversion, so the conclusion should be tested, adjusted and supplemented. The principle of transformation is to turn unfamiliar and difficult problems into familiar, easily solved and solved problems, and to turn abstract problems into concrete and intuitive problems; Turn complex problems into simple ones; Turn the general into a special problem; Turn practical problems into mathematical problems and so on, so that problems can be easily solved.
Common conversion methods
① Direct transformation method: directly transform the original problem into a basic theorem, a basic formula or a basic graphic problem;
(2) method of substitution: using "method of substitution" to transform formulas into rational formulas or algebraic expressions into idempotents, and to transform complex functions, equations and inequalities into basic problems that are easy to solve;
③ Number-shape combination method: study the relationship between quantity (analytical formula) and spatial form (figure) in the original problem, and obtain the transformation path through mutual transformation;
(4) Equivalent transformation method: the original problem is transformed into an equivalent proposition that is easy to solve, so as to achieve the purpose of reduction;
⑤ specialization method: transform the form of the original problem into a specialized form, prove the specialized problem, and make the conclusion suitable for the original problem;
⑥ construction method: "construct" a suitable mathematical model and put it