Lecture Notes on Sine Theorem 1 I. teaching material analysis
Sine theorem is the first section of the first chapter of the textbook "Solving Triangle" published by People's Education Press, and it is also an important content in triangle theory, which is closely related to the basic relationship between the sides and angles of triangle learned in junior high school. Before this, the students have learned sine function and cosine function, and they have enough knowledge reserves. It is the theoretical basis for solving triangles in subsequent courses and a tool for solving many measurement problems in real life. Therefore, mastering the sine theorem can lay a solid foundation for learning to solve triangles and can be used flexibly in practical applications.
Second, the teaching objectives
According to the analysis of the contents of the above textbooks, taking into account the psychological characteristics of students' existing cognitive structure and their original knowledge level, the following teaching objectives are formulated:
Knowledge goal: understand and master the proof of sine theorem, and use sine theorem to solve triangles.
Ability goal: explore the proof process of sine theorem, draw conclusions through induction, and master a variety of proof methods.
Emotional goal: Through the deduction of sine theorem, let students feel the beauty of neat symmetry of mathematical formulas and the practical application value of mathematics.
Third, teaching focuses on difficulties.
Teaching emphasis: the content of sine theorem, the proof of sine theorem and its basic application.
Teaching difficulties: the exploration and proof of sine theorem, judging the number of solutions when the diagonal solutions of two sides and one of them are known.
Fourth, the analysis of teaching methods
According to the characteristics of the content of this course, students' cognitive rules, the knowledge of this course follows the guiding ideology of taking teachers as the leading factor and students as the main body, adopts the teaching method of exploring together with students, puts forward the generative mode of teaching, takes problems as the reference object, and stimulates students' curiosity and thirst for knowledge in learning mathematics, thus making students' thinking gradually deepen from the beginning of problems to the conclusion of guesses, the exploration of guesses and the derivation of theorems, and strengthening their mastery of the content with examples and exercises. That is to guide students to master the thinking method of "observation-conjecture-proof-application". Students' independent, cooperative and inquiry learning methods can make them actively participate in mathematics learning activities and cultivate their sense of cooperation and inquiry spirit.
Teaching process of verbs (abbreviation of verb)
The knowledge teaching in this section adopts the generative mode:
1, problem situation
There is a tourist attraction. In order to attract more tourists, I want to build a sightseeing cableway between two adjacent mountains in the scenic spot. It is known that the distance from the top of the mountain A to the bottom of the mountain C is 65,438+0,500 m, the included angle between the two peaks measured at the foot of the mountain is 450, and the included angle between the foot of the mountain and the top of the mountain A measured at the other peak B is 500m. How long does it take to build a cableway?
Problem mathematics can be symbolized and abstracted into mathematical graphics. That is, AC= 1500m, ∠C=450 and ∠B=300 are known. Ask AB=?
This problem can be solved indirectly by using the height of the auxiliary line BC.
Question: Is there a direct one-step solution based on the data provided?
Thinking: We know that in any triangle, there is an angle relationship between a big side and a big corner, and between a small side and a small corner. Then can we get an accurate quantitative representation of the relationship between edges and angles?
2. Inductive proposition
We discuss the quantitative relationship between sides and angles from a special right triangle: in the triangle ABC as shown in Figure Rt, according to the definition of sine function.
Lecture Notes on Sine Theorem 2 I. teaching material analysis
Solving triangle is not only the basic content of high school mathematics, but also has strong application. In this curriculum reform, it was preserved and became an independent chapter. This part of the content should belong to the chapter of trigonometric function in the knowledge system, and it can also belong to an aspect of vector application in the research method. In a sense, this part is one of the typical contents of solving geometric problems by algebraic method. This course "Sine Theorem", as the first lesson of the unit, is based on the students' existing knowledge of trigonometric functions and vectors, and through the quantitative exploration of the relationship between the angles of a triangle, the sine theorem (an important tool for solving triangles) is discovered and mastered. Through this part of the study, students can experience "observation-guess-proof" in the process of abstracting from "practical problems" to "mathematical problems". At the same time, in the process of solving problems, feel the power of mathematics, and further cultivate students' interest in learning mathematics and their awareness of "using mathematics".
Second, the analysis of learning situation
The school where I teach is a rural middle school in our county. Most students have a weak foundation, and their application awareness and skills of "some important mathematical ideas and methods" are not high. But most students have a high interest in mathematics, and prefer mathematics, especially the content that is closely related to real life like this course. I believe that all the students can actively cooperate and have good performance.
Third, the teaching objectives
1, knowledge and skills: in the created problem situation, guide students to discover the content of sine theorem, deduce sine theorem and simply use sine theorem to solve some simple triangle problems.
Process and method: Students participate in the exploration of problem-solving schemes and try to find the optimal solution by observing, guessing, proving and applying thinking methods, thus causing students to think about some mathematical models in the real world.
Emotion, attitude and values: cultivate students' mathematical thinking methods to explore mathematical laws reasonably, and reflect the universal connection and dialectical unity between things through the connection between plane geometry, trigonometric function, sine theorem and vector quantity product. At the same time, through the discussion and solution of practical problems, students can experience the sense of achievement in learning, enhance their interest and initiative in mathematics learning, and exercise the spirit of inquiry. Establish the concept of "Mathematics is related to me, mathematics is useful, I want to use mathematics, and I can also use mathematics".
2. Teaching emphases and difficulties
Teaching emphasis: the discovery and proof of sine theorem; Simple application of sine theorem.
Teaching difficulty: proof and application of sine theorem.
Fourth, teaching methods and means.
In order to better achieve the above teaching objectives and promote the change of learning methods, I am going to adopt the "problem-based teaching method" in this class, that is, teachers organize teaching with problems as the main line, use multimedia and physical projectors to stimulate interest, highlight key points, break through difficulties, improve classroom efficiency, and guide students to participate in the process of problem solving through the combination of independent inquiry and mutual cooperation, experience success and failure, and gradually establish a perfect cognitive structure.
Teaching process of verbs (abbreviation of verb)
In order to complete the teaching goal I set, solve the key points smoothly, break through the difficulties, and at the same time, based on the principle of being close to life, students and the times, I designed such a teaching process:
(A) create a scene to reveal the theme
Question 1: It's a quiet night, the moon is high and the clouds are light. When you look up at the night sky and enjoy this beautiful night, do you want to know: How far is the unreachable moon from us?
167 1 year, two French astronomers first measured that the distance between the earth and the moon was about 385400km. Do you know how they measured the distance at that time?
Question 2: In today's high-tech era, if you want to know the height of a mountain, you don't need to measure it yourself. You only need a plane flying horizontally to get through the top of the mountain. Do you know why? Also, how does the traffic police measure the speed of the car on the expressway? It is not difficult to solve these problems. As long as you learn this chapter well, you can master its principles. (The blackboard title is "Solving Triangle")
[Design Description] Quote the introduction of this chapter in the textbook, create conflicts between knowledge and problems, and stimulate students' interest in learning this chapter.
(B) special start, find the law
Question 3: In junior high school, we learned the chapter "acute trigonometric function and right triangle solution". The teacher wants to test your strength. Please solve such a problem according to junior high school knowledge. At Rt⊿ABC, sinA=, sinB=, sinC=. From this, can you express all the sides and angles in this right triangle with one expression?
Guide and inspire students to discover sine theorem under special circumstances.
(C) analogy induction, strict proof
Question 4: This question belongs to junior high school, which is relatively simple and not exciting enough. Now if I embarrass you, I'll let you be a teacher once. If a student accidentally writes Rt⊿ABC in the condition as acute angle ⊿ABC, and nothing else has changed, do you think this conclusion is still valid?
[Design Description] At this time, let the students do it themselves. If you feel that you have difficulty in solving the problem, students can also study in groups before and after the table or at the same table, and encourage students to prove this conclusion in different ways. During the inspection, students with different methods were shown on the blackboard. If there are no students who use vectors, the teacher will guide them and remind them whether they can use vectors to complete the proof.