Secondly, talk about learning objectives and requirements.
Through the study of this section, students should master.
(1): coordinate representation of plane vector product.
(2): the distance formula between two points on the plane.
(3): Necessary and sufficient condition of vector ordinate.
And their simple applications, the above three points are also the focus of this lesson. The difficulty of this lesson is the necessary and sufficient condition and flexible application of the ordinate representation of vectors.
Three: Oral Teaching Method
In the teaching process, I mainly adopted the following teaching methods:
(1) heuristic teaching method
Because it is relatively easy to deduce the key coordinate expression in this class, I am going to let students deduce the coordinate expression of two vector quantity products themselves in this class, and then guide students to find several important conclusions: for example, the formula for calculating modulus, the formula for the distance between two points on the plane, and the necessary and sufficient conditions for the vertical vector coordinate expression.
(2) Explanatory teaching methods
Mainly to clarify the concept and relieve students' doubts in concept understanding; When explaining examples, demonstrate the problem-solving process!
The main means of assisting teaching (powerpoint)
(3) discussion method
Mainly through the mutual communication between students, deepen the understanding of difficult problems, improve students' self-study ability and the ability to discover, analyze, solve problems and innovate.
Four: methods of speaking and learning
Students are the main body of the classroom, and all teaching activities should be carried out around students, so as to stimulate students' interest in learning, enhance communication with students in the classroom, and achieve the purpose of finding and solving problems in time. Through intensive lectures and more practice, students' enthusiasm for autonomous learning can be fully mobilized. For example, let the students deduce the coordinate formula of the product of two vectors themselves, and guide the students to deduce four important conclusions! And in specific problems, let students establish the idea of equations and solve problems better!
Five: Talking about the teaching process
I'm going to take this course like this:
First of all, ask the question: what quantities do we need to know to calculate the product of two non-zero vectors?
Continue to ask questions: If you know the coordinates of two non-zero vectors, can you use the coordinates of these two vectors to represent the quantitative product of these two vectors?
Guide students to derive the coordinate expression formula of plane vector product. On the basis of this formula, students can also be guided to draw the following important conclusions:
Calculation formula of (1) module
(2) The distance formula between two points on the plane.
(3) Coordinate representation of cosine of included angle between two vectors.
(4) Necessary and sufficient conditions for vertical scalar representation of two vectors.
The second part is the explanation of examples, through which students can be more familiar with the formula and apply it.
Example 1 is the example 1 on page 22 of this book. This problem is a coordinate formula that directly uses the product of plane vectors. The purpose is to make students familiar with this formula, and on the basis of this problem, find the included angle between these two vectors. The purpose is to familiarize students with the coordinate expression of cosine of the included angle between two vectors. Example 2 is a direct proof of straightness. Although simple, it embodies an important proof method. This method should be mastered by students. This example is actually an application of the necessary and sufficient conditions of the vertical coordinate expressions of two vectors: whether the quantitative product of two vectors is zero is one of the important methods to judge whether the corresponding two straight lines are vertical.
Example 3 is slightly modified on the basis of Example 2. The purpose is to let students use formulas to solve problems and let them have the idea of establishing equations here.
Combined with practice, students can skillfully use formulas and master what they have learned today.
Course of Plane Moving Point Trajectory
First, the teaching objectives
Knowledge and skills
1, and further master the basic method of solving the moving point trajectory equation.
2. Experience the intuition and effectiveness of mathematics experiment, and improve the operation ability of geometry sketchpad.
(2) Process and method
1. Cultivate students' observation ability, abstract generalization ability and innovation ability.
2. Experience the thinking process from perceptual to rational, from image to abstract.
3. Strengthen the methods of analogy and association, and understand the idea of combining equations with numbers and shapes.
Emotions, attitudes and values
1. Feel the dynamic beauty, harmonious beauty and symmetrical beauty of the moving point trajectory.
2. Establish a sense of competition and cooperation, feel the sense of success brought by cooperation and exchange, establish self-confidence, and stimulate the courage to ask questions and solve problems.
Second, the teaching focus and difficulties
Teaching emphasis: using analogy and association to explore the trajectory under different conditions.
Teaching difficulties: the transition between graphics, characters and symbols.
Third, teaching methods and means
The teaching method is a combination of observation and discovery, inspiration and guidance, and cooperative inquiry. Inspire and guide students to actively think and standardize their thinking, help them optimize their thinking process, and on this basis, provide opportunities for students to communicate, help them organize and clarify their thinking, and express their mathematical thinking clearly and accurately.
The teaching method adopts network classroom, four people and one computer, and multimedia teaching method. Through the above teaching methods, on the one hand, the process of knowledge generation is reproduced, and through multimedia dynamic demonstration, the obstacles in the formation of old knowledge and new knowledge (static to dynamic) are broken; On the other hand, it saves time, improves classroom teaching efficiency and stimulates students' interest in learning.
Teaching mode The classroom mode of quality education in key middle schools is "creating situations, stimulating emotions, actively discovering and actively developing".
Fourth, the teaching process
* 1, creating scenes and introducing themes.
In life, we can see the shadow of trajectory curve everywhere.
This is a beautiful night view of the city.
Demonstration Many people think that the trajectory of celestial bodies is conical.
Research shows that the more celestial bodies there are, the more kinds of trajectories there are.
There are also many beautiful trajectory curves in the demonstration building.
Design intention: let students feel that mathematics is around us and feel the trajectory.
The dynamic beauty, harmonious beauty and symmetrical beauty of curves stimulate learning interest.
* 2. Stimulate emotions and guide exploration.
The ladder leaning against the corner slipped. If there is a person standing on the ladder, we can't help thinking, did this person fall straight? Or did you draw a beautiful curve and fly out? Let's turn this problem into a mathematical problem, that is, 20 questions on page 88 of the first volume of the new textbook, that is, the example here is1;
For example 1, the length of the line segment is, and the sum of the two endpoints slides on the axis and the axis respectively, so as to find the trajectory equation of the midpoint of the line segment.
Step 1: Let the students verify the trajectory with the help of the drawing board.
Step 2: Let the students solve the trajectory equation.
Method 1: Set, and then
Youde,
simplify
Method 2: Set and obtain.
simplify
Method 3: Assuming that the distance from a point to a fixed point is equal to a fixed length,
According to the definition of circle;
Step 3: Review the general steps to solve the trajectory equation.
(1) Establish an appropriate coordinate system.
(2) Set the coordinates M(x, y) of the fixed point.
(3) List the constraint conditions p(M) related to the moving point.
(4) Coordinate simplification, f(x, y)=0.
(5) Proof
Among them, the most critical step is to seek equivalence relation and coordinate equivalence relation according to the meaning of the problem.
Design intention: Here, with the help of the animation function of the geometry sketchpad, I let the students intuitively, vividly and dynamically feel that the trajectory of the moving point is a circle, and then let the students solve the trajectory equation. Finally, teachers and students review the general steps of finding trajectory equation together, master literal translation and definition skillfully, and experience the thinking process from perceptual to rational, from image to abstract.
3. Active discovery and development
According to the above example 1, if a person stands in the middle of a ladder, he will make a beautiful arc and fly out. Students will naturally think, what if people don't stand in the middle, but stand at will? Ask students to explore the trajectory when m is not the midpoint.
Step 1: Use the network platform to show the trajectory that students get (teachers consciously integrate together).
Design intention: With the help of mathematical experiments, teachers' doubts and interests can be restored to students, so that students can find their own doubts in the process of practice, and it is easier to stimulate students' enthusiasm for learning and encourage students to learn actively.
Step 2: Break down the action and ask the students three questions:
Question 1: What is the size relationship between BM and MA when the position of M is different?
Question 2. What other common forms reflect the relationship between BM and MA?
Question 3: Can this quantitative relationship be expressed by the ratio of 1?
Step 3: Show the math problems summarized by students.
1, the length of the line segment AB is 2a, the two endpoints B and A slide on the X axis and Y axis respectively, and the point M is the point on AB. If satisfied, find the trajectory equation of point M. ..
2. The length of the line segment AB is 2a, and the two endpoints B and A slide on the X axis and Y axis respectively, and the point M is the point on AB. If it is satisfied, the trajectory equation of point m can be obtained.
3. The length of the line segment AB is 2a, and the two endpoints B and A slide on the X axis and Y axis respectively, and the point M is the point on AB. If so, find the trajectory equation of point m ... (explain what trajectory it is)
Step 4: Finish the questions summarized by students in class 1, and finish questions 2 and 3 after class.
4. Cooperate to explore and innovate.
To change the moving mode of point A and point B, we should also consider the trajectory of the midpoint, and the teacher should give appropriate guidance (here point A is fixed and point B is moving).
Students mainly list the following kinds of movements: circle, ellipse, hyperbola and parabola, and get some corresponding motion trajectories.
5. Arrange homework to achieve expansion.
1, describe the trajectory figure obtained by the above students with words and symbols (imitation example 1), and find the trajectory equation.
2. Given a (4 4,0), point B is a moving point on a circle, and the vertical line in AB intersects with straight line OB at point P, so as to find the trajectory equation of point P. ..
3. Given a (2 2,0), point B is a moving point on a circle, and the vertical line in AB intersects with straight line OB at point P, so as to find the trajectory equation of point P. ..
If the vertical line in the above question is changed to the point P where the general vertical line intersects with the straight line OB, please verify the trajectory of the point P with the sketchpad.
The following are some trajectories that students get after class.
After class, a classmate asked, what happens if the X axis and the Y axis are not perpendicular? How to make a fixed-length segment slide on it?
It can be said that I haven't thought about these problems of students before, which has greatly touched me, and at the same time prompted me to further study the geometric sketchpad and improve my ability. Here, I realize that teachers are no longer just candles, but more like lights, illuminating others and myself at the same time.
The following is the trajectory diagram when the X axis and the Y axis are not perpendicular.
Verb (abbreviation for verb) explains design description:
(A), teaching materials
Plane moving point trajectory is an inquiry course in senior two, and the problem of trajectory has a profound life background. Finding the trajectory equation of a plane moving point involves basic knowledge such as set, equation, triangle and plane geometry. , which is permeated with the idea of movement and change, equations and the combination of numbers and shapes. It is an important content of middle school mathematics, and it is also one of the key points of mathematics examination in college entrance examination over the years.
(2), school situation, learning situation
School situation: Our school is a provincial-level standard school and a provincial-level demonstration high school with relatively complete hardware facilities.
Yes, every classroom has the function of multimedia teaching. In addition, there are two online classrooms and a student electronic classroom.
Reading room, and you can surf the internet at any time.
Learning situation: Most students have computers at home and can surf the Internet at any time. The students got a foundation of geometric sketchpad.
Through the training of this operation, students can quickly draw basic conic curves such as circle, ellipse, hyperbola and parabola.
Line. Students have a certain grasp of the basic methods of solving trajectory equations, but they are not familiar with words, figures and symbols.
There are still great differences in the conversion among the three languages, and the development of cooperation and communication consciousness is unbalanced.
Need to be strengthened.
(3) study law
Observation, experiment, communication, cooperation, analogy, association, induction and summary
(4) Teaching process
1. Create a scene and introduce a theme.
2. Stimulate emotions and guide exploration.
Mathematical problems are abstractly summarized from ladder sliding problems.
Step 1: Let the students verify the trajectory with the help of the drawing board.
Step 2: Let the students solve the trajectory equation.
Step 3: Review the general steps to solve the trajectory equation.
3. Active discovery and development
Explore the trajectory when m is not the midpoint.
Step 1: Use the network platform to show the trajectory obtained by students.
Step 2: Break down the action and ask the students three questions:
Step 3: Show the math problems summarized by students.
4. Cooperate to explore and innovate.
To change the moving mode of point A and point B, we should also consider the trajectory of the midpoint, and the teacher should give appropriate guidance (here point A is fixed and point B is moving).
Students mainly list the following kinds of movements: circle, ellipse, hyperbola and parabola, and get some corresponding motion trajectories.
5. Arrange homework to achieve expansion.
(5) Teaching features:
With the help of network and multimedia teaching platform, students can do experiments by themselves, find and solve problems, and at the same time show their learning situation in time, so that everyone can study together and evaluate the effect together. At the same time, it saves time and improves classroom efficiency.
The whole teaching process embodies four unifications: the unity of learning book knowledge and engaging in practice, the unity of book learning and modern information technology learning, the unity of book knowledge and resource expansion, and the unity of classroom learning and extracurricular practice.
The students in this class are energetic, interested and cooperative. They maintain a good interaction with me, and there will be some disputes from time to time. They asked me some new questions, which reflected my shortcomings and promoted my progress and improvement. Teaching and learning between teachers and students is like a mirror, reflecting each other and making progress together.
inverse function
Teaching objectives:
1. Understand the concept of inverse function, and make clear the relationship between the domain and value of original function and inverse function.
2. Can find the inverse function of some simple functions.
3. In the process of trying and exploring the inverse function, deepen the understanding of the concept, sum up the general steps of finding the inverse function, and deepen the understanding of mathematical thinking methods such as function and equation, combination of numbers and shapes, and from special to general.
4. Further improve the profundity of students' thinking, cultivate students' ability of reverse thinking, analyze problems from a dialectical point of view, and cultivate their ability of abstract generalization.
Teaching emphasis: the method of finding inverse function.
Teaching difficulty: the concept of inverse function.
Teaching process:
teaching activities
Design intent 1. Create situations and introduce new lessons.
1. Review the questions
① the concept of function
② the meaning of each variable in y = f (x)
2. In physics class, students learned the functional relationship between displacement and time of uniform linear motion, that is, S=vt, t= (where velocity V is constant), and in S=vt, displacement S is a function of time t; In t=, time t is a function of displacement s, in this case, we say that t= is the inverse function of function S=vt. What is the inverse function and how to find it is the content of this lesson.
Step 3 write on the blackboard
The introduction of new courses from practical problems has stimulated students' interest in learning and embodied the teaching objectives. This can not only dispel the mystery of the concept of "inverse function", but also let students know the necessity of learning this concept.
Second, case analysis, organizational inquiry
1. problem group 1:
(Give the sum of functions by projection; Images with ())
(1) What is the relationship between the images of these two groups of functions? What is the relationship between these two sets of functions? (answer: the image of sum is symmetrical about the straight line y=x; The image of AND () is also symmetrical about the straight line y = x, which is an operation to find the number cube and the root of the number cube. They are mutually beneficial operations. Similarly, sum () is a reciprocal operation. )
(2) From this, can we find that Y is X?
(3) Is it a function? What does it matter?
(4) What is the connection?
2. The second set of questions:
(1) Is the function y=2x 1(x is the independent variable) and the function x=2y 1(y is the independent variable) the same function?
(2) Is the function (x is the independent variable) the same as the function x=2y 1(y is the independent variable)?
(3) What is the relationship between the domain () of the function and the domain () of the function?
3. The concept of infiltration inverse function.
The teacher points out that such functions are reciprocal functions, and then the teacher and students explore their characteristics together. )
Starting from the functions that students are familiar with, the concept of inverse function is abstracted, which conforms to students' cognitive characteristics and is conducive to cultivating students' abstract generalization ability.
Through these two groups of questions, we pave the way for the derivation of the concept of inverse function, use old knowledge to deduce new knowledge, and design questions in the "zone of recent development" to make students have an intuitive and rough impression of inverse function and lay the foundation for the further abstraction of the concept of inverse function.
Third, teacher-student interaction, inductive definition
1. (According to the above example, both teachers and students have summed up the definition of inverse function. )
In the function y=f(x)(x∈A), let its range be C. According to the relationship between X and Y in this function, we use Y to represent X and get x = j (y). If any value of y in c corresponds to it through x = j (y), then x is a function of independent variable Y. Such a function x = j (y)(y ∈C) is called the inverse function of function y=f(x)(x∈A). It reads like this: Considering the habit of "using x to represent independent variables and y to represent functions", the x and y in are reversed.
2. Guide analysis:
1) The inverse function is also a function;
2) The corresponding rule is reciprocal operation;
3) "If" in the definition means that any function y=f(x) does not necessarily have an inverse function;
4) The definition domain and value domain of function y=f(x) are the definition domain and value domain of function x=f(y) respectively;
5) Functions y=f(x) and x=f(y) are reciprocal functions;
6) understand the symbol f;
7) Reasons for exchanging variables x and y 。
3. Transform the correspondence between x and y twice.
(The independent variable X in the original function is equivalent to the function value Y in the inverse function, and the function value Y in the original function is equivalent to the independent variable X in the inverse function. )
4. The relationship between function and its inverse function
Function y=f(x)
function
Domain of definition
A
C
range
C
A
Fourth, the application of problem solving, summing up the steps
1. (projection example)
Example 1 Find the inverse function of the following function
( 1)y = 3x- 1(2)y = x 1
Example 2 Find the inverse function of a function.
After the teacher writes the examples on the blackboard, the students sum up the steps to find the inverse function. )
2. Summarize the steps of finding the inverse function of the function:
X = f (y) is obtained from y=f(x) at 1
Exchange of x and y in 2 x = f(y).
3 Write the domain of the inverse function.
(Abbreviated as: inverse solution, exchange, writing the domain of inverse function) Is there an inverse function in Example 3( 1)?
The inverse function of (2) is _ _ _ _ _.
(3)(x & lt； 0) is _ _ _ _ _ _ _.
On the basis of the above exploration, the definition of inverse function is revealed, so that students can have a targeted understanding of the characteristics of the definition, and then have a deeper understanding of the definition, which conflicts with their own presupposition and experiences the inverse function. In the process of analyzing and defining, students can experience functions and equations, learn general and special mathematical ideas, and have a better grasp of the symbolic language of mathematics.
Through animation demonstration and table comparison, students' definition of inverse function is raised from perceptual knowledge to rational knowledge, so as to digest and understand.
Through the explanation and analysis of specific examples, it plays an exemplary role in the steps and methods of solving problems for students, and summarizes them in time to cultivate students' habit of analyzing and thinking and ability of summarizing.
The design of the topic follows the requirements of different levels, from cognition to understanding, from mastery to application, step by step, which reflects the reflective understanding of the definition. Students think and practice, and teachers and students analyze and correct together.
V. Consolidation and strengthening, evaluation and feedback
1. Given that the function y=f(x) has an inverse function, find its inverse function y =f( x).
(1)y=-2x 3(xR) (2)y=-(xR and x)
(3) y=(xR, and x)
2. Given the function f(x)=(xR) and the inverse function of x (x), find the value of f(7).
Fifth, reflect on the summary and ask questions again.
This lesson mainly studies the definition of inverse function and the steps to solve inverse function. What are the characteristics of images with two functions that are inverse functions? Why is there such a feature? We will study it in the next section.
Let the students talk about the learning experience of this class, and the teacher will give guidance in due course. )
Further strengthen the concept of inverse function and get the inverse function correctly. Feedback students' mastery of knowledge and evaluate students' implementation of learning objectives. In practice, students can take various forms such as class games and group competitions to arouse their enthusiasm. "The problem is the core of mathematics." Students come into the classroom with questions and come out with new questions.
Sixth, homework
Exercise 2.4 Number 1, Number 2
Further consolidate what you have learned.
Instruction design description
The problem is the heart of mathematics. The formation of a concept is spiraling, and it generally goes through the process from concrete to abstract, from perceptual to rational. This teaching plan introduces the inverse function through a concrete example in physics, and then further induces and analyzes the images of several functions, and finally forms the concept.
The concept of inverse function is a difficult point in teaching, because it is abstract. After two substitutions, it uses abstract symbols. Without the support of concepts such as one-to-one mapping and inverse mapping, it is difficult for students to master the concept of inverse function in essence. Therefore, we boldly use teaching materials to reveal the image relationship between two functions that are inverse functions in advance, and then explore the reasons and find the law. The program starts from the problem, studies the essence, and then draws the concept. This is the order of mathematical research, which conforms to the cognitive law of students and is helpful to the establishment and formation of concepts. In addition, the analysis of concepts and the distribution of exercises are also very accurate, which can meet the multi-level needs of students through different levels of questions and play the role of evaluation and feedback. Through functional equation analysis, reciprocal inquiry, animation demonstration, table comparison, student discussion and other forms of teaching links, students' desire for exploration is fully mobilized. In the process of exploration and analysis,