The application of the combination of numbers and shapes can be roughly divided into two situations: either clarifying some properties of shapes with the help of the accuracy of numbers, or clarifying some relationship between numbers with the help of geometric intuition of shapes, that is, the combination of numbers and shapes includes two aspects: the first situation is "solving shapes with numbers" and the second situation is "helping numbers with shapes". "Solving shapes by numbers" means that some shapes are too simple to see any laws by direct observation, and it is necessary to assign values to the shapes, such as side length and angle.
The basic idea is: Hua, a famous mathematician in China, once said: "The combination of numbers and shapes is good in all aspects, and everything will be fine if it is separated." "Number" and "shape" reflect two attributes of things. The combination of numbers and shapes is mainly exponential one-to-one correspondence. The combination of numbers and shapes is to combine abstract mathematical language and quantitative relationship with intuitive geometric figures and positional relationships. Through the combination of abstract thinking and image thinking, we can simplify complex problems and concretize abstract problems, thus achieving the goal of optimizing problem-solving methods.
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Key points of application of combination of numbers and shapes
1, the combination of numbers and shapes is a common way of thinking to solve mathematical problems. The combination of numbers and shapes can make some abstract mathematical problems intuitive and vivid, change abstract thinking into image thinking, and help to grasp the essence of mathematical problems. In addition, due to the combination of numbers and shapes, many problems are easy to solve and the solutions are simple.
2. The so-called combination of numbers and shapes is the idea of solving mathematical problems through mutual transformation between numbers and shapes according to the corresponding relationship between numbers and shapes, so as to realize the combination of numbers and shapes.
3. Looking at the college entrance examination questions for many years, skillfully using the thinking method of combining numbers and shapes to solve some abstract mathematical problems can get twice the result with half the effort. The key point of the combination of number and shape is to study "helping number with shape"
4. The idea and method of combining numbers and shapes are widely used. For example, in solving equations and inequalities, in finding the range and maximum value of functions, in finding complex numbers and trigonometric functions, we can not only find the method of solving problems intuitively, but also avoid complicated calculation and reasoning, which greatly simplifies the process of solving problems. This is more advantageous in solving multiple-choice questions and fill-in-the-blank questions. We should pay attention to cultivating this kind of ideology and try to have a picture in our mind to broaden our thinking horizons.
5. Paper on the idea of combining numbers with shapes: In short, the idea of combining numbers with shapes in mathematics is a mathematical idea to solve mathematical problems. The combination of numbers and shapes is to combine abstract mathematical language with intuitive graphics, to combine abstract thinking with image thinking, and to solve mathematical problems through the correspondence and transformation of numbers and shapes. There are three main types of solving mathematics problems in middle schools: changing "number" into "shape", changing "shape" into "number" and combining numbers with shapes.
Baidu encyclopedia-combination of numbers and shapes