Summary of Mathematics Problem Solving Methods in Senior High School 1, Matching Method
By using the method of constant deformation, some items in the analytical formula are matched to the sum of positive integer powers of one or more polynomials. The method of solving mathematical problems with formulas is called matching method. Among them, the most common method is to make it completely flat. Matching method 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. Factorization is the basis of identity deformation. As a powerful mathematical tool and method, it plays an important role in solving algebra, geometry and trigonometry problems. There are many methods of factorization, such as extracting common factors, formulas, grouping decomposition, cross multiplication and so on. Middle school textbooks also introduce the use of decomposition and addition, root decomposition, exchange 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. We usually refer to unknowns or variables as elements. 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. Discriminant method and Vieta theorem.
The root discrimination of unary quadratic equation ax2+bx+c=0(a, B, C belongs to R, a≠0) and△ = B2-4ac is not only used to judge the properties of roots, but also widely used in algebraic deformation, solving equations (groups), solving inequalities, studying functions and even geometric and trigonometric operations as a problem-solving method. 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 first judged that the obtained results have a certain form, which contains some undetermined coefficients, then the equations about undetermined coefficients are listed according to the problem setting conditions, and finally the values of these undetermined coefficients or some relationship between these undetermined coefficients are found. This method is called undetermined coefficient method to solve mathematical problems. 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. 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 correct anti-design, we need 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
8. 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.
9, 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.
Extended reading: summary of answering skills of Chinese novels in senior high school 1. "Rational plot" question type:
① Summarize the story with concise sentences;
(2) what role does this plot play in the text?
2. "Character analysis" question type:
(1) Point out the specific methods of describing characters in the novel and explain the benefits or functions;
② Briefly summarize the personality characteristics of the characters;
③ Evaluate the characters objectively and fairly.
3. "Look at the environment" question type:
① What are the characteristics and functions of scenery description in this article?
(2) describe the designated environment, analyze its role in the characters or express the theme.
4. "Talk about ideas" question type:
(1) Tell me about the characteristics of the work in material arrangement and analyze its benefits;
② What role does this sentence (paragraph) play in the text structure;
(3) contact the full text and point out what role something plays in the structure of the article.
5. "Speaking method" question type:
(1) What kind of expression method is adopted in this paper and its role in image-building;
(2) How the unique expression in this paper serves the author's expression;
③ What are the characteristics of language use?
④ Appreciate underlined sentences from the perspective of language use;
⑤ What are the characteristics of the characters' language in this article? Please give examples separately.
6. "Ming theme" question type:
Summarize the author's writing intention and the theme of the work in his own words;
What is the main idea of this article and why?
(3) Before and after the statement, are there any contradictions and why?
④ Explain the social significance of the novel.
Reading and answering skills of Chinese novels in senior high school
Classification and function of novel environment
Environment is the stage for people's activities, including natural environment and social environment.
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-the background of the story.
Function:
(1) Expounds the background of people's activities and their growth, and reveals various complicated social relations;
(2) explain the identity of the characters and show their personalities; Or influence or determine character;
(3) Reveal the essential characteristics of society and reveal the theme.
2. Natural environment
-Specific scenes of human activities, such as location, climate, time, scenery, scenes, etc.
Function:
(1) explain the location, climate, time, scenery, scene, etc. The story;
② Depicting the scenery ..., rendering the atmosphere ..., laying the emotional tone and paving the way for portraying the characters;
(3) Contrast personality or some kind of psychology;
(4) promoting plot development;
⑤ Deepen the theme;
6 symbols and tips;
⑦ Set suspense to stimulate readers' interest in reading.
Answering skills in reading Chinese novels in senior high school
Grasp the plot of the story
(1) Three kinds of problems:
① Summarize the story in one sentence or concise statement;
(2) What are some things written in the article, please summarize them in turn;
③ Summarize part of the novel (including pointing out one of the four parts: beginning, development, climax and ending).
(2) Analysis of the storyline of the novel can start from the following aspects:
① Seize the scene;
2 looking for clues;
③ Clarify the structure of the novel.
(3) When analyzing the plot of the novel, we should pay attention to two points:
① The development and change of plot is the embodiment of the development of contradictions and conflicts, and the analysis of novel plot must grasp the main contradictions and conflicts;
(2) Analyzing the plot is not the purpose of appreciating the novel, but a means to understand the character and grasp the theme of the novel. Therefore, in the process of analyzing the plot, we should always pay attention to understanding its role in the formation of characters and revealing the theme of the novel.
Reading and answering skills of Chinese novels in senior high school.
Appreciate the story.
This kind of topic is mainly about plot conception and its functional proposition.
Frequently asked questions:
1._ _ What role does the scene play in the novel?
2. What role does something or a character play in the novel?
Think about solving problems:
Content function+structure function
Define plot conception as the purpose of expressing characters, and combine the general functions of plot:
1, which explains the environment of the characters' activities.
2. Set suspense to stimulate readers' interest in reading.
3. Pave the way for the next plot development.
4. Take care of the previous article.
5. Promote the development of the plot.
6. Describe the role.
7. Express the main idea or deepen the theme.
8. Play the role of clues.
9. lay a good foundation.
First, create suspense and be fascinating; The second is the care before and after; The third is to set off the side and lay the groundwork; The fourth is to summarize the above and point out the significance of the problem; Five clues. Then answer according to the requirements of the topic and the article.
Organize language expression as required:
_ _ The plot (thing) has the function (structure) of … in the text, highlighting … and expressing … (content).
Evaluation of plot arrangement
① As far as the full text is concerned, it is full of twists and turns.
Function: Fascinating and gripping, enhancing the drama and readability of the story.
(2) As far as the beginning and the end are concerned, there is a beginning-end echo.
Function: make the structure compact and complete.
For example, "Sleeping Beauty" begins with the disappearance of the female A-corner, creating suspense, and ends with the audience applauding, exposing the mystery, making the unexpected plot complete, showing the character's personality and highlighting the theme. Another example is Snowy Night, in which snowflakes are written at the beginning and the end, which exaggerates the bleak atmosphere and hints at the fate of the characters.
(3) As far as the beginning is concerned, there is flashback (put the end at the beginning)
Such as "Blessing", first write the death of Xianglinsao, and then write how Xianglinsao was forced to death by feudal ethics step by step. Make a suspense.
As far as the ending is concerned, there is an abrupt end, leaving a blank.
Such as "calligrapher".
In addition, there are unexpected, tragic and comic styles.