Several ingenious methods to learn junior high school mathematics well
First of all, classify some error information in the exam:
(1) Regret mistake
It's a question that you can do clearly, but you do it wrong.
For example,? The mistakes in the review questions? It is caused by mistakes in examining questions and misreading figures. ? Calculation error? It is caused by a calculation error; ? Plagiarism mistake? I did it right on the draft paper, but I made a mistake when I copied it on the test paper and missed it; ? Misexpression? Is your answer correct but inconsistent with the expression of the topic requirements, such as mixing units?
② Contradictory mistakes
The understanding is not thorough enough and the application is not smooth enough; The answer is not rigorous and complete; Do it right the first time and correct it wrong; Or do something wrong for the first time and then correct it; A problem can't be done halfway, and so on.
③ The fault of omission.
Because no, I answered wrong or guessed right, or didn't answer at all. This is a problem without thinking, understanding and application.
Generally speaking, the ratio of these three types of errors is 2: 7: 1. You can also analyze the proportion of these three types of errors yourself. After you come to a conclusion, you will know what the problem is and solve it accordingly.
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Second, the reasons for these errors:
① Passive learning
Many students have strong dependence or inertia, just passively follow the inertia of teachers and have no initiative in learning. It is manifested in making uncertain plans, waiting for class, not previewing before class, not knowing what class the teacher will take, and being busy taking notes and not listening to the lecture. Doorways? I don't really know everything.
2 can't learn the law.
Teachers should generally explain the ins and outs of knowledge points, analyze the connotation of concepts, analyze key and difficult points, and highlight thinking methods. However, some students didn't concentrate in class, didn't hear the main points clearly or didn't listen completely, took a lot of notes and had many problems. After class, I can't consolidate, summarize and find the connection between knowledge in time, but I just catch up with my homework, confuse the problems, have a little knowledge of concepts, laws, formulas and theorems, imitate mechanically and memorize by rote. Some people work overtime at night, are listless during the day, or don't listen at all in class, so they have another set. The result is half the effort, with little effect.
③ Ignore the foundation.
Some? Feel good about yourself? Some students often despise the study and training of basic knowledge, skills and methods, and often only know how to do it, instead of calculating and writing carefully, but they are very interested in difficult problems to show their own? Level? So ambitious, heavy Quantity? Light? Quality? , fall into the sea of questions. Isn't it just a mistake in calculus or a routine homework or exam? Stuck? .
④ Mathematical thinking is not broad enough.
Some students don't summarize the depth, breadth and chapters of knowledge, do they? Multiple angles? Think about it, okay? Generalization? 、? Analogy? 、? Lenovo? 、? Abstract? And other methods and thinking.
(5) rote memorization, unable to transfer knowledge.
Junior high school mathematics is mainly expressed in vivid and popular language. Some students have established a unified mode of thinking, which can only be mechanically operated and form a fixed way. Not to strengthen the transfer of knowledge, but to try to solve a problem and start more? Brain? , thinking? Live broadcast? Get up. Be good at summarizing and forming some similar problems? How many questions do you ask? .
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Third, scientific learning methods:
It's not enough for students to just want to learn, they must also learn? Can you learn? Only by paying attention to scientific learning methods and improving learning efficiency can we turn passivity into initiative.
① Cultivate good study habits.
Good study habits include making plans, self-study before class, paying attention to class, reviewing in time, working independently, solving problems, systematically summarizing and studying after class.
Make a plan and define the purpose of learning. A reasonable study plan is the internal motivation to promote active learning and overcome difficulties. There are both long-term plans and short-term arrangements. In the process of implementation, we must be strict with ourselves and temper our learning will.
Preview before class is the basis for achieving better learning results. Preview can't go through the motions, we should pay attention to quality, try to understand the teaching materials before class, pay attention to the teacher's ideas in class, grasp the key points, break through the difficulties and solve the problems in class as much as possible.
Classroom is the key link to understand and master basic knowledge, skills and methods. Listen carefully to the key points and difficulties in class and record the contents added by the teacher, instead of writing them all down, paying attention to one and losing one.
Timely review is an important part of improving learning efficiency. By reading textbooks repeatedly, we can strengthen our understanding and memory of the basic concept knowledge system, and link the new knowledge we have learned with the old knowledge for analysis and comparison.
Independent homework is a process of further deepening the understanding of all new knowledge and mastering new skills through independent thinking, flexible analysis and problem solving.
Problem-solving refers to the process of understanding the exposed knowledge errors or missing answers due to blocked thinking in the process of completing homework independently, and making the thinking smooth and supplementing the answers through inspiration. Do the wrong homework again, and think twice if you don't understand the wrong place.
Systematic summarization is an important link to master knowledge and develop cognitive ability comprehensively, systematically and profoundly through positive thinking. To sum up, we should, on the basis of systematic review, take teaching materials as the basis, refer to notes and materials, and reveal the internal relationship between knowledge through analysis, synthesis, analogy and generalization, so as to master all the knowledge.
Extracurricular study includes reading extracurricular books and newspapers. Extracurricular learning is a supplement and continuation of in-class learning. It can not only enrich students' cultural and scientific knowledge, deepen and consolidate what they have learned in class, but also satisfy and develop our hobbies and cultivate their ability to study and work independently.
(2) Step by step to prevent impatience.
Because students are young and have limited experience, some students are impatient, some students are greedy and some students want to rely on it for a few days. Sprint? One stroke, one style, some people are complacent with a little achievement, and they will never recover when they encounter setbacks. Learning is a long-term accumulation process of consolidating old knowledge and discovering new knowledge, which can never be achieved overnight. Learning is a gradual and long-term accumulation process. You must have perseverance, determination and some efforts, and you must prevent impatience in order to achieve final success.
③ Study the characteristics of the subject and find the best learning method.
Mathematics is responsible for cultivating students' computing ability, logical thinking ability, spatial imagination ability, and the ability to analyze and solve problems by using what they have learned. It is characterized by a high degree of abstraction, logic and universality, and requires high ability. The specific ways to find it vary from person to person, but the five links of learning: preview, class, review, homework and summary are indispensable.
(4) communicate more, reflect more, solve doubts and doubts, and resolve the points of differentiation.
Communicate with classmates, consult teachers, carry out variant exercises, resolve differences, and achieve the purpose of mastering and applying knowledge flexibly.
As long as you learn the scientific method, you will have perseverance, confidence and hard work to overcome impatience and overcome it? Smart? Communicate more, reflect more, and develop good study habits, so as to successfully pass the learning adaptation period and make rapid progress in future math achievements.
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Four, some suggestions about learning mathematics:
1, take math notes, especially the different aspects of concept understanding and mathematical laws, as well as the extra-curricular knowledge supplemented by teachers.
2. Establish a mathematical error correction book.
3. Memorize mathematical laws and conclusions.
4. establish a good relationship with your classmates and work hard? Little teacher? , formal mathematics learning? Support groups? .
5. Increase extracurricular reading of mathematics and strengthen self-study.
6. Repeatedly consolidate and eliminate forgetting before school.
7. Learn to summarize and classify.
Nine classical problem-solving methods throughout three years' study
1. matching method
Using the method of constant deformation, the method of matching some items of an analytical formula into the sum of positive integer powers of one or more polynomials to solve mathematical problems is called matching method.
The most commonly used matching method is completely flat matching method, which is an important method of constant deformation in mathematics. It is widely used in factorization, simplifying roots, solving equations, proving equality and inequality, finding extreme values of functions and analytical expressions.
2. Factorization method
Factorization is to transform a polynomial into the product of several algebraic expressions, which is the basis of identity deformation. As a powerful tool of mathematics, it plays an important role in solving algebra, geometry, triangles and other problems.
Factorial decomposition method, in addition to extracting common factors, formula method, grouping decomposition method, cross multiplication and so on. What is introduced in middle school textbooks also includes adding items by splitting items, finding root decomposition, exchanging elements, undetermined coefficients and so on.
3. Alternative methods
Method of substitution is a very important and widely used method to solve problems in mathematics. Usually, unknowns or variables are called variables. The so-called method of substitution is to replace a part of the original formula with new variables in a complicated mathematical formula, thus simplifying it and making the problem easy to solve.
4. Discriminatory formula &; Vieta theorem
Univariate quadratic equation ax? +bx+c=0(a, B and C belong to R, A? 0) Discrimination of roots, △=b? -4ac(2 is square) can not only be used to judge the nature of roots, but also be used as a problem-solving method, which is widely used in algebraic deformation, solving equations (groups), solving inequalities, studying functions and even geometric and trigonometric operations.
Vieta's theorem not only knows one root of a quadratic equation, but also finds another root. Knowing the sum and product of two numbers, we can find the symmetric function of the root, calculate the sign of the root of quadratic equation, solve the symmetric equation and solve some problems about quadratic curve. , has a very wide range of applications.
5. undetermined coefficient method
When solving mathematical problems, it is called undetermined coefficient method to judge that the obtained results have a certain form, including some undetermined coefficients, then list the equations about undetermined coefficients according to the problem setting conditions, and finally find out the values of these undetermined coefficients or find out some relationship between them. It is one of the commonly used methods in middle school mathematics.
6. Construction method
When solving problems, we often use this method to construct auxiliary elements by analyzing conditions and conclusions, which can be a figure, an equation (group), an equation, a function, an equivalent proposition and so on. And establish a bridge connecting conditions and conclusions, so that the problem can be solved. This mathematical method of solving problems is called construction method. Using construction method to solve problems can make algebra, trigonometry, geometry and other mathematical knowledge permeate each other, which is beneficial to solving problems.
7. Find the area method
The area formula in plane geometry and the property theorems related to area calculation derived from the area formula can be used not only to calculate the area, but also to prove that plane geometry problems sometimes get twice the result with half the effort. The method of proving or calculating plane geometric problems by using area relation is called area method, which is commonly used in geometry.
The difficulty in proving plane geometry problems by induction or analysis lies in adding auxiliary lines. The characteristic of area method is to connect the known quantity with the unknown quantity by area formula, and achieve the verification result through operation. Therefore, using the area method to solve geometric problems, the relationship between geometric elements becomes the relationship between quantities, and only calculation is needed. Sometimes there may be no auxiliary lines, even if auxiliary lines are needed, it is easy to consider.
8. Geometric transformation method
In the study of mathematical problems, the transformation method is often used to transform complex problems into simple problems and solve them.
The so-called transformation is a one-to-one mapping between any element of a set and the elements of the same set. The transformation involved in middle school mathematics is mainly elementary transformation. There are some exercises that seem difficult or even impossible to start with. We can use geometric transformation to simplify the complex and turn the difficult into the easy.
On the other hand, the transformed point of view can also penetrate into middle school mathematics teaching. It is helpful to understand the essence of graphics by combining the research of graphics under isostatic conditions with the research of motion.
Geometric transformation includes: (1) translation; (2) rotation; (3) symmetry.
9. reduce to absurdity
Reduction to absurdity is an indirect proof method. First, a hypothesis contrary to the conclusion of the proposition is put forward, and then from this hypothesis, through correct reasoning, contradictions are led out, thus denying the opposite hypothesis and affirming the correctness of the original proposition.
The reduction to absurdity can be divided into reduction to absurdity (with only one opposite conclusion) and exhaustive reduction to absurdity (with more than one opposite conclusion).
The steps to prove a proposition by reduction to absurdity can be roughly divided into: (1) reverse design; (2) return to absurdity; (3) conclusion.
Anti-design is the basis of reduction to absurdity. In order to make a correct anti-design, it is necessary to master some commonly used negative expressions, such as:
Yes/no; Existence/non-existence; Parallel/non-parallel; Vertical/not vertical; Equal to/unequal to; Large (small) inch/small (small) inch; Both/not all; At least one/none; At least n/ at most (n-1); At most one/at least two; Only/at least two.
Reduction to absurdity is the key to reduction to absurdity. There is no fixed model in the process of derivation of contradiction, but it must be based on reverse design, otherwise the derivation will become passive water without roots. Reasoning must be rigorous.
There are the following types of contradictions: contradictions with known conditions; Contradicting with known axioms, definitions, theorems and formulas; There are dual contradictions; Contradictions