How can the mathematical ideas of the third year of junior high school be trained?

Mathematics problems can never be written, you have to find your own way, think of our grandson, let us go to the podium to explain the topic, you don't expect to write the old man, the topic is not only the few in the book, geometry should be bypassed, you should always look at the topics you have written, whether you can write or not. If there is a topic that you can't write, you should ask others for advice in time, it is not enough to just look at other people's steps, you also need someone to explain, so that you can learn well. If you encounter a difficult problem, you should think of the auxiliary line, Brother Sun taught us to treat the answer as a known condition to do it in reverse, then the condition you lack depends on how you make the auxiliary line, in this way, you should be able to learn well.

Anyway, that's how I learned.

Hopefully, thank you

You have to do more problems, our teacher said, math problems are to do more problems, do more, practice naturally makes perfect.

It is also necessary to cultivate a kind of thinking ability, think step by step according to the conditions of the topic, be logical, do not mess up the brain, and figure out the intention of the topic. When you encounter a problem that you can't do, you can generally understand it by looking at other people's solution ideas, but it will improve you.

Thinking ability doesn't help much, you still have to think more and think more for yourself.

Yes, but after reading it, you have to think about why others understand it this way.

The key to geometry is to master the theoretical knowledge and then try to use it. If you use it more, you will have your own inspiration. When I encounter a problem that I can't solve, I feel useful to see other people's solutions to this problem, but it's not too big, the key is to know why you do it, and you can ask the teacher the best question about this question.

Categorize discussion ideas. The classification discussion is to divide the teaching objects into different categories according to their essential attributes, that is, according to the commonality and differences of the teaching objects, those with the same attributes are classified into one category, and those with different attributes are classified into another category. Classification is an important means of mathematical discovery.

In teaching, if the knowledge learned is properly classified, a large amount of complex knowledge can be organized.

Numbers and shapes combine ideas. Generally speaking, people call algebra "number" and geometry "shape", number and shape seem to be independent of each other on the surface, but in fact, under certain conditions, they can be transformed into each other, and the quantity problem can be transformed into a graph problem, and the graph problem can also be transformed into a quantity problem.

The combination of numbers and shapes is fully utilized in all grades. In mathematics teaching, the combination of numbers and shapes with numbers has the advantage of making problems intuitively presented, which is conducive to deepening students' knowledge and understanding. When solving math problems, the combination of numbers and shapes is conducive to students to analyze the relationship between quantities in the problem, enrich the appearance, trigger associations, enlighten thinking, broaden ideas, and quickly find ways to solve problems, so as to improve the ability to analyze and solve problems. Grasping the combination of numbers and shapes with ideological teaching can not only improve students' ability to transform numbers and shapes, but also improve students' ability to transfer their thinking.

Methods of Mathematical Thought in Secondary School Mathematics.

Methods of mathematical thought can be divided into three levels from the degree of difficulty of acceptance:

The first is basic and specific mathematics.

methods, such as matching method, commutation method, undetermined coefficient method, inductive method and deduction method, etc.; The second is the logical side of science.

methods, such as observation, induction, analogy, abstract generalization, etc., as well as analytical, synthetic, and counter-evident methods.

methodology; The third is mathematical ideas, such as the idea of combining numbers and shapes, the idea of functions and equations, and the idea of classification and discussion.

Think about the idea of transformation and transformation.

Mathematical methods of thought can also be classified in other ways.

For example, Hu Jiong.

Tao believes that the highest level of basic mathematical ideas is the foundation and starting point of mathematics textbooks, and the whole middle school is taught.

The content follows the trajectory of basic mathematical ideas.

Symbolizing and Transforming Ideas".

Collections and correspondence.

Ideas" and "axiomatic and structural ideas" constitute the basic mathematical ideas at the highest level. He thinks secondary school.

The basic ideas of mathematics refer to:

It has a universal and strong adaptability in the knowledge and methods of mathematics in secondary schools.

Essential thoughts. It is summarized into ten aspects:

Symbolic thoughts, mapping thoughts, reducing thoughts, decomposing thoughts

Transformation ideas, parametric thoughts, inductive thoughts, analogical thoughts, deductive ideas, model ideas.

Methods in Logic:

analytical, synthetic, anti-correct, inductive; Concrete numbers.

Learning methods: matching method, commutation method, pending coefficient method, same method, etc.

Junior high school mathematical thoughts: The first is the idea of combining numbers and patterns, as well as the idea of classification and discussion from special to general, the idea of transformation, the idea of analogy, and the idea of limit, which is not commonly used in junior high school.

Transformation thoughts, the idea of combining numbers and shapes, the idea of wholeness, the idea of equations, the idea of analogy, the idea of special to the general, the idea of categorical discussion, the idea of limit, etc.

Mathematical problem-solving methods include matching method, commutation method, factorization method, undetermined coefficient method, counterproof method, same method, construction method, geometric transformation method, area method, verification and elimination method, screening method, ** method, etc.

The combination of numbers and shapes, the classification and discussion method.

The methods of thinking in junior high school mathematics can be divided into three levels from the degree of difficulty of acceptance:

The first is the basic and specific mathematical methods, such as the matching method, the commutation method, the undetermined coefficient method, the inductive method and the deduction method.

the second is the scientific logical method, such as observation, induction, analogy, abstract generalization and other methods, as well as the analytical method, the synthesis method and the counter-evidence method.

The third is mathematical ideas, such as the idea of combining numbers and shapes, the idea of functions and equations, the idea of classification and discussion, and the idea of reduction and transformation.

For example: 1. The idea of combining numbers and shapes.

The idea of combining numbers and shapes is to analyze the meaning of algebra and the meaning of geometry according to the internal relationship between the conditions and conclusions given by the mathematical problem, and combine the quantitative relationship shown by the problem with the figure (drawing), and use this combination to find the idea of solving the problem and solve the problem.

2. Categorize and discuss ideas.

In mathematics, sometimes according to the conditions given by the problem, there may be a variety of different situations, then it is necessary to integrate all possible situations together through classification discussion to obtain the final result, this method of categorical thinking, is an important mathematical thinking method, but also an important problem-solving strategy.

3. Substitution method.

In the process of solving the problem, the formula of one or a certain letter is regarded as a whole and represented by a new letter to achieve the purpose of simplifying the formula. The commutation method can simplify a more complex formula, reduce the problem to a more basic problem than the original, and achieve the effect of simplifying the complex and turning the difficult into easy.

4. Matching method.

Try to form a flat formula and then make the desired transformation. This method is often used when solving the problem of the maximum value of the quadratic function, solving the most cost-effective practical problem, and maximizing profits.

5. Pending coefficient method.

When the mathematical formula we are studying has a certain form, to determine it, we need to find the value of the letter to be determined in the formula; For this reason, it is necessary to substitute the known conditions into the undetermined formula, and often obtain an equation or system of equations containing the letters to be determined, and then solve this equation or system of equations to solve the problem.

1. The idea of combining numbers and shapes: according to the internal relationship between the conditions and conclusions of mathematical problems, it not only analyzes its algebraic meaning, but also reveals its geometric significance; Combine quantitative relations and graphics skillfully and harmoniously, and make full use of this combination to seek disintegration ideas and solve problems.

2. The idea of classification and discussion: In mathematics, we often need to examine it in various situations according to the differences in the nature of the research object; This method of categorical thinking is an important method of mathematical thinking, and it is also an important problem-solving strategy.

3. The idea of connection and transformation: things are interconnected, mutually restricted, and can be transformed into each other. The various parts of mathematics are also interconnected and can be transformed into each other.

Extended content: The position of mathematical ideas and methods in the high school entrance examination is becoming more and more important, and the proportion is also increasing. Nowadays, the classroom has completely changed to "learning" as the mainstay, supplemented by "teaching".

However, many students' learning ability is still very poor, and they are still in the stage of memorizing and applying formulas, which requires us to not only teach children knowledge, but more importantly, teach children how to learn and how to use. Therefore, mathematical ideas and methods are implemented into classroom teaching, and students' ability to learn mathematics and apply mathematics is gradually cultivated.

Mathematical thinking is mainly trained in practical skills in junior high school. For questions that are not easy to calculate directly, they can be calculated by drawing thinking.

1+1/2+1/4+1/8+1/16+1/32+1/64=？If it is quite troublesome to calculate directly, it can be used as a graph method for mathematical thinking, and the calculation is relatively simple. If you look at a square as a whole 1 and plot the fraction of the square, you will find that you can get the result by subtracting 1 64 from 1.

That is: the original formula = 1-1 64 = 63 64, and the answer is solved. There are also many drawing methods and problems and logical reasoning, and through practical training, the brain's potential for many mathematical thinking can be developed.