As a teacher, we usually need to prepare a lecture draft, which can effectively help us summarize and improve our lecture skills. How should I write a speech? The following is the lecture notes of inverse proportional function of junior high school mathematics collected by People's Education Publishing House for your reference only. Let's have a look.
Lecture Notes on Inverse Proportional Function in Junior Middle School Mathematics 1 I. teaching material analysis
This section is a summary and review lesson of inverse proportional function. Function itself is an important content in mathematics learning, and inverse proportional function is the basic function. Inverse proportional function is a new function after elementary function learning, ranking second among the three major functions in junior high school. It is different from elementary function, but it is based on elementary function, which lays a foundation for learning higher-level functions and dealing with the relationships among functions, equations and inequalities in the future. Through the review of this chapter, students can further understand the meaning of inverse proportional function, understand the image of inverse proportional function, further explore and understand the properties of inverse proportional function according to the image and analytical formula, and solve some simple practical problems with inverse proportional function. Therefore, the study of this lesson is a process of students' re-understanding and integration of the concept, image and essence of functions.
Second, the analysis of teaching objectives
According to the spirit of "student-centered, active classroom atmosphere and fully mobilizing students to participate in the teaching process" in the curriculum reform. In teaching design, I imagine that by using multimedia courseware to create situations, while mastering the knowledge about inverse proportional function, I can stimulate students' interest in learning and desire to explore, and guide students to actively participate in and explore. Therefore, the teaching objectives are determined as follows:
1, knowledge and ability objectives:
(1) Review the concepts, images, properties and other knowledge points of inverse proportional function, and deepen students' understanding and mastery of this chapter of inverse proportional function through the supporting exercises of corresponding knowledge points.
(2) According to the conditions in the problem, we can determine the analytical formula of the inverse proportional function, draw its image, and determine the value range and increase or decrease of the independent variable according to the problem.
2. Process and Methods Objective: Through the variant exploration of related problems, correctly apply the knowledge of inverse proportional function, further experience and form some basic strategies to solve problems, and cultivate practical ability and innovative spirit.
3. Emotion, attitude and values: create teaching scenarios, encourage students to actively participate in the review activities of inverse proportional functions, stimulate their interest in learning, have fun after solving problems, and continue to infiltrate mathematical thinking methods such as the combination of numbers and shapes.
Third, the analysis of teaching priorities and difficulties
Because the study of this lesson is the process of students' re-understanding and integration of the concept, image and essence of functions. It can help students form some basic strategies to solve problems, improve their ability to analyze and solve problems, and cultivate their innovative spirit. Therefore, I am sure that the teaching focus of this class is to further master the concept, image and nature of inverse proportional function and use it correctly. The difficulty in teaching is the flexible application of the nature of inverse proportional function. The application of the idea of combining numbers with shapes.
Fourth, the analysis of teaching methods
According to the characteristics of teaching materials and students' age characteristics, psychological characteristics and cognitive level, I use the methods of cooperative communication and collective inquiry to inspire students to think deeply, actively explore and actively acquire knowledge. At the same time, pay attention to the connection with students' existing knowledge, and give students enough time to explore independently. Through the guidance of teachers, we can stimulate and mobilize students' enthusiasm, let students do more activities and observations in class, take the initiative to participate in the whole teaching activities, organize students to participate in the learning process of "inquiry-discussion-exchange-summary", and make full use of multimedia teaching in teaching to inspire students through teacher-student activities such as demonstration, operation, observation and practice, so that every student can
V. Guidance on learning methods
This course focuses on students' "learning" and requires students to do more work and observe more, so as to help students form a thinking method of analysis, comparison and induction. In comparison and discussion, improve students' ability to actively acquire new knowledge by using what they have learned. Therefore, in the classroom, we should actively guide students to participate in and cooperate with each other to organize teaching, so that students can truly become the main body of teaching, experience the fun of participation, the joy of success, and perceive the wonders of mathematics.
Sixth, the basic idea of teaching design
(1) knowledge combing: mainly use inverse proportional function to explain the meaning of this chapter; Images and properties of inverse proportional function: using inverse proportional function to solve practical problems.
(2) Cooperation and exchange, interpretation and exploration
1. Review the concept of inverse proportional function and its equivalent form. And the corresponding supporting exercises are designed: judging the inverse proportional function and pointing out the k value; Write functional relations with physical knowledge, realize that mathematical knowledge comes from life, and examine students' mastery of inverse proportional function coefficient and independent variable index.
2. Review the image and properties of inverse proportional function and use it to solve problems. The corresponding exercises are also designed: according to the value of k, determine the quadrant where the inverse proportional function is located and its branches (x >；; 0), obtaining the function value (the value of the independent variable) according to the function relationship and the given independent variable (the function value); The value range of m is determined by the relationship between image attributes and k value; Find the inverse proportional resolution function by undetermined coefficient method; It is difficult for students to judge the function value of a given point according to the increase or decrease of the function on the function image and the abscissa of the point.
3. Comprehensive application: Give the schematic diagram of the intersection of the image y= ax+b of the linear function and the inverse proportional function y =, and the intersection points M(2, m) and N(- 1, -4). Find the analytical expressions of inverse proportional function and linear function, and write the range of x in which the value of inverse proportional function is greater than that of linear function according to the image. This kind of topic is very common in the senior high school entrance examination. It is a comprehensive application of linear function and inverse proportional function. The combination of numbers and shapes and the method of undetermined coefficients can improve students' ability of observation, analysis, comprehensive application and reasonable reasoning.
(c) Classroom exercises: run through the whole classroom teaching. See the courseware for details.
(4) Summary:
Students summarize the main contents of this lesson:
1. The meaning of inverse proportional function;
2. Images and properties of inverse proportional function:
3. Combination of numbers and shapes
Let students transform what they have learned in class into students' quality as soon as possible through the summary of knowledge content; Through the summary of mathematical thinking methods, students can understand the position and application of mathematical thinking methods in solving problems more deeply, and gradually cultivate students' good personality quality goals.
(5) Transfer
(6) After-class reflection:
1. In the process of teacher-student interaction in this class, actively create conditions and opportunities for students to express their opinions, so that they can have a successful learning experience, stimulate their interest in learning, enhance their self-confidence and improve their initiative in learning.
2. Try to embody the principle of taking students as the main body and teachers as the leading factor, and "digest" the content of this lesson smoothly in a relaxed and pleasant atmosphere.
3. Instant training-consolidating new knowledge. In order to deepen students' understanding of knowledge, so as to achieve the effect of consolidation and improvement, I specially designed a group of instant training questions, which integrated the exercises in the supporting exercises into the instant training questions, and consolidated new knowledge through students' observation, discussion and research, as well as the guidance of teachers.
4. Existing problems: students' cooperation is not active enough, few students take the initiative to answer questions, and students' enthusiasm is not fully mobilized; Pay too little attention to middle and lower grade students; Teachers talk too much and students don't have enough time to discuss and communicate; The content of classroom teaching is a little too much, and the teaching task has not been completed within the specified time.
Lecture notes on inverse proportional function in junior high school mathematics II. Today, I'm going to talk about the inverse proportional function and its image in Chapter 17 of the second volume of Grade Eight Mathematics.
First, teaching material analysis:
The content of this lesson is to enter the category of function again on the basis of learning plane rectangular coordinate system and linear function, so that students can further understand the connotation of function and feel that there are various functions in the real world. The image and properties of inverse proportional function are the review and comparison of the image and properties of positive proportional function, and also the basis for learning quadratic function in the future. The study of this course is a process for students to re-recognize the image and essence of functions. Because it is the first time for junior two students to contact hyperbola, we should pay attention to guiding students to master the characteristics of inverse proportional function images in teaching, so that students can have an intuitive understanding of inverse proportional function.
Second, the analysis of teaching objectives:
According to the spirit of "taking students as the main body, activating classroom atmosphere and fully mobilizing students to participate in the teaching process" in the new curriculum reform. In teaching design, I imagine that by using multimedia courseware to create situations, while mastering the knowledge about inverse proportional function, I can stimulate students' interest in learning and desire to explore, and guide students to actively participate in and explore.
Therefore, the teaching objectives are determined as follows:
(1) Knowledge objective:
1. Make students understand the concept of inverse proportional function.
2. Enable students to determine the analytical formula of inverse proportional function according to the conditions in the problem.
3. Make students understand the nature of inverse proportional function, draw its image, and point out that the function value changes with the increase or decrease of independent variables according to the image.
4. The analytical formula of the inverse proportional function will be determined by the undetermined coefficient method.
(2) Ability objectives:
Cultivate students' ability to observe, analyze and solve problems independently.
(3) Mathematical thought:
1. Infiltrate the view that mathematics comes from practice and reacts on practice.
2. Let students realize that things change regularly.
(4) Emotional attitude:
Through the study of inverse proportional function images, the intuitive beauty of images reflecting their nature can be infiltrated, which can stimulate students' interest and cultivate their ability to explore knowledge actively.
Third, the focus and difficulty of teaching.
(1) Teaching emphasis: the concept, image and properties of inverse proportion, and the analytical properties of inverse proportion function determined by undetermined coefficient method.
(2) Teaching difficulty: drawing the image of inverse proportional function.
(3) Solutions
(1) Discuss in groups, think positively, analyze problems and find conclusions.
(2) Training, research and summary
Because the image of inverse proportional function has two branches, and the changing trends of these two branches are different, students will find it difficult when they first contact. In order to highlight key points and break through difficulties. I designed and made a multimedia courseware that can dynamically demonstrate function images. Let students operate by themselves, actively participate in and actively explore the properties of functions, and help students intuitively understand the properties of inverse proportional functions.
(A) Inquiry learning 1- function image drawing method
Question 3: How to draw the image of the proportional function?
Review the drawing method of the image of the direct proportional function through question 3, which is mainly divided into three steps: listing, drawing points and connecting lines, laying a foundation for learning the drawing method of the image of the inverse proportional function.
Question 4: How to draw the image of inverse proportional function?
In the teaching process, students can be guided to imitate the drawing method of proportional function images.
The envisaged teaching design is:
(1) Guide the students to draw the image of proportional function by using the method they have learned, discuss and try in groups, and draw the image of function sum by using list method, dot drawing method and connection method;
(2) Teachers' patrol guidance, reflecting some students' typical mistakes in function images with physical projectors, and finding out the mistakes with students and analyzing the reasons;
(3) Then the teacher demonstrates the steps of drawing inverse proportional function images on the blackboard, showing the correct function images and guiding students to observe their image characteristics (hyperbola has two branches).
The second grade students first came into contact with hyperbola, a special function image. Imagine that students may make mistakes in the following links:
(1) In the "list" section, students may get zero when picking points. Here, students can be guided to draw the conclusion that X cannot be zero by combining algebraic methods. It may also be due to improper selection of points, resulting in incomplete and asymmetric function images. Here, students should be instructed that when listing, the value of independent variable X can be selected as a number with equal absolute value and opposite sign, and the corresponding function values with equal absolute value and opposite sign can be obtained accordingly, which can simplify the calculation procedure and facilitate finding points on the coordinate plane.
(2) In the link of "connecting lines", the connecting lines between points drawn by students may have endpoints and cannot be connected by smooth lines. So what we want to emphasize here is that when connecting the selected points, it should be a "smooth curve", which lays the foundation for learning the image of quadratic function in the future. In order to make the function image clear and obvious, students can be guided to choose as many values of the independent variable X and the corresponding function value Y as possible, so as to get more "points" on the coordinate plane and draw curves. So as to guide students to draw correct function images.
(3) The image intersects with the X axis or the Y axis.
Here I think we can bury a foreshadowing, leave a suspense for students, and lay the foundation for learning the properties of functions later.
Fourth, teaching methods:
Junior high school students are active, curious and clever. Grasping students' characteristics, actively adopting vivid and diverse teaching methods and students' extensive and active participation in learning methods will certainly stimulate students' interest, effectively cultivate students' ability and promote students' personality development. Physically, teenagers are active, easily distracted, and love to express their opinions, hoping to get praise from teachers. Therefore, we should grasp this physiological characteristic of students in teaching. On the one hand, we should use intuitive and vivid images to arouse students' interest and make them focus on the classroom. On the other hand, we should create conditions and opportunities for students to express their opinions and give full play to their initiative in learning. In view of the age characteristics, psychological characteristics and cognitive level of the textbook and the students in Grade Two, it is envisaged to adopt the problem teaching method and the contrast teaching method, so as to inspire students to think deeply, actively explore and actively acquire knowledge through step-by-step questioning. At the same time, pay attention to the connection with students' existing knowledge, reduce the difficulty for students to accept new concepts, and give them enough time to explore independently. Through the guidance of teachers, we can stimulate and mobilize students' enthusiasm, let students do more activities and observations in class, take the initiative to participate in the whole teaching activities, organize students to participate in the learning process of "inquiry-discussion-exchange-summary", and make full use of multimedia teaching in teaching to inspire students through teacher-student activities such as demonstration, operation, observation and practice, so that every student can
Five, learning guidance:
This course is based on students' "learning", which requires students to do more hands-on and observe more, and helps students form a thinking method of analysis, comparison and induction. Let students "learn by doing" in comparison and discussion, and improve students' ability to actively acquire new knowledge by using what they have learned. Therefore, in the classroom, we should actively guide students to participate in and cooperate with each other to organize teaching, so that students can truly become the main body of teaching, experience the fun of participation, the joy of success, and perceive the wonders of mathematics.
Finally, I will talk about the teaching process of this course in detail.
Sixth, the teaching process:
(1) Review the introduction-analytic formula of inverse function
Exercise 1: Write the relationship between the following questions:
(1) The relationship between the perimeter c and the side length a of a square.
(2) When the area of a rectangle is 10, the relationship between its length x and width y.
(3) Master Wang wants to produce 100 parts. What is the relationship between his work efficiency X and his working hours T?
Question 1: Please judge which of these relations we wrote are proportional functions.
1 is mainly to review the definition of direct proportional function, and lay a foundation for students to give the definition of inverse proportional function by comparison.
Question 2: Then please take a closer look. Are there any similarities between the other two functions?
Through question 2, the analytical formula of inverse proportional function is deduced, and students are required to give the definition of inverse proportional function by comparing the definition of direct proportional function, which will not only help to review and consolidate old knowledge, but also cultivate students' comparative inquiry ability.
Good afternoon everyone! Today, what I'm going to talk about is the image and properties of inverse proportional function in Chapter 17, Volume 2 of Grade 8 Mathematics published by People's Education Press. In the first class, I will elaborate on teaching material analysis, teaching objectives, teaching priorities, teaching methods and analysis of learning methods, and teaching process.
I. teaching material analysis
The image and nature of inverse proportional function is the focus of inverse proportional function teaching, which requires students to skillfully use it on the basis of understanding. This lesson is the core of the whole chapter. The main content of learning is to draw the image of inverse proportional function, so that students can experience mathematical activities such as drawing, observing, guessing, thinking and induction by enumerating, chasing points and connecting lines, get a preliminary understanding of the characteristics of inverse proportional function images, gradually clarify the intuitive image of inverse proportional function, and provide students with space for thinking activities. It also lays a solid foundation for learning quadratic functions and other functions in the future.
Second, the teaching objectives
According to my understanding and analysis of this lesson, the teaching objectives are as follows:
1, through students' hands-on operation, learn to draw the image of inverse proportional function in plane rectangular coordinate system by tracing points; 2. By observing the image of inverse proportional function, guide students to observe, analyze and summarize the properties of inverse proportional function. 3. In the process of students' independent exploration of the image and nature of inverse proportional function, students can experience the exploration and creation in mathematics activities and enhance their curiosity and thirst for knowledge in mathematics learning.
Third, the focus and difficulties in teaching
Emphasis: The tracing method is used as the image of the inverse proportional function, and the properties of the inverse proportional function are explored by using the image.
Difficulties: How to accurately draw the image of inverse proportional function by grasping the features?
Fourthly, analyze the teaching methods and learning methods.
Modern educational theory requires that "students' learning knowledge should be a process of understanding things for teaching". According to the cognitive structure and psychological characteristics of eighth grade students, I choose "guided inquiry method". Ask questions from shallow to deep, from special to general. Guide students to explore independently, cooperate and communicate. Let students always be in a learning state of positive thinking and active exploration.
According to the requirements of the new curriculum standards, teachers should guide students to participate in learning activities in an organized, purposeful and targeted manner, encourage students to adopt independent inquiry and cooperative communication, and cultivate students' habits and abilities of "hands-on", "brain" and "oral" so that students can truly become the masters of learning.
Teaching process of verbs (abbreviation of verb)
(A) the creation of situations, the introduction of new courses
1, Question 1: What is the shape of the image of the proportional function? How many steps did we draw?
2. Question 2: What is the shape of the image of the inverse proportional function? Do you want to know?
Through the first question, help students recall the method of drawing function images by graph, realize that any function image can be drawn by graph, activate students' original knowledge, and lay a foundation for exploring the drawing method of inverse proportional function images. The second question gives students an imaginary space and stimulates their enthusiasm for participating in classroom learning.
(2) Analogy and association, exploring communication-inverse proportional function image drawing.
1, Question 1: How to draw the images of inverse proportional functions y= and y =- according to the drawing method of images of direct proportional functions that have been learned?
Firstly, according to the students' answers and supplements, the basic steps of drawing inverse proportional function images are obtained: list-drawing points-connecting lines. Then ask the students to try to draw pictures of two functions in groups. In the teaching process, students can be guided to imitate the drawing method of proportional function images.
Students are exposed to hyperbola, a special function image, for the first time. Students may make mistakes in the following links:
(1) In the List section,
Students can get zero when they get points. Here, students can be guided to draw the conclusion that X cannot be zero by combining algebraic methods. It may also be due to improper selection of points, resulting in incomplete and asymmetric function images. Here, when guiding students' lists, we can choose the value of independent variable X as a number with the same absolute value but opposite sign, and then work out the corresponding function values with the same absolute value but opposite sign, which can simplify the calculation procedure and facilitate finding points on the coordinate plane.
(2) In the link of "connection"
The line between points drawn by students may have endpoints, which cannot be connected by smooth lines, or connect points in two quadrants. Therefore, it is particularly important to emphasize that when connecting the selected points, it should be a "smooth curve", which can also guide students to further analyze the analytical formula Y =≠k≠0≠0 of the inverse proportional function by algebraic method, so that the denominator and x can not be zero. From k≠0, y must not be zero, thus verifying the image of inverse proportional function. When two branches extend infinitely, they can approach the X axis and the Y axis infinitely, but they will never intersect. So as to guide students to draw correct function images. It lays a foundation for learning the properties of functions later. The concept of hyperbola is given.
2. Question 2: What is the relationship between comparing the images with letters y= and Y =- and their opinions?
Guide students to observe, compare and discuss in groups, describe in their own language, from perceptual knowledge to rational knowledge, and improve students' abstract generalization ability.
3. Consolidation training: draw an image with functions y= and y =-.
Let the students do it by themselves in groups, so that they can further understand the basic methods of drawing inverse proportional function images, and also increase their perceptual knowledge for later observation and analysis of the properties of inverse proportional function images.
(3) Explore and compare, and find the law-the essence of function image.
Question 1: Observe the images of functions y= and y =-
(1) Find out what are the * * * similarities of the inverse proportional function y=(k≠0) images? What is the difference?
(2) In which quadrants is each functional image located? What factors determine it?
(3) How does y change with x in each quadrant?
Guide the students through the letters in reverse proportion.
Observing and analyzing the digital image, discussing the relationship between the position of the function image and the symbol of the K value, and discussing that the two branches of the inverse proportional function are in the corresponding quadrant, and the Y value increases (or decreases) with the increase (or decrease) of the X value, which is conducive to deepening students' understanding and mastery of nature; According to the observation of the image, students sum up the properties of the inverse proportional function from the obtained image characteristics. Property: The image of (1) inverse proportional function y=(k is constant, k≠0) is a hyperbola.
(2) when k >; 0, the two branches of hyperbola are located in the first quadrant and the third quadrant respectively, and the value of y in each quadrant decreases with the increase of x value.
(3) When k < 0, the two branches of hyperbola are located in the second and fourth quadrants respectively, and the y value of each quadrant increases with the increase of x value.
(4) summary,
Question 1: What did you learn in this class?
Question 2: What are the similarities and differences between inverse proportional function and direct proportional function in image distribution and properties?
In the form of list, guide students to summarize the properties of inverse proportional function, and compare it with the images and properties of direct proportional function vertically to deepen their understanding. Through students' free discussion, summary and generalization of what they have learned in this chapter, students can further understand the inverse proportional function image and its properties, and experience the joy of learning mathematics in communication and share it with the whole class.
(5) Transfer
This link is mainly for students to deepen their understanding and application of what they have learned, and to know the situation in time.
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