(1) Basic knowledge to be taught to students (2) Basic skills to be mastered by students; (3) Ability to solve practical problems; (4) Personality and ideas.
(1) Basic knowledge
For example, in the teaching of "congruent triangles", we should pay attention to clarifying the concept of congruent triangles. The textbook is described by the vivid language of "coincidence", which is not difficult for students to understand, but they often do not pay enough attention to it and do not realize its importance. Because this concept is not clear, in order to influence the understanding of the undefined concept of "correspondence", it is directly related to solving the triangle and similar triangles's learning set theory in senior high school. Therefore, the concepts of congruence and correspondence should be clarified. Conformal: it includes two aspects: the same shape and the same size. Correspondence finds the corresponding angles of the corresponding edges in turn. The key is to determine the corresponding vertex. -methods and laws.
For example, the connotation of the "inclination angle" of a straight line includes: the upward direction of the straight line, the positive direction of the X axis, the minimum angle and the positive angle.
Y So it is necessary to guide students to think: "What is the position of a straight line in rectangular coordinates?"
Am I sure? "() How to determine the direction of reintroducing straight lines (from bottom to top)
X thus creates a demand for "tilt".
O a correct concept needs to be repeated many times before it can be formed. Because of this, the contrast is here.
Very important. (as shown in figure 1)
Comparison methods: right and wrong comparison method, old and new comparison method, similar comparison method, directional comparison method, comprehensive comparison method, etc.
(2) Basic skills
Interpretation of skills: skills are an automatic way of action fixed on individuals and a summary of a series of ways of action.
Generally speaking, it is an action that is completed according to certain procedures and steps. Skills include mental skills (implicit) and action skills (explicit).
Example 1: The general steps to solve a linear equation with one variable are:
Denominator removal-bracket removal-shifting terms-merging similar terms-into the simplest equation ax=b(a≠0).
-divide the two sides of the equation by the coefficient of the unknown quantity-get the solution of the equation
Example 2: Plane geometry language is the basis of solid geometry language. In the introductory teaching of plane geometry and the training of geometric language expression, the drawing method of line segment extension can be taught, so that students can correctly use the following standardized geometric drawing languages:
(1) extended line segment (AB)
(2) Extended line segment (3) Extended line segment (4) Reverse extended line segment.
Example 3: Calculation of space angle and distance in solid geometry;
Structural calculation conclusion
Spatial computing problem, plane problem, plane problem solution, spatial problem solution
Recognition triangle
[Exercise 1]: Summarize the general steps of "proof by mathematical induction".
(3) Basic methods
The basic methods of middle school teaching can generally be divided into two categories:
One: logical thinking method-the method of studying and thinking about problems. Such as observation, experiment, deduction, induction, analogy, transformation, transformation, abstraction and generalization.
The other: problem-solving method-is a method to deal with a specific problem. Such as substitution, elimination, substitution, reduction, formula, undetermined coefficient, image, analysis, synthesis, paradox, comparison, classification, translation, parameters, mapping and other methods.
For example, in the teaching of complex numbers, the basic method is reduction-transforming complex numbers into real numbers to solve them;
Algebraic Representation: z=a+bi —— Algebraic Problem
plural
Triangle Representation: Z = R()- Triangle Problem
Real number problem
Geometric representation of the problem: vector geometry problem
Properties of complex modules
Example 2 lateral area teaching of finding prisms in solid geometry needs to infiltrate the following teaching methods:
Right-angled prism-rectangle
Finding the edge of the S prism is to cut the lateral area of the prism along a side and then display it on a plane-edge prism-parallelogram.
It must be clear here:
(1) Can the side area be calculated without unfolding the right-angle prism? -Just calculate the sum of several rectangular areas by incomplete induction.
(2) Why lateral area? -Using the reduction method, the space problem is transformed into a plane problem.
(3) Why can it be expanded? Why is it rectangular after expansion? -Cultivate students' reasoning ability.
Oblique prism should be clearly visible:
(1) What is the method of Shanghai Stock Exchange in textbooks? -Incomplete induction.
(2) Can the lateral area formula of oblique prism be deduced and converted into the area calculation formula of straight prism? -Yes, as long as the oblique prism is divided into two sections through the straight section, then a straight prism is combined with the straight section as the bottom, and then S- straight can be used for S- oblique, which embodies the idea of simplification and the filling and excavation method in polyhedron (in the plane, the migration of parallelogram area calculation method)
[Thinking 1]: What are the requirements of middle school mathematics syllabus for cultivating students' mathematical ability? (see outline)
(1) computing power
[Thinking 2]: What are the aspects of high school computing power? What are the requirements?
The following calculations are required to be completed quickly, correctly and reasonably:
A. various algebraic operations of numbers and formulas; Elementary transcendental operation; Geometric operation; Analysis operation; Probability and statistical operations.
[Thinking 3]: "What are the operational requirements in the sequence?
(2) Logical thinking ability
Students' mathematical ability is manifested in many aspects, and thinking ability is the core of students' intellectual structure.
Thinking: intuitive thinking, logical thinking, illogical thinking, logical thinking ability, etc.
[Thinking 4]: How to cultivate students' logical thinking ability?
1, in terms of computing power, in order to achieve the purpose of "correct and fast", it is necessary to summarize the corresponding operation rules in various operations and summarize them into a general form.
Method of thinking algebraic expression multiplication
Algebraic expression product polynomial
factoring
Thinking characteristics:-It is a kind of reverse thinking training, with divergent thinking characteristics, but also exploratory.
A general model for solving factorization
Extract common factor
The product of algebraic expressions is divided into polynomials by formulas.
Cross multiplication
Teaching requirements are divided into different levels, and knowledge points are also divided into primary and secondary levels. Only by understanding the position and function of each specific content or knowledge point in the whole teaching material can we distinguish between primary and secondary, and clarify the key points and difficulties.
Example 1: "One-variable quadratic equation"
Key points and main contents: finding root formula, formula, the relationship between root and learning number.
Example 2: As far as the internal relationship between numbers is concerned; Triangle is the basic figure, and other plane figures can be transformed into triangles to learn.
As far as application is concerned, triangular knowledge is often used in subsequent teaching and production practice.
As far as cultivating students' logical thinking ability and reasoning ability is concerned, triangle chapter shoulders a very important basic task-it is the main focus of plane geometry teaching.
Example 6: The chapter of straight line and plane is the key content in three-dimensional teaching.
Line-plane relationship: master and judge by the vertical relationship between line and plane.
▲ Attach importance to the links within and between disciplines.
The connection between the old and the new within the discipline: primary school and junior high school, junior high school and senior high school, the concept of examples (primary school and junior high school), algorithm, associative law, commutative law, parallel concept.
Special attention should be paid to the treatment of "connection points", "discontinuity points" and "deepening points" in knowledge.
The application of auxiliary angles in algebra and geometry, trigonometry and algebra to solve several problems can make mathematical knowledge penetrate and promote each other and cultivate the comprehensive application ability of mathematical knowledge.
What's the point? How to grasp the key, highlight the key points and disperse the difficulties? What should be paid attention to in teaching?
Fourth, strengthen the application of knowledge.
For example, as an application of geometric series, arrange an example of shopping installment payment related to people's daily life in recent years; As an application of arithmetic progression, "Reading Materials" introduces some calculations about savings. In addition, the added application problems also involve the planning of house demolition and construction, the length of the wire wound on the disk surface and so on.
5. Several problems that should be paid attention to in teaching.
(1) Grasp the teaching requirements.
Because this chapter has a wide range of knowledge, it has the characteristics of cross-knowledge. Under the influence of the educational idea of "one step in place" for exam-oriented education, the teaching requirements of this chapter are easy to rise, and the comprehensive training of "College Entrance Examination" is carried out prematurely, thus affecting the study of basic content and increasing the burden on students.
In fact, learning is a deepening process. As a chapter of senior one, we should devote ourselves to laying a good foundation, conducting preliminary comprehensive training, and consolidating and improving the content of this chapter through continuous application in subsequent study. Finally, in the general review of senior three mathematics, the mastery of this chapter has risen to a new height through systematic combing of knowledge and further comprehensive training.
So this chapter should pay special attention to some easy places. For example, when learning the recurrence formula of a series, don't engage in argumentation and calculation problems involving the deformation of the recurrence formula, as long as you can find the first few items of the series according to the recurrence formula; Don't involve too many skills when learning the summation of series;
(2) consciously review and deepen what you have learned in junior high school.
The new textbooks, like the current middle school textbooks, are basically lined up because of the tight class hours and other reasons, and most of the knowledge in junior middle schools has no systematic and in-depth learning opportunities. The content of junior high school is the necessary basis for learning senior high school mathematics, so it is particularly important to review and deepen the content of junior high school consciously when learning senior high school content. This chapter is the third chapter of high school mathematics, which is closely related to junior high school mathematics and has the widest connection with junior high school mathematics. Therefore, we should communicate junior high school mathematics and senior high school mathematics as much as possible in teaching. For example:
In the general formula of arithmetic progression and geometric progression and the formula of the sum of the first n terms, the relationship between a 1, an, n, d and Sn is involved, and we often use the known quantity in the formula to represent the unknown quantity. In this process, we should consciously review the deformation of the equation and remind and correct the mistakes that are easy to appear in the deformation in time. When solving an unknown number according to related formulas and known conditions (for example, when solving an item), it is often necessary to list the equations or equations before solving them. In this process, let students know that our problem is actually to solve an equation or system of equations, and then analyze which of them are known quantities and how to solve them. Through this conscious analysis, we not only reviewed the knowledge of understanding equations and equations. We should also understand its application and cultivate the consciousness of solving problems with equations or equations;
(3) appropriately strengthen the connection between the content and function of this chapter.
Strengthening this connection properly is not only conducive to the integration of knowledge, deepening the understanding of sequence, and solving problems related to sequence by using the viewpoint and method of function, which in turn can deepen students' understanding of function. For example, before this, the function that students came into contact with was generally a function with continuous change of independent variables, but after this chapter comes into contact with the function with discrete change of independent variables, we can further understand the general definition of function and prevent the negative cognitive transfer of students that may be caused by the previous content arrangement;
The connection between this content and function involves the following aspects.
1. The connection between the concept of sequence and the concept of function.
The function corresponding to a sequence is a function whose domain is a set of positive integers (or a finite subset of the first n numbers), and it is a function in which independent variables are discretely valued at equal distances. In this sense, it enriches the scope of students' concept of contact function.
But sequence and function cannot be equal. Sequence is a series of function values of corresponding function. Based on the above relationship, series can also be represented by images, so we can use the intuition of images to study the properties of series. The general term formula of sequence is actually the analytical expression of corresponding factor. Recursive formula of sequence is also an expression of corresponding function, because as long as the value n of an independent variable is given, the corresponding f(n) can be determined by recursive formula. This, in turn, shows that, as a function, there is not necessarily an analytical formula to directly express the relationship between the dependent variable and the independent variable.
2. The relationship between arithmetic progression and linear function and quadratic function.
According to arithmetic progression's general term formula, every term an of arithmetic progression with non-zero tolerance is a linear function of the number n, so we can know arithmetic progression by using the properties of linear functions. For example, according to the property that the image of a linear function is a straight line and this straight line is uniquely determined by two points, it is easy to understand why two terms can determine a arithmetic progression.
In addition, the formula of the sum of the first n terms of arithmetic progression whose first term is a 1 and the tolerance is d can be written as follows:
That is, when Sn is a quadratic function of n, we can use the viewpoint and method of quadratic function to understand the problem of finding the sum of the first n terms of arithmetic progression. For example, according to the image of quadratic function, we can know the increase and decrease of Sn, extreme value and so on.
(4) Pay attention to cultivating students' ability to use observation, induction, conjecture and proof.
It is a very important learning ability to learn mathematics by observing, inducing, guessing and proving. In fact, in the process of exploring and solving problems, we often start with observation, find out the characteristics of problems, and form a preliminary idea to solve problems; Then explore by induction and put forward a guess; Finally, the proposed conjecture is tested by proof (or counterexample). It should be pointed out that there are not many textbooks that can comprehensively train the above research methods in high school mathematics, but this chapter has provided such training opportunities many times, so we should make full use of them in teaching and don't let them go easily.
The use of () symbol conforms to national standards.
In order to facilitate international communication, the new national standard of quantity and unit stipulates that natural number set n = {0, l, 2.3,}, that is, the natural number begins with o, which is different from the long-standing idiom and will make us feel uncomfortable. However, in order not to violate the above provisions, we should change the past idioms in teaching and call the number set {1, 2, 3, ...} a positive integer set, which is recorded as N+.