Learning math well and solving problems is the key. In the process of solving problems, we should not only strengthen the necessary training, but also master certain rules and skills of solving problems.
1. Mathematical thinking plays an important role in solving problems.
The usual procedures in problem-solving learning are: reading mathematical knowledge and understanding concepts; Reflect on examples and teachers' explanations, and think about the methods, skills and standardized processes of solving problems; Then do math exercises.
Basic questions should be practiced in procedures and speed; Typical problems try to solve a problem and develop mathematical thinking; Finally, we should summarize, reflect and correct mistakes in time, and exchange and learn good solutions and skills. Paulia, a famous math educator, said: "If you don't reflect, you will miss an important and meaningful aspect of solving problems."
In teaching design, teachers should have a clear understanding of mathematical thinking methods, so as to better explore the role of topics, guide students to find solutions and skills to summarize topics, and improve their ability to solve problems.
1. Concepts of functions and equations
The idea of function and equation is the most basic idea in middle school mathematics. The idea of function is to analyze and study the quantitative relationship in mathematics from the point of view of movement change, establish the functional relationship or construct the function, and then analyze and solve the related problems by using the image and nature of the function. The idea of equation is to analyze the equivalence relation in mathematics, construct an equation or an equation, and analyze and solve problems by solving or using the properties of the equation.
2. The idea of combining numbers with shapes
Numbers and shapes can be changed under certain conditions. For example, some algebraic problems and trigonometric problems often have geometric background, and we can solve the related algebraic trigonometric problems with the help of geometric features; Some geometric problems can often be solved by algebraic methods through quantitative structural characteristics. Therefore, the idea of combining numbers and shapes plays an important role in solving problems.
3. The idea of classified discussion
The idea of classified discussion is very important because it is logical, because it covers a wide range of knowledge points, and because it can cultivate students' ability to analyze and solve problems. The fourth reason is that it is often necessary to discuss various possibilities in practical problems.
The key to solve the problem of classified discussion is to break the whole into parts and reduce the difficulty of local discussion. Common types: type 1: discussion caused by mathematical concepts, such as the positional relationship between real numbers, rational numbers, absolute values, points (straight lines and circles) and circles; The second category: discussions caused by mathematical operations, such as whether both sides of inequality are multiplied by a positive number or a negative number; The third category: the discussion caused by the restrictive conditions of properties, theorems and formulas, such as the discussion caused by the application of the root formula of the quadratic equation of one variable; The fourth category: the discussion caused by the uncertainty of graphic position, such as the discussion caused by related problems in right, acute and obtuse triangles. The fifth category: classification discussion caused by the influence of some letter coefficients on the equation, such as the influence of the number of letters in quadratic function on the image, the influence of quadratic term coefficient on the image opening direction, the influence of linear term coefficient on the vertex coordinates, and the influence of constant term on the intercept.
The idea of classified discussion is a way of thinking to classify mathematical objects and seek answers. Its function is to overcome the one-sidedness of thinking and consider problems comprehensively. Classification principle: classification is neither heavy nor leakage. The steps of classification: ① determine the object of discussion and its scope; (2) Determine the classification criteria for classified discussion; (3) classified discussion; (4) Summary and comprehensive conclusion. Pay attention to the dynamic problem, and you must draw a dynamic diagram first.
4. The concept of transformation and transformation
Conversion is one of the most basic mathematical ideas in city middle school mathematics. The combination of numbers and shapes embodies the transformation of numbers and shapes. The idea of function and equation embodies the mutual transformation between function, equation and inequality; The idea of classified discussion embodies the mutual transformation between the part and the whole, so the above three ideas are also the concrete manifestations of transformation and transformation.
However, transformation includes equivalent transformation and non-equivalent transformation, and equivalent transformation requires that the cause and effect in the transformation process are sufficient and necessary; There is only one case of unequal conversion, so the conclusion should be tested, adjusted and supplemented. The principle of transformation is to turn unfamiliar and difficult problems into familiar, easily solved and solved problems, and to turn abstract problems into concrete and intuitive problems; Turn complex problems into simple ones; Turn the general into a special problem; Turn practical problems into mathematical problems and so on, so that problems can be easily solved.
However, transformation includes equivalent transformation and non-equivalent transformation, and equivalent transformation requires that the cause and effect in the transformation process are sufficient and necessary; There is only one case of unequal conversion, so the conclusion should be tested, adjusted and supplemented. The principle of transformation is to turn unfamiliar and difficult problems into familiar, easily solved and solved problems, and to turn abstract problems into concrete and intuitive problems; Turn complex problems into simple ones; Turn the general into a special problem; Turn practical problems into mathematical problems and so on, so that problems can be easily solved.
Common conversion methods are as follows
(1) Direct transformation method: the original problem is directly transformed into a basic theorem, a basic formula or a basic graphic problem.
(2) method of substitution: Using "method of substitution" to transform formulas into rational formulas or algebraic expressions into idempotents, and to transform more complicated functions, equations and inequalities into basic problems that are easy to solve.
(3) Number-shape combination method: study the relationship between quantity (analytical formula) and spatial form (figure) in the original problem, and obtain the transformation path through mutual transformation.
(4) Equivalent transformation method: the original problem is transformed into an equivalent proposition that is easy to solve, so as to achieve the purpose of reduction.
(5) specialization method: transform the form of the original problem into a specialized form, prove the specialized problem, and make the conclusion suitable for the original problem.
(6) Construction method: "Construct" an appropriate mathematical model and turn the problem into an easy-to-solve one.
(7) Coordinate method: Using coordinate system as a tool to solve geometric problems through calculation is also an important way of transformation method.
The guiding ideology of transformation and transformation?
(1) What problem should be transformed into an object?
(2) Where is the transition, that is, the transition to the goal?
(3) How to transform, that is, the transformation method.
Transformation and transformation are the core of all mathematical thinking methods.
Second, the basic methods to solve mathematics problems in middle schools
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(1) observation method: intuitively and purposefully discover the laws, properties and solutions of mathematical objects.
(2) Experimental method: The experimental method is to create some mathematical objects that are beneficial to observation with purpose and simulation, and to visualize and simplify complex problems through observation and research. It has the important advantages of strong intuition, clear characteristics, trial solution and experimental conclusion.
2. Comparison and classification
(1) comparison method
It is a way of thinking to determine the similarities and differences of things. In mathematics, two mathematical objects must have a certain relationship before they can be compared. We often compare the similarities and differences between two kinds of mathematical objects or make a comprehensive comparison.
(2) Classification
Classification is a way of thinking based on the comparison and similarities and differences of mathematical objects, which classifies objects with the same nature into one category and objects with different properties into different categories. As shown in the above figure, the classification of k of a linear function is greater than zero and less than zero when it is not equal to zero, which embodies the principle of no repetition and no leakage.
3. Special and General
(1) specialization method
The method of specialization is to narrow the scope from a given area, even to a special value, special point, special figure, etc., and then consider the solution and rationality of the problem.
(2) Generalized method
4. Association and conjecture
(1) analogy association
Analogy is a way of thinking that another thing may have certain attributes according to the same or different attributes between two objects or things.
New knowledge can be discovered through analogy and association; Through analogy and association, find out the methods and ways to solve mathematical problems;
(2) Inductive conjecture
Newton said: Without bold conjecture, there would be no great invention. Guess can find truth and judgment; Conjecture can foresee the methods and ideas of proof. Junior high school mathematics is mainly to observe the conditions of propositions and draw conclusions, or to put forward solutions and methods to solve problems through observing conditions and conclusions.
Induction is a thinking process of drawing general conclusions from similarities or similarities contained in similar things. There are complete induction and incomplete induction. The conjecture obtained by complete induction is correct, and the conjecture obtained by incomplete induction may be correct or wrong, so it needs to be proved as a conclusion. The key is to make a reasonable and well-founded guess.
5. Substitution and formula
(1) replacement method
When solving a mathematical problem, we regard a formula as a whole and replace it with a variable, thus simplifying the problem. This is called substitution. The essence of substitution is transformation, the key is to construct elements and set elements, and the theoretical basis is equivalent substitution. The purpose is to change the research object, move the problem to the knowledge background of the new object, standardize non-standard problems, simplify complex problems and become easy to deal with.
Substitution method is also called auxiliary element method and variable substitution method. By introducing new variables, scattered conditions can be linked, implicit conditions can be revealed, or conditions can be linked with conclusions. Or turn it into a familiar form to simplify complicated calculation and derivation.
When using substitution method, we should follow the principle of facilitating operation and standardization. Pay attention to the selection of the new variable range after substitution, and make sure that the new variable range corresponds to the value range of the original variable, which cannot be reduced or expanded. You can observe the formula first, and you can find that there is always the same formula in this formula for method of substitution, and then use a letter instead of them to calculate the answer. Then, if there is this letter in the answer, bring the formula in and the calculation will come out.
(2) Matching method
Matching method is a technique of directional deformation of mathematical formula (matching into a "complete square"), and the relationship between known and unknown is found through the formula, thus simplifying the complex. When making a formula, it is necessary to make appropriate predictions and use the skills of "dividing", "adding", "matching" and "gathering" reasonably, so as to complete the formula. Sometimes called "matching method". The most common formula is identical deformation, so that the mathematical formula appears completely square. It is mainly suitable for discussing and solving known or unknown quadratic equations, quadratic inequalities, quadratic functions and quadratic algebras. The most basic formula used in the matching method is the binomial complete square formula (A+B) 2 = A 2+2AB+B 2. By using this formula flexibly, various basic formula forms can be obtained.
6. Construction method and undetermined coefficient method
(1) construction method The so-called construction method refers to the concepts and methods in mathematics, which can be defined and realized in a fixed way through limited steps. Common ones are constructors, graphs and identities. Adding auxiliary lines in plane geometry is a common drawing method. There are three ways to solve problems by construction method: direct construction, changing conditional construction and changing conclusion construction.
(2) The undetermined coefficient method: the polynomial is expressed as another new undetermined coefficient form, thus an identity is obtained. Then, according to the properties of identity, the equation or equation that the coefficient should satisfy is found, and then the coefficient to be solved is found by solving the equation or equation, or the relationship that some coefficients satisfy is found. This method of solving problems is called undetermined coefficient method.
7. Formulas and reduction to absurdity
(1) formula method
The method of solving problems by using formulas. The most commonly used method in junior high school is to find the root of a quadratic equation with one variable by formula; The method of complete square formula, etc. For example, the following set of problems is the application of the complete square formula:
(2) The reduction to absurdity is an "indirect proof method", that is, the title is affirmed and the conclusion is denied, which leads to contradictions, the correctness of the conclusion of the proposition is affirmed and the proposition is proved.
Third, methods and skills to solve new questions in middle school mathematics
1. Mathematical inquiry questions
The so-called exploratory problem is to explore the corresponding conclusions and prove them from the given conditions of the problem, or to explore the necessary conditions and ways to solve the problem from the given requirements of the problem.
Conditional inquiry questions: One of the problem-solving strategies is to regard the questions and conclusions as known, and at the same time make inferences to find out the corresponding required conditions in the process of deduction.
Conclusion exploration question: usually refers to the conclusion is uncertain and not unique, or the conclusion needs analogy, promotion, promotion, or special circumstances need to be summarized. You can guess first and then prove it; You can also seek to prove the conclusion under specific circumstances; Or direct deduction.
Law inquiry: actually, it is to explore a variety of methods to solve problems and formulate a variety of strategies to solve problems.
Activity-based problem exploration: let students participate in certain social practice and solve problems through in-class and out-of-class exploration.
Popularize and explore questions: a simple question can be popularized to produce new conclusions, which is common in junior high school teaching. For example, the judgment of parallelogram can produce many new generalizations, on the one hand, it is its own generalization, on the other hand, it can be extended to rhombus and square.
Exploration is the lifeline of mathematics, and solving exploration problems is a creative thinking activity. The exploration of a mathematical form is by no means the result of a single way of thinking, but the connection and infiltration of multiple ways of thinking, which can make students dare to question, ask questions, reflect and popularize in the process of learning mathematics. Through exploration, experience the process of mathematical discovery, mathematical inquiry and mathematical creation, and experience the happiness brought by creation.
2. Mathematical situation problem
Situation problem is to put forward mathematical ideas and methods in the situation according to the life reality, stories, history, games and mathematical problems in a period of time. This kind of questions are often lively and interesting, which stimulates students' strong research motivation, but at the same time, mathematical situational questions have the characteristics of large amount of information and strong openness, which requires students to extract mathematical problems from the scene and experience the mathematical process of studying practical problems with the help of mathematical knowledge.
For example, when the teacher is talking about the mixed operation of rational numbers,
3. Open mathematical problems
The open problem of mathematics is a new kind of problem compared with the traditional closed problem, which is characterized by insufficient conditions or no definite conclusion. Because of this, the strategies for solving open-ended problems are often varied.
(1) Open mathematics questions generally have the following characteristics.
Uncertainty: The questions raised are often uncertain and general, and the background is also described in general terms, so we need to collect other necessary information to solve the problem.
② Inquiry: There is no ready-made problem-solving model, and some answers may be easily found by intuition, but in the process of solving problems, we often need to think and explore from multiple angles.
③ Incompleteness: The answers to some questions are uncertain and there are various answers, but what matters is not the diversity of the answers themselves, but the reconstruction of students' cognitive structure in the process of seeking answers.
Divergence: In the process of solving, new problems can often be brought out, or the problems can be generalized to find more general and general conclusions. Often put forward through practical problems, students must use mathematical language to mathematize it, that is, establish a mathematical model.
⑤ Development: It can arouse the curiosity of most students, and all students can participate in the solution process.
⑥ Innovation: It is difficult for teachers to use injection teaching, and students can naturally take the initiative to participate. Teachers' roles in problem solving are demonstrator, enlightener, encourager and collaborator.
(2) the classification of mathematics open questions
Starting from the four elements (conditions, basis, methods and conclusions) that constitute the mathematical problem system, it can be qualitatively divided into four categories; If the answer sought is the condition of a mathematical problem, it is called conditional open problem; If the answer sought is a basis or method, it is called a policy open question; If the answer sought is a conclusion, it is called a conclusive open question; If the conditions, solving strategies or conclusions of a mathematical problem need to be set and searched by the solver in a given situation, it is called a comprehensive open problem.
Collect materials from students' study life and familiar things, and design various forms of mathematics open questions, aiming at opening students' thinking and potential learning ability. The opening of mathematics is entitled that students of different levels have created opportunities to learn mathematics well. The application of various problem-solving strategies effectively develops students' innovative thinking, cultivates students' innovative skills and improves students' innovative ability.
(3) Opening up the teaching characteristics of "carrier" with mathematics.
① Open teacher-student relationship: Teachers and students become problem-solving collaborators and researchers.
② Open teaching content: Open questions often have incomplete conditions or conclusions, which need to be analyzed and studied by collecting data, leaving room for innovation in mathematics.
③ Openness of teaching process: Because the openness of research content can arouse students' curiosity, and because of the openness of questions, there is no ready-made problem-solving model, so there will be room for imagination for all students to participate in imagination and answer.
(4) The educational value of open questions.
It is beneficial to cultivate students' good thinking quality;
Contribute to the formation of students' subjective consciousness;
It is conducive to the participation of all students and the realization of democracy and cooperation in teaching;
It is beneficial for students to experience success, build confidence and enhance their interest in learning;
It is helpful to improve students' ability to solve problems.
4. Mathematical modeling problems (junior high school mathematical modeling problems can also be regarded as mathematical application problems)
The new curriculum standard of mathematics points out that students should apply what they have learned and solve practical problems to meet the basic needs of social daily life and productive labor. One of the learning purposes of junior high school mathematics is to cultivate students' ability to solve practical problems, requiring students to analyze and solve mathematical problems in production and life, and to form the consciousness and ability to be good at applying mathematics. Judging from the mathematics propositions of senior high school entrance examinations in various provinces and cities, more attention is paid to the examination of students' ability to flexibly use mathematical knowledge to solve practical problems. It can be said that cultivating students' ability to solve practical problems is one of the basic ways to enable students to use their mathematical knowledge to solve practical problems.
Three Types of Math Application Problems in Junior Middle School
(1) Explore the application of conclusion mathematics
According to the conditions given in the proposition, it is required to find one or more correct conclusions.
(2) interdisciplinary mathematics application problems
① Mathematics and physics
② Mathematics and biochemistry
The above two problems are related to biology and chemistry, which embodies the application of mathematics in biochemistry.
In a word, the application of mathematics well examines students' reading comprehension ability and daily life experience, and at the same time examines students' abstract generalization and modeling ability as well as their judgment and decision-making ability after obtaining information. The hot topics of mathematics application in senior high school entrance examination mainly include life, statistics, measurement, design, decision-making, sales, open exploration, interdisciplinary and so on. The senior high school entrance examination has a good guiding role in strengthening students' application consciousness and ability. This requires us to be good at tapping the potential application functions of textbook examples and exercises in our usual teaching. The typical mathematical problems in textbooks are skillfully returned to the prototype of life production, and the practical background is created, which is transformed into practical problems with profound mathematical connotation, thus enhancing the application consciousness and developing the mathematical modeling ability.
Fourth, master junior high school math problem-solving strategies to improve math learning efficiency.
(1) Carefully analyze the problem and find the breakthrough point to solve it.
Because of the complexity of mathematical problems, students are easily influenced by fixed thinking, which will have a great influence on problem-solving thinking. To this end, teachers should give students correct guidance, help students adjust their thinking, carefully re-analyze the topic, and find the right starting point, so that the problem will be solved. For example, AB=DC and AC=DB are known. Proof: ∠ A = ∠ D.
This problem is a classic problem to prove congruence, mainly to train students' ability to integrate known conditions and observe and read pictures. But to prove ∠AOC=∠DOB from the intuitive point of view of graphics, such an idea will only fall into the trap of topic setting. Therefore, when reviewing this topic, teachers should guide students to fully consider the two known conditions of the topic and remind students to add some auxiliary lines appropriately.
(2) Use your imagination and take advantage of regional advantages to win by surprise.
Area problem is a common problem in mathematics. There are profound mathematical ideas in the definition of area and related laws. If students can fully understand the charm and master mathematical reasoning thinking, it is possible to solve other mathematical problems with the help of area. Because the area of geometric figures is closely related to line segments, angles, arcs, etc. The area method can not only prove the equivalence of various geometric figures, but also prove several geometric problems such as equal line segments, unequal line segments, equal angles and proportional formulas. Example 1. If e and f are the midpoint of side AB and side CD of rectangular ABCD respectively, and rectangular EFDA is similar to rectangular ABCD, the aspect ratio of rectangular ABCD is () (a)1:2 (b) 2:1(c)1:2 (d
According to the ABove known information, the ratio of the width AD to ab of rectangular ABCD is the similarity ratio of rectangular EFDA to rectangular ABCD. Solution: Let the similarity ratio of rectangular EFDA and rectangular ABCD be K. Because E and F are the midpoint of rectangular ABCD, S rectangular ABCD=2S rectangular EFDA. So s rectangle EFDA: s rectangle ABCD=k2. So k = 1: 2. That is, the length-width ratio of rectangular ABCD is1∶ 2; Therefore, choose (c).
This problem uses the property that the area ratio of similar polygons is equal to the square of the similarity ratio, and skillfully solves the problem of the aspect ratio in similar rectangles. In fact, the process of forming problem-solving ideas with the help of area is the process of students' thinking transformation.
(3) Take special values skillfully to simplify the generation.
Although junior high school mathematics is basic mathematics, it does not mean that it is not difficult. Especially in quality education, junior high school mathematics pays more and more attention to the cultivation of mathematical thinking from the perspective of cultivating students' comprehensive quality and ability. Therefore, the setting of many math problems is quite difficult to adjust, which makes math problems more complicated. A single way of thinking or solving problems will be more difficult in front of some topics. If we study some mathematical problems in a certain range and consider all the values one by one, there will be many problems and even get into trouble. In this case, it is the key to solve the problem to avoid the conventional solution and jump out of the established mathematical thinking.
Example 2. Decomposition factor: x2+2xy-8y2+2x+ 14y-3.
Train of thought analysis: This problem is a binary polynomial, so it is good to solve it from the conventional way of thinking. However, from the perspective of cultivating students' thinking ability, teachers can guide students to explore other problem-solving methods on the basis of conventional solutions. If one of the unknowns is set to 0 from the angle of taking a special value skillfully, it can be temporarily hidden and factorized according to the formula of another unknown, so as to achieve the purpose of converting binary into one.
Solution: let y=0 and get x [sup] 2 [/sup]+2x-3 = (x+3) (x-1); Let x=0, and we get:-8y2+14y-3 = (-2y+3) (4y-1). When the coefficients of the first term decomposed twice are 1 and 1; -2、4。 It can be seen that 1×4+(-2)× 1 is exactly equal to the coefficient of the xy term in the original formula. So it is: x2+2xy-8y2+2x+14y-3 = (x-2y+3) (x+4y-1).
In fact, using the special value method is also called the zero point method. This method can play a great role in factorization and help students find other methods to solve problems. Generally speaking, the steps are as follows: a, factorize the result obtained by setting one letter in the polynomial to 0; B, factorizing the result obtained by setting another letter in a plurality of items to 0; C, synthesizing the results of the first two steps to obtain the decomposition result of the original polynomial. However, it should be noted that the constant term of the first factor of two factorizations must be equal. For example, in this question, 3 of x+3 is equal to 3 of -2y+3, and-1 is equal to-1 of 4y. Otherwise, combining the results of these two steps, you will be at a loss.
(4) ingenious transformation and transition solution
When solving mathematical problems, we should not only comprehensively analyze the known conditions, but also be good at excavating the hidden conditions in the problems, and skillfully use the relationship between knowledge in mathematics to solve the problems from a comprehensive and brand-new perspective.
For example, it is known that AB is the diameter of a semicircle, its length is 30 cm, and points C and D are bisectors of the semicircle. Find the graphic area surrounded by chord AC, AD and arc CD.
This problem needs to be solved is the area of an irregular figure. Perhaps the thinking of most students is to connect CDs and turn them into a corner and an arch. The sum of the two areas is the problem to be solved. At this time, teachers should guide students to learn to use the known conditions of radius to help them connect the other two OC and OD auxiliary lines, and convert the area of irregular graphics required to be solved by the topic into the area of fan-shaped OCD, so as to make the problem-solving ideas clear at a glance.
To sum up, junior high school mathematics problem-solving has strong flexibility. Some math problems have more than one solution, but many solutions. Some math problems can't be solved by conventional methods and need special methods. Therefore, we should pay attention to its flexibility and skill when solving mathematical problems. Problem-solving skills are very important in the senior high school entrance examination and cannot be ignored. Junior high school math teachers should attach importance to the study of problem-solving skills, encourage students to think differently, discover problem-solving skills, improve problem-solving efficiency and enhance their ability to learn math.