Teaching plan of trigonometric function in senior high school mathematics: trigonometric function from any angle I. Teaching objectives
1. Master the definitions of sine, cosine and tangent functions at any angle (including the judgment of domain and sign); Understand the definitions of cotangent, secant and cotangent function at any angle.
2. Experience the emergence and development of the concept of trigonometric function, understand the tool function of rectangular coordinate system, and enrich the experience of combining numbers with shapes.
3. Cultivate students' materialistic epistemological viewpoint of seeing the essence through phenomena, and penetrate the dialectical materialistic worldview of mutual connection and transformation of things.
4. Cultivate students' scientific attitude of seeking truth from facts.
Second, the key points, difficulties and key points
Emphasis: Definition, domain and symbol judgment of sine, cosine and tangent functions at any angle.
Difficulties: Understand trigonometric function as a function with real numbers as independent variables.
Key points: how to establish rectangular coordinate system; Six-to-six certainty (? Determine, the ratio is also determined) and dependence (ratio and? Change with the change of).
Third, teaching ideas and methods
In teaching, we should pay attention to dealing with traditional textbooks with the new curriculum concept. Students should not only accept, memorize, imitate and practice mathematics learning activities, but also explore, practice, cooperate and communicate independently, read independently and interact with teachers and students. Teachers should play the roles of organizers, guides and collaborators, guide students to participate, reveal the essence and experience the process.
According to the content of this class, the cognitive characteristics of senior one students and my own teaching style, this class adopts? Enlighten exploration, combine teaching with practice? Methods of organizing teaching.
Fourth, the teaching process
[Counseling clues:
Recognized in retrospect: the concept of function, the definition of acute triangle function (the relationship between the angles of acute triangle) Question situation: Can it be extended to any angle? Stones from Other Mountains: Establishing Cartesian Coordinate System (Why? ) Optimize cognition: learn the acute trigonometric function in rectangular coordinate system. Exploration and development: study six ratios of any angle (relationship with angle: certainty and dependence, meet the definition of function? ) customization: definition of trigonometric function at any angle: examples and exercises for element analysis of trigonometric function (determination of corresponding law, definition field, range and positive and negative symbols) review and summary homework]
(A review of the introduction, memories, and recognition.
Cut to the chase and face all the students' questions:
In junior high school, we learned the trigonometric function of acute angle. In the first few lessons, we extended the acute angle to any angle and studied the angle system and arc system. What should we learn in this course?
Explore trigonometric functions from any angle (blackboard writing topic), please recall and explain clearly:
(Scenario 1) What is a function? Or how is a function defined?
Let the students recall and say their names. The definition of projection display specification is corrected and emphasized by the teacher according to the answer:
Traditional definition: suppose there are two variables x and y in a changing process. If for each value of x, y has a unique and definite value corresponding to it, then it is said that y is a function of x, x is called an independent variable, and the range of the independent variable x is called the domain of the function.
Modern definition: Let A and B be non-empty number sets. If any number in set A has a unique number f(x) corresponding to it according to some corresponding relation f, then it is called mapping? :A？ B is a function from set A to set B, written as: y= f(x), x? A, where x is called the independent variable, and the range of the independent variable x is called the domain of the function.
Teaching plan of trigonometric function in senior high school: inductive formula of trigonometric function 1 teaching goal
1. Knowledge and skills
(1) The inductive formula of trigonometric function can be derived from the definitions of trigonometric function and trigonometric function line in the unit circle.
(2) Using inductive formula, the simplified evaluation problem of trigonometric function with arbitrary angle can be transformed into the simplified evaluation problem of trigonometric function with acute angle.
2. Process and method
(1) experienced the process of exploring the quantitative relationship intuitively from geometry, and cultivated students' mathematical discovery ability and generalization ability.
(2) Through the exploration and application of inductive formula, cultivate the ability of transformation and improve students' ability to analyze and solve problems.
3. Emotions, attitudes and values
(1) Instruct students through video, cultivate students' self-study ability and give full play to their initiative.
(2) In the process of exploring inductive formulas, cooperative learning is used to cultivate students' inquiry ability and research spirit.
2 Key points and difficulties
Teaching emphasis: exploration? The inductive formula of -a? What are the inductive formulas of +a and -a? On the basis of the discovery process of -a inductive formula, teachers guide students to deduce.
Teaching difficulties: the geometric relationship between +a, -a and the terminal position of angle A, and the coordinate relationship (intersection with the unit circle) caused by the terminal position relationship, and the inductive formula is deduced by using the definition of trigonometric function of arbitrary angle. Research roadmap? .
3 teaching means and methods
Video-guided learning, problem-based teaching method, cooperative learning method, combined with multimedia courseware
4 teaching process 4. 1 first class teaching activities 1 topic introduction
The concept of angle extends from acute angle to arbitrary angle, so the definition method of acute angle trigonometric function defined in junior high school is introduced into the definition method of arbitrary angle trigonometric function, so that students can understand that the thinking structure of today's lesson is: the problem of arbitrary angle trigonometric function is transformed into the coordinate problem of the research point, and the coordinate of the point is determined by the position of the terminal edge, so that students can deduce the inductive formula? Research roadmap? Create conditions.
Looking back on the formula 1, it is emphasized that its function is to transform the evaluation problem of trigonometric function of arbitrary angle into 0? ~360? Evaluation of trigonometric functions to determine the scope of the whole class is 0? ~360? Problems related to trigonometric functions of angles.
Then solve the problems in the video: (discussion for 3 minutes, random roll call, feedback)
sin390？ ，sin480？
sin600？ , sin (-30? )
Use multimedia to demonstrate the use of video? Symmetry? Methods to solve the trigonometric function value and deduce 0? ~360? Trigonometric function value table of special angle.
Activity 2 Derivation of Activity Formula 4
Use the above introduction to discuss a and? -One? +a，2？ -a's terminal relationship.
Explain a and a again according to the content in the video. The terminal relation of -a, Question: How to express an angle which is symmetrical with the terminal edge of angle A and symmetrical with Y axis about the origin? (Communicate with each other, and the team leader will collect questions from the team members)
Answer relevant questions and show symmetry to the media.
For the derivation of Formula 2 in the video, (play the clip again and show the chart on ppt) ask the students about their self-study, and the group leader will organize the students to derive Formula 2 and Formula 3.
Activity 3 requires students to participate in the self-discussion of Formula 2 and Formula 3.
Let the students prove it themselves. It is best to use charts, guided by the group leader, so that the group can reach a * * * understanding and focus on reflecting the problem (students draw tables on the blackboard during discussion) (5 minutes)
Call the group leader, report the discussion and show the discussion results.
Show inductive formula with ppt, and emphasize the road map of learning inductive formula of trigonometric function: the relationship between angles? Symmetrical relationship? Coordinate relationship? The relationship between trigonometric function values.
Ready to supplement the explanation is:
(1) For 2? Understanding of -a and -a trigonometric functions;
(2) The scope of application of A in the formula is not only applicable to acute angles, but we often need to convert it into acute angles when solving;
③ The function of inductive formula is extended from the angle of terminal symmetry.
Activity 4 Practice Simple Application
Example 1. Use the formula to find the following trigonometric function values.
(Examples of textbooks are omitted)
Students discuss with each other, and * * * report the learning situation with the team leader who has completed (5 minutes).
Designed to solve the problem of evaluating sin330 in video? Tell the students that the formula is flexible in use. In fact, there is no specific order, but we can summarize a general step with the idea of classification.
Supplementary exercise: sin(-240? ) (3 minutes)
Activity 5 lecture summary
Open summary
In knowledge, I learned four groups of inductive formulas; At the level of thinking method: inductive formula embodies the idea of transforming from unknown to known; The inductive formula reveals the relationship between two trigonometric functions, and the terminal edges of these two trigonometric functions have some symmetry. It mainly embodies the mathematical thought of combination of reduction and number and shape.
In retrospect, among your team members, which students do you think performed better and which students need more efforts? What do they mainly need to improve after class? (5 minutes)
Activity 6 Job Level Job
1, read the textbook and experience the thinking method in the process of deducing the inductive formula of trigonometric function;
2. Required Textbook 23 pages 13
3, choose to do the problem
(1) Can it be deduced from any two sets of formulas 2, 3 and 4 to another set of formulas?
(2) Angle? And horns? What is the special positional relationship between the terminal edges of? Can you explore the relationship between their trigonometric functions?
The inductive formula of 1.3 trigonometric function
Class design class record
The inductive formula of 1.3 trigonometric function
1 first class teaching activities 1 project introduction
The concept of angle extends from acute angle to arbitrary angle, so the definition method of acute angle trigonometric function defined in junior high school is introduced into the definition method of arbitrary angle trigonometric function, so that students can understand that the thinking structure of today's lesson is: the problem of arbitrary angle trigonometric function is transformed into the coordinate problem of the research point, and the coordinate of the point is determined by the position of the terminal edge, so that students can deduce the inductive formula? Research roadmap? Create conditions.
Looking back on the formula 1, it is emphasized that its function is to transform the evaluation problem of trigonometric function of arbitrary angle into 0? ~360? Evaluation of trigonometric functions to determine the scope of the whole class is 0? ~360? Problems related to trigonometric functions of angles.
Then solve the problems in the video: (discussion for 3 minutes, random roll call, feedback)
sin390？ ，sin480？
sin600？ , sin (-30? )
Use multimedia to demonstrate the use of video? Symmetry? Methods to solve the trigonometric function value and deduce 0? ~360? Trigonometric function value table of special angle.
Activity 2 Derivation of Activity Formula 4
Use the above introduction to discuss a and? -One? +a，2？ -a's terminal relationship.
Explain a and a again according to the content in the video. The terminal relation of -a, Question: How to express an angle which is symmetrical with the terminal edge of angle A and symmetrical with Y axis about the origin? (Communicate with each other, and the team leader will collect questions from the team members)
Answer relevant questions and show symmetry to the media.
For the derivation of Formula 2 in the video, (play the clip again and show the chart on ppt) ask the students about their self-study, and the group leader will organize the students to derive Formula 2 and Formula 3.
Activity 3 requires students to participate in the self-discussion of Formula 2 and Formula 3.
Let the students prove it themselves. It is best to use charts, guided by the group leader, so that the group can reach a * * * understanding and focus on reflecting the problem (students draw tables on the blackboard during discussion) (5 minutes)
Call the group leader, report the discussion and show the discussion results.
Show inductive formula with ppt, and emphasize the road map of learning inductive formula of trigonometric function: the relationship between angles? Symmetrical relationship? Coordinate relationship? The relationship between trigonometric function values.
Ready to supplement the explanation is:
(1) For 2? Understanding of -a and -a trigonometric functions;
(2) The scope of application of A in the formula is not only applicable to acute angles, but we often need to convert it into acute angles when solving;
③ The function of inductive formula is extended from the angle of terminal symmetry.
Activity 4 Practice Simple Application
Example 1. Use the formula to find the following trigonometric function values.
(Examples of textbooks are omitted)
Students discuss with each other, and * * * report the learning situation with the team leader who has completed (5 minutes).
Designed to solve the problem of evaluating sin330 in video? Tell the students that the formula is flexible in use. In fact, there is no specific order, but we can summarize a general step with the idea of classification.
Supplementary exercise: sin(-240? ) (3 minutes)
Activity 5 lecture summary
Open summary
In knowledge, I learned four groups of inductive formulas; At the level of thinking method: inductive formula embodies the idea of transforming from unknown to known; The inductive formula reveals the relationship between two trigonometric functions, and the terminal edges of these two trigonometric functions have some symmetry. It mainly embodies the mathematical thought of combination of reduction and number and shape.
In retrospect, among your team members, which students do you think performed better and which students need more efforts? What do they mainly need to improve after class? (5 minutes)
Activity 6 Job Level Job
1, read the textbook and experience the thinking method in the process of deducing the inductive formula of trigonometric function;
2. Required Textbook 23 pages 13
3, choose to do the problem
(1) Can it be deduced from any two sets of formulas 2, 3 and 4 to another set of formulas?
(2) Angle? And horns? What is the special positional relationship between the terminal edges of? Can you explore the relationship between their trigonometric functions?
Teaching plan of trigonometric function in high school mathematics: images and properties of trigonometric function I. Analysis of teaching content
This thematic unit is divided into three parts. The first part reviews trigonometric formulas, the second part reviews the images and properties of trigonometric functions, and the third part reviews sine and cosine theorems. This lesson is the second part? Close the door? Students are expected to develop their knowledge and ability in a spiral way. Therefore, the focus of this lesson is the perfect combination and flexible application of the properties of images and trigonometric functions. The difficulty lies in the process of knowledge transformation and flexible application, and the improvement of students' comprehensive ability to solve problems by using knowledge.
Second, the proposition trend
In recent years, the college entrance examination has lowered the requirements for trigonometric transformation, but strengthened the examination on the images and properties of trigonometric functions, because the properties of trigonometric functions are an important part of learning functions, the basis of learning advanced mathematics and applied technology, and a tool to solve practical problems in production, so the properties of trigonometric functions are the focus of this unit review. In review, we should make full use of the idea of combining numbers with shapes and combine images with nature. It is not only helpful to master the image and nature of the function, but also to skillfully use the thinking method of combining numbers with shapes to obtain the properties of the function and describe the image of the function by using the properties of the function.
Third, the design concept and thought
The core idea of flipping the classroom is to let? Knowledge transfer takes place outside the classroom, and knowledge internalization takes place in the classroom? So we need to rebuild the learning process. Information transmission? By students before class. Teachers not only provide videos, but also provide online tutoring. ? Absorption and internalization? It is done through interaction in class. Teachers can know students' learning difficulties in advance and give effective guidance in class. The mutual communication between students is more conducive to promoting the absorption and internalization of students' knowledge. Compared with traditional ideas, the roles of classroom and teachers have changed. Teachers are more responsible for understanding students' problems, guiding students to use knowledge, giving play to the role of organizers, guides and collaborators, guiding students to participate, revealing the essence and experiencing the process.
Fourthly, the analysis of students' learning situation.
In recent years, the admission score of Qingdao No.2 Middle School has been significantly improved. Run a school that students need for development? ,? Every student is a good student? Under the guidance of advanced educational concepts, students' comprehensive ability has been continuously improved. This year's students are the second graduating class since the establishment of No.2 Middle School Branch. Class 3.2 of Senior Three is a new class I took over after the placement of Senior Two, and the overall level of the class has improved rapidly.
Verb (abbreviation of verb) teaching goal
1. Organize the images and properties of sine, cosine and tangent functions through video before class.
2. Be able to use the image and nature of trigonometric function to design and solve problems flexibly, further understand the idea of combining numbers with shapes, and improve the flexibility of students' thinking.
3. Through independent thinking and the analysis of small lecturers, improve students' initiative and participation in learning, and improve the ability of cooperative inquiry.
Sixth, the teaching process.
Video before class:
1. Play the Apple version of trigonometric functions created by Lv Liang and Liu Yujia, and review the images and basic properties of trigonometric functions.
【 Design Intention 】 Use familiar pop songs to arouse students' learning enthusiasm.
2. Images and properties of self-combing trigonometric functions.
Function y=sin xy=cos xy=tan x
Images over a period of time
Domain of definition
range
odevity
periodism
Symmetry symmetry center:
Symmetry axis: symmetry center:
Symmetry axis: symmetry center:
Symmetry axis:
Monotonicity is increased in _ _ _ _ _ _ _ _. When x = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ takes the minimum value-1. X = _ _ _ _ _ _ _ _ _ _ _ _ and y takes the maximum value of 1; When x = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ minimum value of y is-1.
【 Design Intention 】 In the form of tables, students can independently consolidate the basic knowledge of the three basic elementary functions, build a performance platform for small lecturers in the classroom, and lay a solid foundation for the achievement of goal 2 of this class.
(3) The symmetric center of the function is.
(4) Shift the image of the function to the left by one unit, and then shorten the abscissa of each point on the obtained image to the original multiple, and keep the ordinate unchanged to obtain the image of the function, then the monotonic increasing interval of the function is.