Lecture Notes on the Concept of Function 1 I. Contents of the lecture:
The concept of quadratic function and related exercises in the first section of chapter 6 of the ninth grade mathematics volume II of Jiangsu Education Press.
Second, teaching material analysis:
1, the position and function of teaching materials
This lesson is to learn the concept of quadratic function on the basis that students have learned a function, a proportional function and an inverse proportional function. Quadratic function is the last concrete function learned in junior middle school, and it is also the most important function, which occupies a large proportion in the middle school examination questions over the years. At the same time, quadratic function is closely related to quadratic equation and quadratic inequality that we have learned before. Further research on quadratic function will provide new methods and ways for its solution, and make students understand the important idea of "combination of numbers and shapes" more deeply. The concept of quadratic function in this lesson is the basis of learning quadratic function, which paves the way for learning quadratic function images later. Therefore, this lesson occupies an important position in the whole textbook.
2, teaching objectives and requirements:
(1) Knowledge and skills: enable students to understand the concept of quadratic function, master the method of listing the relationship of quadratic function according to practical problems, and understand how to determine the range of independent variables according to practical problems.
(2) Process and method: review old knowledge, introduce practical problems, and experience the process of exploring the concept of quadratic function to improve students' ability to solve problems.
(3) Emotion, attitude and values: Through observation, operation, communication and induction, we can deepen our understanding of the concept of quadratic function, develop students' mathematical thinking, and enhance their desire and confidence in learning mathematics well.
3. Teaching emphasis: Understand the concept of quadratic function.
4. Teaching difficulty: Determine the range of resolution function and independent variable from practical problems.
Third, the teaching design method:
1, starting with the creation of situations, and gestating the teaching process through knowledge reproduction.
2. Starting from students' activities, through innovation, using the teaching process.
3. Understand the teaching process through exploration and research and deep thinking.
Fourth, the teaching process:
(a) review of issues
1. What is a function? What functions have we learned before?
(Linear function, proportional function, inverse proportional function)
2. What are their forms?
(=x+b，≠0； =x，0； = , ≠0)
3. What is the independent variable of the linear function (=x+b)? What is a function? What is a constant? Why is there a condition of ≠0? What effect does value have on function properties?
The purpose of reviewing these questions is to help students understand the concepts of independent variables, functions and constants, deepen their understanding of the definition of functions, and emphasize the conditions for comparing quadratic functions with A ≠0.
(B) the introduction of new courses
Function is to study the relationship between two variables in a certain change process. We have learned the direct proportional function, the inverse proportional function and the linear function. Look at the relationship between two variables in the following three examples. (Computer demonstration)
Example 1, (1) When the radius of a circle is r(c), what is the relationship between the area s (c) and the radius?
Solution: s = π r (r >; 0)
Example 2. What is the relationship between the area () and the length x () of a rectangular field surrounded by a fence with a perimeter of 20?
Solution: = x (20/2-x) = x (10-x) =-x+10x (0
Example 3: Assuming that the annual interest rate of RMB one-year fixed deposit is X, after one-year maturity, the bank will automatically transfer the principal and interest on a one-year basis. If the deposit amount is 100 yuan, what is the relationship between the principal and interest after two years (excluding interest tax) and X?
Solution: = 100( 1+x)
= 100(x+2x+ 1)
= 100 x+200 x+ 100(0 & lt； x & lt 1)p = " " & gt； & lt/x & lt； 1)>
The teacher asked: What are the similarities and differences between the functions listed in the above three examples and linear functions?
The design intention is to let students list the relationship through concrete examples, inspire students to observe and think, and summarize the relationship between quadratic function and linear function: (1) Analytic functions are algebraic expressions (indicating that this function has the same characteristics as linear function). (2) The maximum number of independent variables is 2 (which is different from linear function).
(C) explain the new lesson
The above functions are different from the linear functions, proportional functions and inverse proportional functions we have learned, so we call these functions quadratic functions.
Definition of quadratic function: A function in the form of =ax2+bx+c (a≠0, A, B and C are constants) is called a quadratic function.
Consolidate the understanding of the concept of quadratic function;
1, emphasizing "the form is", that is, the function name is defined by the form. A quadratic function is a quadratic polynomial about x (the algebraic expression about x must be an algebraic expression).
2.In =ax2+bx+c, the independent variable is X, and the range of values is all real numbers. But in practical problems, the range of independent variables is the value that makes practical problems meaningful. (e.g. 1, r >；; 0)
3. Why do you need a≠0 in the definition of quadratic function?
(If a=0, AX2+BX+C is not a quadratic polynomial about X)
4. In Example 3, in the quadratic function = 100x2+200x+ 100, a= 100, b=200 and c = 100.
5. Can b and c be zero?
As can be seen from the example 1, both b and c can be zero.
If b=0, then = ax2+c;
If c=0, then = ax2+bx；;
If b=c=0, then = ax2.
Note: The above three forms are special forms of quadratic function, and =ax2+bx+c is the general form of quadratic function.
The design intention emphasizes the understanding of the concept of quadratic function, which helps students to better understand and master its characteristics and pave the way for the next judgment of quadratic function.
Judge: Which of the following functions is a quadratic function? Which are not quadratic functions? If it is a quadratic function, point out a, b and c.
( 1)= 3(x- 1)+ 1(2)
(3)s=3-2t (4)=(x+3)- x
(5) s= 10πr (6) =2+2x
(8) = x4+2x2+ 1 (it can be pointed out that it is a quadratic function about x2).
By studying the concept of quadratic function in design intention theory, students can realize what kind of function quadratic function is in practice and apply theoretical knowledge to practical operation.
(4) Consolidate exercises
1. It is known that the sum of two right angles of a right triangle is 10c.
(1) When the length of a right-angled side is 4.5c, find the area of this right-angled triangle;
(2) Let the area of this right triangle be Sc2, and one of the right angles be xc, so as to find S-off.
Functional relations in x.
The topic of design intention gradually transits from concrete data to expressing relational expressions with letters, which allows students to experience the process from concrete to abstract, thus reducing the learning difficulty of students.
2. It is known that the side length of a cube is xc, the surface area is Sc2 and the volume is Vc3.
(1) Write the functional relationships between S and X and V and X respectively;
(2) Which of these two functions is the quadratic function of x?
For practical problems with simple design intent, it is easy for students to list the functional relationships and distinguish which is the quadratic function. Through the practice of simple topics, let students experience the joy of success, stimulate their interest in learning mathematics and establish confidence in learning mathematics well.
3. Let the height of the cylinder be h(c) as a constant, the radius of the bottom surface be rc, the perimeter of the bottom surface be Cc, and the volume of the cylinder be Vc3.
(1) Write c about r respectively; The functional relationship between v and r;
(2) Are both functions quadratic?
The problem of design intention requires students to memorize the formulas of cylinder volume and bottom circumference, which is equivalent to doing a review here and linking it with what they learned today.
The fence is 30 long, and a rectangular flower bed is enclosed by the fence. Write the functional relationship between the flower bed area (2) and the length x, and point out the range of independent variables.
The design intention is a little more complicated than the previous question, aiming at enabling students to use their brains and think positively, so that students can "jump".
(5) Expansion and extension
1. Known quadratic function = AX2+BX+C, when x=0, = 0; When x= 1, = 2; X= - 1，= 1。 Find a, b, c and write the resolution function.
The design intention is to slightly penetrate the simple problem of finding the quadratic resolution function by the undetermined coefficient method, paving the way for the next class.
2. Determine the value in the following function
(1) If the function = x 2-3+2+x+1is a quadratic function, the value of must be _ _ _ _
(2) If the function = (-3) x 2-3+2+x+ 1 is a quadratic function, the value of must be _ _ _ _
This topic focuses on reviewing the characteristics of quadratic function: the highest degree of independent variable is 2, and the coefficient of quadratic term is not 0.
(6) Summing up thinking:
What did you learn from this course? What else is unclear?
The purpose of the design is to let students talk about the harvest of this class, cultivate students' good habits of self-examination and self-summary, and organize and systematize their knowledge. And from it, we can know what students are not clear about, so as to supplement them in future teaching.
(7) Transfer:
Required questions:
1. The side length of a square is 4. If the side length increases by x, the area increases. Find out the functional relationship about x .. Is this function a quadratic function?
2. Saw a square with a side length of xc at each of the four corners of a rectangular wooden board with a length of 20c and a width of 15c, write the functional relationship between the area (c2) of the remaining wooden board and the side length x(c) of the square, and indicate the range of independent variables.
Choose to do the problem:
1. It is known that the function is a quadratic function, and the value of is obtained.
2. Try to draw the images of quadratic function =x2 and =-x2 in the plane rectangular coordinate system.
In the design intention homework, there are two kinds of topics: required topics and selected topics. Implementing hierarchical teaching to reflect the new curriculum standards. Everyone learns valuable mathematics, and different people get different development. In addition, the fourth question is added to stimulate students' interest in learning quadratic function images.
Five, thinking about teaching design
On the premise of achieving the teaching goal
Based on modern educational theory
With the help of modern information technology
Running through a principle-the principle of taking students as the main body
Highlight a feature-the feature of fully encouraging praise.
Infiltrate a kind of consciousness-the consciousness of applied mathematics
Lecture 2 of the Concept of Function is conducive to improving teachers' theoretical literacy and ability to master teaching materials, and also to improving teachers' language expression ability, so it has attracted the attention of teachers and boarded the elegant hall of educational research. The following is a draft of the concept of function I have compiled, hoping to help everyone!
Hello, examiners, I am today's examinee X, and the topic of my speech today is "the concept of function".
The new curriculum standard points out that mathematics curriculum should be geared to all students, meet the needs of students' personality development, let everyone get a good mathematics education, and let different people get different development in mathematics. Today, I will carry out this concept and start my lecture with teaching material analysis, analysis of learning situation and analysis of teaching process.
First of all, talk about textbooks.
First of all, let me talk about my understanding of the textbook. "The concept of function" is the content of chapter 2. 1 of Beijing Normal University Edition, and the content of this lesson is the concept of function. Function content is a main line of high school mathematics learning, which runs through the whole high school mathematics learning. It is also a bridge to communicate algebra, equations, inequalities, sequences, trigonometric functions, analytic geometry, derivatives and so on, and it is also the basis for further study of advanced mathematics in the future. The process of function learning has experienced intuitive perception, observation and analysis, induction and analogy, abstract generalization and other thinking processes, and students' mathematical thinking ability can be improved through learning.
Second, talk about learning.
Next, talk about the actual situation of students. The new curriculum standard points out that students are the main body of teaching, and it can be said that becoming a teacher who meets the requirements of the new curriculum standard is a compulsory course. At this stage, students already have certain analytical ability and logical reasoning ability. Therefore, it is relatively easy for students to learn this lesson.
Third, talk about teaching objectives.
According to the above analysis of the teaching materials and the grasp of the learning situation, I have formulated the following three-dimensional teaching objectives:
Knowledge and skills
By understanding the concept of function, we can point out the definition domain, corresponding law and value domain of specific functions, and correctly use the "interval" symbol to represent the definition domain and value domain of some functions.
(2) Process and method
Through examples, we further understand that function is an important mathematical model to describe the dependence between variables. On this basis, learn to describe functions with sets and corresponding languages, understand the role of correspondence in describing the concept of functions, and further deepen the mathematical thinking method of sets and corresponding.
Emotions, attitudes and values
Feel the joy of successful independent exploration and stimulate the interest in learning mathematics.
Fourth, talk about the difficulties in teaching.
I think a good math class must highlight the key points and break through the difficulties from the teaching content. The establishment of teaching focus is definitely inseparable from the content of my class. Then, according to the teaching content, it can be determined that the teaching focus of this lesson is: the modeling idea of function and the three elements of function. The teaching difficulties of this course are: the meaning of symbol "y=f(x)", the interval representation of function definition domain and value domain, and the abstraction of function concept from concrete examples.
Verb (abbreviation of verb) oral teaching method and learning method
Modern teaching theory holds that in the teaching process, students are the main body of learning, teachers are the organizers and guides of learning, and all teaching activities must be centered on students' initiative and enthusiasm. According to this teaching idea, combined with the content characteristics of this class, students' psychological characteristics and cognitive rules, I take the problem as the main line and adopt teaching methods such as inspiration, lecture, group cooperation and independent inquiry.
Sixth, talk about the teaching process.
Below I will focus on the teaching process I designed.
(A) the introduction of new courses
The first is the import link. Question: How much do you know about functions? How does junior high school define functions? Can you give me an example? This leads to the theme of this lesson "Function Concept".
Using the concept of junior high school function to import can narrow the distance between students and new knowledge and help students further improve the knowledge system of knowledge framework.
(2) Explore new knowledge
Next is the most important part of exploring new knowledge in teaching. I mainly use the methods of explanation, group cooperation and independent inquiry.
First, show life examples with multimedia.
(1) the relationship between the altitude of the mountain and the temperature;
(2) The relationship between distance and time when the car is driving at a constant speed;
(3) The relationship between boiling point and air pressure.
Guide students to analyze and summarize the above three examples, what are their similarities, and judge whether the relationship between the two variables in each example is a functional relationship according to the concept of function learned in junior high school.
Default: ① There are two non-empty data sets A and B; ② There is a certain correspondence between the two data sets; ③ For each X in number set A, there is a unique and definite Y value corresponding to it in number set B according to a certain correspondence F. ..
Next, through the similarities of the above examples, combined with teaching materials, guide students to think about the concept of inductive function. Organize students to read textbooks, and pay attention to the following questions during reading.
Question 1: What is the concept of function? What are the similarities and differences in the definition of function between junior high school and senior high school? What does the symbol "X" mean?
Question 2: What are the three elements that make up a function?
Question 3: What is the concept of interval? What is the relationship between interval and set? How to express the interval on the number axis?
After ten minutes, organize students to communicate with each other in the class.
Presupposition: the concept of function: given two groups of non-null numbers A and B, if there is a unique number f(x) corresponding to any number X in set A according to a certain correspondence F, then this correspondence F is called a function defined on geometry A, which is denoted as F: A → B, or y=f(x), X ∈ A. At this time, X is called an independent variable, and set A is called a definite function.
The three elements of a function include: definition domain, value domain and corresponding rules.
Interval:
In order to let students understand the essence of function concept more deeply, they should ask questions at this time.
Question 1: What are the similarities and differences between junior high school and senior high school?
It is emphasized that the essence of a function is that there is a definite correspondence between two groups of numbers, and it is one-to-one or many-to-one, not one-to-many.
Question 2: What does the symbol "y=f(x)" mean? Can "y=g(x)" represent a function?
It should be emphasized that the symbol "y=f(x)" is a function symbol and can be represented by any letter. F (x) represents the function value corresponding to X, and a number is not the product of F and X. ..
Question 3: What form can communication F take?
In the process of interpretation, it should be emphasized that the corresponding relation f can be an analytical formula, an image or a table.
Question 4: Can the three elements of a function be missing? Point out what are the three elements in the three examples.
In the process of explanation, it is emphasized that the three elements of the function are indispensable.
Question 5: Use interval to represent the domain and value domain of three examples.
Design intention: In this process, I completely give the class to the students, and the teacher plays the role of organizer and guide. Using the enlightening principle, students can think independently, operate by hand, and discuss with students in the process, which strengthens the communication between students and is conducive to cultivating students' sense of cooperation and inquiry ability.
(3) Classroom exercises
The next step is to consolidate and improve ties.
Organize students to list several examples of functions in their own lives, describe them with definitions, point out the domain and value domain of functions and express them with intervals.
The setting of such questions enables students to further consolidate their knowledge and gradually master it skillfully.
(4) Summarize the homework
At the end of the course, I will ask: What did you get today?
Guide students to review: the concept of function, the three elements of function and the expression of interval.
The homework I designed for this class is as follows:
1. Find the value of the following function.
(1) It is known that f(x)=5x-3 and f (x)=4.
(2) Known
Find g(2).
2. As shown in the figure, the cross section of the irrigation channel is isosceles trapezoid, the bottom width is 2m, the channel depth is 1.8m, and the slope is 45.
(1) By using analytical expressions, the area a of water in the cross section is expressed as a function of water depth h.
(2) Determine the definition domain and value domain of the function
(3) Try to draw the image of the function
This design can make students understand the core of this class and pave the way for the expression of learning function in the next class.