Effective learning methods of mathematical geometry
First, gradually improve the ability of logical argumentation.
The proof of solid geometry is irreplaceable by any part of mathematics. Therefore, there has been a saying of solid geometry in the college entrance examination over the years. When demonstrating, we must first be rigorous and accurately understand any definition, theorem and inference. The symbolic representation is completely consistent with the theorem, and only when all the conditions of the theorem are met can the relevant conclusions be deduced. Don't jump to conclusions without complete conditions. Secondly, when demonstrating the problem, we should use analytical methods, that is, gradually find the sufficient conditions for the conclusion to be established, move closer to the known, and then use comprehensive methods (? Push out method) written in the form.
Second, based on textbooks, lay a solid foundation
A shortcut to learning solid geometry is to learn theorem proving in textbooks, especially the proof of some key theorems. The content of the theorem is very simple, that is, the explanation of the relationship between lines, lines and surfaces, and surfaces. However, theorem proving is generally more complicated or even abstract when beginners learn. Grasp the content of the theorem deeply, and make clear what the function of the theorem is, where it is used and how to use it.
Third, cultivate spatial imagination.
In order to cultivate spatial imagination, some simple models can be made at the beginning of learning to help imagination. For example, a cube or a cuboid. Find the relationship between line, line and face, and face to face in a cube. By observing the position relationship of points, lines and surfaces in the model, I gradually cultivate my imagination and recognition ability of spatial graphics. Secondly, we should cultivate our own painting ability. You can start with simple figures (such as lines and planes) and simple geometries (such as cubes). The last thing to do is to establish a three-dimensional concept, so that you can imagine the graphics of space and draw them on a plane (such as paper and blackboard), and you can also draw them on a plane according to it? Stereo? Graphics, imagine the real shape of the original space graphics. Spatial imagination is not a rambling fantasy, but based on assumptions and geometry, which will give spatial imagination wings.
Fourth,? Transformation? Application of ideas
Personally, I think the main way to solve the problem of solid geometry is to make full use of it? Transformation? This kind of mathematical thought, it is very important to find out what has changed, what has not changed and what is the connection in the process of transformation. For example:
(1) The angle formed by two straight lines with different planes is converted into the included angle of two intersecting straight lines, that is, parallel lines leading to two straight lines with different planes through any point in space. The angle formed by the diagonal and the plane is converted into the angle formed by the straight line, that is, the angle formed by the projection of the diagonal and the diagonal on the plane.
(2) The distance of a straight line in different planes can be converted into the distance between a straight line and a plane parallel to it, or into the distance between two parallel planes, that is, the distance of a straight line in different planes can be converted into the distance between a straight line and a plane and the distance between planes. And surface distance can be transformed into line-surface distance, and then into point-surface distance, and point-surface distance can be transformed into point-line distance.
(3) Plane parallelism can be transformed into line parallelism, and line parallelism can also be transformed into line parallelism. Line-to-line parallelism can be obtained by line-to-surface parallelism or surface-to-surface parallelism, and the two can be transformed into each other. Similarly, surface verticality can be transformed into line verticality, and then into line verticality.
Effective learning suggestions of mathematical geometry
First, master every knowledge point skillfully.
All the knowledge points in mathematics are the key for us to solve geometric problems.
In teaching, we don't ask every student to memorize these knowledge points, but ask students to understand them skillfully, remember the knowledge points according to the figures and apply them flexibly to the exercises. If you are not proficient in knowledge, you can't find the entrance to the geometry problem, let alone use it flexibly. Because mathematics is a discipline with rigorous thinking, and geometry embodies this. When solving geometric problems, every step and every link must be based on sufficient reasons, such as conditions, definitions, axioms, theorems, inferences and so on.
Second, through the training of basic questions, consolidate knowledge points.
Just because we have mastered the basic knowledge points does not mean that we have learned geometry. Because math problems are flexible, we must learn to change constantly and skillfully use our knowledge points in the process of solving geometry problems, so as to truly master the knowledge points.
Third, carefully examine the questions, find the right starting point and use the knowledge points flexibly.
When we are proficient in knowledge points, we should feel relaxed about the most basic knowledge questions.
So if you want to learn the geometry part of mathematics well, you need to accumulate some knowledge points and then use them flexibly. This requires us to be familiar with the key points of solving common problems, refine a big new problem into a small new problem, and then use knowledge points to break through, so as to get a breakthrough in solving new problems. When you haven't found a practical way to solve new problems, you should be good at grasping the focus that may help you solve new problems.
One of the two parallel lines is vertical, so is it vertical to the other? Inference, to achieve the analysis of the whole problem, but also let us learn a kind of knowledge integration and infiltration.
Fourth, summarize, focus on training easy-to-mistake questions and strengthen knowledge points.
This work is not only a teacher's business, but also requires students to finish it independently.
When students summarize the topics, classify the topics they have done, and know what types of problems they can solve, what common problem-solving methods they have mastered, and what types of problems they can't do, they can really master the tricks of this subject and really do it. Let it change, I will never move? . If this problem is not solved well, after entering senior two and senior three, some students will do problems every day, but their grades will drop instead of rising. The reason is that they do repetitive work every day, and many similar problems are repeated, but they can't concentrate on solving the problems that need to be solved. Over time, the problems that can't be solved have not been solved, and the problems that can be solved have also been messed up because of the lack of overall grasp of mathematics.
Matters needing attention in learning mathematical geometry
(1) For straight lines and their equations, we must first grasp two breakthrough points as a whole: ① Clarify basic concepts. In the straight line part, the most important concepts are the slope and inclination of the straight line and the relationship between the slope and inclination. Tilt angle? The value range of is breakthrough [0,? ), when the tilt angle is not equal to 90? When the slope k=tan. ; When the inclination angle = 90? Slope does not exist. ② The equations of straight lines have different forms, and students should classify and summarize them from different angles. Angle 1: According to whether the slope of the straight line exists or not, the equations of the straight line can be divided into two categories. Angle 2: From an oblique angle? In [0,? /2)、? =? /2 and (? /2,? ), know the characteristics of straight lines. On this basis, we broke through and put five different forms of linear equations into it. The conditions and limitations of the breakthrough of different forms of linear equations are different and should be summarized.
(2) For the part of linear programming, we must first understand the region represented by the linear programming equation. Here we can use the origin method, if the conditions are met, then this area contains the origin; If the origin does not meet the conditions, the represented area does not contain the origin.
(3) For the circle and its equations, we should remember the meanings of the standard equation and the general equation of the circle respectively. For the study of the circle part, we should expand all the knowledge related to the circle that we learned in junior high school, including the concepts of inscribed circle, circumscribed circle, circumferential angle and central angle of a triangle, as well as the positional relationship between points and circles, the positional relationship between circles and the characteristics of inscribed regular polygons of circles, etc. Only in this way can we master all the knowledge related to the circle more completely.
(4) For ellipse, parabola and hyperbola, we should understand the source of focus, the directrix equation and the related concepts such as focal length, vertex, eccentricity and path from their two definitions. The focus of each conic curve is on the X axis and the Y axis, which should be mastered separately.
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