分类讨论思想在函数中的应用（Application of the idea of classification discussion in functions）

1、分类讨论思想在函数中的应用（Application of the idea of classification discussion in functions）Application of the idea of classification discussion in functionsHairdressing middle school, Lin YijieI. Analysis of teaching contentThe idea of classification discussion is an important method in solving mathematical pr

2、oblems, and it runs through the whole content of mathematics in middle school. Need math problems by means of classification discussion idea to solve, the classification caused by reasons, can be summarized as: the mathematical concept is related to the definition of classification; mathematical the

3、orems and formulas or operational properties, using the classification rule is given; a variety of circumstances or a variety of possible mathematical problems solving the knot theory; contains the variables in the mathematical problem, these parameter values will lead to different results. The appl

4、ication of classified discussions tends to simplify complex problems. The most important principle in classification thinking is no heavy, no leakage. The process of classification can train students to think carefully and methodically, while classified discussion can promote students ability to stu

5、dy problems and explore rules. The idea of classified discussion occupies an important position in the junior middle school mathematics in senior high school entrance examination is hot and difficult, there are many discuss thought reflects the knowledge and skills of mathematics in junior middle sc

6、hool, can be found either in algebra or geometry, definition, theorem proving, their distribution in the concept of operation rules (properties), graphics (like) the nature and solve specific problems, generally speaking, the use of classified discussion of ideas and methods to solve the problems of

7、 two categories: one is involved; or function or algebraic equation, according to the different values of the letters, discuss the problem respectively in different ranges within. Second, according to the geometry of the points and lines appear in different locations, one by one to discuss and solve

8、 problemsThe function always occupies a very important position in mathematics curriculum, especially in junior high school stage, it not only has important function of fundamental and practical application, but also for the further study of students also play an important role, is one of the most i

9、mportant content of the mathematics curriculum. In the function teaching material, it contains rich mathematical ideas, such as transformation thought, model thought, combination of figures and shapes, classification thought, etc. these insights will play an important role in the future of mathemati

10、cs learning and students life. From the point of view of classification, we should discuss the value of the independent variable, the solution of the analytic formula, the content of the nature and the problem of function solving.This section uses the specific knowledge of the function to penetrate

11、the classification and discuss the application of ideas.Two. Study situation analysisThis class is the students with learning function, the plane right angle coordinate and function concept, a function (proportional function), basic inequality and inequalities and a function of the contact, inverse

12、proportion function on the further penetration of classification discussion idea, so as to improve the students ability, cultivation of mathematical thinking. Grade eight students have the abstract thinking ability and the classification of certain discussion, but some have just learned the basic kn

13、owledge of function, there are still some problems on the part of the knowledge is not solid, do not understand the profound and comprehensive application abilityThree. Analysis of teaching methods and methodsBased on the above analysis, the problem of this lesson is how to permeate in teaching, esp

14、ecially how to grasp the degree of infiltration. Classification thinking, unlike general mathematical knowledge, can be mastered through the teaching of several classes. It is based on students age characteristics, students at all levels of learning and knowledge of the characteristics of the gradua

15、l infiltration, spiral up, and constantly enrich their own connotations.Should first pay attention to control the difficulty of teaching, step-by-step, gradual deepening of the principles and adopt flexible and effective teaching methods for students in mathematics learning process, by analogy, obse

16、rvation, analysis, synthesis, abstraction and generalization, spiral, active application of classification form be arranged thoughtFour. (1) teaching objectives:1. knowledge and skills goals: combining function specific problems, the initial grasp of classification methods and principles;2. emotion

17、and attitude: comprehend, classify, discuss, solve the problem of function, and feel the charm of mathematical thinking;3., process and method: through observation, comparison, analysis, discussion and induction, let students gradually understand the necessity of classification discussion, learn cla

18、ssification and improve ability.(two) heavy and difficult points: the necessity and method of classification;Five. Teaching processguide1. plane rectangular coordinate system, P x is a semi shaft axis, O and OP=2, as the origin of coordinates, the coordinates for the point P is;1) deformation of the

19、 plane rectangular coordinate system, P is a point on the X axis, O and OP=2, as the origin of coordinates, the coordinates for the point P is;2) plane rectangular coordinate system, P is a point on the O axis, as the origin of coordinates, and the OP=a (a0), then the coordinates of point p;Teacher

20、asked: 1) what is the difference between variant 1 and motif? Are the two answers the same?2) whats the difference between variant 2 and 1? Through the comparison of these 3 questions, what should we pay attention to in solving the problem?(introducing the topic classification, discussing the applic

21、ation of ideas in functions)Design intention: through the change type teaching, guides the student to discover the classification need positively, understands the classification discussion necessityexample 1 in the Cartesian coordinate system, O is the coordinate origin, and A (-3, 3).(1) line segme

22、nt OA=;(2) P to the point on the axis, if POA is OA with a waist of the isosceles triangle, is to meet the conditions of the P total points, their coordinates are;Deformation: now (2) changed to: P is the point on the axis, if POA is an isosceles triangle, is to meet the conditions of the P total po

23、ints, their coordinates are.Induce：Teacher: 1) what is the cause of the problem 2 and how many answers to the problem?2) is your answer complete? What should we pay attention to in the course of the discussion?Design intent: to guide students to carefully take the initiative to find the topic backgr

24、ound and diversity of conditions is not clear to the necessary classification discussion, attention to discuss no leakage of.2 cases of figure, side length of the square of 4 ABCD, P from D point, along the D-C-B-A to 1 units / sec, request an area of Y Delta APD on time x (s) of the formula, and dr

25、aw the function of image.Teacher: 1) whether the analytical formula of this problem can be used a formula can be expressed out? Should be discussed in several cases?2) what should we pay attention to when solving the problem of motion?3) what are the reasons for this classification?Solution: when p

26、movement in segments DC, 0x = 4,When p movement in segments BC, 4x = 8,When p movement on segment BA, 8x12,Design intention: the subject is adapted from the ninth questions of group A, the purpose of which is to guide the students to find out the possible positions of the fixed points in the movemen

27、t and the non uniqueness of the expressions, so as to discuss the classificationcase 3 the lines are x and Y respectively. The axis is at A and B two points, and O is the origin(1) for the delta area of AOB;(2) O point can draw a straight line to a AOB into an area of two 1:2? If so, how many pictur

28、es can I draw? Write the functional relation of such a straight lineTeacher: 1) we should draw the rough picture;2) what knowledge points have been investigated in question 1?3) 2 to the intersection of two lines containing K represented by formula, involving the parameter calculation, please be sur

29、e to master;4) what are the reasons for this classification?Solution: (1) make x=0, then y=-2; make y=0, then x=3,* A (3, 0), B (0, -2) * AO=3, BO=2(2) there are 2. A straight line for y=kx (k = 0), cross the line AB at CBecause C is the intersection point of the two line, so there isSolutiontherefo

30、reWhenWhenTo sum up, there are two lines satisfying, respectivelyDesign intention: this topic is adapted from the complex exercises C group eighteenth questions, combined with common examination questions, the purpose of guiding students to actively discover the diversity of the conclusions, leading

31、 to the needs of classification.Example 4 the approximate image of a straight line and a hyperbola in the same Cartesian coordinate system may be ()T: this question should be discussed in terms of K symbols.Design intention: to guide students to discover the need for classifying arguments as a result of the nature of their own needs.summaryEach has its mathematical conclusion conditions, using each mathematics method often has its applicable scope, the mathematical problems we encountered today in some of the background diversity, conditions are no