1. Analysis of the processing method of function content In the whole middle school stage, the learning of function begins from the compulsory education stage, while the systematic learning focuses on the initial grade of high school. Compared with the past, the requirements of curriculum standards for functional content have changed greatly. 1. Emphasize the background of the function and the understanding of its essence. Whether introducing the concept of function or learning three function models, the curriculum standard requires fully showing the background of function and learning knowledge from concrete examples. In previous textbooks, function is regarded as a special mapping, and students' understanding of the concept of function is based on their understanding of the concept of mapping. Students should not only face several abstract concepts that appear at the same time: correspondence, mapping and function, but also clarify their relationships. Practice shows that it is very difficult to understand these abstract concepts and their relationships at the cognitive development level of senior high school students. Starting from the actual background examples of functions, it is more beneficial for students to establish the concept of functions by strengthening the generalization process of concepts. On the one hand, rich examples are not only the background of concepts, but also concrete examples of understanding abstract concepts; On the other hand, under the problem situation created by examples, students can fully experience the process of abstract generalization and understand the concept connotation. 2. The application function of reinforcement function thinking method is an important mathematical model to describe the changing law of the real world. Therefore, functions are widely used in the real world. Strengthening the application of function not only highlights the idea of function model, but also provides more application carriers, so that the abstract function concept has more concrete content support. For example, the newly added contents are "growth of different functional models" and "dichotomy". In the former, by comparing the growth differences of function models, students can grasp the characteristics of different function models more deeply, and when facing simple practical problems, they can choose or establish appropriate function models according to their characteristics to reflect the dependence between variables in practical problems; The latter fully embodies the relationship between function and equation, and is one of the methods to solve the approximate solution problem of equation from the viewpoint of function. Through the study of dichotomy, students can deepen their understanding of the essence of function concept, learn to look at and solve problems from the perspective of function, and gradually form the consciousness of establishing links between different knowledge. Second, the basic idea of function content Function content includes: the concept and nature of function, basic elementary function (ⅰ), function and equation, function model and its application. With the clue of understanding the essence of the concept of function, these contents can be organically organized into a whole, and students can use this as a carrier to understand the concept of function step by step. Clues of content organization: the understanding of the essence of function concept is not directly given, but introduced by the background example induction teaching material. Because of the high abstraction of the concept of function, it takes a long process for students to truly understand the concept of function, and it is necessary to provide students with opportunities to understand and consolidate the concept of function at different levels and from different angles. Firstly, after summarizing the definition of function on the basis of analyzing the characteristics of typical examples, we have a preliminary understanding of function by discussing its representation and basic properties. They have enriched their understanding of the concept of function from two aspects: the form of expression and the law of change. Then, based on three basic elementary functions, the concept of function is consolidated. After learning the definition and basic properties of functions, the discussion of general concepts turned to the study of specific functions. The concepts and properties of exponential function, logarithmic function and power function are the concretization of general function concepts and properties. With a specific function as the carrier, under the guidance of the general concept of function, the study of its nature embodies the process of "concreteness-abstraction-concreteness" and deepens the understanding of the concept of function. Finally, from the perspective of application, once again consolidate and improve the understanding of the function. A standard to truly understand a concept is to see if it can be used to solve problems. The final function application of the textbook includes dichotomy, the growth difference of different function models, the establishment of function models to solve practical problems, etc., in the hope of improving students' understanding of function concepts in the process of "using". 2. The main way to break through the difficulties: the process of embodiment, strengthening the understanding of the concept of connection function runs through the learning of function content, which is also the difficulty of teaching and learning. What methods should be adopted to break through this difficulty and help students better understand the concept of function? Students should go through the process of generalization for such abstract concepts as constituent functions. Generalization is to distinguish and fix some attributes of an object or relationship. This requires us to fully demonstrate the generalization process, fully mobilize students' rational thinking and guide them to actively observe, analyze and summarize when compiling teaching materials. Three representative examples are selected from the textbook. Firstly, the dependence between variables in the first two examples is analyzed in detail by using set and the corresponding language, and a demonstration of how to analyze the function relationship is given to students. Then, ask the students to imitate themselves and give the analysis of the third example. Finally, through "thinking", students are asked to summarize the same attributes of three examples and establish the concept of function. In such a process from concrete (background example) to abstract (function definition), students can better understand the concept connotation through their own thinking, from analyzing a single example to summarizing the same characteristics of a class of examples. As the core concept of middle school mathematics, function has internal relations with many concepts of middle school mathematics, which provides many angles and opportunities for understanding the concept of function. Therefore, strengthening the connection between function and other mathematical knowledge is the inherent requirement of function concept teaching. For example, there are many representations of functions. Strengthening the connection and transformation between different representations, so that students can learn to flexibly choose representation methods according to the characteristics of specific problems is a means to promote understanding. The textbook gives the results of six tests of three students and the corresponding class average scores through examples, and requires the analysis of the learning situation of three students. The key to solve this problem is to transform table representation into image representation according to the characteristics of table representation and image representation of functions. Another example is that functions are closely related to real life. When compiling textbooks, we should pay attention to strengthening the connection between functions and real life, such as introducing concepts from background examples and arranging a certain number of application questions (attenuation of carbon 14, earthquake magnitude, solution acidity, etc. ) In examples and exercises, these all reflect the external relationship between functions and real life. Another example is to introduce dichotomy from the perspective of solving equation problems from the perspective of function, reflecting the relationship between function and equation, and so on. Third, several key problems in the compilation of function content 1. How to choose examples Whether it is to strengthen the conceptual background or highlight the connection and application of knowledge, the important factor that can achieve good results is to choose appropriate examples. So, how to choose examples to help students learn? For background examples and application examples playing different roles, the standards are not exactly the same. But generally speaking, first, the background knowledge of the examples should be as simple as possible to avoid the influence of the complexity of the background on the understanding of mathematical knowledge itself; Second, the examples should be rich, which is conducive to a comprehensive and accurate understanding of knowledge and will not produce deviations; Third, examples should be close to students' life and have certain times, so as to stimulate students' interest in learning. For example, when introducing the concept of function, the textbook chooses to use analytical expressions to represent the problem of projectile flight, images to represent the problem of Antarctic ozone hole, and tables to represent Engel coefficient. The first question is familiar to students majoring in physics, and the last two questions are often mentioned in daily life. The background is relatively simple, and students will not downplay the study of function concepts because they need to know too much background knowledge. What's more, these three problems contain different forms of function expression. Using them to summarize the concept of function can eliminate some cognitive biases that may exist in junior high school learning and make students realize that no matter what the expression is, as long as there is a y corresponding to each x, it is a function, which is the essential feature of the function. Another example is to write an analytical formula between the fare and the mileage according to the rules for setting the bus fare, and use the analytical formula to make a "ladder fare table" that we often see on the bus for the conductor, which is not only close to students' life, but also has practical application value and can stimulate students' interest and enthusiasm in learning. 2. How to develop the concept and break through the difficulty of the concept of function? We can adopt the method of explicit process to strengthen the connection in the whole learning of function content. Then, concretely speaking, how to expand the concepts of monotonicity and dichotomy of functions when consolidating the understanding of function concepts from three directions, so that students can master them, so as to achieve the purpose of consolidating the understanding of function concepts? The essence of function is to study the changing law of function. The most intuitive acquisition of this rule comes from images, and the ups and downs of images are monotonic. The problem is how to help students rise from geometric intuition to strict mathematical definition. Similarly, the dichotomy also needs to go through a process from intuitive understanding to mathematical definition. Therefore, it is necessary to divide the process from intuition to strict mathematical definition into several levels, so as to build a step of understanding for students and enable them to gradually acquire concepts. For example, when introducing the monotonicity of functions, first, the images of linear functions and quadratic functions are given, and their image characteristics, that is, rising or falling, are observed; Then use the question "How to describe the' rise' and' fall' of the function image" to guide students to describe the image characteristics in natural language; Finally, we think about "how to use the analytical formula f(x)=x2 to describe' with the increase of x, the corresponding f(x) decreases' ……", and transform the description of natural language into the description of mathematical symbol language, thus popularizing the mathematical definition of monotonicity. Through these three steps, the concept of monotonicity is developed by combining number and shape, which not only helps students to obtain the concept through their own efforts, but also understands the concept from both number and shape. 3. The main points and ways of using information technology in function content There is no doubt that information technology is used in mathematics courses. Similarly, the use of information technology is also one of the most concerned issues in textbook compilation. Then, what are the contents suitable for using information technology in the function, how to use them, and how to use them in textbooks? Information technology has powerful image function, data processing function and good interactive environment. Taking advantage of these advantages, the main points that information technology can use in this part of the function are: finding the function value, making the function image, studying the function properties, fitting the function and so on. Using some commonly used software, such as excel and Geometer's Sketchpad, it is easy to make function images, which is of great significance to discuss the growth differences of different function models, and the growth differences can be intuitively found from several images. Using calculator can solve the problem of large amount of calculation of dichotomy, thus paying more attention to the idea of dichotomy and understanding the relationship between function and equation; The interactive environment of computer provides a powerful platform for students to explore independently and enriches learning methods. For example, when discussing the properties of exponential function and logarithmic function, we can fully show the dynamic change process of the image, so as to seek "invariance" in the change and discover the properties of the function. When compiling teaching materials, on the one hand, give tips in places suitable for information technology, such as "You can use the computer …"; On the other hand, some topics of information technology application are introduced in detail through expanding columns, such as "drawing function images with computers", focusing on the methods of making function images with commonly used software, "exploring the properties of exponential functions with information technology", giving the situation of inquiry, requiring students to discover laws by using information technology themselves, "collecting data and establishing function models" and introducing how to fit functions with information technology, and so on. Through these methods, teachers and students can be provided with opportunities and space to use information technology.