This is one of the basic mathematical ideas, which is mainly reflected in the second chapter of the first volume of Algebra, "Basic Knowledge of Algebra".

For example, let the number A be a and the number B be represented by an algebraic expression: (1) 2 times the sum of two numbers A and B: 2 (a+b) (2) the difference between 2 times A and 5 times B: 2a-5b.

Second, the idea of combining numbers with shapes.

The combination of numbers and shapes is one of the most important and basic thinking methods in mathematics, and it is an effective idea to solve many mathematical problems. "Less is not intuitive, but more is difficult to be nuanced" is a famous saying of Professor Hua, a famous mathematician in China, which highly summarizes the role of the combination of numbers and shapes. The following contents in mathematics textbooks reflect this idea.

1, the one-to-one correspondence between points on the number axis and real numbers.

2. One-to-one correspondence between points on the plane and ordered real number pairs.

3. The relationship between function and image.

4. The sum, difference, multiplication and division of line segments (angles) should make full use of numbers to reflect shapes.

5. Solving triangles, finding angles and side lengths, and introducing trigonometric functions are how to solve problems by algebraic methods.

6. In the chapter "Circle", the definition of circle, the positional relationships between points and circles, straight lines and circles, and circles and circles are all treated as quantitative relationships.

7. The second statistical method in preliminary statistics is to draw statistical charts, which are used to reflect the distribution and development trend of data. In fact, it is through the "shape" to reflect the data dressing situation, development trend and so on. In fact, it is to embody the characteristics of numbers through "shape", which is a direct application of the idea of combining numbers and shapes in practice.

Third, change ideas (return to ideas)

In the whole junior middle school mathematics, the idea of transformation has been running through it. Transforming thinking is to transform an unknown (to be solved) problem into a solved or easy-to-solve problem, such as simplifying the complex, changing the difficult to the easy, changing the unknown to the known, and changing the high order to the low order. It is one of the most basic problem-solving ideas and one of the basic thinking methods of mathematics. The following contents reflect this idea:

1, the solution of the fractional equation is to transform the fractional equation into the quadratic equation that I learned before. Here, the new problem to be solved has become a solved problem, which embodies the transformation idea.

2. Solve the right triangle; Turn the non-right triangle problem into a right triangle problem; Turn practical problems into mathematical problems.

3. It is proved that the sum of the internal angles of a quadrilateral is 360 degrees, that is, a quadrilateral is transformed into two triangles. At the same time, the idea of transformation is also used to discuss the sum of the inner angles of polygons.

Fourth, the idea of classification.

The classification of rational numbers, algebraic expressions, real numbers, angles, triangles and quadrangles, the positional relationship between points and circles, straight lines and circles, and the positional relationship between circles are all discussed through classification.