Classified discussion: it is a logical method to solve problems and also a mathematical idea, which is of great help to simplify the research object and develop people's thinking. The so-called classified discussion means that when the objects given by the question cannot be studied uniformly, the research objects need to be classified according to certain standards, and then the conclusions of each category are obtained by studying each category separately, and finally the answer to the whole question is obtained by synthesizing all kinds of results. Classification discussion is essentially a mathematical strategy of "breaking the whole into parts, breaking them one by one, and then breaking the whole into parts".

The idea of combining numbers and shapes: that is, combining mathematical relations with graphics to solve problems. It is often related to the following contents: (1) the correspondence between real numbers and points on the number axis; (2) correspondence between function and image; (3) The correspondence between curve and equation; (4) Concepts based on geometric elements and geometric conditions, such as complex numbers and trigonometric functions; (5) The structure of a given equation or algebraic expression has obvious geometric significance. Such as equation.

Holistic substitution thinking: Holistic substitution is a method to solve problems by using holistic thinking. Its basic idea is to consider some objects in the problem as a whole, so as to find out the internal relations of the problem and find out the solutions to the problem. The key to solving problems with holistic thinking is to choose and determine the whole. See:/edit/gzbw/wsdb/G3/20060928165438+.

Change ideas

The idea of reduction is to turn the unknown into the known, the complex into the simple and the difficult into the easy. For example, fractional equations are transformed into integral equations, algebraic problems are transformed into geometric problems, and quadrilateral problems are transformed into triangular problems. The methods to realize this transformation are: undetermined system.

Number theory method, collocation method, whole generation method and the transformation idea from dynamic to static and from abstract to concrete.

Example 1 The chicken and the rabbit are in the same cage. There are 50 heads in the cage, 140 feet. How many chickens and rabbits are there?

The essence of analytic transformation is a constant problem, and the known components can be deformed first. Each chicken has two feet and each rabbit has four feet, which is a self-evident known ingredient in the question. Now, the known ingredients in the question are distorted: "At your command", each chicken is required to hang one foot (golden rooster shape) and each rabbit hangs two front feet (Yue Bai shape of Jade Rabbit). Then, there are 50 heads in the cage, but there are only 70 feet left. And at this time, the number of chickens is equal to the number of feet, and the number of rabbits is different from that of rabbits. A rabbit has an extra foot. Now there are 50 heads and 70 feet, which means there are 20 rabbits and 30 chickens.

In which unit, it depends on the mastery, because different topics have different methods, and it is difficult to draw a clear picture.