Function is one of the important basic concepts in mathematics. The further study of mathematical analysis, including limit theory, differential calculus, integral calculus, differential equations and even functional analysis, takes function as the basic concept and research object. Other disciplines, such as physics, also use the basic knowledge of function as a tool to study and solve problems. The teaching content of function contains extremely rich dialectical thoughts, which is a good material for students' dialectical materialism education. The thinking method of function also permeates the whole process of middle school mathematics and other disciplines.
Function is the main content of middle school mathematics. It is closely related to many contents of middle school mathematics. "Function and its image" in junior high school algebra belongs to the content of function, and exponential function, logarithmic function and trigonometric function in senior high school mathematics are the main parts of function content. By studying these functions, we can understand their properties, images and preliminary applications. The limits of the following contents, the preliminary knowledge of calculus, etc. Is all the contents of the function. This sequence can be regarded as an integer function. The point pairs (n, an) reflected by arithmetic progression's general term are all distributed on the image of straight line Y = KX+B, and arithmetic progression's first n terms and formulas can also be regarded as the quadratic function relation about n(n∈N), and geometric progression's content also belongs to the whole function of exponential function type. Other mathematics content in middle school is also related to function content.
Because students have learned the definition of function from the viewpoint of variables in junior high school, and learned several simplest functions in detail, they are no strangers to functions. Therefore, when redefining the function in senior high school, it is important to let students realize its advantages, fundamentally reveal the essence of the function, and let students actively distinguish between the function and the decomposition function. Understanding this point is of great significance to the study of the properties of subsequent functions. Function is the closest combination point with junior high school mathematics. If the contents of algebra in junior high school are not learned well or forgotten too much, there will be obstacles in learning this chapter. Many contents in this chapter are taught on the basis of junior high school, such as the concept of function. Before teaching, we should review the main contents of junior middle school functions and their images, including the concept of functions, the description of function images, the properties of linear functions and quadratic functions, etc. Another example is the expansion of the concept of exponent. Without the basic knowledge of positive integer exponential power, zero exponential power and negative integer exponential power, the exponential power of rational number can not be given, and the nature of operation is the same. Therefore, we should pay attention to the connection with the relevant contents learned in junior high school in this chapter and do a good job in the connection and transition of junior high school mathematics.
Finally, pay attention to the cultivation of the idea of combining numbers with shapes. Images occupy a considerable proportion in the content of functions, and function images play an important role in studying the properties of functions. By observing the changing trend of function images, the properties of functions can be summarized. The relationship between the function of the inverse function and the function image is also induced by the changing characteristics of the image. The properties of exponential function and logarithmic function are given by the function image itself. Therefore, special attention should be paid to the use of function images in teaching, so that students can not only observe the corresponding properties from the images, but also have a way of proving them when learning properties. In the teaching process, we should pay attention to cultivate students' skills in drawing some simple function images, remember some common sketches of function images, and form the habit of using function images to explain the nature of functions and analyze problems.
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A: (1) module 1: straight lines and circles.
It mainly embodies the basic ideas and methods of analytic geometry, and uses coordinate method to study the positional relationship between straight lines and circles in plane geometry, reasoning and demonstration, quantitative calculation and other issues.
(2) Module 2: Conic Curve
This paper studies the basic concepts, definitions, equations and properties of three kinds of curves: ellipse, hyperbola and parabola. And use mathematical software to study more conic problems.
(3) Module 3: The application of space geometry and vector method in solving these problems. Including: the line-line relationship of space, the line-plane relationship of space, the plane relationship of space, and the key and difficult knowledge of cylinder, cone and sphere.
Mathematics is a subject that studies numbers and shapes in the objective world, while geometry is an organic combination of numbers and shapes, and it is the combination point of numbers and shapes. At the same time, it is also the foundation of modern mathematics and the starting point of learning advanced mathematics knowledge, so the teaching of geometry in middle school mathematics is extremely important.
To learn geometry well, students must set up the viewpoint of movement and think about analytic geometry from the viewpoint of movement. For example, curves are trajectories of moving points, and the translation of straight lines and coordinate systems must be considered from the perspective of the formation process of things' motion. At the same time, we should think about the coordinates of points and the equation of curves, especially the relationship between the coordinates of moving points and the equation of curves. When you enter analytic geometry, you enter a world of movement and change. How to show students the changing world of sports? Modern information technology provides us with the performance effect of displaying dynamic pictures, which makes it possible for us to dynamically display the formation process of curves, which is more conducive to students' forming sports views and dialectical materialist thinking methods, and promotes the improvement of students' innovative ability and thinking ability.
Based on the current high school mathematics textbook Geometry in our province, this study aims to explore the mathematical thinking method of geometry and show the knowledge and connection of analytic geometry systematically, vividly and vividly. The making of courseware mainly studies the formation process of moving point trajectory of straight line, circle, ellipse, hyperbola and parabola, the characteristics of graphics and the changing process of the position relationship between straight line and conic curve.