Teaching objectives
1, knowledge and skills
(1) Understand and master the definition range, range, periodicity, (small) value, monotonicity and parity of sine function;
(2) Be able to skillfully use the properties of sine function to solve problems.
2. Process and method
Let students explore the properties of sine function through the image of sine function on R; Explain examples, summarize methods and consolidate exercises.
3. Emotional attitudes and values
Through the study of this section, cultivate students' innovation ability, exploration ability and induction ability; Let students experience the joy of their own exploration success and cultivate self-confidence; Make students realize that transforming "contradiction" is an effective way to solve problems; Cultivate students' scientific attitude of seeking truth from facts and persistent research spirit.
Emphasis and difficulty in teaching
Key point: the properties of sine function.
Difficulties: the application of sine function.
teaching tool
projector
teaching process
Create situations and reveal topics.
Students, mathematics-We have studied functions and mastered several angles to discuss the properties of a function. Do you remember any of them? Last class, we learned the image of sine function y=sinx on R. Let's discuss its properties according to the image.
Explore new knowledge
Let the students look at the projection, carefully observe the image of sine curve and think about the following questions:
What is the domain of (1) sine function?
(2) What is the range of sine function?
(3) What is its maximum value?
(4) How to divide its positive and negative intervals?
(5)? What is the solution set of (x)=0?
Teachers and students come to the conclusion together:
1. domain: the domain of y=sinx is R.
2. Scope: Draw out the sine function line in the unit circle. Conclusion: |sinx|≤ 1 (bounded).
See the sine function line (figure) to verify the above conclusion, so the value range of y=sinx is [- 1, 1].
2. Excellent teaching plan for senior two mathematics.
Teaching purpose:
1. Understand the property theorem and inverse theorem of the vertical line in the line segment, master the relationship between these two theorems, and use these two theorems to solve geometric problems.
2. Understand the trajectory of the median vertical line.
3. Cultivate students' action, image and abstraction in combination with the teaching content.
Teaching focus:
Introduction, proof and application of the property theorem and inverse theorem of line segment.
Teaching difficulties:
The relationship between the property theorem of the median perpendicular of a straight line segment and the inverse theorem.
Teaching focus:
1. All points on the perpendicular bisector are equidistant from both ends of the line segment.
2. All points with equal distance to both ends of a line segment are on the middle vertical line of this line segment.
Teaching AIDS:
Projector and projection film.
Teaching process:
First, ask questions.
What are the property theorems and inverse theorems of 1. angular bisector?
2. How to make the midline of a line segment?
Second, the new lesson
1. Please make the middle vertical line EF of line segment AB in the exercise book (please do it on the blackboard).
2. Take any point P on EF, connect PA and PB, and measure PA=? ，PB=？ What is the relationship between these two values?
Through students' observation and analysis, the result is PA=PB. Try a little p or PA=PB. Guide the students to guess that all points on EF are equal to points A and B, and then ask the students to describe this conclusion as a proposition (shown by slides).
Theorem: A point on the vertical line of a line segment is equal to the distance between the two endpoints of this line segment.
This proposition is obtained by drawing, observing and guessing, and it must be proved to be true in theory before it can be used as a theorem.
As shown in the figure, the straight line EF⊥AB, the vertical foot is C, and AC=CB, and the point P is on EF.
Verification: PA=PB
How to prove PA=PB? Students only need to prove RT δ PCA ≌ RT δ PCB.
Proof: ∵PC⊥AB (known)
∴∠PCA=∠PCB (vertical definition)
In Δ δδPCA and Δ δδPCB.
∴δpca≌δpcb(sas)
That is: PA=PB (the corresponding sides of congruent triangles are equal).
Conversely, if PA=PB, P 1A=P 1B, p, P 1 on what line?
After P 1 crosses p, make a straight line EF, and after AB to c, you can prove Δ pap1≌ PBP1(SSS).
∴EF is the bisector δδPAB of the vertex angle of an isosceles triangle.
∴EF is the middle vertical line of AB (the three-line unity property of isosceles triangle)
∴P, P 1 is on the perpendicular line of AB, so the inverse theorem of the above theorem is obtained (to inspire students to narrate) (to demonstrate with slides).
Inverse theorem: the point where the two endpoints of a line segment are equidistant is on the middle vertical line of this line segment.
According to the above theorem and inverse theorem, we can know that the straight line MN can be regarded as a set of all points from two points A and B to the same distance.
The middle vertical line of a line segment can be regarded as the set of all points with the same distance between the two endpoints of the line segment.
For example, three (slide show)
Example: As shown in figure ABC, the perpendicular lines of AB and BC intersect at point P, which proves that PA=PB=PC.
Prove: ∵ Point P is on the perpendicular of AB line.
∴PA=PB
Similarly, PB=PC
∴PA=PB=PC
From the example PA=PC, we can know that point P is on the perpendicular line of AC, so the perpendicular lines of three sides of a triangle intersect with point P, and the distance from this point to the three vertices is equal.
Four. abstract
The key to the correct application of these two theorems is to distinguish their conditions and conclusions, strengthen the analysis before proof and find out the method of proof. The function of the theorem is to prove that two line segments are equal or the points are on the middle perpendicular of the line segments.
3. Excellent teaching plan for senior two mathematics.
Teaching objectives
1. Master the product of plane vectors and its geometric significance;
2. Master the important properties and operation rules of plane vector product;
3. Understand that the problems of length, angle and verticality can be solved by the product of plane vectors;
4. Master the conditions of vertical vector.
Emphasis and difficulty in teaching
Teaching emphasis: the definition of quantity product of plane vector
Teaching difficulties: the definition of plane vector product, the understanding of operation law and the application of plane vector product.
teaching tool
projector
teaching process
Comment introduction:
The necessary and sufficient conditions for the theorem vector of vector * * * and the line of non-zero vector * * * are that there is only one non-zero real number λ, so that = λ.
Course summary
(1) Let the students review what they have learned in this lesson. What are the main mathematical thinking methods involved?
(2) In the learning process of this class, there are still some places you don't quite understand, please ask the teacher.
How did you do in this class? What was your experience?
Homework after class
P 107 Exercise 2.4A Group 2 and Group 7 Questions
Summary after class
(1) Let the students review what they have learned in this lesson. What are the main mathematical thinking methods involved?
(2) In the learning process of this class, there are still some places you don't quite understand, please ask the teacher.
How did you do in this class? What was your experience?
4. Excellent teaching plan for senior two mathematics.
Textbook analysis
1. Knowledge content and structure analysis
Set theory is an important foundation of modern mathematics. In high school mathematics, the set of preparatory knowledge is closely related to other contents, which is the basis of learning, mastering and using mathematical language. Set theory and its reflected mathematical ideas have been applied in more and more fields. Starting from the set that students are familiar with (natural number set, rational number set, etc.). ), the textbook gives the meaning of elements and sets with examples. Students can abstract and use concrete examples.
2. Analysis of the significance of knowledge learning
Through the learning process of self-inquiry, we can understand the meaning of set, understand the relationship between elements and set, choose appropriate language to describe different specific problems, and feel the significance and function of set language.
3. Teaching suggestions and learning guidance
Because there are many new concepts and symbols in this section, although the content is relatively simple, it is not appropriate to speak too fast. While explaining concepts, students should read more textbooks and communicate with each other. On this basis, they should understand concepts and be familiar with the use of new symbols. We should arouse students' enthusiasm through questioning, independent exploration, cooperation and exchange, and self-summary.
Analysis of learning situation
In junior high school, students have learned some sets or trajectories of points, such as: a point set (circle) whose distance to a fixed point on the plane is equal to a fixed length; The set of points with equal distances to the two endpoints of a line segment (the perpendicular bisector of a line segment). This is helpful for students to learn the knowledge in this lesson, but now what we want to popularize is this "set", which is not only a set of points or a set of figures, but also a "set of designated objects". Set language is the basic language of modern mathematics. The use of this language not only helps to express mathematical content concisely and accurately, but also can be used to describe mathematical content.
Teaching objectives
1. Knowledge and skills
(1) Through autonomous learning, students initially understand the concept of set, the relationship between elements and sets, the certainty, mutual difference and disorder of set elements, and the commonly used number sets and their notation;
(2) Master the common representations of sets-enumeration and description.
2. Process and method
By understanding the meaning of set and the "subordinate" relationship between elements and set through examples, we can choose appropriate languages (such as natural language, graphic language and set language) to describe different specific problems, improve the ability of language transformation and abstract generalization, and establish the consciousness of expressing mathematical content in set language.
3. Modality and value
On the basis of mastering basic concepts, we can solve related problems, gain a sense of achievement in mathematics learning, improve students' ability to analyze and solve problems, and cultivate students' application consciousness.
Important and difficult
1. Teaching emphasis: basic concepts and expression methods of set.
2. Teaching difficulties: choose the appropriate method to correctly represent the set.
Teaching idea
Through examples and several sets that students are familiar with, this paper introduces the concept of set, and then gives the representation method of set. Students can master the content of this lesson through self-experience, self-study and self-summary. The teaching process is arranged according to the links of "asking questions, students discussing, summarizing, acquiring new knowledge and self-testing".
teaching process
Preparation before class:
Make a preview plan for students in advance: a. preview the chapter about set in junior high school mathematics; B preview this section and try to find the connection with the past; Collect examples of using collectibles in life.
Introduction to the new lesson: Students, today we are going to learn collective knowledge. In primary school and junior high school, we have come into contact with some sets, such as the set of natural numbers, the set of rational numbers and the inequality X-7.
Let's play a game in three groups, shall we? Let's compete with each other to answer questions and comment on each other's advantages and disadvantages, shall we? Students should say: OK when they are aroused! )
The process of teaching and learning:
Presupposition question design intention teacher-student activities teacher activities
Students in Group One, Group Two and Group Three, did you get any inspiration from reading the examples (1) to (8) on the second page of the textbook? Put forward a vague question, leaving three groups of students with broader thinking space. Stimulate thinking and interest. The teacher nudged and corrected the wrong answer direction in time. (ideal answer: we have learned a lot about sets. We will give some examples of sets. )
Students are divided into three groups to answer in turn. Can you tell what they have in common? Pave the way for the definition and meaning of set and cultivate students' generalization ability. Guide students to draw correct conclusions. Finally, an accurate definition is given: we call the object of study an element; The whole composed of some elements is called a set (abbreviated as set). Students discuss in groups and take turns to answer. Can you tell the relationship between elements and sets? How to express it? What is the forehead symbol? Through the students' own summary, we can remember the relationship between elements and sets more deeply. The teacher guides the students to get accurate answers. (ideal answer: the set is a whole, the elements are individuals, and the set is made up of elements. Sets are represented by capital letters, such as a; Elements are represented by lowercase letters, for example, a. If A is an element of set A, say A belongs to set A and mark it as A. If A is not an element in set A, say A does not belong to set A and mark it as A.) Students discuss and answer in turn.
You can pick out the mistakes in each other's answers and compete. What methods do we usually use to describe sets? How to express it? Guide students to know two common representations of set. The teacher guides and corrects me. (ideal answer: enumeration: the method of enumerating the elements of a set one by one and enclosing them with curly braces "{}" is called enumeration. Description: The method of representing a set with the * * * same characteristics of the elements contained in the set is called description. The specific method is: write the general symbols and the range of values (or changes) of the elements in this set in curly brackets, then draw a vertical line, and write the * * * same characteristics of the elements in this set after the vertical line. The students went to the blackboard to answer and practice. Who can try to talk about the characteristics of the elements in the set? Expand knowledge, let students have a loving and rational understanding of the characteristics of elements, and develop their inquiry thinking. Teacher's guidance. (ideal answer: once the elements are given, they are definite, definite, different and mutually different, and there is no order and disorder.
That is, (1) certainty: for any element, either it belongs to a specified set or it does not belong to the set, there must be one of them.
(2) Mutuality: the elements in the same set are different from each other.
(3) Disorder: arbitrarily change the arrangement order of elements in a set, and they still represent the same set. ) Students discuss and answer. What do you mean two groups are equal? Deep understanding set. The teacher gave the answer. If the elements that make up two sets are the same, we say that the two sets are equal. ) Students discuss and answer.
5. Excellent teaching plan for senior two mathematics.
1. Preview the textbook and introduce questions.
Preview the textbooks P 54 ~ P 57 according to the following outline and answer the following questions.
(1) How to get samples from the "Inquiry" in the textbook P55?
Tip: put these cookies in an opaque bag, stir them well, then touch them and don't put them back.
(2) What are the most commonly used simple random sampling methods?
Tip: draw lots and random numbers.
(3) What do you think are the advantages and disadvantages of drawing lots?
Tip: The advantage of drawing lots is simple and easy, which is more convenient when there are few individuals in the group, but it is not suitable when there are many individuals in the group.
(4) Which direction can I read when reading by random number method?
Tip: You can read from the left, right, up and down directions.
2. Summary, the core must be remembered.
(1) Simple random sampling: Generally speaking, a group contains n individuals, from which n individuals are selected one by one (n≤N). This sampling method is called simple random sampling if the probability of each individual in the group being drawn is equal every time.
(2) There are two common simple random sampling methods-lottery method and random number method.
(3) Generally speaking, the lottery method is to divide n individuals in the population, write the numbers on the digital labels, put the digital labels in a container, stir them evenly, extract one digital label at a time, and extract it continuously for n times to get a sample with a capacity of n. 。
(4) Random number method is sampling with random number table, random number dice or computer-generated random number.
(5) The advantage of simple random sampling is that it is simple to operate and effective when the total number is small.
[Problem thinking]
(1) In simple random sampling, is the possibility of an individual being drawn related to the number of times?
Tip: In simple random sampling, every individual in the population is equally likely to be drawn every time, no matter how many times it is drawn.
(2) What are the similarities and differences between lottery and random number?
Tip:
similar
① All of them belong to simple random sampling, and the number of individuals in the sampling population is limited;
(2) Extract from the population one by one.
discrepancy
① The lottery method is simpler than the random number method;
② The random number method is more suitable for a large number of individuals in a group, while the lottery method is suitable for a small number of individuals in a group. Therefore, when there are a large number of individuals in the group, the random number method should be chosen, which can save a lot of manpower and cost.