Feng Chunxiang of liyang city Daibu Middle School

Textbook: New Curriculum Standard Textbook-Compulsory 5

Subject: Sine and Cosine Theorems

[Abstract]: The teaching experiment of "Situation, Question, Reflection and Application" under the guidance of dialectical materialism epistemology, modern mathematics view and constructivism teaching view aims to cultivate students' awareness of mathematical problems, develop the habit of finding and asking questions from a mathematical perspective, form independent thinking, improve students' ability to solve mathematical problems, and enhance students' innovative consciousness and practical ability. Creating mathematical situation is the premise, asking questions is the key, solving problems is the core, and applying mathematical knowledge is the purpose, so the situation should conform to the students' "nearest development zone". "sine and cosine theorem" has a wide range of application values, and we create situations according to actual needs in teaching.

[Keywords:]: sine and cosine theorem; Solve the triangle; Mathematical situation

First, teaching design

1, teaching background

In the teaching practice in recent years, we found such a strange phenomenon: most students think that mathematics is very important, but it is difficult; Learning is bitter, too abstract and too boring. If you don't go to school, you don't pay attention to it, and there will be less opportunities to use mathematics in the future. Many students rely entirely on the teacher's explanation, so they can't teach themselves, dare not ask questions, and don't know how to ask them. This shows that students can't learn mathematics, have fear of mathematics and have no confidence. How can such a mentality innovate mathematics? Even if there is innovation, it is not proportional to the cost of students, killing too much happiness and personality. Constructivism advocates situational teaching and believes that most learning should be related to specific situations. Only by solving the problems related to the real world can the constructed knowledge be richer, more effective and easier to migrate. In 2003, we conducted a teaching experiment of "creating mathematical situations and asking mathematical questions". After a period of teaching experiments, most students have been able to adapt to this learning style, think positively and dare to put forward their own concerns and ideas. They gradually changed from passively accepting knowledge in the past to actively exploring and seeking knowledge, which enhanced their interest in learning mathematics.

2. teaching material analysis

"Sine and Cosine Theorem" is the main content of the second section of the first chapter of the standard experimental textbook "Compulsory Mathematics 5" in senior high school. It is one of the two important theorems to solve the oblique triangle problem, and it is also a direct extension of the Pythagorean theorem in junior high school. It is the concrete application of trigonometric function knowledge and plane vector knowledge in triangles, and it is an important tool to solve other mathematical problems that can be transformed into triangle calculation problems and practical problems in production and life, so it has wide application value. This lesson is the second lesson in the teaching of sine theorem and sine and cosine theorem. Its main task is to introduce and prove the sine and cosine theorem, which belongs to the "theorem teaching class" in terms of class type. Bruner pointed out that students are not passive and passive recipients of knowledge, but active and active knowledge explorers. The role of teachers is to create a situation for students to explore independently, guide students to think and participate in the process of knowledge acquisition. Therefore, doing a good job in the teaching of "Sine and Cosine Theorem" can not only review and consolidate old knowledge, enable students to master new and useful knowledge, experience dialectical views such as connection and development, but also cultivate students' application consciousness and practical operation ability, as well as their ability to ask and solve problems and other research-based learning.

3. Design concept

Constructivism emphasizes that students will not walk into the classroom with empty heads. In daily life, in the past study, they have formed a wealth of experience, from the daily necessities of life around them to the operation of the stars in the universe, from natural phenomena to social life, almost all of them have their own views. Moreover, even if they have not been exposed to some problems and have no ready-made experience, when problems are presented to them, they can often form some kind of explanation based on relevant experience and their own cognitive ability. Moreover, this explanation is not all a wild guess, but a logical assumption based on their experience background. Therefore, teaching can't ignore these experiences of students, start a new stove and load new knowledge from the outside, but take students' existing knowledge and experience as the growth point of new knowledge and guide students to "grow" new knowledge and experience from the original knowledge and experience.

Therefore, according to the "situation-problem" teaching mode, along the main line of "setting situation-asking questions-solving problems-applying reflection", we take exploring and asking mathematical problems from the situation as the starting point of teaching, and organize teaching with "problems" as the red line, forming a "situation-problem" learning chain in which questioning and solving problems are mutually triggered, so that students can truly become questioners and problem solvers. According to the above spirit, the following designs are made: ① Create realistic problem situations as the background for asking questions; (2) Enlighten and guide students to ask practical problems they care about, gradually transform and abstract practical problems into transitional mathematical problems, and use sine and cosine theorems when solving problems, thus triggering students' cognitive conflicts, revealing the necessity of solving oblique triangles, and giving students the motivation to further explore and solve problems. Then guide the students to grasp the mathematical essence of the problem and extend it to a general mathematical problem: knowing the two sides of a triangle and their included angles, find the third side. (3) In order to solve the problems raised, guide students to "grow" new knowledge and experience from the original knowledge and experience, get two right triangles by making the vertical line of BC, and then get the expression of sine and cosine theorem by using Pythagorean theorem and acute trigonometric function, thus guiding students to carry out strict logical proof. When proving, the key is to inspire and guide students to clarify the following two points:

The first is the starting point of proof.

;

The second is how to transform vector relation into quantity relation. ④ Students independently use the proved conclusions to solve the problems raised in the course.

Second, the teaching process

Type 1: Solve triangles and related problems.

1.( 1) is in. If,,, the area of is.

Variant: If known, can you find the other three elements?

Example 1. Given, found.

Variant: (Question Training 4) In the middle, the side length is known.

Example 2. (Original example 4. ) The opposite sides of three internal angles are known respectively, and the size of the angles is calculated.

Variant: (Question Training 3) If the ratio of three sides of a triangle is equal, then the maximum angle of the triangle is equal to.

Type 2: the problem of judging the shape of triangle

2. In, if, it is (shape).

Example 3. Try to judge the shape.

Student exercises:

1. Known, if, then.

2. In, if, then the shape is (shape).

3. In, known, and then.

4. In, the triangle is known to have been solved.

Third, teaching reflection.

Creating mathematical situation is the basic link of "situation, problem, reflection and application" teaching. Teachers must comprehensively consider students' physical and mental characteristics, knowledge level, teaching content, teaching objectives and other factors, compare available situations, and choose situations with better educational functions.

Starting from the application needs, it is one of the commonly used methods to create mathematical situations of cognitive conflict. "Sine and Cosine Theorem" has a wide range of application values, so the mathematical situation used in teaching is created from the application requirements of this course. This situation comes from the textbook 1 chapter 1, the application examples of sine and sine cosine theorems. Practice shows that it is an effective way to create situations by transforming examples and exercises in textbooks into situations. As long as teachers can conduct in-depth, detailed and comprehensive research on the teaching materials, it is not difficult to find that there are many materials available in the teaching materials.

The teaching mode of "situation, question, reflection and application" advocates organizing teaching activities with questions as the "red line" and asking questions with students as the main body. How to guide students to ask questions is the key to successful teaching. Teaching experiments show that whether students can ask mathematical questions is not only influenced by their own factors such as their mathematical foundation, life experience and learning style, but also restricted by external factors such as their environment and teachers' attitude towards problems. Therefore, teachers should not only pay attention to creating suitable mathematical situations (not only with rich connotations, but also with the induction, inspiration and exploration of "questions"), but also really change their attitude towards students' questioning and improve their guidance level. On the one hand, they should encourage students to ask questions boldly; on the other hand, they should properly handle the questions raised by students. Pay attention to the results of students' learning and pay more attention to the process of students' learning; Pay attention to students' mathematics learning level, and pay more attention to students' emotions and attitudes in mathematics activities; Pay attention to whether or not to create a situation for students to experience the process of mathematical activities, cultivate students' awareness of mathematical problems by "questioning and asking questions" and improve students' ability to ask mathematical questions, which is the starting point and destination of teaching activities.