First, the teaching content:

This lesson mainly explores practical problems, constructs mathematical models, discovers sine theorem by using mathematical experiment conjecture, proves it in theory, and finally applies it simply.

Second, teaching material analysis:

1, the position and function of teaching materials: this section is arranged in the first chapter of the curriculum standard experimental textbook for ordinary high schools. Compulsory Mathematics 5》(A version A) is arranged after senior two students have learned trigonometry knowledge, which is obviously the application of trigonometry; At the same time, as a theorem in the triangle, it is also a direct extension of the right triangle in junior high school, and the application of the theorem itself (the application of the theorem is specially studied in the next section) is very extensive. Therefore, it is necessary to do a good job in teaching this section, so that students can feel the idea and method of "analogy-conjecture-confirmation" through the exploration, discovery and confirmation of sine theorem in any triangle, and realize the transition from qualitative research to quantitative research.

2. Emphasis and difficulty in teaching: the key point is the discovery and confirmation of sine theorem; The difficulty is to prove it by the method of triangle circumscribed circle.

Third, the teaching objectives:

1, knowledge target:

Master sine theorem and understand the verification process.

2, ability goal:

(1) through the exploration of practical problems, cultivate students' ability to observe, ask questions, analyze and solve problems by mathematical methods.

(2) Improve students' cooperative ability and mathematical communication ability.

(3) Cultivate students' innovative consciousness and ability.

3, emotional attitudes and values:

(1) Through students' independent exploration, cooperation and communication, and personal experience of the discovery of mathematical laws, students' innovative quality of being brave in exploration, good at discovery and not afraid of difficulties is cultivated, their psychology of learning success is enhanced, and their interest in learning mathematics is stimulated.

(2) Cultivate students' patriotism and sense of responsibility for studying hard for the motherland through examples with social significance.

Fourth, teaching ideas:

This course adopts inquiry-based classroom teaching mode, that is, in the teaching process, under the guidance of teachers, on the premise of students' autonomy and cooperation, with the discovery of sine theorem as the basic exploration content and the surrounding world as the reference, it provides students with sufficient opportunities to freely express, question, explore and discuss problems, so that students can apply what they have learned to individual, group and collective attempts to solve problems, solve doubts and dispel doubts. Let students learn in "activities", develop in "initiative", increase knowledge in "cooperation" and innovate in "inquiry". The design idea is as follows:

Verb (abbreviation of verb) teaching process;

(A) the creation of problem situations

Let's put some pictures about military subjects before class, and give an example at the beginning: one day, my nuclear submarine A was patrolling a certain sea area, and suddenly I found an enemy boat B sailing 40 northwest at a speed of 30 knots. After research, it was decided to launch a torpedo at it to give a deterrent blow. It is known that the speed of torpedo is 60 knots. How to determine the launch angle to hit the enemy ship?

Design a practical problem that students are interested in, attract students' attention, and let them immediately enter the role of researchers! ]

(2) Inspire and guide students to observe problems with mathematical methods and establish mathematical models.

Using geometric sketchpad to simulate and demonstrate the whereabouts of torpedoes and enemy ships, a triangular problem is abstracted in the process of discussing the launching angle of torpedoes:

1. Investigate the range of angle A and recall the essence of "big side versus big angle"

2. Ask the students to guess the exact angle of Angle A, from AC=2BC, so that B=2A.

So as to abstract a prototype:

3. There is an error between the actual angle of the measured angle A and the estimated angle, which leads to contradiction:

How can qualitative research be transformed into quantitative research?

4. Further modify the formula in the prototype to inspire students to imagine boldly: and so on.

[Intuition first, guessing the way, causing students to think positively in the conflict! ]

(3) Using the research method of "special case to general" to guide students to guess mathematical laws.

Ask questions:

1, how to test the above equation? Stimulate students' thinking, start with a special case (right triangle) that they are familiar with, and screen out the equations that can be established.

2. Does this conclusion apply to any triangle? Instruct students to verify the general triangle with tools such as scale, compass and calculator.

3, let the students always test the results, draw a guess:

In a triangle, the angle satisfies the relationship with the opposite side.

["Special case → analogy → conjecture" is a common scientific research idea! ]

(D) Let students make various attempts to explore the methods of theoretical verification.

Ask questions:

1, how to turn a conjecture into a theorem? Make students pay attention to the difference between conjecture and theorem and strengthen the rigor of students' thinking.

2. How to verify the theory? Cultivate students' changing ideas and confirm them by changing them into familiar right triangles.

3. Can you find out their proportion? In order to test whether students have mastered the above research ideas. Demonstrate with geometric sketchpad, find the ratio and break through the difficulties.

4. Turn conjecture into theorem, use it to solve the problems raised at the beginning of the class, and carry out appropriate ideological education.

[Students become discoverers and creators! Let students enjoy the joy of success! ]

(5) Review and summarize, and assign homework.

1, sine theorem has the beauty of symmetry and harmony.

2. "Analogy → experiment → conjecture → confirmation" is a common thinking and method to study problems.

Thinking after class: Are there any other quantitative relationships between the angles in the triangle?

Six, blackboard design:

sine law