Since the second half of the 20th century, the great development of mathematics application is one of the remarkable characteristics of mathematics development. In today's era of knowledge economy, mathematics is moving from behind the scenes to the front. The combination of mathematics and computer technology enables mathematics to directly create value for society in many aspects, and also opens up broad prospects for the development of mathematics. For a long time, China's mathematics education failed to pay full attention to the relationship between mathematics and practice, and between mathematics and other disciplines. Therefore, in recent years, the practice of mathematical modeling in universities and middle schools in China shows that the teaching activities of mathematical application meet the needs of society, which is conducive to stimulating students' interest in learning mathematics, enhancing students' awareness of application and expanding their horizons. Under such a curriculum concept, the curriculum standard B version of People's Education Publishing House has brought us a spring breeze. It is not only a simple change of words, but also a prominent embodiment of teaching thought. A large number of learning activities such as "mathematical inquiry" and "mathematical modeling" are set up in the whole set of teaching materials, which provide the practical background of the basic content and reflect the application value of mathematics. These courses, which embody the application of mathematics, further create favorable conditions for students to form active and diverse learning methods, stimulate students' interest in learning mathematics, and encourage them to form the habit of independent thinking and active exploration in the learning process.
Below, the author makes the following teaching design and attempt on the content of function (the first class).
Textbook analysis
1. The position and function of this class
Function is one of the important basic concepts in mathematics. The basic courses of advanced mathematics that students further study, including limit theory, differential calculus, integral calculus, differential equations and functional analysis, all take 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. It is a re-understanding of the concept of function, that is, to understand the general definition of function with the idea of set, which is based on the preliminary discussion of the concept of function, the expression method of function relationship and the position of image in junior high school. The deepening and improvement of function and application research is also the basic knowledge needed for further participation in industrial and agricultural production and construction in the future. The study of this chapter plays a decisive role in middle school students' mathematics learning, not only in knowledge, but also in mathematical modeling, which will also be a lifelong benefit.
2. Teaching emphases and difficulties
Emphasis: Perceptual function is an important mathematical model to describe the dependence between two variables, and the concept of function is understood on the basis of mapping.
Difficulty: the understanding of function symbol y=f(x).
Teaching objectives
1. Knowledge and skill objectives:
(1) Help students to establish the background of mathematical concepts through different life examples, so as to correctly understand the concept of functions.
(2) Functions can be described by sets and corresponding languages, and the elements that constitute functions, that is, domains and corresponding rules, can be understood; Further understand the meaning of correspondence law.
2. Process and method objectives:
Understanding function is an important mathematical model to describe the dependence between variables. On this basis, learn to describe functions with sets and corresponding languages, and reproduce the process of generating function knowledge. Experience the process of solving practical problems with mathematical ideas, methods and knowledge in mathematical modeling.
3. Emotional attitudes and values goals:
By creating real life scenes, students can be close to real life and pay attention to social reality; Feel the function of correspondence in describing the concept of function, stimulate students' interest in learning mathematics, cultivate students' sentiment and cultivate their spirit of scientific exploration.
teaching process
First, create a problem situation.
Teacher: When we were in junior high school, we learned the concept of function and knew that we could use function to describe the dependence between two variables. Today, we will learn more about functions and their constituent elements. Let's look at a few examples:
Question 1: A shell landed after launching and hit the target 26s later. The firing height of the shell is 845m, and the variation law of the height h(m) from the ground with time t(s) is h= 130t-5t2. The following questions were raised:
(1) 1s, 10s, 20s, how high is the shell off the ground?
(2) When is the shell highest from the ground?
(3) Can you point out the range of variables T and H? Represented by set a and set b respectively.
(4) For any time t in set A, according to the corresponding relation h= 130t-5t2, is there a unique height H in set B?
Student: Because I have a junior high school foundation, I can tell the answers to the first three questions quickly. Question (4) The teacher inspires students to describe the dependency between a set and the corresponding language: within the range of T, any given T has a unique height H corresponding to it according to the given analytical formula.
Starting with the life problems of multimedia presentation, this paper reproduces the concept of describing functions with variables in junior high school, which lays the foundation for defining functions with sets and corresponding viewpoints in the future. ]
Question 2. The temperature change of a city meteorological observatory on 24-hour test day is shown in the figure.
(1) What's the temperature at 8 am?
(2) Can you point out the range of variables T and θ? Represented by set a and set b respectively.
(3) For each time t in set A, according to the image, is there a unique temperature θ corresponding to it in set B?
Student 1 A: the temperature at 8 o'clock in the morning is about 0. c； The value range of t is [0,24];
The range of θ is [-2,9].
Answer 2: For each time t in set A, there is a unique temperature θ in set B according to the image.
Then the teacher asked the students to review the changes in their family life in the past ten years. What aspects of consumption have changed greatly? What aspects of consumption have not changed much?
[The students' enthusiastic response further aroused their enthusiasm and experienced the process of abstracting practical problems into mathematical models. This is actually a process of advocating doing and using mathematics and paying attention to the formation and development of students' knowledge. ]
What data do you think should be used to measure the quality of family life? The slide shows the change of Engel coefficient with time (year), which shows that the quality of life of urban residents in China has changed significantly since the Eighth Five-Year Plan.
t
9 1
92
93
94
95
96
97
98
99
00
0 1
r
53.8
52.9
50. 1
49.9
49.9
48.6
46.5
44.5
4 1.9
39.2
37.9
After reading the chart, imitate 1 question and 2 questions, and describe the relationship between Engel coefficient r and time t (year) in the table.
Student induction: For any time t (year) in the table, there is a unique Engel coefficient r corresponding to it according to the table.
Second, explore new knowledge.
Students discuss the * * * characteristics of the above examples in groups, and come to the conclusion that they all involve two sets of non-empty numbers A and B, and there is a certain correspondence, so that for each number X in A, Y and X in B have a unique correspondence according to this correspondence.
[Practical problems lead to concepts, stimulate students' interest, give students space to think and explore, let students experience the process of mathematical discovery and creation, and improve their ability to analyze and solve problems. ]
Definition of 1. function
Let a and b be non-empty number sets. If any number X in set A has a unique value Y corresponding to it according to a certain correspondence F, then this correspondence is called a function on set A ... Note where. Domain: the value range of x (number set a) is called the domain of function; If the independent variable takes the value a, the value y determined by the law f is called the function value of the function at a. Range: the set of function values {y/y=,} is called the range of the function.
Teachers and students recall the concept of function introduced in junior high school, which is expressed as follows:
Suppose there are two variables in a process of change. If each value has a unique value correspondence, it is said to be an independent variable and a function.
We see that the function is defined from the viewpoint of movement change, which reflects people's understanding of it in history. This definition is intuitive and easy to accept. Therefore, it is appropriate to introduce the concept of function in junior high school according to the principle of content arrangement from shallow to deep, and strive to conform to students' cognitive laws. ]
Teacher: The corresponding laws of functions are usually represented by symbols. Function symbols indicate that any domain is obtained under the action of "corresponding rules". In a relatively simple case, the corresponding rule can be expressed by an analytical formula, but in many problems, the corresponding rule needs to be expressed by several analytical formulas, and sometimes it can't even be expressed by one analytical formula. What's that look?
Health: It should be expressed in other ways (such as lists and pictures).
Students discuss in groups and pay attention to several aspects of function definition: (Teacher's blackboard)
(1), directionality;
(2) keywords "arbitrary x" and "unique number f(x)".
(3)A and B are nonempty number sets;
(4) Any element in A has a unique element in B; But the elements in B may or may not have a unique corresponding element in A, which is obviously within the scope.
When explaining concepts, teachers consciously use fonts of different colors on the multimedia screen to highlight key points and mobilize students' non-intellectual factors to understand concepts. ]
2. Question 4:
(1) Is the corresponding rule below a function on a given set?
①R, g: the reciprocal of the independent variable;
②R, h: the square root of the independent variable;
③R, s: the square of the independent variable t minus 2.
(2) Are the following functions the same?
①f(x)=x2，x∈R；
②s(t)=t2，t∈R；
③g(x-2)=(x-2) 2，x∈R。
Health: Determine two elements of a function: the domain and the corresponding law.
The discussion about the interaction between teachers and students shows that functions are represented by symbols, but this symbol did not appear when learning functions in junior high schools. It should be noted that:
(1), is said to be a function, not equal to the product of sum;
② Not necessarily analytical;
③ Sum is different.
3, example teaching:
Take 1 as an example. A watermelon stall sells watermelons, which cost 4 cents per catty under 6 kg and 6 cents per catty above 6 kg. Please give the functional relationship between the weight x and the selling price y of watermelon.
Solution: By analytical method, the analytical expression of this function can be divided into two cases:
At that time,; At that time,
Teacher: This kind of function is called piecewise function, and we can also express it in an image way. Ask the students to draw the image of this function.
Teacher: Can this functional relationship be expressed in tabular form? Inconvenient. Because there are too many grades of watermelon weight, it is not easy to make a complete list
Third, consolidate exercise 1: The following figure can be used as a function image ().
Exercise 2: Which of the following functions is the same as the function?
Fourth, class summary.
The research and study of this class is here. Please review the exploration and gains of this class.
Born in 1, we know the definition of function: Let A and B be non-empty number sets, then the mapping from A to B is called A to B.
Function, written as, where,.
We know that there are three representations of functions: analytical method, tabular method and graphical method.
3. We know three elements of a function: domain; Scope;
The correspondence law in is the basis of the function, and the correspondence law is the core of the function.
In this class, we have group activities such as discussion, cooperation and communication, and personally experience the process of abstracting practical problems into mathematical models and explaining and applying them. We feel that mathematics is everywhere around us.
Teacher: Good point! These are the key points of our class. I hope to see the results of your independent thinking and exploration in the future and show your research style.
Verb (abbreviation of verb) modeling operation
(1) a nail 1.50 cents each, and the money for buying a nail is yuan. Please list the function relationship between and and draw the image of the function.
(2) mailing the parcel, and the postage per kilogram of the parcel is 2 yuan. After the postal distance exceeds 100km, 20 cents will be charged for each additional 1km. Find out the functional relationship between postage and the number of kilometers traveled by the package.
Ask the students to record the weather forecast for a week and list the functional relationship between the daily maximum temperature and the date.
teaching evaluation
First, pay attention to the formation process of function concept and understand the true meaning of mathematics.
As we all know, mathematical concepts are directly or indirectly abstracted from the objective world, and their definitions are mostly given by the method of "problem scenario-extracting essential attributes-extending to general". The concept of function in this lesson is that under the guidance of teachers, students appear as explorers and participate in the process of revealing the law of concept formation, so that their thinking has gone through a cognitive process from concrete to abstract and summarizing the essence of things. By understanding the hidden thinking method in the process of knowledge formation, students not only get the concept of function, but also broaden their thinking space, understand the true meaning of mathematics, and train their generalization ability while mastering the concept.
Second, the problem design is open and novel, infiltrating mathematical thinking methods.
As we all know, students' original knowledge and experience are the basis of learning, and students' learning is a self-generating process based on original knowledge and experience. Before learning the concept of function, students have been exposed to function in junior high school. Teachers are good at using analogy in teaching, grasping the advantages and disadvantages of the two concepts of function in junior high school and senior high school, so that students can understand the organic connection between knowledge and feel the integrity of mathematics. On the basis of students' cooperation and communication, students summed up several aspects of function definition, which were permeated with transformation ideas and inductive methods.
Third, tap teaching material resources and expand students' exploration space.
As we all know, mathematics textbooks are the embodiment of mathematics curriculum standards and the selection of mathematics knowledge system, which is very convenient for teachers and students to use. In this course, teachers don't just stay on the surface of textbooks, but study and familiarize themselves with textbooks seriously, make full use of various teaching resources, organize students to explore, and cultivate students' inquiry ability. This well-designed inquiry activity can stimulate students' enthusiasm for learning mathematics and improve their ability to explore and study problems.
Fourth, improve the way of teaching and learning, so that students can take the initiative to learn.
Enriching and improving students' learning methods is the basic idea pursued by senior high school mathematics curriculum. Students' mathematics learning activities should not be limited to memorizing, imitating and accepting concepts, conclusions and skills. Independent thinking, independent exploration, hands-on practice, cooperative communication and reading self-study are all important ways to learn mathematics. In this kind of teaching, there are both teachers' teaching and guidance and students' independent exploration and cooperation. Teachers in the whole class pay attention to students' subjective participation, leaving students with appropriate space and time for expansion and extension, stimulating students' interest in learning mathematics and forming good study habits.
Fifth, attach importance to mathematical modeling activities and cultivate students' application consciousness.
When talking about the application of mathematics, Friedenthal, a famous mathematics educator, pointed out that "we should understand the application of mathematics from two aspects: we should not only attach importance to extracting mathematical concepts and principles from practical problems, but also attach importance to using mathematical concepts and principles to deal with practical problems in turn"; In order to apply school mathematics to different situations more widely, mathematization should be the main way of mathematics teaching. In this lesson, teachers guide students to find problems from actual situations through mathematical modeling activities, and boil them down to mathematical models to form mathematical problems (that is, mathematization of actual problems). At the same time, it broadens students' horizons and realizes the scientific value, application value and humanistic value of mathematics.