Du Junyi, a middle school affiliated to Beijing Capital Normal University
The design of linear function and quadratic function in compulsory No.1 middle school 2.2 is an important feature of the B version of the textbook, and it is also one of the symbolic contents that reflect the importance attached by the B version to the transition from junior high school to senior high school. In the teaching practice of using textbooks, I think that some changes in the design of examples and exercises may be more conducive to realizing the value and role of this content in senior high school mathematics learning. The following are my immature thoughts, which I hope to discuss with my colleagues.
1 Basic understanding of teaching and learning in this section;
1. 1. Analysis of teaching content
Mathematical analysis of 1. 1. 1
As a simple and basic elementary function, linear function and quadratic function are very important in both junior high school and senior high school, and they are also the most closely related contents in the concrete mathematics content of junior high school and senior high school.
On the one hand, in real life, both linear function and quadratic function are important function models, so they have been studied in junior high school, but mainly at the level of "seeing". In high school, linear function is closely related to linear equation in analytic geometry. Quadratic function is an important function model to understand the concept, monotonicity and parity of function from the perspective of mapping. Quadratic function is a concrete model example to solve axisymmetric function; Quadratic function is also an appropriate and important carrier to reveal the relationship among functions, equations and inequalities (with moderate difficulty); In addition, in the middle school stage, in the process of learning functions by using derivative tools, the symbols of derivative functions of a large number of functions are often related to the symbols of quadratic functions, so it is also the knowledge basis for using derivative tools to solve the problems related to monotonicity, maximum value and extreme value of functions.
Because quadratic function makes students learn quadratic function again in high school, it mainly highlights the difference of function research methods. Junior high school mainly understands from the visual angle of image, while senior high school emphasizes abstract analysis from algebraic characteristics of resolution function to study and understand the essence of cognitive function, so as to fully understand and understand the complementarity of stippling and resolution function analysis in studying the image nature of function.
The matching method used in the structural feature analysis of quadratic resolution function is an important algebraic deformation technique, which will be applied in analytic geometry in the future. Moreover, the image properties are analyzed from the structural characteristics of analytic algebra, which paves the way for studying and analyzing the geometric characteristics of curves from the algebraic characteristics of equations in analytic geometry in the future, and has made some infiltration in skills and ideas.
As far as its role in senior high school mathematics is concerned, among the basic elementary functions, linear function and quadratic function are one of the very good carriers.
1. 1.2 education analysis
Junior high school mathematics knowledge is little, shallow, easy and narrow. High school mathematics knowledge is extensive, which is the promotion and extension of junior high school mathematics knowledge and the perfection and sublimation of junior high school mathematics knowledge. There are higher requirements for senior high school students in learning methods, self-study ability and thinking habits. But senior one students were junior high school students yesterday and senior high school students today. Knowledge was a junior high school textbook yesterday, which is relatively easy and simple. When senior one started school, it was a high school textbook, which became abstract and difficult to understand. The steps are too high and there is no buffer transition. After students enter high school, many students quickly show their inadaptability to high school mathematics learning. Quadratic function is a good carrier for students to understand the difference and connection between learning content and learning methods in junior high school and realize the connection between junior high school and senior high school.
1.2 Determination of teaching objectives
Teaching objectives 1.2. 1
① Take two important function models, linear function and quadratic function, as carriers, and learn the general methods to study the properties of functions. Through the review and improvement of these two functions, we can communicate the internal relationship between junior and senior high school mathematics and realize a smooth transition.
(2) Highlight the idea of combining numbers with shapes, and further clarify the relationship between algebraic features of resolution function and shape features of function images, such as: the geometric meaning of parameters of linear function and the influence of parameters of quadratic function on images; Further improve the operation level, such as: the formula and decomposition of quadratic formula, the solution of equations (the analytical formula for solving undetermined coefficients); Look at mathematics from the viewpoint of further improving the relationship between students, such as the relationship between linear function, quadratic function and linear equation, quadratic equation and linear inequality and quadratic inequality.
(3) Deepen the understanding of function symbols and understand the function of function symbol language in mathematical expression.
④ In the reading, communication and learning of the properties of linear function and quadratic function, we can gain the happiness of reviewing the past and learning new things.
1.2.2 key points and difficulties:
This section focuses on understanding the general method of learning function and the relationship between number and shape; Algebraic operation skills; Relationships among equations, inequalities and functions.
The difficulty in this section is to understand and use the relationship between number and shape, as well as the understanding and application of functional symbol language.
1.3 Analysis of the learning situation in the teaching of linear function and quadratic function
The images and properties of linear function and quadratic function are the core content of senior high school entrance examination. After the baptism of preparing for the third grade, students have a good understanding of linear function and quadratic function. However, from the understanding and understanding of the concept of function and the essence of image, it is flawed. It is difficult for students to separate the properties of analytic functions. On the one hand, the previous research on images was basically at the level of looking at pictures and speaking. On the other hand, students are not trained enough in formula calculation in junior high school, and the skills of algebraic operation deformation are not skilled. Although students know something about the image of function in coordinates in junior high school, their understanding of the relationship between resolution function and image needs to be deepened gradually. Therefore, it is not without obstacles for students to meet the learning requirements of linear function and quadratic function in senior high school, so teachers need to do appropriate guidance, inspiration and explanation.
1.4 Selection of teaching methods and means
Good mathematics teaching should be based on learners' life experience and their own knowledge background, and provide students with opportunities to fully carry out mathematical practice activities and exchanges, so that they can truly understand and master mathematical knowledge, ideas and methods in the process of independent exploration, and at the same time gain rich experience in mathematical activities. Students should be the masters of learning, and teachers should be the organizers, guides, collaborators and researchers of students' learning mathematics.
Therefore, this lesson mainly uses heuristic inquiry, that is, through the design of a series of questions, to guide and inspire students to learn independently after class. By asking questions, exchanging discussions, and dispelling doubts and doubts in class, students' thinking will be gradually deepened, and students' knowledge and understanding of "the nature and application of quadratic function images" and "the general method and application of studying functions" will be promoted.
2 Design of teaching process
2. 1 Design of teaching activities
1. arrangement before class:
Teach yourself this section before class, and try to think about and answer the questions on the self-study outline, and mark or record the unanswerable questions and problems in the reading process.
2. Communication and discussion in class:
Teachers ask students to report their thinking results according to the order of questions in the reading outline, other students make comments, and teachers make appropriate comments and supplements (mainly encouragement and praise).
3. Classroom exercises:
After dealing with the problems in the self-study syllabus, let the students do practical exercises to test, consolidate and remedy the knowledge, methods and ideas they have learned.
2.2 Pre-class preparation (reading and learning question string design)
2.2. 1 problem selection and design principles
1. Reviewing quadratic function in junior middle school is helpful for students to extend and improve their knowledge and understanding of quadratic function;
2. Taking concrete quadratic decomposition function and function image as the carrier, it is helpful for students to understand and use abstract function symbol language;
3. It is helpful for students to know and understand the general ideas and methods of research function from the actual operation process;
4. It helps students to feel the magical power and value of mathematical thinking in their thinking and exploration of problems;
5. It can satisfy the desire of excellent students to learn and explore.
2.2.2 Design and intention of the questions in the self-study syllabus
Question 1: Please point out the basic properties of linear function images (domain, range, monotonicity and parity) according to the linear resolution function, and explain the influence of parameters on the basic properties of linear function images.
Design intent
Question 2: Why does a linear function look like a straight line? How to understand the word "inclination"? How does the change of slope reflect the inclination of a straight line?
Design intent
Question 3: (Exercise A-5) In what range do the following functions take values?
( 1) (2)
Design intent
Question 4: Read the textbook example 1 and think about the following questions:
1. In the analysis of an example, is it unnecessary or meaningful to get the vertex from the formula in step (1) and the intersection of the image and the axis from the equation in step (2)? Please compare and analyze the two ways of "getting an understanding of the image through the abstract analysis of analytic function" and "getting an intuitive understanding of the function image through enumeration and drawing", and try to explain their respective functions and significance in studying the properties of the function image.
2. Example 1 What is the basis for listing, plotting and drawing in step (3)? That is why the points obtained in this way are points on the function image.
3. How to understand the process of proving image symmetry in step (4) of Example 1? Is it okay to take unlimited measures in this process? On the other hand, if a function is satisfied with everything in the domain, what is the nature of the image of this function? How to give an explanation? What do you think is the significance of introducing functional symbols into high school mathematics from the symbolic language expression proved above and the description of the concepts of monotonicity and parity of functions learned earlier?
4. Step (5) of example 1 is to observe the image and get the monotone interval of the function. If there is no list drawing, can we get monotonous interval from the analysis of resolution function? How do you explain that your monotonicity conclusion is correct?
Design intention guides students to think deeply. ① Students gradually discovered the method of learning function in senior high school, and began to pay attention to the abstract analysis of algebraic characteristics of analytic functions, so as to have a preliminary understanding and mastery of image properties. ② Promote students' knowledge and understanding of the relationship between function image and resolution function. ③ Let students feel the necessity of introducing more mathematical symbolic languages into senior high schools. An equation replaces the lengthy Chinese expression of the symmetry of the function image, and simply and clearly shows the relationship among independent variables, function values and motion changes.
There is no essential difference between Example 2 and Example 1, but the positive and negative quadratic terms of quadratic function are changed, which is somewhat redundant. So this question is only based on the reading of 1 example.
If students are selected to repeat the exercises in Example 1, it is better to abstract Example 1 from a concrete quadratic function to a general quadratic function, replace the constant coefficient with a letter coefficient, and discuss the positive and negative of the quadratic coefficient. These recommendations are summarized below.
Question 5: Complete 2.2. 1 Exercise A-5 and 2.2.2 Exercise A-3, and think about the relationship among functions, equations and inequalities. What role does function play in solving inequality?
The design intention urges students to discover the internal relations among functions, equations and inequalities.
Question 6: How to treat "Generally speaking, any quadratic function can be formulated as
, among which,
Thus, the properties of quadratic function are summarized. "What is the understanding of this passage? What do you think it provides for you to solve the related problems of quadratic function? Method or conclusion or something else?
Design intention urges students to learn what to pay attention to in reading. The answer to this question can reflect students' different cognitive strategies and lay the foundation for classroom communication. Communicating students' concerns in classroom communication helps students to reflect on their cognitive strategies, thus improving their learning methods and strategies.
(Note: Australian educator Biggs (1987) defined three cognitive strategies. The first is shallow orientation. Usually mechanical learning. Study focuses on seemingly important titles and elements, and try to remember them. They think memorizing details is the best way to learn. The second is deep-seated orientation. It is often based on curiosity and meaning, and the learning content is combined with personal meaning scenes and existing knowledge. The third is achievement orientation. The purpose is similar to shallow orientation, product-centered, and manifested as the strategy of following the teacher's teaching. )
Question 7: Do you think the textbook P59 is satisfied with the properties of quadratic function? What other attributes can be added? Please summarize and sort out a project with the most satisfactory nature.
The design intention makes students realize what aspects should be paid attention to in the study of function properties. Definition domain, value domain, monotonicity, parity, symmetry of image, intersection of image and axis, etc.
Question 8: Under what circumstances can we use the undetermined coefficient method to find the resolution function?
The design is intended to remind students to pay attention to the understanding of some words in the textbook.
Question 9: Complete the textbook example1; Exercise a-5; After practicing B- 1, how to set a quadratic resolution function is more convenient to calculate when calculating the quadratic resolution function of undetermined coefficients?
The design intention is to understand the values and meanings of different expressions of the resolution function.
2.3 Design choice and intention of classroom exercises
Exercise 1: Try to complete the following set of questions (adapted from the function in Example 2 of the textbook), and reflect after completing the questions.
(1) Given a function, the size of the trial comparison value is not calculated.
The design intention is to deepen students' analysis and understanding of monotonicity of functions.
(2) Given the function, try to compare the values.
(3) Given the function, try to compare the values.
Design Intention (2)(3) This is a variant of a question (1), which enables students to understand the influence of parameters on the properties of functions. (5) Strengthen the application of classification ideas.
(4) The quadratic function satisfying downward opening is known, and the numerical values are compared.
The design aims to deepen students' understanding of abstract functions and enhance their understanding of symbolic language.
(5) The known function is monotonically decreasing, satisfying. Try to compare the values.
Design Intention This is an extension of question (4). It is more difficult to remove the background of quadratic function, which deepens students' understanding of symbolic language expression of function symmetry.
Reflection: Please point out the differences and connections of the above five small problems, and summarize the general methods of comparing quadratic function values abstractly.
The design intention is to train students' ability of induction and abstract generalization and grasp the essence of the problem. As long as the opening direction and symmetry axis remain unchanged, the relative size will not change. In addition, the quadratic function image is an important schematic diagram for studying axisymmetric functions.
Exercise 2:
(1) Exercise B-3: Find the domain and value of the following function by collocation method:
① ②。
The problem of meaning selection is not a quadratic function at first glance, but if it is observed that it is a quadratic function under the radical, the definition domain can be found by solving the unary quadratic inequality according to the quadratic function image, which highlights the internal relationship between the function and the equation and inequality. If the quadratic function under the root sign is found in the value range of the definition domain, the value range of the function will be obtained, which requires a lot of thinking and mastery of the image and properties of the quadratic function of one variable, which embodies the general function of the collocation method. The difficulty of students lies in the root number, and the difficulty lies in not grasping the main contradiction under the root number. The answer to the second question is also very interesting. There is only one element in both the domain of definition and the domain of value, which in turn enables students to deepen their understanding of the concept of function.
(2) Find the range of the function and tell which interval it is increasing function. On which interval is the decreasing function? Draw an image of the function.
After the design intends to define the value of the independent variable of the quadratic function, the function is no longer a quadratic function, it is only a part of the quadratic function image, and the local properties are observed in the whole. This problem strengthens the understanding of the concept that a function is a mapping between two groups of numbers, and highlights the need to pay attention to the definition domain when learning functions.
(3) Find the definition and value domains of the following functions by the collocation method:
( 1); (2)
The design intention enables students to deepen their understanding of the problem that can be transformed into a quadratic function of one variable, and also enables students to realize the role of matching method in solving problems. In addition, we should learn the quadratic function, discuss its properties in a limited range of values, and let students define the function from the corresponding angle between sets in high school.
2.4 After-class exploration and research
With the exploration and research of the textbook P6 1 as the theme, this paper writes a short paper on image translation transformation, and asks to explain the relationship between algebraic change of resolution function and image translation change (whether the internal coordinate explanation can be found from an external correspondence). In order to find the translation relationship between function images, what kind of deformation processing is often needed for resolution functions?
Exploration and research on the following issues in the revised 4 textbook:
Explore functions and; Use; The image relationship with.
Design intention guides students to explore and reveal the reasons behind mathematical phenomena. The modification of 4 is more helpful for students to discover the inherent laws of abstraction and gain experience in image translation and transformation.
This paper is selected from the Proceedings of the Sixth Experimental Work Seminar of Senior High School Mathematics B Edition.