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Classification discussion often appears in the form of the final finale in math problems. Need to pay attention to the following points:
1, familiar with the right angle of a right triangle, the waist and angle of an isosceles triangle, and the symmetry of a circle. According to the special properties of graphics, find suitable objects to discuss and solve one by one. When discussing the existence of isosceles triangle or right triangle, we must follow certain principles, do not omit them, and finally synthesize them.
2. The location of the discussion point. Be sure to see clearly the range of the point, whether it is on a straight line or on a ray or line segment.
3. The correspondence of graphs mostly involves congruence or similarity of triangles, and the possible correspondence of diagonal and edge is discussed in classification.
4. If there are absolute values and squares in algebraic deformation, we should pay attention to the selection of open-close symbols.
5. The value or range of the inspection point. This part mainly examines the classification of the range of independent variables, and pays great attention to the nature, conditions and scope of the theorem in solving problems.
6. If there is an intersection between the function image and the coordinate axis in the function topic, it must be discussed that this intersection point is the intersection point with which half axis of which coordinate axis.
7. When the motion mode changes (for example, moving from one line segment to another), the written function should be discussed in sections.
It is worth noting that after listing all the possibilities that need to be discussed, it is necessary to carefully examine whether every possibility will exist and whether it needs to be abandoned.
Most commonly, if a quadratic equation has two unequal real roots, then we have to see whether both roots can be preserved.
Four Secrets
Breakthrough 1: I can't do it, find similarity, have similarity, and use similarity.
The finale involves many knowledge points and it is difficult to transform knowledge. Students often don't know how to start, and they should always look for similar triangles according to the meaning of the question.
Cut-in point 2: the graph or basic graph needed to construct the theorem.
In the process of solving problems, it is sometimes necessary to add auxiliary lines, which almost always follow the principle of constructing graphs or some common basic graphs required by theorems.
Cut-in point 3: approaching invariants
When the movement of the figure changes, the position, size and direction of the figure may change, but in this process, there are often some two line segments, or some two angles, or some two triangles corresponding to the position or quantitative relationship does not change.
Point 4: Find the information of multiple solutions in the topic.
There are more than one situation in which a graph changes in motion, which may meet the conditions, that is, two or more solutions. How to avoid missing answers is also a headache for candidates.
In fact, all the information can be found in the topic, which requires us to dig deep into the topic, in fact, it is a repeated and serious examination of the topic.
03 answering skills
1, accurate positioning to prevent "picking sesame seeds and throwing watermelon"
In your mind, you must give a time limit to the finale or several "difficulties". If you exceed the upper limit you set, you must stop. Go back and check the previous questions carefully, try to ensure that you are foolproof in choosing and filling in the blanks, and check the previous solutions as much as possible.
2. Solving the mathematical finale is a problem.
The first question is not a problem for most students; If you can't understand the first question, don't give up the second question easily.
The process will be written as much as possible, because the mathematical solution is graded step by step, the handwriting should be neat and the layout should be reasonable;
Try to use more geometric knowledge, less algebraic calculation, try to use trigonometric functions, and less the properties of similar triangles in right triangles.
04 finale skills
Looking at the senior high school entrance examination papers all over the country, the key questions of mathematical synthesis are 22 questions and 23 questions. We might as well divide them into function synthesis questions and geometry synthesis questions.
(A) Functional comprehensive questions
First, given the rectangular coordinate system and geometric figure, we can find the analytical formula of the (known) function (that is, we know the type of the function before solving it), and then we can study the figure, find the coordinates of points or study some properties of the figure.
The functions known in junior high school are:
① Linear function (including proportional function) and constant function, and their corresponding images are straight lines;
② Inverse proportional function, whose corresponding image is hyperbola;
③ Quadratic function, whose corresponding image is parabola. The main method to find the analytic expression of known functions is the undetermined coefficient method, and the key is to find the coordinates of points, while the basic methods to find the coordinates of points are geometric method (graphic method) and algebraic method (analytical method).
(2) Geometry synthesis questions
Firstly, the geometric figure is given and calculated according to the known conditions, and then the moving point (or moving line segment) moves, resulting in the corresponding changes in line segment and area.
Find the analytical expression of the corresponding (unknown) function (that is, I don't know what the resolution function is before I find it) and find the definition domain of the function. Finally, I will explore and study according to the functional relationship, which generally includes:
Under what conditions is the figure an isosceles triangle, a right triangle, a quadrilateral, a diamond, a trapezoid, etc.
Explore what conditions two triangles meet, such as similarity;
Explore the positional relationship between line segments, etc.
Explore the relationship between areas and find the value of X and the value of independent variables when a straight line (circle) is tangent to a circle.
The key to finding the unknown resolution function is to list the equivalent relationship between independent variables and dependent variables (that is, to list the equations containing X and Y) and write it in the form of y=f(x).
Generally, there are direct method (directly listing the equations containing X and Y) and compound method (listing the equations containing X, Y and the third variable, and then finding the functional relationship between the third variable and X, substituting and eliminating the third variable to get the form of y=f(x)), and of course, parametric method, which has exceeded the requirements of junior high school mathematics teaching.
In junior high school, the methods of finding equivalence relations are mainly using Pythagorean theorem, parallel line cutting proportional line segments, triangle similarity and equal area. Finding the domain is mainly to find the special position (limit position) of the graph and solve it according to the analytical formula.
The final exploration problem is ever-changing, but it is essential to analyze and study the graph. Find the value of x by geometric and algebraic methods.
When solving mathematical comprehensive problems, we should keep in mind the combination of numbers and shapes, turn big problems into small problems, and don't forget the potential conditions. Classification discussion should be rigorous, equation function should be instrumental, calculation and reasoning should be rigorous, and innovation quality should be improved.
Skills of doing mathematical finale;
★ Answering skills for the grand finale of junior two mathematics.
★ What are the skills for solving the final math questions in junior high school?
★ Share the idea of solving the final problem of junior high school mathematics.