People's education printing plate eighth grade first volume mathematical function concept teaching plan
Teaching material analysis:
As the core content of elementary mathematics, function runs through the whole elementary mathematics system. Function plays a connecting role in senior high school mathematics, and it inherits and deepens the concept of function in junior high school. In junior high school, we only stayed in several simple types of functions, and regarded functions as the dependence between variables. In senior high school, we added the concept of functions. Corresponding? This chapter runs through the idea of function, from special to general, the idea of combining numbers and shapes, from perceptual to rational, and the idea of mathematical modeling. The study of these contents will undoubtedly have a far-reaching impact on students' future study.
Teaching objectives:
1. Knowledge and skills:
(1) Understand the concept of a function, (you can describe a function with sets and corresponding languages, understand the three elements that make up a function, and find the domain of a simple function);
(2) Can it be used correctly? Interval? The symbol of represents some sets.
2. Process and method: Through students' own analysis, abstraction and generalization of practical problems, abstraction, generalization and regression are cultivated.
Ability to accept knowledge and model;
3. Emotion and values: The introduction of familiar life cases has stimulated the interest in learning mathematics and enhanced the application of mathematics.
Consciousness, innovative consciousness. Cooperate with each other to learn, enhance the sense of cooperation and realize the importance of cooperative learning.
Teaching methods: based on inspiration and inquiry, supplemented by discussion.
Learning methods: observation and analysis, independent inquiry, cooperation and communication.
Teaching emphasis: Understand the actual background of functions and describe functions with sets and corresponding languages.
Difficulties in teaching: Understand the actual background of functions and describe functions with sets and corresponding languages.
Teaching process:
First, review the introduction:
1. Discussion: What are the variables in this example? What is the relationship between variables?
2. Review the definition of junior high school function:
In a changing process, there are two variables X and Y, and for each value of X, Y has a unique and definite value corresponding to it. At this point, y is a function of x, x is an independent variable and y is a dependent variable.
Representation methods include: analytical method, list method and image method.
Second, introduce conceptual scenarios:
Thinking 1: (textbook P 15) Give three examples:
A. A shell hits the target after 26 seconds, with a shooting height of 845 meters. The variation law of the height h (m) of the shell from the ground with time t (s) is as follows.
In recent decades, the ozone in the atmosphere has decreased rapidly, resulting in the problem of ozone hole. The curve in the figure shows the change of ozone hole area over Antarctica. (See textbook P 15)
C. Engel coefficient (food expenditure amount? Total expenditure reflects the quality of life of people in a country. ? Eight-five? Engel's coefficient of urban residents in China since the plan is as follows. (See table P 16 in the textbook)
Discussion: What are the variables in the above three examples? What is the range of variables? What is the correspondence between the two variables? What do these three examples have in common?
Induction: The relationship between the three instance variables can be described as follows: For each X in number set A, it corresponds to the unique Y and IT in number set B according to a certain correspondence F, which is recorded as:
Third, the understanding of the concept:
Definition of 1. function:
Let a and b be two sets of non-empty numbers. If any number x in set A has a unique number corresponding to it according to a certain correspondence F, it is called a function from set A to set B, and it is recorded as:
Where x is the independent variable, the range A of x is the definition domain, the value Y corresponding to the value of x is the function value, and the set of function values is the range. Obviously, this range is a subset of set B.
note:
① ? y=f(x)？ Is a function symbol, which can be represented by any letter, such as? y=g(x)？ ;
② Functional symbols? y=f(x)？ F(x) in in represents the function value corresponding to x, a number, instead of f multiplied by x.
Thinking 2: What are the three elements that make up a function?
A: Definition domain, correspondence and range.
1 Among the following four images, the one that is not a function image is ().
2. Set,, gives the following four graphs, in which () can represent the functional relationship with m as the domain and n as the range.
Induction: (1) linear function y=ax+b (a? The domain of 0) is r, and the range is also r;
(2) quadratic function (a? The domain of 0) is r, and the range is b; When a>0, range; When a￠0, range.
(3) The definition domain and value domain of the inverse proportional function are.
2. Interval and writing:
Let a and b be two real numbers, a.
(1) The set of real numbers x satisfying inequality is called a closed interval, which is expressed as [a, b];
(2) The set of real numbers x satisfying inequality is called an open interval, which is expressed as (a, b);
(3) The set of real numbers x satisfying inequality is called semi-open and semi-closed interval, which is expressed as:
The real numbers A and B here are both called the endpoints of the corresponding interval. (See table P 17 in the textbook for the representation of the number axis)
Symbol? Reading? Infinite? ; ? -Reading? Negative infinity? ; ? +reading? Positive infinity? . We express the set satisfying the real number x as
.
Small test knife:
r，{x|x？ 1}、{ x | x & gt5}、{x|x？ - 1}、{ x | x & lt0}
(Students do, teachers modify)
3. Conceptual application:
Example 1. Known function,
( 1);
(2) when a >; The value of 0.
(For the answer, see P 17 Case 1)
Exercise. Given the function f(x)=x2+2, find f(-2), f(-a), f(a+ 1), f(f(x)).
Answer: f (-2) = 6f (-a) = a2+2f (a+1) = a2+2a+3f (f (x)) = x4+4x2+6.
Example 2 Known functions.
( 1); (2) Calculation:
Solution: (1) by.
(2) the original formula
Comments: The discovery of laws can make us realize ingenious calculation. Correctly exploring the conclusion of the previous question is the key to answering the latter question.
Fourth, the effect acceptance and summary:
Classroom test
1. Use intervals to represent the following sets:
2. Given the function f(x)=3x+5x-2, find the values of f(3), f(-), f(a) and f(a+ 1);
3. Textbook P 19 Exercise 2.
4. If =+x+ 1 is known, then = _ _ 3+_ _ _ _; f[]=_57_____。
5. If known, then =? 1 .
(2) Summary:
What does the actual background of the function say?
What do you think is the essence of the concept of function? How to understand the correspondence of functions?
What kind of set can be expressed by interval?
Task:
Sports 1.2A group, No.4, No.5 and No.6;
Reflections on the teaching of mathematical functions in the first volume of the eighth grade
Function is one of the most important contents in high school mathematics, which runs through the whole high school mathematics learning and is a lifelong mathematics learning process. Its importance is mainly reflected in:
1, the function itself originated from real life, such as natural science and even social science, and has a wide range of applications.
2. Function itself is an important content of mathematics, and it is a bridge to communicate algebra, geometry and trigonometry. It is also the basis and method for further studying advanced mathematics in the future.
3. The function part contains many important mathematical methods, such as the idea of function, the idea of equation, the idea of classified discussion, the idea of combining numbers with shapes, the idea of reduction, method of substitution, the method of determining coefficients, the matching method and so on. These thinking methods are the basis of further learning mathematics and solving mathematical problems, and they are also the parts that students should pay attention to in the teaching process.
This part of the knowledge of function is a major difficulty in teaching, mainly because the concept is abstract, which is quite difficult for students to understand and even more difficult to accept. Change? Words. What is the main research? Variable? With what? Variable? The relationship between them requires us to look at and contact related problems from the perspective of variables and the key points of action changes, which is quite different from the thinking characteristics of learning knowledge from a static point of view in junior high school, so function has become the first obstacle for freshmen to enter high school. Some students graduated from high school and didn't understand the concept of function clearly.
In fact, in the knowledge of learning function, the concept of function is the most important and the most difficult place, and it is easy to break through the later learning. The main content of current mathematics textbooks is the technical form of mathematical knowledge. The same is true of the concept of function, both traditional and modern definitions are abstract mathematical forms. In mathematics teaching, learning formal expression is the basic requirement, but it should not be limited to formal expression, and should emphasize the understanding of the essence of mathematics, otherwise lively mathematical thinking activities will be submerged in the ocean of forms. The teaching of mathematical knowledge should return to its origin and try to reveal the concept, law, conclusion, development process and essence of mathematics. The more abstract the mathematical concept is, the more so. Therefore, the teaching of function concept should not be scripted, but should pay attention to the reorganization of knowledge. Try to prompt the essence of the concept of function, so that students can really understand it, find it useful and are willing to learn.
People who read the first volume of the eighth grade math function concept teaching plan also read:
1. Eight Grade One Mathematics Inequality Teaching Plan
2. The application practice of the first volume of the eighth grade mathematics.
3. The first volume of the eighth grade mathematics exercises the group of one yuan and one time inequalities.
4. Reflections on the teaching of first-order functions and first-order inequalities in junior two mathematics.
5. Tutoring materials for second-year mathematics: one-dimensional linear inequality group