Learning methods of girls in senior high school mathematics solid 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-line parallelism can be obtained by line-plane parallelism or plane-plane parallelism, and they can be transformed into each other. Similarly, surface verticality can be transformed into line verticality, and then into line verticality.
Problem solving skills of solid geometry in senior high school
Proof strategy of 1, parallel and vertical positional relationship
(1) Judging from the nature of the known idea and the idea to prove it, that is, combining analytical method and comprehensive method to find the idea to prove the problem.
(2) Adding auxiliary lines (or faces) according to the nature of problem setting conditions is one of the commonly used methods.
(3) The three perpendicular theorem and its inverse theorem are used most frequently in the college entrance examination questions, so it should be given priority to prove that the straight line is vertical.
2. Calculation methods and skills of spatial angle.
Main steps: one post, two certificates and three calculations; If you use vectors, it is a proof and two calculations.
(1) Angle formed by two straight lines on different planes ① Translation method: ② Complement method: ③ Vector method:
(2) The angle formed by a straight line and a plane
(1) To calculate the angle between a straight line and a plane, the key is to make a vertical line, find a projection and convert it into the same triangle for calculation, or use a vector for calculation.
② Calculate by formula.
(3) dihedral angle
① Practice of plane angle: (1) Definition method; (2) Three vertical theorems and their inverse theorem methods; (3) Vertical plane method.
(2) Calculation method of plane angle:
(i) Find the plane angle, and then calculate it by triangle (solving triangle) or by vector; (ii) Projection area method; (3) Vector included angle formula.
3. Calculation methods and skills of spatial distance.
(1) Find the distance from a point to a straight line: the perpendicular of a point to a straight line is often determined by using the three perpendicular theorem, and then it is solved in the related triangle, or the distance from a point to a straight line is found by using the method of equal area.
(2) Find the distance between two straight lines on different planes: generally, find the common vertical line first, and then find the length of the segment of the common vertical line. If you can't do the common perpendicular directly, you can convert it into a line-plane distance solution (in this case, you don't need the college entrance examination).
(3) Find the distance from a point to a plane: generally, find (or make) a plane perpendicular to the known plane passing through the point, make a vertical line through the plane of the point by using the properties of the vertical plane, and then calculate; Can you use it too? Triangular cone product method? Find the distance directly; Sometimes when it is difficult to find the distance from a known point to a plane directly, we can convert the distance from a point to a plane into the distance from a straight line to a plane, so? Transfer? Ask from another angle? Distance from point to plane? . Finding the distance from a straight line to a plane and the distance from a plane to a plane are generally converted into the distance from a point to a plane.
High school mathematics solid geometry formula
It is not difficult to learn how to build several, and spatial imagination is the key. Point, line and cube is a hundred gardens.
Points are used as attribution on the surface of lines, and lines are used as inclusion in the surface. Based on the four axioms, deductive calculus has skills.
Two straight lines in space intersect in parallel and have different planes. The straight lines are parallel and in the same direction, and the equiangular theorem enters the space.
Make sure that the straight line is parallel to the surface and find a parallel line in the surface. It is known that the straight line is parallel to the surface, and the intersection line is the surface to find the intersection line.
To prove that the plane is parallel to the plane, find two intersecting straight lines in the plane. If the straight line is parallel to the plane, don't look.
It is known that the plane is parallel to the plane, and it is inevitable that the line is parallel to the plane; If it intersects all three sides, two parallel lines are obtained.
Make sure that the straight line is perpendicular to the surface, and the two intersecting straight lines are on the vertical plane of the straight line. These two lines are perpendicular to the same plane and extend parallel to each other.
Two sides are perpendicular to the same line, and one side is parallel to the other. In order to make a surface perpendicular to a surface, a surface intersects the perpendicular of another surface.
The face is perpendicular to the right angle and the line is perpendicular to the heart.
While projecting with four lines, find a vertical line obliquely. Lines are ingeniously perpendicular to each other, and the theorem of three perpendicular lines is elegant.
Spatial distance and included angle are transformed into plane in parallel, and the answer is found in triangle.
The introduction of new vector tools opens a new chapter in calculation and proof. Establishing a space system to find coordinates, vector operation is simpler.
There is no end to knowledge innovation, and knowledge is speculative and courageous to climb.
Polyhedron and rotator, the continuation of the above content. Play a new role as a carrier, and the positional relationship is all in the room.
The basic formula is the basis for calculating the volume by calculating the area. Regular bodies use formulas, and irregular bodies rely on transformation.
Develop a good division method and turn it into a new world.
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