I. teaching material analysis (I) Status and Role

"Power Function" is selected from the compulsory course of the new mathematics textbook for senior one 1 Chapter 2, Section 3. As one of the basic elementary functions, it not only has a wide range of practical applications, but also plays a role in connecting the past with the future. From the overall arrangement of teaching materials, the purpose of learning and understanding power function is to enable students to acquire more systematic function knowledge and research methods, and lay a good foundation for learning trigonometric functions and other functions in the future. I have learned three power functions: Y = X, Y = X2, Y = X- 1 in junior high school.

This section is a further generalization, induction and development of junior high school related content, and it is a high sublimation of power-related knowledge. After this section, exponential function, logarithmic function and power function will be organized scientifically, reflecting the spirit of organization and systematization full of mathematics. Let students know how to learn a kind of function systematically. In this class, students should experience the research method and transfer it to the study of other functions.

(2) Analysis of learning situation

(1) Students have been exposed to functions, and they have established the consciousness of using the definition range, range, parity and monotonicity of functions to study functions, and initially formed the ability of cooperative inquiry into mathematical problems.

(2) Although students have learned to draw exponential function and logarithmic function images by sketching, they still lack perceptual knowledge of drawing power function images.

(3) The level of students is uneven, and individual differences are obvious.

Second, the target analysis

The new curriculum standard points out that "three-dimensional goal" is a closely related organic whole.

(A) Teaching objectives

(1) knowledge and skills

Let students understand the concept of power function and draw the graph of power function.

② By combining these images, students can understand the changes and properties of power function images.

(2) Process and method

① Let students observe and summarize the nature of power function, and cultivate students' ability of generalization, abstraction and map recognition.

② Make students understand the mathematical thinking method of combining numbers and shapes, and cultivate students' ability of finding, analyzing and solving problems.

(3) Emotional attitudes and values

① Eliminate students' strangeness to power function through familiar examples, thus leading to concepts, attracting students' attention and stimulating students' interest in learning.

② Using multimedia to understand the changing law of power function images, so that students can realize the role of modern technology in the process of mathematical cognition, thus stimulating students' desire for learning.

③ Cultivate students' ability to discover the general consciousness from the special and study the parity of functions by using images. And guide students to discover the symmetrical beauty in mathematics, so that students can have fun in drawing and recognizing pictures.

(2) Key points and difficulties

According to my understanding of the content of this lesson, I will focus on:

Emphasis: Understand the concept and properties from five specific power functions.

Difficulty: summarize the nature of power function from its image.

Thirdly, the analysis of teaching methods and learning methods

teaching method

The teaching process is a process in which teachers and students participate together. Teachers should be good at inspiring students to learn independently, fully arouse their enthusiasm and initiative, effectively infiltrate mathematical thinking methods and strive to improve students' quality. According to this principle and the teaching goal to be achieved, and to stimulate students' interest in learning, I adopted the following teaching methods.

1, guided discovery comparison method

Because there are five kinds of power functions, students can draw the image of the function first, observe its analytical formula and image, look for similarities and differences from the perspective of formula and shape, and compare them, so as to understand the concept of power function and the images and properties of the five power functions more deeply.

2. Assisting teaching with information technology.

Because multimedia information technology can be vivid and easy to attract students' attention, multimedia production can be used to introduce situations and guide students to take this course. Then, the geometric sketchpad is used to draw images of five power functions, creating a rich environment for students to combine numbers and shapes, helping students to understand the concept of power functions and the influence of exponential changes of power functions on the shape and monotonicity of function images, thus summarizing the properties of power functions.

3. Practice and consolidate the discussion learning method

In this way, we can highlight the key points and solve the difficulties, so that students can not only think independently and deeply, but also have extensive exchanges and cooperation with their classmates. In this way, students will have a deeper understanding of these five power functions, their ability to analyze and solve problems will be further improved in this process, and the overall learning atmosphere of the class will be stronger.

(2) study law

This lesson is mainly to summarize the characteristics of power function model, explore the image of power function, observe and find its related properties, and then change the observation angle to find the characteristics of odd-even function. Emphasize the process of hands-on operation, observation, discovery and induction.

Because it is difficult for students to find the power function in the first quadrant, students are guided to concretize abstract problems in the teaching process, and dynamically evolve with the help of multimedia to form a more complete knowledge structure.

Fourthly, the analysis of teaching process.

(A) Teaching process design

(1) Create a situation and ask questions. The new curriculum standard points out: "Let students learn mathematics in concrete and vivid situations". In this class, we ask questions from familiar life situations. The design of questions has changed the traditional design method with clear purpose, given students a space to think, and fully reflected the students' dominant position.

Question 1: What are the common features of functions in the following questions? Is it an exponential function?

By discussing and summarizing the students, we can draw the following conclusions: P = W, S = A2, v=a, A = S 1/2, V = T- 1.

At this time, it may be difficult for students to observe. The teacher suggested that you can use X to represent the independent variable and Y to represent the function value. The above function becomes:

Are powers of several independent variables. Are all alike in shape

The function of.

Secret topic: In this class today, let's learn power function.

(A) the main content of the classroom

The concept of (1) power function

① Definition of power function.

Typically, functions

It is called a power function, where x is an independent variable and a is a constant.

② The difference between power function and exponential function.

Power function-the base is an independent variable and the exponent is a constant;

Exponential function-Exponential is an independent variable and the base is a constant.

(2) Images and properties of several common power functions.

Draw the following common power function images by students and fill in the table according to the images.

Summarize the identity of * * * functions according to the contents in the above table and the pictures. Let the students communicate, and the teacher organizes the students to summarize the nature according to their answers.

The design intention of the above questions: the combination of numbers and shapes is an important mathematical thinking method, including the idea of helping shapes with numbers and helping numbers with shapes. Ask students to practice through problem design, and clarify the essence of power function with vivid lines.

Teacher's comment: the nature of power function.

① All power functions are defined at (0, +∞), and the images pass (1, 1).

② If a > 0, the image of the power function passes through the origin, and it is increasing function in the interval [0, +∞).

(3) If a < 0, the power function is a decreasing function in (0, +∞). In the first quadrant, when X approaches the origin from the right, the image is infinitely close to the Y axis on the right side of the Y axis. When x tends to +∞, the image is infinitely close to the X axis above it.

④ When a is odd, the power function is odd; When a is even, the power function is even.

Based on problem design, students can get five power function images from the exponential function, logarithmic function and drawing points they have learned, but it is much more complicated to draw power function images than exponential function and logarithmic function images, because power functions will change greatly with the slight change of power index. Therefore, before drawing points, students should be guided to explore the properties of several special power functions. For example, analyze the definition domain and parity of function, draw an image according to the research results and tracing points, let students observe the image characteristics, and get the corresponding function properties from the image characteristics, so that students can fully understand the research methods of the system. At the same time, students will have more difficulty in inductive nature than exponential function and logarithmic function. Therefore, we only need to know their images and basic properties in teaching, and we don't need to expand and introduce the general power functions too much. In teaching, we adopt the arrangement from concrete to general, and then from general to concrete.

Through the participation of students, students can deeply understand the main contents and thinking methods of this lesson, so as to deepen their knowledge again.

(3) Consolidation and deepening of in-class training.

The selection of examples and exercises should be combined with students' cognitive inquiry, consolidate the key knowledge of this lesson, and combine knowledge application. This lesson mainly selects two examples.

Example 1 is an example in the textbook: it is proved that f(x)=x 1/2 is a increasing function at (0, +∞). This topic first judges the monotony interval and monotonicity of the function from the perspective of "shape", and then demonstrates the monotonicity of the function from the perspective of "number" with the definition, thus cultivating students' mathematical thinking and professional quality of solving problems by combining number and shape.

Example 2 is a supplementary example, which mainly cultivates students' ability to construct a function according to style and solve problems by using the properties of the function, thus deepening students' understanding of power function and its properties. Note: Because students are not very familiar with power function, they should deliberately reflect the power function y = x 1 in their comments. 3 is the drawing method of increasing function and Y = X-5/4, that is, once again let students understand the basic idea of drawing images to solve problems according to analytical formulas.

(4) Summary, review and reflection. Summary is not only a simple knowledge review, but also a summary of knowledge, methods and experiences by giving full play to students' dominant position. I designed three questions:

(1) What did you learn from this lesson?

(2) What have you learned from this lesson?

(3) What skills have you mastered through this lesson?

(2) Homework design is divided into compulsory questions and multiple-choice questions. The required questions reflect the knowledge level of students in this course. Topic selection is an extension of the content of this course, focusing on the extension and coherence of knowledge and emphasizing the application of what you have learned. Through homework setting, students at different levels can get the joy of success and see their potential, thus stimulating students' full interest in learning and promoting the formation of a learning atmosphere of independent development and cooperative inquiry. I designed the following homework:

(1) required questions

(2) Choose to do the problem

Blackboard design

The blackboard writing should basically reflect the contents and methods of the whole class, reflect the classroom process, and concisely reflect the knowledge structure and its interconnection; Can guide teachers' teaching process and guide students to explore knowledge; By using slides to assist blackboard writing, the class time is saved and the class process is more coherent.

Evaluation and analysis of verbs (abbreviation of verb)

The evaluation of students' learning results is of course important, but what is more important is the evaluation of students' learning process. I use the combination of timely comment, delayed comment and mutual evaluation of students to comprehensively examine the development of students' knowledge, concepts and abilities. In the process of questioning and exploring, I evaluate whether students have a positive emotional attitude and tenacious rational spirit, evaluate whether students' inductive guessing ability has been developed in the process of conceptual reflection, and investigate whether students have a complete training of power function through consolidation exercises, and make adjustments and supplements in time. The above is my understanding and design of this class. Please criticize and correct me.

thank you

Excellent Lecture Notes on Mathematics in Senior One of People's Education Press (2)

Hello, judges and teachers!

I am an undergraduate math player. Today, I'm going to talk about "monotonicity of function and (small) value" in the first class of the third quarter of the first chapter of senior high school mathematics (at this time, I can write it on the blackboard to relieve tension). I will come from teaching material analysis; Analysis of teaching objectives; Teaching methods and learning methods; Teaching process; I will state my design plan for this course from five aspects of teaching evaluation. I urge the expert judges present to criticize and correct me.

I. teaching material analysis

1, the position and function of teaching materials

(1) This lesson is mainly about the monotonicity of functions;

(2) Learning on the basis of learning the concept of function and laying a foundation for the learning of basic elementary functions, therefore, it plays an important role in connecting the past with the future in teaching materials; (You can look at the chapters before and after this topic to write)

(3) It is a hot and difficult issue in the college entrance examination over the years.

(just change it according to specific topics, and delete non-hot and difficult issues)

2. Textbooks are heavy and difficult.

Key point: the definition of monotonicity of function.

Difficulties: Proof of Monotonicity of Functions

Breakthrough of important and difficult points: On the basis of students' existing knowledge, through careful observation and thinking, through group cooperation and exploration, the breakthrough of important and difficult points can be realized. (This must be available)

Second, the teaching objectives

Knowledge Goal: Definition of Monotonicity of (1) Function

(2) Proof of monotonicity of function

Ability goal: to cultivate students' comprehensive analysis and abstract generalization ability, and to understand the reduction thought from simple to complex and from special to general.

Emotional goal: to cultivate students' spirit of exploration and sense of cooperation.

(This teaching goal design pays more attention to the teaching process and emotional experience, based on the diversification of teaching goals. )

Third, the analysis of teaching rules

1, analysis of teaching methods

"There must be laws in teaching, but not in teaching", and proper methods will be effective. Teachers are the organizers, guides and collaborators of teaching in the new curriculum standards, and students' enthusiasm and initiative should be fully mobilized in the teaching process. Based on this principle, I mainly adopt the following teaching methods in the teaching process: open inquiry, heuristic guidance, group discussion and feedback evaluation.

2. Analysis of learning methods

"It is better to teach people to fish than to teach them to fish", and the most valuable knowledge is about methods. As the theme of teaching activities, the state and degree of students' participation in the learning process is the most important factor affecting the teaching effect. In the choice of learning methods, I mainly use: independent inquiry, observation and discovery, cooperation and exchange, induction and summary.

(The first three parts should be controlled within three minutes and can be deleted appropriately. )

Fourth, the teaching process

1, introduce the new with the old.

Let the students draw the images of the first function f(x)=x and the second function f (x) = x 2 through the small research before class, observe the characteristics of the function images and make a summary. Through group discussion and induction in class, students are led to find that the teacher's conclusion is that the image of a linear function f(x)=x rises linearly in the defined domain, while the image of a quadratic function f (x) = x 2 is a curve, which falls at (-∞, 0) and rises at (0, +∞). (Add gestures appropriately to make it look more natural)

2. Create problems and explore new knowledge.

Then the question is raised. Can the expression of quadratic function f (x) = x 2 be used to describe the image of the function at (-∞, 0)? The teacher summed up and combined with books to reveal the definition of monotonicity of function, and emphasized that monotonicity of function can be judged by difference method.

Let the students imitate the expression just now to describe the image of quadratic function f (x) = x 2 at (0, +∞), and find some students to answer, thus standardizing the students' mathematical terms.

Let students learn the definition of monotone interval of function independently, and lay a good foundation for the next example study.

3. Give examples and apply what you have learned.

Example 1 mainly consolidates and applies the monotone interval of the function, and finds out the monotone interval of the function by observing the image with the function defined at (-5,5). This example is mainly based on students' individual answers. After the students answer, correct their answers through mutual evaluation and check their mastery of the monotonous interval of the function. It is emphasized that monotonous intervals are generally written in the form of half opening and half closing.

After explaining the examples, students can complete Exercise 4 after class by themselves, and test their learning effect by answering questions collectively.

Example 2 applies monotonicity of function to other fields, and proves Boyle theorem of physics through monotonicity of function. This is a hot and difficult issue in the college entrance examination over the years. This example should be proved by the teacher's performance, so as to standardize the steps of summary and proof. When comparing two differences with three and simplifying four, it is important to simplify f(x 1)-f(x2) into the form of sum-difference product quotient, and then compare it with 0.

After the students are familiar with the proof steps, do exercise 3 after class and find several students to perform on stage in groups. Other students will complete the proof steps by themselves below and evaluate each other through self-evaluation.

4. Summary

In this lesson, we mainly studied the definition and proof process of function monotonicity, and paid attention to cultivating students' exploration spirit and cooperation consciousness in the teaching process.

5, homework layout

In order to let students learn different kinds of mathematics, I will assign homework in layers:

6. Blackboard design

I try to summarize the main points of this lesson concisely, so that students can see it at a glance.

(The most important part of this part takes six to seven minutes. The definitions and examples must explain the students' activities. )

Teaching evaluation of verbs (abbreviation of verb)

This lesson is based on students' existing knowledge. In the teaching process, students' enthusiasm and initiative are fully mobilized through independent inquiry and cooperative communication, and feedback information is absorbed in time. Through students' self-evaluation and mutual evaluation, internal motivation and external stimulation can coordinate and promote students' mathematical literacy.