First, combine numbers with shapes.
According to the internal relationship between the conditions and conclusions of mathematical problems, this paper not only analyzes its algebraic significance, but also reveals its geometric significance, so as to skillfully and harmoniously combine quantitative relations with figures, and make full use of this combination to seek answers and solve problems.
Second, the idea of connection and transformation.
Things are interrelated and mutually restricted. Can be transformed into each other. All parts of mathematics are also interrelated and can be transformed into each other. When solving problems, if we can properly handle the mutual transformation between them, we can often turn the difficult into the easy and simplify the complicated. Such as: substitution transformation, known and unknown transformation, special and general transformation, concrete and abstract transformation, partial and whole transformation, dynamic and static transformation and so on.
Third, the idea of classified discussion.
In mathematics, we often need to investigate the research object in different situations according to its different nature. This classified thinking method is an important mathematical thinking method. It is also an important problem-solving strategy.
Fourth, the undetermined coefficient method
When the mathematical formula we are studying has a certain form, we only need to find the value of the letter to be determined in the formula to determine it. Therefore, if the known conditions are substituted into the formula in a certain form, the square or equations containing the letters to be found will often be obtained, thus solving the problem. The undetermined coefficient method is an important mathematical problem-solving method, which is widely used in algebraic constant deformation and research function.
Verb (abbreviation of verb) matching method
It is an important deformation skill in junior high school algebra to construct an algebraic expression in a plane way and then transform it as needed. Matching method plays an important role in decomposing factors, solving equations and discussing quadratic functions.
Sixth, alternative methods.
In the process of solving problems, in order to solve problems further, a certain (or some) letter formula is taken as a whole and represented by a new letter. Method of substitution can simplify a complex formula, and simplify the problem into a more basic problem than the original one, so as to achieve the purpose of simplifying the complex and turning the difficult into the easy.
VII. Analytical Methods
When studying or proving a proposition, we trace back to the known conditions from the conclusion, that is, we deduce the sufficient conditions for its establishment from the conclusion. If the establishment of this condition is not obvious, we will take it as a conclusion and further study the sufficient conditions for its establishment until we reach the known conditions (or known facts), so that the proposition can be proved. This method is called old analysis. This kind of thinking process is often called "grasping the fruit and finding the cause". In junior high school, the idea of solving problems and proving problems only by analysis is generally not required to answer or prove propositions by analysis.
Eight, comprehensive method
When studying or proving a proposition, if the direction of reasoning is to start from known conditions (or known facts) and draw conclusions step by step, this method is called synthesis method. This kind of thinking process is usually called "free guidance". The method we usually use to solve or prove problems is synthesis.
Nine, deductive method
Deduction is a kind of reasoning method that deduces from general things that special things also have certain properties. In short, the reasoning method from general to special is called deductive innovation. The main form of deduction is syllogism, which consists of major premises and conclusions. The theoretical basis of syllogism is logical axiom. Color in junior high school is a proposition that deductive reasoning answers or proves that number is not.
Induction
Induction is a kind of reasoning method to deduce general things from special things with certain properties. In short, the reasoning method from special to general is called induction, also called inductive reasoning. Can be divided into complete induction and incomplete induction.
XI。 similar
Among many objective things, some have similar properties. According to some of their properties, the reasoning method that they may be the same or similar in other properties is called analogy, which is also called analogical reasoning. Analogy can be special to special, or general to general reasoning.